# Fractal Metasurfaces and Antennas: An Overview for Advanced Applications in Wireless Communications
## 3 Fractal Reflectarray Antennas
### Review of Existing Fractal Reflectarray Elements
#### 3.1.2 Fixed-Size Fractal Patches  
In [45], a new type of miniaturized reflectarray cell was introduced, which uses a fixed-size Minkowski patch. The phase tuning is obtained by using a variable-length slot, cut in the ground plane at the patch center (see Figure 9a). The fractal construction procedure is explained in Figure 9b. The initial element (i.e., \(n=0\)) is a segment of length L\({}_{\text{g}}\) called the \"generator\". In the first iteration (\(n=1\)), the length of the generator is divided by three, then the central part is replaced by a rectangle with a base equal to w(L\({}_{\text{g}}\)/3), where w is the penetration factor varying between zero and one. The same procedure is applied iteratively to each side of the first-order fractal. The main advantage of this approach is related to the fixed size of the radiator, which allows one to exploit the miniaturization ability of fractals. Both the first- and the second-order Minkowski fractal patches were considered in [45]. The analysis showed that the resonant frequency of the cell decreases as  
Figure 8: Multi-slice fractal patches [33]: (**a**) construction of the fractal element; (**b**) simulated reflection phase versus patch length of a fractal patch with two slots and four slots (\(\boxplus\)\(f=10\) GHz); (**c**) linearized phase curves of two-slot multi-slice patch for different patch lengths. the slot length \(L_{s}\) increases (Figure 9c). Therefore, the fractal unit cell proposed in [45] can reduce the resonance frequency, resulting in a miniaturization effect. However, as a side effect, the slot in the ground plane produces unwanted back-radiations. A novel fixed-size Minkowski fractal patch was proposed in [46] for the design of small reflectarray elements. The inherent miniaturization capability of the fractal geometry is fully exploited in [46], leaving the patch length unchanged and using the fractal scaling factor to realize the phase tuning of the reflectarray cell. A square patch generator with the side length \(L\) is iteratively modified as described below (Figure 10a). In the initial iteration, a smaller square with a side length of SL is extracted from the center of each side of the generator patch. Here, S represents the fractal scaling factor, which varies from zero up to \(L/3\). The fractal construction is reiterated by applying the above operation to each side of the first-order fractal structure, obtaining a more complex self-similar shape (Figure 10a). Since the resonant side length of the patch is equal to \(L_{n}=(1+2S)^{n}\)\(L\) (\(n=\) iteration number, \(L_{n}=\) resonant length of \(n^{\text{th}}\) iteration patch), the cell can be miniaturized. To validate it, a fractal-shaped reflectarray element, embedded in a 0.3\(\lambda\)\(\times\) 0.3\(\lambda\) cell, was designed at 10 GHz to achieve a rather good phase agility of more than 300\({}^{\circ}\). Ultimately, the fractal element proposed in [46] can offer, at the same time, good phase ranges, very small sizes (about 30% smaller than those of a standard variable-size reflectarray configuration), and lower reflection losses. An alternative reflectarray unit cell was proposed in [47] (see Figure 11). Its layout is essentially derived from the first-iteration fixed-length Minkowski patch proposed in [46]. Figure 11c shows the patch geometry construction, which begins with a square element of dimensions L \(\times\) L. Unlike the Minkowski patch, a smaller square of side SL is removed only from the center of the two lateral sides (i.e., the resonant sides, when a TE\({}^{\text{x}}\) incident wave is assumed), resulting in a linearly polarized patch along the y-axis (see Figure 11a). The reflection-phase tuning is achieved by varying the fractal scaling factor S from 0 up to 0.45, leaving both the patch length and the distance between adjacent patches unchanged. In Figure 11b, it can be observed how the miniaturized reflectarray cells are able to offer good performance in terms of phase variation.