# Fractal Metasurfaces and Antennas: An Overview for Advanced Applications in Wireless Communications
## 3 Fractal Reflectarray Antennas
### Fractal Reflectarray Elements with Multifunctional Operation  
Reflectarrays represent an interesting alternative in the development of innovative mmw antennas for future telecommunication systems (5G and 6G), being able to offer  
Figure 11: Miniaturized linearly polarized unit cell with a fractal patch with a variable scaling factor [47]: (**a**) top view, (**b**) 3D view, (**c**) fractal construction, (**d**) simulated reflection coefficient for different cell sizes. Figure 10: Minkowski patch with variable scaling factor S: (**a**) fractal construction; (**b**) unit cell geometry; (**c**) measured and simulated reflection-phase curves vs S at 10 GHz; (**d**) photo of the measurement setup with some fractal reflectarray samples (Adapted with permission from [46], IEEE, 2014). high gains thanks to their spatial feeding mechanism [34; 35]. Moreover, reflectarrays can provide several key skills, such as beam scanning, multi-beam or shaped radiation patterns, polarization diversity, and multi-frequency operations, which are useful for enhancing the capacity, coverage, and throughput performance of telecommunication systems. In order to achieve reflectarray unit cells with multifunctional operation, two fractal based cells have been proposed in [49; 50], respectively, for dual-band applications at 28/38 GHz and for dual-band/dual-polarization operation within the Ka band (i.e., 27 and 32 GHz). Thanks to their compact and versatile nature in achieving frequency and/or polarization diversity, both unit cells have been proposed for 5G applications. In particular, the dual-band unit cell proposed in [49] is composed of two pairs of alternately arranged fractal patches (Figure 12a), each designed to operate around a specific resonant frequency (i.e.,\(f_{1}\) and \(f_{2}\)). The single patch layout is based on the first iteration of the fixed-length Minkowski patch originally proposed by the authors in [46]. The patch geometry shown in Figure 12a is characterized by an initial square element of side length \(L_{n}\) (\(n=1\), 2). A smaller \(\text{S}_{\text{n}}L_{n}\times\text{S}_{\text{n}}L_{n}\) square (0 \(<\) S\({}_{\text{n}}<\) 0.45) is cut from the center of the two lateral sides (i.e., the resonant sides), resulting in a linearly polarized element along the y-axis (Figure 12a). The reflection-phase tuning is achieved by independently varying the scaling factor \(\text{S}_{\text{n}}\) of each element while keeping the patch dimension as \(L_{n}\times L_{n}\). As shown in [49], this configuration allows for minimal inter-element spacing and negligible mutual coupling effects between the two frequency bands. This guarantees an independent phase-tuning mechanism for both desired bands. A reflectarray consisting of \(11\times 11\) identical cells (Figure 12c) was printed on a 0.254 mm thick DiClad880 substrate as experimentally validated in [49]. The cell reflection-phase curve was measured in the frequency range from 26.5 to 40 GHz, achieving a phase variation of approximately 300\({}^{\circ}\) around both operating frequencies. The simulations and measurements were in good agreement (Figure 12b), validating the dual-band fractal cell. A similar concept was applied in [50] to design a single-layer dual-band/dual-polarized reflectarray cell. The proposed unit cell, as described in [50], comprises four miniaturized fractal-based elements (Figure 13a). The patches are arranged in two pairs, each operating at two different frequencies within the Ka band (i.e., 27 and 32 GHz). Each pair comprises two linearly polarized patches rotated by 90\({}^{\circ}\) to provide double polarization operations at both frequencies. The scaling factor of each element was independently varied to adjust  
Figure 12: Dual-band fractal unit cell [49]: (**a**) top view and 3D view; (**b**) measured and simulated reflection phase; (**c**) measurement setup and reflectarray prototype. the reflection phase for each frequency/polarization.