## 4 Aperture Design: Parametric Analyses and Performance Evaluation
### Hexagonal Ridged Radiating Elements
Again, this is due to the lower cut-off frequency of the higher-order mode (21.7 GHz) with respect to the three-ridge counterpart (30.7 GHz). Similar values are obtained for the coupling for circular polarization of the six-ridge configuration in (Figure 16f). In this case, there is less variations between the planes. The worst value is obtained at \(60^{\circ}\) for \(\phi=90^{\circ}\) (dotted red line), being \(-5\) dB at higher frequencies. Concerning the scan loss, Figure 17 shows the results of both hexagonal designs within the scan range of \(-60^{\circ}\) to \(60^{\circ}\) for both linear and circular polarization. In terms of linear  
Figure 16: S-Parameters for both hexagonal designs. The dashed lines indicate the cut-off frequencies and the green box indicates the operative frequency band. (**a**) Hexagon with three ridges for linear polarization; (**b**) hexagon with six ridges for linear polarization; (**c**) hexagon with three ridges for circular polarization. S\({}_{11}\) for (\(\theta=0^{\circ}/30^{\circ}\)) is below \(-20\) dB; (**d**) hexagon with six ridges for circular polarization. S\({}_{11}\) for (\(\theta=0^{\circ}/30^{\circ}\)) is below \(-20\) dB; (**e**) hexagon with three ridges for circular polarization; and (**f**) hexagon with six ridges for circular polarization. polarization, both hexagonal apertures (Figure 17a,b) show a decay between 2.5 and 3.5 dB at \(\pm\)60\({}^{\circ}\) in the E-plane for the three considered frequencies (solid lines). These values increase up to 5 dB for the hexagonal aperture with three ridges and up to 6 dB for the hexagonal aperture with six ridges, in both D- and H-planes (dashed and dotted lines). Differently from the quadridge structure, the hexagonal waveguide apertures do not have 90\({}^{\circ}\) rotational symmetry. Therefore, a full set of scan losses and XPD data is provided for both left-hand (LH) and right-hand (RH) circular polarization. For both solutions, the scan loss ranges from 3 to 6 dB (Figure 17c,e). Minor differences are visible between the two circular polarizations. The XPD is reported in Figure 18. The curves for the three-ridged aperture are not symmetrical along the scan range (Figure 18a,c,e) due to the one-fold symmetry of the geometry (the same consideration applies for the scan loss in Figure 17a,c,e). Symmetrical curves with respect to broadside (\(\theta=0\)) are instead observed for the six-ridge solution thanks to its two-fold symmetry. XPD values ranging from 8 to 10 dB are observed at the scan limits. As already discussed for the quadridge, these values are related to the different behaviour of the structure for the two principal polarizations along the scan planes. It should be noted that the complete scan loss and polarizatio  
Figure 17: Scan loss for both hexagonal designs. Concerning to the coupling at the broadside (Figure 19b), the hexagon aperture shows an improvement of 6 (yellow line) and 10 dB (purple line) with respect to the evanescent quadridge (red line) at the central frequency when it is loaded with three and six ridges, respectively. Instead, for high scan angles (Figure 19c), both hexagon apertures have a similar trend up to 1 dB of difference between them. Figure 19d,e shows the scan loss and XPD, respectively, at the highest frequency of the band, 20.2 GHz, when scanning from \(-60^{\circ}\) to \(60^{\circ}\) for the circular polarization for all apertures. Values of 8 and 0 dB for the high scan angles are obtained for the quadridge in both scan loss and XPD, respectively. The evanescent quadridge (yellow line) shows a stable and uniform scan loss for both the right and left circular polarization with a drop of 2 dB within a 13.2% bandwidth up to a \(50^{\circ}\) scan angle. In terms of XPD, values of 14 dB are obtained. Acceptable values have been obtained for the scan loss in both hexagonal apertures (\(-3\) dB) scanning up to \(50^{\circ}\) (red and purple lines).