## 4 Aperture Design: Parametric Analyses and Performance Evaluation
### Quadridge Radiating Elements  
The ARC of the open-ended quadridge waveguide reported in Section 3.1, whose parameters are summarized in Table 1, is reported in Figure 10a for linear polarization. The ARC is the relevant physical parameter because it directly accounts for both the self-reflection of the element itself and all the mutual coupling contributions from the other array elements. The ARC strongly depends on the scan angle/plane because the coupled contributions also depend on the phase shift between cells (as reported above, the phase shift is related to the scan angle). The solid, dashed and dotted lines refer to the E- (\(\phi=0^{\circ}\)), D- (\(\phi=45^{\circ}\)) and H-planes (\(\phi=90^{\circ}\)), respectively. The line colours identify the scanning angles. The vertical black dash-dotted lines indicate the cut-off frequencies of the waveguide and the green box highlights the operative bandwidth. The ARC for only one of the two fundamental degenerate modes is shown due to its 90\({}^{\circ}\) rotational symmetry along the propagation axis. It can be seen that the open-ended quadridge waveguide does not provide a satisfactory matching level even at the broadside (green solid line). This is related to the proximity of the lower cut-off frequency (vertical black dash-dotted line). In addition, when scanning up to 60\({}^{\circ}\) in the E-plane (red solid line), scan blindness is observed at 20 GHz, i.e., the signal is completely reflected at this frequency, reducing the bandwidth even further. The other principal plane (dotted red line) instead shows a reflection level of about \(-5\) dB for the overall band. Two sharp features on the S\({}_{11}\) curves are visible above the upper limit of the frequency band. The first one, located at 20.47 GHz (vertical black dash-dotted line), is due to the higher-order mode of the waveguide. It is related to the waveguide geometry, therefore, it is independent of the scan angle. However, it does not appear at \(\phi=90^{\circ}\) (dotted line) because of the symmetry of the structure, i.e., the fundamental mode is not coupled to the third mode along such a plane (H-plane). The second sharp feature is at 20.61 GHz (black arrow) and this is due instead to the appearance of the grating lobes at this frequency (see Equation (1)). Concerning the coupling for linear polarization, it is only significant in the D-plane, where it reaches \(-6.5\) dB when the scan angle goes up to 60\({}^{\circ}\). These curves are not \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Values (mm)** \\ \hline \(p\) & Period & 7.8 \\ \(a\) & Aperture side & 6.8 \\ \(h_{R}\) & Ridge height & 1.5 \\ \(w_{R}\) & Ridge width & 1 \\ \hline \(l_{E}\) & Evanescent section length & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameter values for the quadridge elements. shown for brevity. However, their effect is included in the circular polarization behaviour discussed hereafter. As far as the ARC for circular polarization is concerned, scan blindness decreases by almost 3 dB (red solid line in Figure 10c) because, as stated above, only one of the two linear components is completely reflected. The results shown for circular polarization have been calculated taking into account an ideal polarizer (the parameters simulated for linear polarization have been combined according to Equations (2) and (3)). \[S_{11}^{Circ} =\frac{1}{2}[S_{11}-S_{22}-j(S_{12}+S_{21})] \tag{2}\] \[S_{21}^{Circ} =\frac{1}{2}[S_{21}-S_{12}-j(S_{11}+S_{22})] \tag{3}\]  
To overcome this scan blindness, a square waveguide section can be placed between the end of the ridges and the aperture (see Figure 11). Its effect on the E-plane response is shown in Figure 12 as a function of the length \(l_{E}\). The same phenomenon, also called spurious  
Figure 10: S-Parameters for both quadridge designs. The dashed lines indicate the cut-off frequencies and the green box indicates the operative frequency band.