# Multi-Beam Circular Polarized Reflectarray on Parabolic Reflector by Variable Rotation Technique
## 3 Design and Optimization of The Rotating Element of the Reflectarray  
We consider a left-handed circular polarized (LHCP) impinging plane wave on a multilayer periodic structure with the unit cell shown in Figure 3. This unit cell is made of two dielectric layers. The bottom 2 mm dielectric layer comprised of Rohacell material is used as separator with the ground plane. The upper 0.127 mm dielectric layer comprised of Arlon material is used to host a conductive cross printed on its lower limit as shown in Figure 3b. The reflectarray cells have a quasi-periodicity, with a period in mm of \(7.6\times 7.6\) (\(0.499\times 0.499\) in wavelengths). This reflectarray element was used  
Figure 2: Scheme of offset parabolic reflector. The feed location and incident angle of the central element are shown. in [22] in order to demonstrate experimentally the proof of the concept of generating two beams in specular directions for CP, and satisfactory results were obtained. The direction of the impinging plane wave is the \(-\mathbf{\dot{z}}^{\prime}\) direction and its electric field is expressed as:  
\[\mathbf{E}_{\text{inc}}^{\text{LHCP}}=E_{0}(\mathbf{\dot{\kappa}}^{\prime}-j\mathbf{ \dot{y}}^{\prime})e^{\mathrm{i}\mathbf{\dot{y}}_{0}z}, \tag{3}\]  
where \(\mathbf{\dot{\kappa}}^{\prime}\) and \(\mathbf{\dot{y}}^{\prime}\) are the unitary vector parallel to the \(x^{\prime}\) and \(y^{\prime}\) axes as shown in Figure 3a. These unitary vectors are parallel to the larger and shorter cross arms of Figure 3a. According to [19] and [23], when the LHCP wave is incident on the periodic structure, the electric field of the reflected wave has two contributions of circularly polarized waves: a co-polar contribution given by an LHCP wave and a cross-polarization contribution given by a right-handed circularly polarized (RHCP) wave. Both circularly polarized waves propagate in the positive direction \(+\mathbf{\dot{z}}^{\prime}\),  
\[ \begin{array}{l}\mathbf{E}_{\text{ref}}=\frac{1}{2}e^{-j2\alpha}E_{0}\Big{(} R_{x^{\prime}x^{\prime}}-R_{y^{\prime}y^{\prime}}\Big{)}(\mathbf{\dot{\kappa}}^{ \prime}+j\mathbf{\dot{y}}^{\prime})e^{-\mathrm{i}\mathbf{\dot{y}}_{0}z^{\prime}}\\ \quad\quad\quad+\frac{1}{2}E_{0}\Big{(}R_{x^{\prime}x^{\prime}}+R_{y^{\prime} y^{\prime}}\Big{)}(\mathbf{\dot{\kappa}}^{\prime}-j\mathbf{\dot{y}}^{\prime})e^{-\mathrm{i} \mathbf{\dot{y}}_{0}z^{\prime}}\end{array}  \tag{4}\]  
where \(R_{x^{\prime}x^{\prime}}\) and \(R_{y^{\prime}y^{\prime}}\) are the reflection coefficients for the co-polar components in directions of the \(x^{\prime}\) and \(y^{\prime}\) axes, respectively, as shown in Figure 3a (i.e., in the directions of the crossed arms) and \(\alpha\) is the angle of rotation of the crosses shown in Figure 3a. In (4) it is assumed that cross-polarization reflection coefficients \(R_{x^{\prime}y^{\prime}}\) and \(R_{y^{\prime}x^{\prime}}\) are negligible. This assumption is based on the symmetry of the reflectarray element. For example, let us consider an impinging plane wave with its electric field linearly polarized in \(x^{\prime}\). In the cross arm oriented in the \(x^{\prime}\) direction, due to the symmetry of the arm regarding the \(x^{\prime}\) axis, there is no reason to expect a contribution to the reflected electric field with \(y^{\prime}\) polarization. The same rationale can be applied for the cross arm oriented in the \(y^{\prime}\) direction. As will be shown later, this assumption leads to an equation which predicts the level of the cross-polarization contribution of a circularly polarized wave with enough accuracy for practical purposes. Note that the first contribution in (4) is the co-polar reflected electric field of an LHCP wave, since the impinging waves propagate in the \(-\mathbf{\dot{z}}^{\prime}\) direction and the reflected wave propagates in \(+\mathbf{\dot{z}}^{\prime}\) direction. The second contribution in (4) is the cross-polarization reflected electric field of an RHCP wave.