## 3 Waveguide Design: Cut-Off Analyses
### Quadridge Waveguide  
The geometry of the quadridge waveguide [13; 14] and its parameters can be seen in Figure 3. As written above, the waveguide's side \(a\) is 6.8 mm. The iso-level curves of the cut-off frequencies for the two fundamental degenerate modes \(f_{c_{1,2}}\) and the higher-order mode \(f_{c_{3}}\) are shown in Figure 4a,b, respectively, as a function of the width \(w_{R}\) and the height \(h_{R}\) of the ridges. Even if the waveguide supports two degenerate modes (one for each polarization) the intervals \(f_{c_{1,2}}\) and \(f_{c_{3}}\) can be referred to as monomodal bandwidths.  
The dashed line corresponds to the lower limit of the operative bandwidth (\(F_{min}=17.7\) GHz). Therefore, the useful ridge parameter combinations is on the right-hand side of Figure 4a (\(f_{c_{1,2}}<F_{min}\)). On the other hand, the dash-dotted line in Figure 4b shows the upper limit of the operative bandwidth (\(F_{max}=20.2\) GHz). The dashed curve is reported again in Figure 4b to obtain the parameter region where only the two fundamental degenerate modes are propagating within the frequency band (17.7-20.2 GHz). It can be seen that low values of \(w_{R}\) provide a larger available range of ridge height, \(h_{R}\). However, such regions should be further reduced considering a manufacturing constraint of 1 mm for the minimum ridge width. An optimal value (OV) has been chosen to be almost equally spaced from both cut-off frequency limits. The resulting dimensions are \(h_{R}\) equal to 1.5 mm and \(w_{R}\) equal to 1 mm, providing a cut-off frequency of 17.6 GHz (see Figure 4a) for the fundamental degenerate modes \(f_{c_{1,2}}\) and 20.5 GHz (see Figure 4b) for the third mode \(f_{c_{3}}\). The performance of this waveguide as a radiating element will be shown in Section 4.1.  
Figure 4: Cut-off frequency map for the quadridge waveguide. The green region indicates all possible valid solutions for the operative bandwidth. The point labelled OV refers to the optimal value based on both the manufacturing constraints and equal spacing between the frequency limits for Fmin and Fmax (see Section 4.1).  
Figure 3: Proposed quadridge waveguide aperture for square lattice array.  
From the previous analysis, it is apparent that the quadridge configuration does not provide large margins between the operative band and the corresponding cut-off frequencies. This is consistent with the significant bandwidth limitations for this type of element shown in Section 4.1. For this reason, the hexagonal waveguide with three and six ridges is studied in the next subsection. It should be noted that Figure 3(a,b) can be also used to perform the design of quadridge waveguides with different operative frequencies (not necessarily Ka downlink). The reported cut-off data allow for a wider combination of parameters.  
The electrical field distribution for the two fundamental degenerate modes (Mode 1 and Mode 2) and the higher-order mode (Mode 3) of the quadridge are shown in Figure 5[27]. The electrical field distribution is computed using CST Studio Suite 2021 instead of alternative numerical codes for ridge waveguides [28]. Because the aperture has two-fold symmetry, the first two modes have the same propagation constant, also called degenerate modes (the first mode is aligned along the horizontal axis, Figure 4(a), while the second mode is on the vertical axis, Figure 4(b)).