# Enhanced Efficiency and Reduced Side Lobe Level Convex Conformal Reflectarray
## 3 Feed Design Procedure
Note that if step or slope discontinuities are used and the antenna is fed by the \(TE_{11}\) mode, only the modes with \(m=1\) (i.e., \(TE_{1n}\) and \(TM_{1n}\)) are excited and, therefore, considered in the previous sum. The aim of the optimization process is to determine the radius \(R\) of the aperture, the number \(N\) of modes and the set of \(a_{i}\) that allows to maximize the CRA gain and to minimize its cross-pol and SLL. This is obtained through the minimization of the function as follows:  
\[f(\mathbf{a})=\alpha_{1}(G^{ideal}-G(\mathbf{a}))+\alpha_{2}X_{pol}(\mathbf{ a})+\alpha_{3}SLL(\mathbf{a}) \tag{2}\]  
where \(G^{ideal}\) is the gain of an equivalent uniform aperture, and \(G(\mathbf{a})\) is the actual antenna gain; \(\alpha_{1}\), \(\alpha_{2}\) and \(\alpha_{3}\) are weights varying between 0 and 1 to give more or less emphasis to one term or to another. Finally, \(\mathbf{a}\) is the vector collecting the complex amplitudes that have to satisfy the further condition that the overall input power is 1 W (i.e., \(\sum_{i=1}^{N}|a_{i}|^{2}=1\)). The radius \(R\) and the number of modes \(N\) are handled in a different way; for what concerns \(R\), its increase improves the system performance, but a too-large aperture will correspond to a too-bulky horn, and therefore, \(R\) is selected as the best compromise between these two opposite constraints. The choice of \(N\) is made instead, optimizing the antenna performance with an increasing number of modes and stopping when the improvement is below a fixed threshold. Following these criteria, the best value for \(R\) results to be 18 mm, while only the first two modes (i.e., \(TE_{11}\) and \(TM_{11}\)) significantly contribute to the antenna radiation pattern. Once these parameters are determined, it is possible to proceed with the optimization. The maps in Figure 2 show the iso-level curves of the system gain, maximum cross-polarization and side lobe level discrimination that are reported as a function of the magnitude (\(|\gamma|\)) and phase (\(\angle\gamma\)) of the ratio \(\gamma\) of the \(TM_{11}\) to the \(TE_{11}\) modal coefficient: \(\gamma=a_{TM11}/a_{TE11}\), evaluated at \(f_{0}\) (30 GHz). The best compromise, by considering the overall operative frequency band \(29\div 31\) GHz, is represented by \(\gamma=0.67e^{|250^{\circ}}\). According to the curves in Figure 2, this choice will guarantee that the whole antenna is characterized by a gain of at least 33 dB (see plot on the top of Figure 2), a maximum SLL (in the horizontal plane) approximately equal to \(-23\) dB and a cross polarization level of around \(-37\) dB. The numerical and experimental characterizations of the entire antenna system confirm these expected results. It is interesting to note that, from the design maps reported in Figure 2, it is possible to derive that a Potter horn solution (\(TE_{11}\) and \(TM_{11}\) modes in phase in power ratio \(84\%/16\%\)[29]) would lead to worse performance (gain around 30.5 dB, maximum cross-polarization protection 28 dB). Moreover, such a horn, with aperture radius \(R=18\) mm, would present six modes with \(m=1\) above cut-off at 30 GHz (i.e., \(TE_{(1,r)}\) and \(TM_{(1,r)}\) with \(r=1,2,3\)); then, a quite long slope would be necessary to minimize the excitation of the other modes. Figure 2: _Cont._  
The second step of the developed procedure involves the horn actual synthesis. In principle, different horn architectures can be exploited, but the smooth wall one is preferred since it can be more easily realized and it is less sensitive to manufacturing tolerances. The geometry was obtained by applying an optimization scheme aimed to obtain a suitable profile for the horn generatrix, starting from a linear one. To this end, a set of \(N_{p}+2\), equidistant points \(P_{n}\) are initially located on the generatrix profile, where the initial point (\(P_{0}\)) and the final one (\(P_{N_{p}+1}\)) are placed to fit the radii \(R_{w}\) of the waveguide connected to the horn and \(R\) of its circular aperture (see Figure 3).