# Multi-Beam Circular Polarized Reflectarray on Parabolic Reflector by Variable Rotation Technique
## 2 Required Phase Shift to Tilt Dual CP Focused Beams in Specular Directions  
Figure 1 shows the side view of a reflectarray on an arbitrary reflector surface with the phase center of the feed at the source point. In this section, we present an easy technique for computing the required phase distribution to be added in reflected waves by the rotating elements in order to tilt dual CP focused beams in specular directions. Ray optics theory is considered in this formulation. Let us consider the impinging wave on the \(i\)-th element of the reflectarray with propagation vector given by \(\mathbf{k}_{\text{inc}}=k_{0}[\sin(\theta_{\text{inc}})\cos(\phi_{\text{inc}}) \mathbf{\hat{x}}^{\prime}+\sin(\theta_{\text{inc}})\sin(\phi_{\text{inc}}) \mathbf{\hat{y}}^{\prime}-\cos(\theta_{\text{inc}})\mathbf{\hat{z}}^{\prime}]\) where \(k_{0}\) is the vacuum wavenumber at the working frequency and \(\theta_{\text{inc}}\), \(\phi_{\text{inc}}\) are the spherical angular coordinates of the impinging wave in the local reference system on the \(i\)-th element (see \(x^{\prime}\)-\(z^{\prime}\) axis in Figure 1). According to the law of reflection, a reflected wave will be produced with the propagation vector \(\mathbf{k}_{\text{ref}}=k_{0}[\sin(\theta_{\text{inc}})\cos(\phi_{\text{inc}}) \mathbf{\hat{x}}^{\prime}+\sin(\theta_{\text{inc}})\sin(\phi_{\text{inc}}) \mathbf{\hat{y}}^{\prime}+\cos(\theta_{\text{inc}})\mathbf{\hat{z}}^{\prime}]\). In this way, the phase of the reflected wave will be \(\mathbf{k}_{\text{ref}}\cdot\mathbf{r}\), where \(\mathbf{r}\) is the vector position of the \(i\)-th element of the reflectarray with respect to the global reference system shown in Figure 1 (see the \(x\)-\(z\) axis in Figure 1). Now, let us consider a desired reflected wave with propagation vector given by \(\mathbf{k}_{\text{beam},1}=k_{0}[\sin(\theta_{\text{beam},1})\cos(\phi_{\text{ beam},1})\mathbf{\hat{x}}^{\prime}+\sin(\theta_{\text{beam},1})\sin(\phi_{ \text{beam},1})\mathbf{\hat{y}}^{\prime}+\cos(\theta_{\text{beam},1})\mathbf{ \hat{z}}^{\prime}]\), where \(\theta_{\text{beam},1}\) and \(\phi_{\text{beam},1}\) are the spherical angular coordinates of the desired beam direction in the local reference system on the \(i\)-th element (see again the \(x^{\prime}\)-\(z^{\prime}\) axis in Figure 1). In order to obtain the desired reflected wave with the desired beam direction, the \(i\)-th element should introduce to the reflected wave. \[\mathbf{\Delta k}\cdot\mathbf{r}=\left(\mathbf{k}_{\text{beam},1}-\mathbf{k}_ {\text{ref}}\right)\cdot\mathbf{r}. \tag{1}\]  
On the other hand, let us consider a reflected wave with propagation vector \(\mathbf{k}_{\text{beam},2}=k_{0}[\sin(\theta_{\text{beam},2})\cos(\phi_{\text {beam},2})\mathbf{\hat{x}}^{\prime}+\sin(\theta_{\text{beam},2})\sin(\phi_{ \text{beam},2})\mathbf{\hat{y}}^{\prime}+\cos(\theta_{\text{beam},2})\mathbf{ \hat{z}}^{\prime}]\) (Figure 1), where \(\theta_{\text{beam},2}\) and \(\phi_{\text{beam},2}\) are the spherical angular coordinates of the specular beam direction in the local reference system on the \(i\)-th element (see again the \(x^{\prime}\)-\(z^{\prime}\) axis in Figure 1). According to Figure 1, this direction of the reflected wave is the specular direction of the direction given by \(\mathbf{k}_{\text{beam},1}\) with respect to the propagation vector \(\mathbf{k}_{\text{ref}}\). In order to obtain this reflected wave with this specular beam direction, the \(i\)-th element should introduce to the reflected wave the additional phase given by:  
\[-\mathbf{\Delta k}\cdot\mathbf{r}=\left(\mathbf{k}_{\text{ref}}-\mathbf{k}_ {\text{The parabolic surface focuses the beams in CP while the VRT is applied to introduce the additional phases given in (1) and (2) to the phase of the reflected waves to produce two focused beams in specular directions. We consider an offset parabolic reflector shown in Figure 2 with parabola vertex at the coordinate origin.