## 4 Aperture Design: Parametric Analyses and Performance Evaluation
### Hexagonal Ridged Radiating Elements  
Differently from the quadridge, where the optimum value (OV) of the ridge dimensions was selected on the basis of the cut-off analysis only, the hexagonal structures allow for a larger range of parameters (see Section 3.2). For this reason, full-wave analyses have been carried out to determine the OVs for the three and six-ridge configurations. Figure 15 shows the contour plots of ARC averaged (from broadside direction to maximum scan angle) as a function of the ridge parameters for a 5% (Figure 15a,b) and 13.2% (Figure 15c,d) bandwidth. This allows to check which ridge dimension combination provides the best starting point in terms of matching. In this case, a similar performance is obtained whether the design considers the whole band or only a percentage of it. As expected, the design curves for a 5% bandwidth provide a larger useful region in terms of parameter combina  
Figure 14: XPD for both quadridge designs. tions. For both configurations, where the optimal (blue) region is apparent, the selected OVs are shown in Table 2. The manufacturing constraint of 1 mm for the ridge width has also been considered. The optimum values for the hexagon with three ridges (Figure 15c) are 1 mm and 1.75 mm for width and height, respectively. The resultant cut-off frequencies are 17.4 GHz and 30.7 GHz for the fundamental and higher-order modes. In the case of six ridges (Figure 15d), optimum values are 1 mm and 1.7 mm for width and height, respectively. In this case, cut-off frequencies are 17.4 GHz and 21.7 GHz for the fundamental and higher-order modes. The S-Parameters of both hexagonal designs in the E-, D- and H-planes at different scan angles are reported in Figure 16a,b. No scan blindness is visible due to the high cut-off frequency of the third mode. The worst value is around \(-4\) dB in the H-plane when scanning to the 60\({}^{\circ}\) limit (dotted red line). For smaller scan angles (green and yellow lines), the six-ridge solution provides an improvement of about 2 dB in the E- and D-planes above 18.5 GHz. Scattering parameters for the other fundamental mode (S\({}_{22}\), i.e., the other polarization) are similar to S\({}_{11}\), therefore they are not reported for brevity. S\({}_{21}\) for linear polarization (not shown) is in the order of \(-10\) dB and its effect is accounted for in the circular polarization data. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Values (mm)** \\ \hline \(p\) & Period & 9 \\ \(a\) & Aperture side & 8 \\ \hline \(h_{R1}\) & Ridge height (three ridges) & 1.75 \\ \(w_{R1}\) & Ridge width (three ridges) & 1 \\ \hline \(h_{R2}\) & Ridge height (six ridges) & 1.7 \\ \(w_{R2}\) & Ridge width (six ridges) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design parameter values for the ridged hexagonal apertures. Figure 15: ARC averaged on the scan range as a function of the ridge parameters. OV indicates the optimum value based on the manufacturing constraints. (**a**) Hexagon with three ridges in a 5% bandwidth; (**b**) hexagon with six ridges in a 5% bandwidth; (**c**) hexagon with three ridges in a 13.2% bandwidth; and (**d**) hexagon with six ridges in a 13.2% bandwidth. As far as circular polarization is concerned, both hexagonal designs (Figure 16c,d) have similar behaviours. The three-ridge solution has a flatter behaviour due to the higher cut-off frequency of the third mode. The overall levels are similar to the evanescent quadridge in Figure 10d. The coupling for circular polarization for the hexagon with three ridges is shown in Figure 16e. The values are below \(-10\) dB for scanning angles up to \(30^{\circ}\) for each principal plane. A value of about \(-5\) dB is obtained at the scanning angle limit for \(\phi=90^{\circ}\) (dotted red line). Comparing Figure 16c,d, it can be observed that the six-ridge solution exhibits a worse performance at the upper frequency limit of the band (i.e., some bandwidth reduction).