# Enhanced Efficiency and Reduced Side Lobe Level Convex Conformal Reflectarray
## 3 Feed Design Procedure  
As it is well known, the use of an optimal illumination system is central for increasing the gain of a reflector [27] or a reflectarray [1,4]. However, the feed is designed or selected \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline  & \multicolumn{2}{c}{**SLL, E-Plane [dB]**} & \multicolumn{2}{c}{**SLL, H-Plane [dB]**} & \multicolumn{2}{c}{**Gain [dB]**} & \multicolumn{2}{c}{**Efficiency [\%]**} \\
**Freq.** & **Opt.** & **Ref.** & **Opt.** & **Ref.** & **Opt.** & **Ref.** & **Opt.** & **Ref.** \\
**[GHz]** & **Feed** & **[20]** & **Feed** & **[20]** & **Feed** & **[20]** & **Feed** & **[20]** \\ \hline
29.2 & \(-12.5\) & \(-9.6\) & \(-21\) & \(-17.4\) & \(32.7\) & \(29\) & \(46.5\) & \(18.9\) \\ \hline
30 & \(-17.3\) & \(-15.1\) & \(-22.8\) & \(-19.2\) & \(33.4\) & \(29.7\) & \(51.8\) & \(21.1\) \\ \hline
30.8 & \(-16\) & \(-14.9\) & \(-25.2\) & \(-19.6\) & \(33.5\) & \(29.2\) & \(50.4\) & \(17.9\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measured performance of the proposed CRA and comparison with the configuration in [20]. Figure 1: Side view of an off-set convex reflectarray. (as most frequently occurs in RA cases) firstly by defining the constraints it must satisfy and then evaluating its stand-alone performance. Moreover, in [28], numerical results are shown as the first demonstration on a co-design procedure, where the feed horn for an off-set reflector is optimized, also taking, in some way, into account the presence of the reflector itself: at each step of the optimization process, a different horn configuration is simulated with an accurate but also highly computationally expensive FEM approach and then the obtained radiation pattern is used in the physical optics based simulation of the reflector. The innovative procedure described in the following is based on the idea of designing an ad hoc feed to optimize the entire antenna system performance: it differs not only from what it is commonly shown in the literature on RAs, but also from what it is proposed in [28], where numerical results are shown as first demonstration of a co-design procedure, according to which the feed horn for an off-set reflector is optimized, also taking, in some way, into account the presence of the reflector itself since the way in which the optimization of the entire antenna system is performed is different. In the present case, the definition of the optimum feed for enhancing the CRA performance cannot be managed as a classical optimization problem in which the entire antenna system is simulated with a full-wave approach since it would require a huge computational time. For this reason, it is necessary to decouple the overall electromagnetic problem into two main steps: contrarily to what is done in [28], first a full-wave analysis of the convex reflectarray, with a simplified model of the feed, is performed and used to optimize the performance of the entire system; then, the actual structure for the feed, with the same features, is derived. In the first step of the procedure, the antenna configuration in Figure 1 is considered, but the feed is substituted with an open-ended circular aperture, with radius \(R\), located in the same position. The overall system is simulated with CST Microwave Studio, computing the radiation patterns \(\mathbf{E}_{i}(\theta,\varphi)\) for single mode incidents separately on the open-ended circular aperture. The total radiation pattern is, therefore, equal to the sum of \(\mathbf{E}_{i}(\theta,\varphi)\) weighted by the relevant complex amplitude of the modes on the circular aperture \(a_{i}\), i.e., the following:  
\[\mathbf{E}_{tot}(\theta,\varphi)=\sum_{i=1}^{N}a_{i}\mathbf{E}_{i}(\theta, \varphi) \tag{1}\]  
where the sum is limited to the number \(N\) of modes above the cut-off at the horn aperture, and \(i\) represent both the azimuthal and radial indexes, \(m\) and \(n\). This simplified model does not consider the effect of the external profile of the horn and the modal coupling at the aperture.