## 1 Introduction
Let us consider the \(x\) and \(y\) components of an incident electric field \(E_{\mathrm{0x}}^{\mathrm{inc}}\) and \(E_{\mathrm{0y}}^{\mathrm{inc}}\) on a reflectarray panel (i.e., the \(z=0\) plane in Figure 1). If the reflectarray is an X-polarized antenna, then the \(\mathcal{R}\) set is define as follows (see Equation (16) in [9]):  
\[\mathcal{R}\equiv \{E_{\mathrm{cp}}^{\mathrm{rad}}\in\mathbb{C}:\exists\varphi(x,y) \in\mathbb{R}\text{ with } \tag{1}\] \[E_{\mathrm{cp}}^{\mathrm{rad}}=jk_{0}\frac{e^{-jk_{0}\sigma}}{2 \pi r}\Big{[}\cos^{2}(\phi)+\sin^{2}(\phi)\cos(\theta)\Big{]}\] \[\iint_{S_{\mathrm{BA}}}E_{\mathrm{0x}}^{\mathrm{inc}}(x,y)e^{i \varphi(x,y)}e^{ik_{0}(\sin(\theta)\cos(\phi)x+\sin(\theta)\sin(\phi)y)}dxdy\},\]  
where \(E_{\mathrm{cp}}^{\mathrm{rad}}\) depicts the main component (i.e., co-polar component) of the radiated electric field, \(\{r,\theta,\phi\}\) are the spherical coordinates of the observation point in the reflectarray coordinate system (see Figure 1), \(k_{0}\) is the working wavenumber in the vacuum, \(\varphi(x,y)\) is the phase introduced by the reflectarray element, \(S_{\mathrm{RA}}\) is the surface of the aperture of the reflectarray panel, and \(j\) is the imaginary unit. On the other hand, given thresholds \(T_{\mathrm{min}}(\theta,\phi)\) and \(T_{\mathrm{max}}(\theta,\phi)\) to the magnitude of \(E_{\mathrm{cp}}^{\mathrm{rad}}(\theta,\phi)\), the \(\mathcal{M}\) set of requirement templates is defined as follows (see Equation (14) in [9]):  
\[\mathcal{M}\equiv \Big{\{}E_{\mathrm{cp}}^{\mathrm{rad}}\in\mathbb{C}:T_{\mathrm{ min}}\leq\Big{|}E_{\mathrm{cp}}^{\mathrm{rad}}\Big{|}\leq T_{\mathrm{max}} \Big{\}}. \tag{2}\]  
In the intersection approach, the search for intersections between both sets is carried out using the MAP. The simplest form of the MAP originated from [PERSON] [24], whose method was later expanded in [25] to address the case of two closed convex sets. (For a more detailed study on alternating projection methods, we refer interested readers to [26]). However, if any set is non-convex, then convergence to an intersection point (if one exists) is not assured. In such scenarios, it is feasible to observe convergence to pairs of points which yield a local minimum of the distance between the two sets [26]. One of the points in the pair is known as a trap point [27], and these are produced due to a lack of convexity in the sets. Therefore, the AGP method has to be considered in order to take into account the non-convexity of sets. In [9], it is shown that \(\mathcal{R}\) and \(\mathcal{M}\) are non-convex sets in general. To avoid trap points, the authors of [9] applied three algorithms to the intersection approach and compared the convergence results with the [PERSON] method (i.e., the simplest form of the MAP), showing significant improvements. These algorithms rely on separating hyperplanes and the decomposition method in polar cones [27]. The first algorithm works to avoid trap points and is expected to converge to an intersection point. The second algorithm only works to avoid trap points. Therefore, the authors of [9] proposed a third hybrid algorithm where the algorithm based on decomposition in polar cones is applied during the initial iterations (to avoid trap points), followed by the algorithm based on separating hyperplanes for the remaining iterations. However, 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 several trap points. This trap situation depends on the initial point used to start working with the algorithms. In [28], a progressive synthesis via the MAP in different stages was proposed, with a first stage where a small reflectarray panel with a few reflectarray elements is synthesized using the successive projections method.