# Optimization Environment Definition for Beam Steering Reflectarray Antenna Design
## 2 Antenna and Optimization Environment Description
### Performance Parameters  
For each antenna configuration, it is possible to use the Aperture Field Method to compute the radiation pattern of the antenna. This is an approximated approach that provides good results in the identification of the Radiation Pattern for RAs. With respect to a complete FEM solution, the performance worsening is acceptable, and the computational time is a couple of orders of magnitude lower in the aperture field method [25]. Figure 3: Definition of Beam Deviation Factor. Figure 2: Reflection characteristics of the selected patch: the upper diagram shows the reflection amplitude as function of the patch size, while the bottom part shows the angle variation induced by the reflection. Each element of the planar reflector is identified by its position, assuming a reference system located in the plane of the reflector and centered with respect to the antenna:  
\[\vec{r}_{mn}=\left\{ \begin{array}{c}x_{mn}\\ y_{mn}\\ 0\end{array} \right\} \tag{1}\]  
Hence, the distance between each patch and the center of the array is:  
\[r_{mn}=\sqrt{x_{mn}^{2}+y_{mn}^{2}} \tag{2}\]  
The feeder location is identified with spherical coordinates: in this way, the radial coordinate \(d_{f}\) is representative of the amount of energy radiated outside the reflector, and the tilting angles represent the position of the offset feeder that is oriented towards the center of the array. Due to the fact that generally the feeder is tilted only with the \(\theta\) coordinate, its vector position is:  
\[\vec{r}_{f}=\left\{ \begin{array}{c}-d_{f}\sin\theta_{f}\\ 0\\ d_{f}\cos\theta_{f}\end{array} \right\} \tag{3}\]  
Thus, the distance between the feed and each patch is:  
\[r_{fmn}=\sqrt{x_{nm}^{2}+y_{mn}^{2}+z_{f}^{2}} \tag{4}\]  
The two important elements that are used for the evaluation of the radiation pattern are the radiated field from the feeder and the reflection properties of each patch that are characterized by the two parameters \(q_{E}\) and \(q_{H}\). In order to evaluate the field received by the reflector, it is necessary to evaluate the angles \(\phi_{F_{m}n}\) and \(\theta_{F,mn}\) of each patch, seen from the feeder:  
\[\phi_{F,mn}=\arccos\left(\frac{x_{mn}+d_{f}\sin\theta_{f}}{\sqrt{(x_{mn}+d_{f} \sin\theta_{f})^{2}+y_{mn}^{2}}}\right) \tag{5}\]  
\[\theta_{F,mn}=\arccos\left(\frac{d_{f}^{2}+|\vec{r}_{mn}-\vec{r}_{f}|^{2}-r_{ mn}^{2}}{2d_{f}\sqrt{(x_{mn}+d_{f}\sin\theta_{f})^{2}+y_{mn}^{2}}}\right) \tag{6}\]  
The field received by each patch, expressed in spherical coordinates in the feeder reference system, is:  
\[\mathbf{E}^{F}=\left( \begin{array}{c}E_{\theta}^{F}\\ E_{\phi}^{F}\end{array} \right)=\left( \begin{array}{c}j\frac{k_{0}}{2\pi r_{ fmn}}e^{-jk_{0}r_{fmn}}\cdot\cos^{q_{fc}}\theta_{F,mn}\cdot\cos\phi_{F,mn}\\ -j\frac{k_{0}}{2\pi r_{fmn}}e^{-jk_{0}r_{fmn}}\cdot\cos^{q_{fh}}\theta_{F,mn} \sin\phi_{F,mn}\end{array} \right) \tag{7}\]  
where \(q_{fc}\) and \(q_{fh}\) are the two parameters that characterize a feed horn. Then, this field projected in Cartesian coordinates of the feeder reference system is:  
\[\mathbf{E}^{F}=\left( \begin{array}{c}E_{x}^{F}\\ E_{y}^{F}\\ E_{z}^{F}\end{array} \right)=\left( \begin{array}{c}\cos\theta_{mn}\cos\phi_{ mn}E_{\theta}^{F}-\sin\phi_{mn}E_{\phi}^{F}\\ \cos\theta_{mn}\sin\phi_{mn}E_{\phi}^{F}+\cos\phi_{mn}E_{\phi}^{F}\\ -\sin\theta_{mn}E_{\theta}^{F}\end{array} \right) \tag{8}\]Finally, it is possible to calculate the field in the reference system of the reflector:  
\[\mathbf{E}_{mn}^{R}=\left( \begin{array}{c}F_{mn,x}^{R}\\ E_{mn,y}^{R}\end{array} \right)=\left( \begin{array}{c}\cos\theta_{f}E_{x}^{F}- \sin\theta_{f}E_{z}^{F}\\ -E_{y}^{F}\end{array} \right) \tag{9}\]  
The field calculated is the one that each patch receives as input from the feeder. Then it is possible to calculate the reflected field. With this aim, the geometrical characteristics of the patch (design variable of the antenna design problem) should be considered, since they affect the reflection coefficient.