## 4 The 3D Array: Methodology and Numerical Results
### The 3D Array
At \(\varphi=45^{\circ}\), the worst performances are recorded, limiting directivity FoV to \(\pm 24^{\circ}\). Axial ratio FoV in Figure 13b performs slightly better in the \(\pm 31^{\circ}\) range. As previously stated, the maximum LHCP directivity among all the steering angles is 42.3 dBiC. It is used as reference value for computing directivity loss \(\Gamma\) for the 3D array. For the 3D array, only T1 is shown in Figure 13, since T2 is a scaled version of T1 and the same performances were simulated. Overall, the array has directivity FoV of \(\pm 70^{\circ}\) in the worst case for \(\varphi_{s}=0^{\circ}\) (violet line). Directivity FoV for the best case when \(\varphi_{s}=45^{\circ}\) is \(\pm 77^{\circ}\). The reduction in directivity FoV with respect to the linear case (\(\pm 81^{\circ}\)) is explained by the fact that the 3D array is not conformal in the azimuth. Although not exactly matching the performance obtained in the linear case, configuration T1 still shows a remarkable improvement from the planar case, considerably extending the directivity and total field of view of \(\pm 46^{\circ}\). Moreover, axial ratio FoV is also strongly improved from \(\pm 31^{\circ}\) of the planar array to \(\pm 76^{\circ}\). Analyzing directivity loss \(\Gamma\), T1 has a maximum directivity of 37.55 dBiC which is translated in a loss of \(\Gamma=4.75\) dB. Discussing these results, the planar reference array has higher directivity at zenith but also higher scan losses through the scan range. From a link budget perspective, the maximum throughput is higher for the reference case, but it is reduced strongly as soon as the steering angle diverges from zenith, causing a drop in performances. On the other hand, the presented 3D architecture has a lower throughput due to the loss in the directivity but constant performance in the extended FoV region. A bigger array, T2, in Figure 12b is therefore built by considering the footprint of the planar array. As previously mentioned, the number of elements is increased to \(N=6180\). Same performances are found for the T2 case with directivity loss reduced to 2.79 dB and a maximum directivity of 39.51 dBiC, which is very close to the original requirements of 40 dBiC. As last, in Figure 14, the FoV map for both directivity (Figure 14a,c) and axial ratio (Figure 14b,d) is shown. The contour highlights the requirements of \(-3\) dB for the directivity scan loss and 3.5 dB for the axial ratio. The achieved total FoV for both the architectures is depicted through the red rectangle, \(\pm 24^{\circ}\) for the planar and \(\pm 70^{\circ}\) for the 3D array. In Figure 15, it can be clearly noticed how the achieved FoV is strongly broadened for the 3D array. As an example, spherical directivity LHCP at \(\theta_{s}=70^{\circ}\) and \(\varphi_{s}=45^{\circ}\) is shown for the planar and 3D array in T2 configuration. In the zoom area, the spherical directivity is normalized with respect to its maximum among all the steering angles (42.3 dBiC for the planar array and 39.51 dBiC for the 3D T2 array). Contour lines are depicted every \(-3,-10,-20\) dB to highlight the directivity roll-off. Even if in the 3D architecture the pointing angle is not fully centered in the \(-3\) dB region, the beam shape is still well symmetric. On the other hand, in the planar array the beam is strongly wider and not anymore in the desirable \(-3\) dB region.