# Cascaded Frequency Selective Surfaces with Matryoshka Geometry for Ultra-Wideband Bandwidth
## 4 Results Analysis
### Single-Layer FSS
One observes in Figure 13 that at the frequency of 1.75 GHz, which is the first resonance frequency, there is enough electric current through all the rings of the geometry to allow the signal wave to pass. This is the frequency in Figure 12 at which \(|S(2,1)|\) attains \(-0.37\) dB. The incident signal wave also passes through the FSS at 3.8 GHz in Figure 13, which is the second resonance frequency. At 2.7 GHz in Figure 13, it is possible to see that the electric  
Figure 13: Electric current distribution of the FSS with band-pass matryoshka geometry. current in the unit cell attains the lowest intensity, so the signal wave cannot pass through the FSS, which is the reason why the lowest point in the frequency response is attained in Figure 12.  
The electric current distribution for reject-band FSS is shown in Figure 14. At 0.5 GHz, the unit cell is not sufficiently excited by the electric current, allowing the signal wave to pass through the FSS. At the first resonance frequency in Figure 14, 1.75 GHz, a high electric current density is observed in the outer ring of the unit cell, contributing to the FSS reflecting the signal wave. However, at 2.7 GHz, even with a high electric current density in the outer ring of the unit cell, the signal wave passes through the FSS. That happens because the inner ring has a low current density and a frequency higher than the one in the previous example. The signal wave will only be reflected when the inner ring has a sufficient electric current density, as seen at 3.8 GHz in Figure 14.  
The results below were obtained by individually measuring the signals that interacted with the FSSs and the horn antennas. They were normalized by the reference measurements to minimize the losses caused by the environment. Figures 15 and 16 show the results obtained with the measurement for an FSS with matryoshka geometry at different inclinations, for different values of the angle \(\varphi\) and polarizations. The signal loss is greater for angular inclinations close to 90\({}^{\circ}\), when the situation is equivalent to a measurement without FSS since, in this case, the incident plane wavefront is perpendicular to the FSS plate. Analyzing the frequency response of the FSS, it can be noted that the greater the horizontal angular inclination of the measurement window regarding the incident plane wavefront, the greater the displacement of the curves to the left in relation to the frequency response with a horizontal inclination of 0\({}^{\circ}\), where the FSS plate is perpendicular to the incident wavefront. It can also be noted that, although the frequency responses for the two polarizations of the incident wave are not identical--as expected due to the symmetric matryoshka geometry--it is possible to identify resonance frequencies at 1.8 GHz and 3.6 GHz at 0\({}^{\circ}\), both for \(x\) and \(y\) polarizations, as shown in the black trace of Figures 15 and 16. When the wavefront incidence angles are considered, one can observe that the first resonance frequency varies from 1.81 GHz for y polarization at 30\({}^{\circ}\), as shown in the blue trace of Figure 15, to 1.88 GHz for \(x\) polarization at 45\({}^{\circ}\), as shown in the green trace of Figure 16. Meanwhile, the second resonance frequency varies from 3.65 GHz for \(y\) polarization at 30\({}^{\circ}\), in the blue trace of Figure 16, to 3.70 GHz for \(x\) polarization at 45\({}^{\circ}\), in the green trace of Figure 12:0  
Figure 12:1 Electric current distribution of the FSS with band-reject matryoshka geometry.