# Cascaded Frequency Selective Surfaces with Matryoshka Geometry for Ultra-Wideband Bandwidth
## 4 Results Analysis
### Cascaded FSS  
The characterization of the cascaded FSS was conducted following the same procedures as the previous FSSs. Figure 19 shows the frequency response graph of a cascaded FSS with two layers aligned with different spacing between them, incrementing millimeter by millimeter. It is possible to see that as the spacing increases, the bandwidth uniformly increases between 1.7 GHz and 3.8 GHz, reaching up to 2 GHz. The simulated frequency response, in terms of the modulus of the transmitting parameter \(S(2,1)\) as a function of the incident wave frequency, presents the highest bandwidth equal to 2.1 GHz at spacings of 18 mm, 19 mm, and 20 mm between the two parallel layers, while the lowest bandwidth equal to 1.1 GHz occurs for a spacing equal to 4 mm. The increase in bandwidth was saturated for a spacing of around 18 mm for the highest bandwidth and 4 mm for the lowest bandwidth. An analysis was conducted for the electrical current distribution of the cascaded FSS, presented in Figure 20. At 0.5 GHz, we observe the band-reject allowing the wave signal to pass through the first layer, but the band-pass reflects the signal due to the low electric current distribution, preventing the wave signal from reaching the receiving antenna. At 1.75 GHz in Figure 20, the band-reject has a high electrical current distribution throughout the unit cell and the FSS reflects the signal before reaching the second layer. That is why at this frequency in the blue line in the frequency response of Figure 19, we can observe that the 1.75 GHz is below \(-10\) dB. At the frequency of 2.7 GHz in Figure 20, the current distribution in the band-reject seems to be sufficient, but as previously in Figure 14, the inner ring has a low current density through which the wave signal can pass from the first layer. However, it is prevented by the second layer due to a low electric current distribution in the band-pass. The last frequency analyzed in Figure 20 was 3.8 GHz, where despite the second layer with the band-pass having a high enough electric current distribution to allow  
Figure 19: Simulated frequency response of a cascaded FSS with matryoshka geometry and different values of spacing between the layers. the wave signal to pass through, the first layer with the band-reject is reflecting with the high electric current density in the inner ring. The experimental results, on the other hand, did not show such a uniform pattern in the frequency response. In Figure 21, frequency response curves are presented for spacing between the parallel FSSs varying from 4 to 4 mm for the \(x\) polarization. The same procedure was repeated for \(y\) polarization to generate Figure 22. From these results, it is possible to see that the measured frequency responses have a pattern of shifting the resonance frequency relative to the variation in the spacing between the FSS layers. In the frequency response measured for \(x\) polarization, it only varies from 2 GHz to 3.2 GHz for 4 mm and from 1.75 GHz to 3 GHz for 12 mm and higher, as shown in Figure 21. In contrast, for \(y\) polarization, there is a 1.2 GHz bandwidth in the 2 to 3.2 GHz range for a spacing equal to 4 mm, and another with a 2 GHz bandwidth in the range from 1.78 to 3.82 GHz for spacings of 12 mm and higher, as shown in Figure 22.  
The influence of the wavefront incidence angle, \(\varphi\), can also be analyzed by observing Figures 23 and 24. These figures show the experimental frequency response of the cascaded FSS, in terms of \(|S(2,1)|\), for different values of the wavefront incidence angle \(\varphi\), \(y\) polarization, and two different values of the spacing between the layers, equal to 4 mm and 12 mm, respectively. Figure 21: Experimental frequency response of a cascade FSS with matryoshka geometry, \(x\) polarization for the incident wave and different spacing between the layers. Figure 20: Electric current distribution of the cascaded FSS with 12 mm spacing between layers.