# Multi-Beam Circular Polarized Reflectarray on Parabolic Reflector by Variable Rotation Technique
## 3 Design and Optimization of The Rotating Element of the Reflectarray
Since the incident angle \(\theta_{\text{inc}}\) in most elements of the parabolic reflectarray can be approximated by the incident angle at the center of the parabolic reflectarray, the incident angles \(\theta_{\text{inc}}=~{}13.03\text{deg}\) and \(\phi_{\text{inc}}=~{}0\text{deg}\) were considered in the parametric study. Since variations of the lengths L\({}_{1}\) and L\({}_{2}\) produce roughly independent variations on the phase of \(R_{x^{\prime}x^{\prime}}\) and \(R_{y^{\prime}y^{\prime}}\), these lengths can be adjusted to comply with the condition \(R_{x^{\prime}x^{\prime}}=-R_{y^{\prime}y^{\prime}}\). There are many pairs of lengths (L\({}_{1}\), L\({}_{2}\)) which provide \(180^{\circ}\) phase difference. The selected lengths were L\({}_{2}\) = 4.9 mm and L\({}_{1}\) = 6.025 mm. A parametric study of rotating of the cross was performed considering the arms' lengths obtained in the previous adjustment. Figure 4a shows the phases of the reflected electric fields of the co-polar circularly polarized waves with respect to the rotation angle. The phase for RHCP increased with the rotation angle with an opposite sense to the phase for LHCP, as predicted in (5). The amplitude of the reflected electric field of the cross-polar circularly polarized wave for the RHCP incident wave is shown in Figure 4b. Similar results were obtained for the LHCP incident wave. In this figure we notice that the amplitude of the reflected electric field of the cross-polar circular polarized wave was only negligible when the rotating angle was about 0, 90, or \(-90\) degrees (i.e., a pure RHCP reflected wave is produced). However, for 45 or \(-45\) degrees it was far from being negligible. This behavior can be explained considering the views of the periodic lattice for two values of the rotational angle: 0 and 45 degrees. We can compare the reflected waves of a linearly polarized incident wave whose components were parallel to the cross arms for rotating angle values of 0 and 45 degrees. If for this case we compare the geometry visualized in Figure 5a,b it is obvious that we had two different periodic structures and therefore it could happen that the geometry of Figure 5c would not give a 180 deg phase shift between the linear co-polar reflection components (i.e., the condition \(R_{x^{\prime}x^{\prime}}=-R_{y^{\prime}y^{\prime}}\) is not satisfied), although the geometry was adjusted to give that shift when the rotating angle was 0\({}^{\circ}\) (i.e., the cross was optimized for a 0\({}^{\circ}\) rotation angle but not for 45 deg rotation). To avoid the issues found in the previous parametric study, a few pairs of lengths (L\({}_{1}\), L\({}_{2}\)) were optimized in order to fulfill the condition \(R_{x^{\prime}x^{\prime}}=-R_{y^{\prime}y^{\prime}}\). The first pair of lengths was optimized when there was no rotation. The obtained values were L\({}_{1}\) = 6.0 and L\({}_{2}\) = 4.9 mm. According to previous discussion and Figure 5, the geometry lattice at the rotation angle of 22.5 and 45 degrees differed significantly from the geometry lattice for the non-rotated cross. So, the values of L\({}_{1}\) = 6.0 and L\({}_{2}\) = 4.9 mm would provide significant phase error values \(\Delta\varphi\) and similar high cross-polarization level to those shown in Figure 4b. So, length optimization was imposed at the rotation angles of 22.5 and 45 degrees to avoid the high values of cross-polarization level. In this way, the second and third pairs of lengths were optimized to fulfill the condition \(R_{x^{\prime}x^{\prime}}=-R_{y^{\prime}y^{\prime}}\) for rotation angles 22.5 and 45.0 degrees. The obtained values were L\({}_{1}\) = 6.14, L\({}_{2}\) = 4.91 mm for the rotation angle of 22.5 deg and L\({}_{1}\) = 6.24, L\({}_{2}\) = 4.92 mm for the rotation angle of 45.0 deg. We checked that the optimized values of the pair (L\({}_{1}\), L\({}_{2}\)) for rotation angles of 22.5 and 45.0 degrees also roughly fulfilled the condition \(R_{x^{\prime}x^{\prime}}=-R_{y^{\prime}y^{\prime}}\) for rotation angles of \(-\)22.5 and \(-\)45.0 degrees.