## 2 Antenna Synthesis and Modeling
### Radiating Element Analysis  
The radiating element selected for the array implementation is depicted in Figure 5. It is a multilayer, dual-polarized, slot-coupling-fed radiator comprising two stacked circular patches in order to span the required 11.9% bandwidth at 29.25 GHz. The stack-up, top to bottom, comprises a larger 2.79 mm diameter circular patch, etched on the bottom of a 0.127 mm thick Rogers RT-5880 (\(\epsilon_{r}=2.2\), \(\tan\delta=0.0009\)) substrate [34]; a 0.508 mm thick  
Figure 4: Capability of scanning of the proposed architecture. (**a**) Contour plot showing SLL in scanning for the optimized antenna; SLL computation is limited to a scan angle 57\({}^{\circ}\) from broadside since eventual higher lobes out of this scan would fall outside Earth’s surface. (**b**) 3D polar plot for a scan angle \(\theta_{0}=52.25^{\circ}\), \(\phi_{0}=27^{\circ}\); yellow crosshair indicates highest SLL position within antenna footprint, red crosshair highest SLL outside footprint.  
Figure 3: Attained broadside pattern. (**a**) Pattern drawn in polar coordinates; yellow crosshair shows the location of maximum-level side lobe; a small inset with a zoomed area shows a close-up of the broadside region (\(\theta=0^{\circ},\phi=0^{\circ}\)) of the radiation pattern. (**b**) \(\phi=0^{\circ}\) plane, (**c**) \(\phi=15^{\circ}\) plane. Due to the symmetry of the array lattice, the cuts \(\phi=30^{\circ}\), \(\phi=60^{\circ},\ldots\phi=150^{\circ}\) are identical to the cut \(\phi=0^{\circ}\), while the cuts \(\phi=15^{\circ}\), \(\phi=45^{\circ},\ldots\phi=165^{\circ}\) are identical to the cut \(\phi=15^{\circ}\).  
Rogers RT-5880 layer; a smaller 2.045 mm diameter circular patch, etched on the top of a 0.254 mm thick Rogers RO4350B (\(\epsilon_{r}=3.66\), \(\tan\delta=0.0037\)) substrate (Figure 5a) [35]; a ground plane with a pair of C-shaped feeding slots; a 0.245  mm thick Rogers RT6006 (\(\epsilon_{r}=6.15\), \(\tan\delta=0.0027\)) substrate [36]; the feeding micro-strip which, finally, leads to coaxial connectors (Figure 5a). The two feed points guarantee polarization agility. The full description of the element is out of the scope of this paper. The reader can refer to [37] for the design of multilayer, slotted feed radiating elements. Element performances, in terms of matching and patterns, are shown in Figure 5c,d. For brevity, results concern only one of the two feeding ports. A slight asymmetry in the pattern on one of the two planes is due to the asymmetry in the slot position beneath the patches.  
Furthermore, Monte Carlo simulations were performed to quantify the sensitivity of the antenna's radiation characteristics to manufacturing tolerances, adopting the same statistical framework used in the SLL study. Realistic variability ranges for both material properties and fabrication processes were applied. The resulting reflection-coefficient statistics appear as a family of \(|S_{11}|\) curves; Figure 5c shows the mean response and the \(\pm 1\sigma\) envelopes. An analogous analysis for the radiation pattern produces a set of gain profiles, but, in this case, the dispersion is negligible. Indeed, all patterns differ by at most a fraction of a decibel, hence only the mean curve for the two principal radiation cuts is reported in Figure 5d.