## 3 Phase-Only Synthesis Through Sequential Stages with Different Illuminations  
First, let us define the intensity of illumination as the magnitude of the tangential electric field on the reflectarray panel (i.e., \(z=0\) plane). Figure 1 shows a reflectarray with \(N_{x}\times N_{y}\) reflectarray elements at a size of \(a\times b\), with different coordinate systems and the feeder coordinate system \(\left\{x_{\text{F}},y_{\text{F}},z_{\text{F}}\right\}\), and reflectarray coordinate system \(\left\{x_{\text{F}},y_{\text{F}},z_{\text{F}}\right\}\) both having spherical coordinates \(\left\{r_{\text{F}},\theta_{\text{F}},\phi_{\text{F}}\right\}\) and \(\left\{r,\theta,\phi\right\}\), respectively. The reflectarray elements are arranged in a rectangular grid, and they are illuminated by a feeder (typically a horn antenna). Different levels of illumination are shown, from a strong intensity at the center to a weak intensity at the edge of the reflectarray panel. Although there is a wide variety of models of illumination of a reflectarray panel with a feeder antenna [30; 31; 32; 33; 34], all models predict a gradually decreasing intensity of illumination from the center to the edge of a reflectarray panel. This typical behavior of the intensity of illumination can be used in the synthesis process in the following iterative way:  
1. The entire synthesis process is divided into several sequential stages. In each stage, the taper of illumination is gradually decreased compared with the previous stage until using the illumination given by the specific feeder for the reflectarray design. 2. The synthesis algorithm operates by using a given phase-shift distribution as the initial point until the process converges. 3. The resultant phase-shift distribution is used as the initial point in the next stage. Since the illumination intensity is gradually decreased in each stage, the reflectarray elements far from the center receive lower illumination intensities than the elements located around the center in previous stages. In this way, the elements far from the center have less weight in the synthesis process, and consequently, the number of degrees of freedom of the problem is reduced in the earlier stages. According to the previous section, when the number of reflectarray elements is reduced, fewer local extrema are expected. Therefore, dividing the process into different stages helps to avoid a wide variety of local extrema compared with solving the entire problem without stage division. In the context of non-convex sets, a lower the number of local extrema involves a lower number of trap points. Let us choose the \(\cos^{q}(\theta)\) model of the radiated electric field generated by the feeder antenna, and let us consider an X-polarized feeder antenna [30]. This model predicts that the incident electric field on the reflectarray panel is given by the following equation [30]:  
\[\mathbf{E}^{\mathrm{F}}(r_{\mathrm{F}},\theta_{\mathrm{F}}, \phi_{\mathrm{F}})=\frac{jk_{0}e^{-jk_{0}r_{\mathrm{F}}}}{2\pi r_{\mathrm{F}}} \{\mathbf{\hat{x}}_{\mathrm{F}}[\cos(\phi_{\mathrm{F}})\sin(\phi_{\mathrm{F} })\cos^{q_{\mathrm{E}}+1}(\phi_{\mathrm{F}})-\] \[\sin(\phi_{\mathrm{F}})\cos(\phi_{\mathrm{F}})\cos^{q_{\mathrm{H }}}(\theta_{\mathrm{F}})]+\hat{\mathbf{y}}_{\mathrm{F}}\Big{[}\sin^{2}(\phi_{ \mathrm{F}})\cos^{q_{\mathrm{E}}+1}(\theta_{\mathrm{F}})+\cos^{2}(\phi_{ \mathrm{F}})\cos^{q_{\mathrm{H}}}(\theta_{\mathrm{F}})\Big{]}\] \[-\hat{\mathbf{z}}_{\mathrm{F}}\sin(\theta_{\mathrm{F}})\sin(\phi_ {\mathrm{F}})\cos^{q_{\mathrm{E}}}(\theta_{\mathrm{F}})\} \tag{19}\]  
where \(\{r_{\mathrm{F}},\theta_{\mathrm{F}},\phi_{\mathrm{F}}\}\) are the spherical angular coordinates of the observation point in the feeder coordinate system and \(\{\mathbf{\hat{x}}_{\mathrm{F}},\hat{\mathbf{y}}_{\mathrm{F}},\hat{\mathbf{z}} _{\mathrm{F}}\}\) are their unit vectors (see Figure 1). When an axially symmetric pattern is desired, the same \(q\) power is selected (i.e., \(q_{\mathrm{E}}=q_{\mathrm{H}}=q\)).