## 4 Design for Dual Polarization  
The designed bifocal antenna will be illuminated by a cluster of dual-polarized horns. Each horn will generate two adjacent beams in orthogonal polarization in a plane forming around 60\({}^{\circ}\) with respect to the plane \"\(v\) = 0\", in order to produce a complete cellular coverage. The discrimination of the orthogonal polarizations will be performed on the main RA, using reflectarray cells that enable independent phase control in each polarization. Two different phase distributions will be implemented on the main RA (one for each polarization), while the sub-RA will present the same phase distribution for both polarizations (the one shown in Figure 5a). The phases required on the main RA for the orthogonal polarization can be obtained by adding a progressive phase term to the bifocal phases in Figure 5b. The phase increment at each reflectarray cell can be computed starting from the following expression, which provides the phase distribution of the reflected field on the main RA surface to generate a collimated beam in the direction (\(\theta_{b}\), \(\varphi_{b}\)):  
\[\Phi\ (x_{i},y_{i})=-k_{0}\ \text{sin}\ \theta_{b}\ (x_{i}\ \text{cos}\ \varphi_{b}+y_{i}\ \text{sin}\ \varphi_{b}), \tag{3}\]  
Figure 8: Amplitude (in dB) of the incident field on the main RA when the antenna is illuminated from: (a) F\({}_{1}\), (b) F\({}_{3}\) and (c) F\({}_{5}\). Figure 9: (a) Cluster of horns used to illuminate the antenna, including the location of the foci, (b) pattern contours of 40 dBi and 47.5 dBi for the beams produced by the bifocal DRAA at 20 GHz. where \(x_{i}\) and \(y_{i}\) are the coordinates of the \(i\)-element with respect to the main-RA center, and \(k_{0}\) is the propagation constant in a vacuum. Equation (1) can be adapted to work with the \((u_{b},v_{b})\) coordinates associated to the direction of maximum radiation:  
\[\Phi\;(x_{i},y_{i})=-k_{0}\;(x_{i}\cdot u_{b}+y_{i}\cdot v_{b}). \tag{4}\]  
Then, the difference in the required phase shift between two beams whose maximums are located at coordinates \((u_{b1},v_{b1})\) and \((u_{b2},v_{b2})\) will be:  
\[\Delta\Phi\;(x_{i},y_{i})=-k_{0}\;[x_{i}\cdot(u_{b2}-u_{b1})+y_{i}\cdot(v_{b2} -v_{b1})]. \tag{5}\]  
The increment of phase that must be added to the bifocal phases on the main RA to generate the adjacent beams in the orthogonal polarization has been computed cell by cell, using Equation (5) and considering the \((u,v)\) coordinates associated to the maximums of the beams produced by the central feed. According to Figure 9b, the maximum of the central beam is located at \((u_{b1}=0.326,\,v_{b1}=0)\), so the position of the adjacent beam in the orthogonal polarization must be \((u_{b2}=0.332,\,v_{b2}=0.01)\) in order to produce a triangular spot lattice. The resulting progressive phase distribution to be added to the bifocal phases for the original polarization is depicted in Figure 10a. The pattern contours of 40 dBi and 47.5 dBi for the beams generated at 20 GHz by the bifocal DRAA in the two polarizations are presented in Figure 10b, where illumination from the same cluster of horns (now operating in dual polarization) shown in Figure 9a is considered. The proposed design method is valid for operation in dual linear polarization (LP) or dual circular polarization (CP). In the first case, the reflectarray cells on both the sub-RA and the main RA can be composed of orthogonally-arranged sets of three parallel dipoles, as those shown in Figure 11a. This reflectarray cell allows introducing a different phase shift in each LP (with the incident electric field oriented in the direction of the dipoles) by properly adjusting the lengths of the corresponding group of dipoles. The dipoles are printed on a Diclad 880B dielectric substrate that is backed by a metallic ground plane. The thickness of the substrate is 1.524 mm, the permittivity is \(\epsilon_{\rm r}=2.3\) and the loss tangent is tan\(\delta=0.001\).