# Cascaded Frequency Selective Surfaces with Matryoshka Geometry for Ultra-Wideband Bandwidth
## 4 Results Analysis
### Single-Layer FSS  
The curves shown in Figures 11 and 12 were obtained considering horizontal and vertical polarizations for the incident wave. They show the simulated frequency response of the FSS in terms of the modulus of the parameter \(S\), both when transmitting, \(|S(2,1)|\), and reflecting, \(|S(1,1)|\), the incident plane wave. As expected, due to the matryoshka geometry being symmetrical on all sides, perfectly identical results were obtained in both polarizations, where the signal transmitted in vertical polarization is identical to the signal transmitted in horizontal polarization  
Also, as expected, when implementing the negative of a band-stop matryoshka geometry, we have a band-pass matryoshka geometry with an inverted frequency response, in which the reflected signal of the band-pass structure presents the same resonant frequencies as the signal transmitted from the band-stop structure. However, for this analysis, we consider its lowest points of \(|S(2,1)|\) as resonance frequencies so that it is possible to observe their effects in the cascaded structure. Figure 11: Simulated frequency response of the FSS with band-reject matryoshka geometry for incident horizontal and vertical polarizations. Figure 12: Simulated frequency response of the FSS with band-pass matryoshka geometry for incident horizontal and vertical polarizations. The resonant frequencies of the matryoshka geometry FSS were 1.75 GHz and 3.8 GHz, as shown at the lower points of the blue and red lines in Figure 11. Meanwhile, the FSS with negative matryoshka geometry, although having the same resonance frequencies of 1.75 GHz and 3.8 GHz at their peaks, shows its lowest frequencies at 2.75 GHz and 4.15 GHz, as shown in the red and blue lines of Figure 12.  
It is important to note that although the geometry in Figure 12 is the exact negative of the geometry in Figure 11, the frequency responses of the FSSs are not exactly reversed, even in the simulation. In the simulation, all the unit cells are identical and without any manufacturing errors, but the FSSs behave as different types of filters. The matryoshka geometry FSS behaves as a band-reject filter, blocking frequencies within the bandwidth range and letting frequencies outside the bandwidth range pass through. In the resonance frequency, the lower the value of the frequency response, the better the band-reject filter blocks the signal. The negative of the matryoshka FSS behaves as a band-pass filter, passing signal waves on frequencies within the bandwidth range and attenuating frequencies outside that bandwidth range. For the resonance frequency, the closer to 0 dB the frequency response is, the better the FSS lets the signal wave pass through. This is why, in Figure 11, the frequency responses presented lower values at 1.75 GHz and 3.8 GHz compared with Figure 12.  
A simulation was carried out of the electric current distribution for both FSS layers. The FSSs present different electric current paths in the metallic surface due to electromagnetic coupling between the geometry trails. The FSS geometry on the metallic surface generates a resonant effect that results in microwave trapping and absorption. The attenuation of EM waves is generated as the electric field interacts with the electrons and induces currents due to the presence of conductive metal. In Figures 13 and 14, we observe the electric current distribution in the single layers. For a band-pass FSS, if there is current excitation in the unit cell, the signal passes through the FSS at a chosen frequency. If the unit cell is excited by an electric current, then the FSS reflects the signal wave. The reverse is true for a band-reject FSS when no excitement in the unit cell lets the signal pass through, while a current excitation causes the FSS to reflect the signal at a chosen frequency. Figure 13 shows a 0.5 GHz frequency electric current distribution for which the signal is blocked by the FSS due to insufficient current excitation.