## 4 Design for Dual Polarization
The cell dimensions have been set to P\({}_{\rm X}=\) P\({}_{\rm Y}=7.5\) mm (\(\uplambda\)/2 at 20 GHz) in order to avoid grating lobes. The width of the dipoles is 0.5 mm, and the separation between the centers of adjacent parallel dipoles is 1 mm. Figure 10: Design for dual polarization: **(a)** phase increment (in degrees) required with respect to the initial polarization and **(b)** pattern contours of 40 dBi and 47.5 dBi for the beams produced at 20 GHz by the bifocal DRAA, using solid lines for polarization 1 and dashed lines for polarization 2. The surface reflection at the reflectarray cell has been modeled using a matrix formulation that relates the Cartesian components of the incident and reflected tangential electric field. The reflectarray cell is characterized by the reflection matrix (\(R\)), formed by the co-polar and cross-polar reflection coefficients associated to both polarizations:  
\[\left( \begin{array}{c}E_{X}^{ref}\\ E_{Y}^{ref}\end{array} \right)=\left( \begin{array}{cc}R_{XX}&R_{XY}\\ R_{YX}&R_{YY}\end{array} \right)\left( \begin{array}{c}E_{X}^{inc}\\ E_{Y}^{inc}\end{array} \right) \tag{6}\]  
where the co-polar reflection coefficients for the \(X\) and \(Y\) polarizations are represented by \(R_{XX}\) and \(R_{YY}\), respectively, and the cross-polar coefficients are \(R_{XY}\) and \(R_{YX}\), respectively. The values of the co- and cross-polar coefficients have been obtained by means of an in-house electromagnetic code that applies the Method of Moments in the spectral domain (SD-MoM). The SD-MoM analysis code assumes that the reflectarray cell is placed in a periodic environment (which is a common approach for the analysis and design of reflectarray antennas [14; 15; 16; 17; 18; 19; 20]) and takes into account the real incidence angles on the cell. More details about the in-house SD-MoM code and its validation can be found in [28]. The dual-polarization capability of the reflectarray cell can be explained as follows. The lengths of the dipoles oriented in the direction of the \(x\)-axis will control the phase shift introduced in \(X\)-polarization (the phase of the \(R_{XX}\) coefficient), while the lengths of the dipoles oriented in the direction of the \(y\)-axis will adjust the phase shift introduced in \(Y\)-polarization (the phase of \(R_{YY}\)). The lengths of the \(x\)-oriented and \(y\)-oriented dipoles will be separately adjusted to provide the required phase shift in each LP, considering that the lengths of the lateral dipoles are scaled by a factor of 0.78 with respect to the lengths of the central dipoles (\(l_{\rm A1}=0.78l_{\rm A2}\) and \(l_{\rm B1}=0.78l_{\rm B2}\)). The use of orthogonal sets of dipoles results in a very low coupling between the \(X\) and \(Y\) polarizations [28]. The independent phase control in each polarization provided by the reflectarray cell allows the proposed antenna to generate two independent beams in orthogonal polarization with the same feed. For a correct design of the reflectarray antenna, a common requirement is that the phase of the co-polar reflection coefficients (\(R_{XX}\) and \(R_{YY}\)) must present a smooth variation in a range of at least 360\({}^{\circ}\), in order to be able to implement the phase distributions shown in Figure 5. The phase of the \(R_{XX}\) and \(R_{YY}\) coefficients and the amplitude of the cross-polar coefficients (\(R_{XY}\) and \(R_{YY}\)) at 20 GHz under oblique incidence \(\theta_{i}=30^{\circ}\), \(\varphi_{i}=30^{\circ}\) (which are about the maximum incidence angles in the main RA) have been depicted in Figure 11b vs. the length of the central dipole in each polarization (note that the lengths of the lateral dipoles are also varied, according to the relation \(l_{\rm A1/B1}=0.78l_{\rm A2/B2}\)). As can be seen, the reflectarray cell provides a satisfactory performance, with around 400\({}^{\circ}\) phase range in both polarizations and cross-polar amplitudes below \(-\)23 dB for most of the dipole lengths (with a maximum of \(-\)18 dB).