# Dual-Band Single-Layer Fractal Frequency Selective Surface for 5G Applications
## 3 The Unit Cell  
Many FSS designs consist of the repetition of a unit cell element replicated in an organized manner, typically following a bidimensional square matrix structure. Then, when the objective is to analyze the performance of the surfaces composed by a squared matrix of replicated unit cells, the design of such a unit cell is the first step to reach a valid design. The design of the unit cell is described in this section, with its geometrical parameters shown in Figure 2. The unit cell is made out of a combination of a pair of Sierpinski triangles organized as a bow tie, connected by a central metal strip. Both triangles in the bow tie are stage 2 fractal Sierpinski triangles, horizontally mirroring each other. In Table 1, all the required design parameters are defined. Moreover, it contains the final sizes reached after the design process explained further in this paper. The distances between the bow ties are defined as followed:  
* \(dx\) is the distance from the center of the metal strip and the corners of Sierpinski triangles to the side;
* \(dy\) is the distance in between the bottom and the top of two neighboring Sierpinski triangles. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Explanation** & **Value (mm)** \\ \hline \(w\) & height of the bow tie & 87 \\ \(t\) & width of the central element & 0.5 \\ \(dx\) & horizontal distance between elements & 3 \\ \(dy\) & vertical distance between elements & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the unit cell and the values of them in the final design. Figure 1: Different stages for the Sierpinski triangle [18]. Figure 2: Unit cell design. The main difference between the bow tie from [11] and the one proposed here is the possibility to vary the width \(t\) of the added central metal strip. This element is an excellent granular tuning parameter, which helps in fine tuning both band-stop central frequencies and bandwidths around them. The simulation results of the tuning can be found in Section 4.2. Another benefit of adding this metal strip is the effect it has on the production of the physical FSS. The connection point of the two triangles is essential, and the contact area is very small. If the two are not connected as expected, the resonance could be completely different. Then, the presence of the metal strip makes it easier for prototyping: one can simulate almost everything, but prototyping implies taking into account construction tolerances, and the vortex of the triangle is not easy to build perfectly. Regarding the two triangular elements, these are fractal objects that must be developed by iteratively growing an initial shape in multiple stages. In order to decide the fractal stage required to reach the objective, initial simulations were performed with unit cells composed by pairs of stage 1, 2, 3 or 4 Sierpinski triangles, described in Figure 3. Frequency responses of these unit cells can be observed in Figure 4. The number of resonances is directly related to the iteration on the Sierpinski triangle. When the design does not iterate on itself (in stage 1), there is only one notch at the lower frequency band in the attenuation frequency response, around 800 MHz. When the stage 2 design is implemented, a second notch at the higher frequency band of 3.5 GHz also appears. This means that, at least, a stage 2 unit cell is needed to obtain the dual-band characteristic. When the design iterates more upon itself to stages 3 and 4, no further improvements can be noted within the bandwidth of interest. These higher iterations increase complexity, and thus make it more challenging to produce prototypes or final products. This is why the final design sticks with the stage 2. Another notable effect of increasing the number of iterations is the shift of the two notches. The fourth iteration has a complete shift to the lower frequencies, but also brings the two notches closer towards each other.