# Multi-Beam Circular Polarized Reflectarray on Parabolic Reflector by Variable Rotation Technique
## 3 Design and Optimization of The Rotating Element of the Reflectarray
According to (4), when \(R_{x^{\prime}x^{\prime}}=-R_{y^{\prime}y^{\prime}}\), a pure reflected LHCP wave is obtained (there is no cross-polarization contribution). Moreover, the added phase shift by the rotating element to the reflected electric field of this pure LHCP wave is \(-2\alpha\). Indeed, when \(R_{x^{\prime}x^{\prime}}=-R_{y^{\prime}y^{\prime}}\), the reflected field is:  
\[\mathbf{E}_{\text{ref}}^{\text{LHCP}}=e^{\mathrm{i}(-2\alpha+\angle R_{x^{ \prime}x^{\prime}})}|R_{x^{\prime}x^{\prime}}|E_{0}(\mathbf{\dot{\kappa}}^{ \prime}+j\mathbf{\dot{y}}^{\prime})e^{-\mathrm{i}\mathbf{\dot{y}}_{0}z^{\prime}}. \tag{5}\]  
Figure 3: (**a**) Definition of the unit cell. The period is 7.6 mm. The components \(x^{\prime}\) and \(y^{\prime}\) of the incident field are parallel to the larger and shorter cross arms of the figure. (**b**) Side view of the periodic multilayer structure. The bottom dielectric layer is made of Rohacell material (permittivity \(\varepsilon_{r}=1.12\) and loss tangent \(\tan\delta=0.002\)) and the upper layer is made of Arlon material (permittivity \(\varepsilon_{r}=2.3\) and loss tangent \(\tan\delta=0.002\)). Note that when there is no rotation, the phase shift of the reflected electric field of the pure LHCP wave is equal to the phase of the \(R_{x^{\prime}x^{\prime}}\) reflection coefficient. The phase of \(R_{x^{\prime}x^{\prime}}\) offers an additional degree of freedom to modify the phase of the reflected wave in (5). This additional degree of freedom can be exploited by means of the use of conductive cross. In the case of conductive cross, roughly independent phase control of reflection coefficients \(R_{x^{\prime}x^{\prime}}\) and \(R_{y^{\prime}y^{\prime}}\) is possible from independent variations of the lengths L\({}_{1}\) and L\({}_{2}\). This independent phase control provides more flexibility in the design process since the condition \(R_{x^{\prime}x^{\prime}}=-R_{y^{\prime}y^{\prime}}\) can be satisfied for several values of the phase of \(R_{x^{\prime}x^{\prime}}\). These several values of the phase of \(R_{x^{\prime}x^{\prime}}\) can be used in order to correct phase errors. For example, a small phase correction can be added on both polarizations to improve the beam shaping. In a similar way, when the impinging wave is RHCP, and \(R_{x^{\prime}x^{\prime}}=-R_{y^{\prime}y^{\prime}}\), the reflected electric field of the contribution of the RHCP wave will suffer a phase shift with opposite sign, \(+2\alpha\), as stated in [15]. When the condition \(R_{x^{\prime}x^{\prime}}=-R_{y^{\prime}y^{\prime}}\) is not exactly fulfilled, else \(R_{x^{\prime}x^{\prime}}=-R_{y^{\prime}y^{\prime}}e^{j\Delta\varphi}\), where an \(\Delta\varphi\) is an error in the phase, then the cross-polarization contribution in (4) will be given by:  
\[\mathbf{E}_{\text{ref}}^{\text{RHCP}}=\frac{1}{2}E_{0}R_{x^{\prime}x^{\prime}} \Big{(}1-e^{j\Delta\varphi}\Big{)}(\hat{\mathbf{x}}^{\prime}-j\hat{\mathbf{y} }^{\prime})e^{-jk_{0}z^{\prime}}. \tag{6}\]  
Note that this cross-polarization contribution does not depend on the rotation angle, and only depends of the error in the phase \(\Delta\varphi\). So, the amplitude of the cross-polarization contribution increases as this phase error \(\Delta\varphi\) increases. Recall that (6) was obtained under the assumption that cross-polarization reflection coefficients \(\text{R}_{x^{\prime}y^{\prime}}\) and \(\text{R}_{y^{\prime}x^{\prime}}\) are negligible. This equation predicts the cross-polarization level in the circularly polarized reflected wave. A parametric study of the rotating element of the unit cell shown in Figure 3 was carried out in order to check the performances of the previously shown co-polar and cross-polarization contributions. This parametric study was carried out with the periodic structure module of the newFASANT suite [21]. In this parametric study, the working frequency was 19.7 GHz.