## 4 Results for Reflectarray Synthesis
This maximum value of compliance was reached when the minimum value of the Frobenius norm was also produced by the same algorithm (i.e., the sequence produced by the hybridized algorithm was located in a region where \(\mathcal{R}\) and \(\mathcal{M}\) were extremely close, and consequently, there were more points \((u,v)\) which satisfied the requirement templates). Thus, the hybridized algorithm offered the best reduction in separation between the pre-projected co-polar component and projected co-polar component of the radiated electric fields, with significantly improving compliance. Figure 5a,b displays the color map of the magnitude of the co-polar component of the radiated electric field with respect to the minimum thresholds \(T_{\text{min}}(\theta,\phi)\) (i.e., the difference given by \(||E_{\text{cp},k}^{\text{rad}}||-T_{\text{min}}(\theta,\phi)\)) in the final iteration (k = 1000). These results are shown (a) for Algorithm 1 ([PERSON]'s method) and (b) for the hybridized algorithm. In this color map, the yellow color indicates a difference of 0 dB or higher, while the blue color indicates a difference of \(-1\) dB or less. Therefore, the found non-compliance regions are shown as non-yellow regions (holes in the radiation pattern). Although the hybridized algorithm yielded a higher fraction of the overall radiation pattern compared with [PERSON]'s method, the results remain insufficient due to a hole (non-compliance region) near \((u,v)=(0.29,-0.02)\) at the central region of the coverage region. Let us implement the synthesis-only phase in sequential stages with different illuminations at this point. Figure 5(a),b illustrates the Frobenius norm which measured the separation between the pre-projected co-polar component and the projected co-polar component of the radiated electric fields versus the number of iterations (i.e., \(J(E^{\text{rad}}_{\text{cp},k})=\|P_{\mathcal{R}}(E^{\text{rad}}_{\text{cp},k}) -E^{\text{rad}}_{\text{cp},k}\|\)) obtained using Algorithm 1 ([PERSON]'s method) and the hybridized algorithm, respectively, plotted against the number of iterations in each stage. In Figure 5(a), Algorithm 1 shows convergent Frobenius norm values in all stages, with these values increasing in each stage. Figure 5(b) shows a significant reduction (roughly 1000 times less) in the separation after 100 iterations, when Algorithm 2 (separating hyperplanes) was activated. This fact can be interpreted as the sequence produced by Algorithm 1 getting stuck in trap points. On the other hand, note that before 100 iterations, the Frobenius norm values increased with respect to the previous stage for both algorithms. This fact was expected because, in later stages, the reflectarray elements far from the center of the antenna are more illuminated, and consequently, they generate radiated electric fields which do not comply with the requirement template of \(\mathcal{M}\). Figure 4: (**a**) Frobenius norm of the difference between the pre-projected and projected co-polar component versus the number of iterations (i.e., \(J(E^{\text{rad}}_{\text{cp},k})=\|P_{\mathcal{R}}(E^{\text{rad}}_{\text{cp},k}) -E^{\text{rad}}_{\text{cp},k}\|\)). (**b**) Fraction of the coverage region which fulfills \(T_{\text{min}}\leq\left|E^{\text{rad}}_{\text{cp}}\right|\leq T_{\text{max}}\). Figure 5: Color map of the difference \(||E^{\text{rad}}_{\text{cp},k}||-T_{\text{min}}(\theta,\phi)\) in the final iteration (k = 1000). Results are shown for (**a**) Algorithm 1 ([PERSON]’s method) and (**b**) the hybridized algorithm. Figure 6(a),b depicts the fraction of the whole coverage region which met the mission requirements versus the number of iterations in each stage, as obtained by (a) Algorithm 1 ([PERSON]'s method) and (b) the hybridized algorithm. In Figure 6(a), slight improvements in compliance can be observed in each stage, with a final convergent compliance of 0.75 in the final stage. In Figure 6(b), slight improvements in compliance can also be observed in each stage, but a final convergent compliance of 0.938 was reached.