# Fractal Metasurfaces and Antennas: An Overview for Advanced Applications in Wireless Communications
## 2 Fractal Antenna Elements
Figure 3a shows a small but long wire antenna based on the use of the Koch fractal curve. The antenna was presented in [27] with the aim to demonstrate how the space-filling properties of fractal shapes can be used to design small antennas. Actually, the reduction in the antenna size is an ultimate goal in a large number of applications, especially those involving mobile terminals. The Koch monopole, described in [27], is created by using the fractal construction method depicted in Figure 3a. The initial iteration, \(K0\), is a straight monopole. The next iteration, \(K1\), is obtained by applying similarity transformations to \(K0\), including scaling by one third and rotations of \(0^{\circ}\), \(60^{\circ}\), \(-60^{\circ}\), and \(0^{\circ}\), respectively. This process is repeated iteratively to obtain the subsequent elements \(Ki\) (\(2\leq i\leq 5\)). Figure 3a shows the five-iteration version of the Koch monopole (i.e., \(K5\)), which has a height \(h\) equal to \(6\) cm, but a total length given by the formula \(l=h\cdot(4/3)^{5}=25.3\) cm. In [26], five antennas have been implemented corresponding to the first five iterations of the Koch curve (see Figure 3a). Both experimental and numerical data have shown that the Koch fractal monopole can improve the performance of conventional monopoles (refer to Figure 3b). This demonstrates the possibility of using \"miniaturized\" fractal antennas that fit into smaller volumes while maintaining efficient performance. Figure 4a shows a fractal antenna based on the Cantor set. The structure was proposed in [28] for designing a wideband antenna for energy harvesting applications. The unique property of fractal antennas to introduce multiple nearby resonances is exploited in [28] to provide wider bandwidths compared to conventional antennas. As illustrated in Figure 1a, the design of a Cantor fractal set is quite simple: the initial straight line is copied and divided into three equal segments, from which the central one is removed; the same procedure is then applied iteratively to the newly generated segments. In [28], the fractal iteration steps are repeated until an antenna bandwidth (BW) of 0.8-2.2 GHz is achieved  
Figure 2: Penta-band Sierpinski monopole: (**a**) layout and sizes, (**b**) input reflection coefficient (Adapted with permission from [26], IEEE, 1998). (i.e., 98% BW). Figure 4 shows a photo of the antenna prototype realized in [28] (Figure 4a) and its input reflection coefficient (Figure 4b). Figure 5 illustrates further examples of fractal antennas [29], including the Hilbert curve-based monopole, the Mandelbrot-based microstrip patch antenna, and the Koch island (or'snowflake') fractal patch. Several Hilbert monopoles were fabricated and tested in [30]. The length of the Hilbert monopole, given by the expression \(L_{n}=\frac{4^{n+1}-1}{2^{n+1}-1}h\) (\(n=\) iteration number and \(h=\) height of the straight monopole--i.e., \(n=0\)), increases with each iteration \(n\), so that the resonant frequency decreases as \(n\) increases (Figure 5a). The results also showed that the antennas become electrically smaller as the fractal iteration increases at a higher rate compared to other fractal geometries. A microstrip antenna with high directivity inspired by the Mandelbrot fractal (Figure 5b) was designed in [31]. The antenna was proposed as an alternative to antenna arrays in order to avoid the use of feed networks. However, the proposed solution does not offer  
Figure 4: Cantor fractal antenna: (a) fabricated prototype, (b) input reflection coefficient (Adapted with permission from [28], EurAAP, 2017). Figure 3: Five-iteration Koch monopole: (a) monopole layout construction, (b) input impedance (Adapted with permission from [27], IEEE, 2000). the possibility of pattern shaping or beam steering, making it particularly attractive in applications for which low-cost, low-weight and fixed-beam radiation patterns are required (e.g., small base station antennas used for WiFi, Industrial Scientific Medical (ISM) bands, the Internet of Things (IoT), etc.).