# Design of a 3-Bit Circularly Polarized Reconfigurable Reflectarray
## 2 Materials and Methods
### Design of 3-Bit Circularly Polarized Reconfigurable Reflectarray  
The aperture efficiency of the feed antenna can be expressed as the product of the capture efficiency, \(\eta_{\mathrm{s}}\), and the irradiation efficiency, \(\eta_{\mathrm{i}}\):  
\[\eta_{\mathrm{a}}=\eta_{\mathrm{s}}\cdot\eta_{\mathrm{i}} \tag{1}\]  
The capture efficiency, \(\eta_{\mathrm{s}}\), and irradiation efficiency, \(\eta_{\mathrm{i}}\), are closely related to the feed performance, feed position, unit performance, and unit distribution of the reflectarray antenna. The normalized power pattern function of the feed antenna is fitted with a mathematical model and can be expressed as  
\[U_{f}(\theta,\varphi)=\cos^{2\eta_{f}}\theta\;(0\leq\theta\leq\frac{\pi}{2}) \tag{2}\]  
where \(\theta\) represents the angle between the direction vector of the feed to the reflector and the direction vector of the main beam of the feed antenna. The size of \(q_{f}\) is related to the width of the main lobe of the feed antenna. The larger the \(q_{f}\) is, the narrower the main lobe  
Figure 5: The operating characteristics of 3-bit circular polarization reconfigurable enfecting unit. Right-hand circular polarization: (**a**) reflection amplitude and (**b**) reflection phase. Left-hand circular polarization: (**c**) reflection amplitude and (**d**) reflection phase. of the feed pattern will become. The normalized power pattern of the microstrip patch unit can also be represented by a similar mathematical model:  
\[U_{e}(\theta,\varphi)=\cos^{2q_{e}}\theta\ (0\leq\theta\leq\frac{\pi}{2}) \tag{3}\]  
For a standard microstrip patch unit, the value of \(q_{e}\) is typically 1. Assuming that H is the vertical height from the phase center of the feed source to the reflectarray, \(s\) is the distance between the main beam pointing point and the position of the reflecting unit, and \(D\) is the maximum aperture size of the reflectarray, then the computational formula of the capture efficiency, \(\eta_{\rm s}\), and irradiation efficiency, \(\eta_{\rm i}\), of the reflectarray antenna can be expressed as  
\[\eta_{\rm s}=\frac{2 qf+1}{2\pi}\int_{0}^{2\pi}\int_{0}^{\frac{D}{2}}\frac{H}{ r^{3}}(\frac{r^{2}_{0}+r^{2}-s^{2}}{2r_{0}r})^{2 qf}\rho d\rho d\varphi \tag{4}\]  
\[\eta_{\rm i}=\frac{4}{\pi D^{2}}\frac{\left[\int_{0}^{2\pi}\int_{0}^{\frac{D}{2 }}\frac{1}{r^{1}+qe}(\frac{r^{2}_{0}+r^{2}-s^{2}}{2r_{0}r})^{df}\rho d\rho d \varphi\right]^{2}}{\int_{0}^{2\pi}\int_{0}^{\frac{D}{2}}\frac{1}{r^{2}+2 qe}( \frac{r^{2}_{0}+r^{2}-s^{2}}{2r_{0}r})^{2 df}\rho d\rho d\varphi} \tag{5}\]  
where the values of \(r_{0}\), \(r\), and \(s\) can be calculated by the coordinates of the main beam-pointing point and the position of the reflecting unit. Therefore, in the design of the reflectarray antenna, the number of antenna units and the size of aperture surface can be determined according to the gain index requirements of the antenna, and then the feed antenna with the appropriate \(q_{f}\) value can be selected. While meeting other conditions, the appropriate feed position can be selected. Finally, the structural design of the reflectarray can be completed. Figure 6: Influence of the oblique incidence angle on the operating characteristics of the unit: xoz plane (**a**) reflection amplitude and (**b**) reflection phase; yoz plane (**c**) reflection amplitude and (**d**) reflection phase. The 3-bit circularly polarized reconfigurable reflectarray designed in this study consists of 64 reflecting units arranged in an \(8\times 8\) configuration, working at a central frequency of 3.5 GHz, and the longest size of the array is 350 mm. The \(q_{f}\) and \(q_{e}\) of the circularly polarized feed and reflecting unit's pattern functions are 4 and 1, respectively. In order to reduce the influence caused by the occlusion of the feed source, the offset feed mode is adopted, and the main beam of the feed source always points to the center of the array. Firstly, the offset feed angle is set to \(0^{\circ}\), and the height of the feed source can be optimized.