## 5 Comparison between Infinite and Finite Approaches
In the infinite approach, a Floquet port has been used instead of the perfect matched layer (PML) because it provides better absorption at higher scan angles [32]. Up to 18 Floquet modes have been considered in both lattices for CST. The number of modes in HFSS has instead been increased to 30 and 42 for the square and triangular lattice, respectively, to achieve an attenuation of 50 dB for the last considered mode [33]. In both infinite models, the distance between the Floquet port and the aperture has been set to 0.25\(\lambda_{0}\) at 17 GHz. Both the S-Parameters for ARC and the active element pattern (AEP) for the scan loss and XPD were computed with both simulators. The unit cell has been replicated in both directions to obtain a finite array model. PML boundary conditions have been applied at a certain distance from the model in all directions including the back. Both the number of elements and the PML distance will be discussed in the convergence analysis reported hereafter. Regarding the S-Parameters, the calculation of the ARC has been performed by summing the self-contribution of the central element and the coupling from all the other ones with the required weight for beam scanning [34]. The excitation of the central element has been considered to compute the embedded element pattern (EEP). This approach has been preferred over the scan of the whole finite array due to the limited number of elements in a row/column. Figure 20a,b show the convergence analysis of the finite array model and its comparison to the infinite CST one. The most critical condition is \(\theta=60^{\circ}\) where the scan blindness is present at about 20 GHz for the quadridge in the square lattice. This phenomenon is difficult to model with a small finite array simulation where the contribution of the farther elements is missing. Nevertheless, both the red and cyan dashed curves (11 \(\times\) 11 and 13 \(\times\) 13, respectively) approach the purple curve (infinite) quite well. The convergence analysis stopped at 13 \(\times\) 13 elements due to peak memory (150 GB) and time consumptions (9 h in a Intel Xeon with two processors at 3 GHz). It can be observed that it is not only necessary to increase the number of elements, but also the PML distance (see yellow and red dashed curves where the PML distance is 0.25\(\lambda\) and 1.5\(\lambda\), respectively, at 17 GHz) to  
achieve convergence between the finite and infinite simulations. Such distance is required to attenuate the evanescent wave interaction occurring at high scan angles. Figure 21 shows the ARC for the E-plane (\(\phi=0^{\circ}\)) for both considered designs using the three simulation methods described above. For the quadridge (Figure 21a), the infinite models are in very good agreement at all scan angles. The finite array simulation (\(11\times 11\)) follows the same trend but with the expected discrepancy at 20 GHz. As for the hexagonal waveguide (Figure 21b), the larger differences occur when the reflection level of the infinite array is below \(-15\) dB. An overall agreement is achieved between the curves, confirming that the results are consistent. This analysis is also useful to assess the minimum number of elements that should be considered for array prototyping. This number is smaller than \(48\times 48\) which was discussed in Section 2. Figure 22 shows the scan loss across the D-plane at three frequencies ranging from 17.7 GHz to 20.2 GHz within a \(\pm 60^{\circ}\) scan angle. The three different methods are presented with solid, dashed and dotted curves (see legend). A remarkable agreement is visible for both designs. A similar comparison is provided for the XPD in Figure 23. Good agreement between the three methods has been achieved for the quadridge. As far as the hexagonal structure is concerned, the infinite methods are in perfect agreement. The finite method shows a consistent trend. However, a significant ripple is visible on such an EEP. Additional simulation efforts should be carried out to achieve a higher degree of convergence.