## 3 Waveguide Design: Cut-Off Analyses
### Hexagonal Ridged Waveguides  
Hexagonal designs are presented to overcome the limitations found in the quadridge structure. Although in triangular lattice the inter-element spacing and the dimensions of the waveguide are larger than in the previous case, it is still necessary to load with ridges. The larger size of the hexagonal waveguide with respect to the quadridge is also helpful as far as manufacturing is concerned. Two configurations have been considered with three [29] and six ridges [18] (Figure 6). The former (three ridges) leads to a simple manufacturing process, whereas better polarization performance is expected from the latter (six ridges) due to its two-fold symmetry. Figure 5: Electrical field distribution for the quadridge modes. Figure 6: Proposed hexagonal ridged apertures for triangular lattice array. As reported in Section 2, the waveguide size \(a\) is 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 7, as a function of the width \(w_{R}\) and the height \(h_{R}\) of the ridges. As far as the three-ridged waveguide is concerned (Figure 6(a),c), it can be seen that with a parameter combination of \(w_{R}=2.5\) mm and \(h_{R}=3\) mm (close to the upper-right corner) the maximum bandwidth approaches 3:1. Therefore, in principle this waveguide can cover both a Ka-band downlink (17.7-20.2 GHz) and Ku-band downlink (10.7-12.75 GHz). This aspect again highlights the importance of the cut-off analyses in Figure 7 as design curves for future design activities. The dashed line corresponds to the lower limit of the operative bandwidth (\(F_{min}=17.7\) GHz). Therefore, the useful ridge parameter combinations are on the right-hand side of Figure 7 (\(f_{c_{1,2}}<F_{min}\)).The dash-dotted line showing the upper limit (\(F_{max}=20.2\) GHz) is only visible in Figure 6(d) for the six-ridge configuration. As mentioned above, the three ridges do not provide limitations related to the higher-order mode (\(f_{c_{3}}>F_{max}\) for all the considered parameter values). The benefits of the triangular symmetry (\(60^{\circ}\) rotational symmetry) in terms of very high cut-off frequencies/resonances for the higher-order mode are also used in [30]. It is apparent that both solutions in Figure 6 allow for a larger range of parameters \(w_{R}\) and \(h_{R}\) with respect to the quadridge (see Figure 4). In this case, the optimal value (OV) has been determined using the full-wave simulations presented in Section 4.2. The electrical field distributions for the two fundamental degenerate modes (Mode 1 and Mode 2) and the higher-order mode (Mode 3) of the hexagonal waveguides with three and six ridges are shown in Figure 8. The hexagonal geometry with three ridges does not have two-fold symmetry. However, the two fundamental modes (1 and 2) have the same propagation constant (Figure 7(a),b). The reason for such a degeneration is shown in Figure 9. It should be noted that Mode 2 (whose field configuration in Figure 7(b) looks quite different from the field configuration for Mode 1 in Figure 7(a)) can be expressed as the difference between two \(120^{\circ}\)--rotated versions of Mode 1 (titled \(1^{\prime}\) and \(1^{\prime\prime}\)). It can also be shown that Mode 1 can instead be expressed as the sum of \(1^{\prime}\) and \(1^{\prime\prime}\). In conclusion, all the presented  
Figure 7: Cut-off frequency map for the hexagonal waveguide. The green region indicates all possible valid solutions for the operative bandwidth. The point labelled OV refers to the optimal value based on the manufacturing constraints and the radiating performance reported in Section 4.2.  
field configurations arise from 120\({}^{\circ}\) rotations of the same mode (or a combination of them), therefore, they have the same propagation constant (degenerate modes). Of course, any other linear combinations of the degenerate modes are possible.