## 5 Conclusions  
When a large reflectarray needs to be synthesized, several trap points (local minima of the distance between both sets) can appear. In this situation, it is possible to fall into oscillatory behavior between the several trap points. One way to avoid this oscillatory behavior is to perform phase-only synthesis with sequential stages of different illuminations. The mathematical formalism which supports this strategy has been shown for the first time in this work. This mathematical formalism is based on the relation between two sets of electric fields: the reflected electric field and the co-polar component of the radiated electric field. The relation of these sets is formed by means of a 2D inverse discrete Fourier transform, which is an isomorphism that preserves the Frobenius norm of each set. Therefore, a reflectarray with a low level of edge illumination will produce lower values for the Frobenius norm of the radiated electric fields than those obtained using a high level of edge illumination. This ensures fewer local minima (trap points) when a low level of edge illumination is used. The phase-only synthesis of satellite reflectarrays was performed using algorithms of AGP to avoid trap points with sequential stages of different illuminations. These AGP algorithms were as follows: [PERSON]'s algorithm (whose convergence is not guaranteed for non-convex sets due to the possible appearance of trap points), an algorithm based on separation hyperplanes (whose sequence of points only leaves trap points and converges to the intersection point), an algorithm based on the decomposition method in polar cones (whose sequence of points leaves trap points), and a hybridized algorithm which works like the decomposition method in polar cones during the first iterations (to leave possible trap points) and the algorithm based on separation hyperplanes during the rest of the iterations (to move even further away from possible trap points and converge to the intersection of the sets). The obtained results were compared with those obtained using a single stage. In the latter case, although the hybridized algorithm provided higher compliance values than those obtained using the von Neumann method, the results were insufficient. This was due to the existence of a hole (non-compliance region) in the center region of the coverage. This hole existed because of the presence of local minima (trap points) in the optimization  
Figure 8: Color map of the difference \(||E_{\text{cp}}^{\text{rad}}||-T_{\text{min}}(\theta,\phi)\) in the final iteration (k = 1000) for the final stage. Results are shown for (**a**) Algorithm 1 ([PERSON]’s method) and (**b**) the hybridized algorithm. problem, which has a high number of degrees of freedom (i.e., a high number of reflectarray elements). Bearing this in mind, the reflectarray was iteratively synthesized in different stages where the illumination gradually increased at the edges of the reflectarray (i.e., the degrees of freedom were gradually increased). After applying the hybridized algorithm with the synthesis process in different stages, the previous hole in the center region of the coverage disappeared. These results demonstrate that the hybridized algorithm presented here, which incorporates sequential stages of different illuminations, markedly outperformed the conventional von Neumann algorithm cited in existing publications for the phase-only synthesis of satellite reflectarray antennas. As a result, the future research directions will be to implement the hybridized algorithm with sequential stages of different illuminations to other applications, such as near-field phase-only synthesis applications. Conceptualization, R.F. and R.E. ; methodology, R.F. and R.E. ; software, R.F. ; validation, R.F. and R.E. ; formal analysis, R.F. and R.E. ; investigation, R.F. and R.E. ; resources, R.F. ; writing--original draft preparation, R.F. ; writing--review and editing, R.F. and R.E. ; visualization, R.F. and R.E. ; supervision, R.E.