# Frequency Selective Surfaces: A Review
## 4 Types of FSSs
### FSS Based on Array Elements
#### 4.1.3 Fractal Based FSSs  
Nowadays, significant research in microwave engineering is reported in the design and development of new FSS designs based on the theory of fractal geometry due to its attractive features [29; 68]. Geometric patterns generate fractals through an iterative procedure for an infinite number of times [69]. However, pre-fractal shapes originate by truncating the iterative method  
Figure 6: (**a**). Convoluted square loop FSS (**b**). Array element with interwoven geometry (**c**) The geometry of the miniaturized element based on meandered lines showing 2 \(\times\) 2 unit cells [64] (**d**) FSS based on meandered elements showing the variation of resonant frequency with the inclusion of meandered stage [65]. (**e**) Bandwidth versus air separation of cascaded FSS [65] (**f**) Evolution process of the FSS unit cell in [67]. after a finite number of iterations. FSS with fractal elements was originally proposed in [62] to cut down the size of the structure by applying the space-filling curves of Hilbert and Minkowski. Later, the innovation in recently reported researches has brought unprecedented prospects for engineers to investigate unlimited possible fractal configurations that had not existed in earlier time. Important performance attributes that are realized by fractal arrays include efficient methodologies to element size reduction, multi-band behavior, low-side lobe designs, and improvement in fast beam forming algorithms by utilizing the recursive property of the fractals [70; 71]. Fractal FSSs are famous for their self-similarity nature, thereby creating multi-band response and compactness. The multi-band frequency response of Sierpinski fractals has lucatively been used in the past for designing dual-band FSSs [72; 73]. Additionally, the impact on the frequency response has been investigated via the scaling factor, iteration level, and width of the metallic patch [29; 74; 75]. It is proved with the increase in the electrical length of the fractal patch, a decrease of the working frequency is achieved, and for that reason, a lessening in the structural dimensions is observed. Figure 7a-h gives an overview of some of the fractal shapes used in recent researches. Fractals occupy a much smaller area because of their efficient space usage. The Koch dipole (Koch curve family) is much smaller than a straight wire dipole whereas both of them resonate at the same frequency. A Koch curve is generated by an iterative process, as shown in Figure 8a [76]. The whole length of a Koch curve at its Nth iteration is (4/3)N of the length at zeroth iteration (a straight line). Therefore, a Koch dipole starting and ending at the same point as its normal dipole counterpart effectively offers a much greater length and thus a lower resonance frequency. In fact, this results in a miniaturized element design. (1) Fractal Dimension and its Generation by Iterated Function System  
To obtain self-similar dimension, the fractal geometry is divided by scaling it down with replicas of itself. If the scaling fraction is \"sf\" and \"N\" copies of the original geometry to be made, and then the self-similarity dimension (SFD) is defined in (8) [76]. \[\text{SFD}=\frac{\text{logN}}{\text{log}(1/\text{sf})} \tag{8}\]  
As an example, four copies of a fractal square can be made with sf = 1/2, nine copies with sf = 1/3, 16 copies sf = 1/4, or N2 copies of sf = 1/N. Using above formula, the SFD is established to be equal to 2. SFD of several fractal geometries can be determined with this approach. A wide range of practical fractal element structures can be generated while using an extremely adaptable method called Iterated Function System (IFS) [71]. IFS is an effective design tool in fractal antenna engineering and fundamentally offers a generic structure for the characterizing and manipulating the fractals. Figure 7: Some typical fractal FSS geometries investigated in different applications.