# Design of a 3-Bit Circularly Polarized Reconfigurable Reflectarray
## 2 Materials and Methods
### Design of 3-Bit Circularly Polarized Reconfigurable Reflectarray
According to the calculation formulas of aperture efficiency (1), (4), and (5), the relationship between the efficiency of the reflectarray antenna and the height of the feed source can be obtained. According to the definition of the far-field condition, the height of the feed antenna needs to be greater than 12 mm. As shown in Figure 7a, when the aperture efficiency is the highest, the height, H, is selected to be 270 mm. Then, the H remains unchanged, and the offset feed angle is optimized again according to the formula, and the result shown in Figure 7b is obtained. It can be seen that the aperture efficiency of the antenna decreases with the increase in the offset feed angle. In order to minimize the occlusion of the feed source and ensure that the antenna has a large aperture efficiency, the final selection of the offset feed angle is \(20^{\circ}\). Assuming that the target direction of the main beam of the designed reflectarray antenna is set as (\(\theta_{m}\), \(\varphi_{m}\)), according to the classical array synthesis theory, in order to meet the requirements of the expected operation of the reflectarray antenna, the electromagnetic wave received and reflected by each unit in the reflectarray needs to form an equal phase plane on the target beam direction (\(\theta_{m}\), \(\varphi_{m}\)). That is, the comprehensive superposed phase, \(\varphi_{r}(i,j)\), of electromagnetic wave reflected by the reflecting unit needs to be satisfied:  
\[\varphi_{r}(i,j)=-k_{0}\big{(}x_{ij}\mathrm{sin}\,\theta_{m}\mathrm{cos}\, \varphi_{m}+y_{ij}\mathrm{sin}\,\theta_{m}\mathrm{sin}\,\varphi_{m}\big{)}+ \varphi_{0} \tag{6}\]  
where (\(x_{ij}\), \(y_{ij}\)) represents the location of the ijth reflecting unit, \(\varphi_{0}\) is a constant reference phase used to optimize the distribution of states among the array units, \(k_{0}=2\pi/\lambda\) is the free space wavenumber, and \(f\) is the operating frequency of the antenna. Assuming that the electromagnetic wave emitted by the feed irradiates to the ijth unit on the reflectarray, introducing a delay phase, \(\varphi_{i}(i,j)\), and the compensating phase provided by the jith unit is the value of \(\varphi(i,j)\), then the three phases satisfy the relationship:  
\[\varphi_{r}(i,j)=\varphi_{i}(i,j)+\varphi(i,j) \tag{7}\]  
Assuming that the distance from the feed source to the ijth reflecting unit is \(d_{ij}\) and the position coordinate of the feed source is (\(x_{f}\), \(y_{f}\), \(z_{f}\)), the delay phase \(\varphi(i,j)\) of the  
Figure 7: The efficiency of a reflectarray antenna varies with the position of the feed: (**a**) height and (**b**) offset feed angle. electromagnetic wave incident through the feed source to the reflector surface can be expressed as  
\[U_{f}(\theta,\varphi)=\cos^{2\eta_{f}}\theta\;(0\leq\theta\leq\frac{\pi}{2}) \tag{2}\]0  
\[U_{f}(\theta,\varphi)=\cos^{2\eta_{f}}\theta\;(0\leq\theta\leq\frac{\pi}{2}) \tag{2}\]1  
From the above formula, it can be seen that the compensating phase, \(\varphi(i,j)\), provided by the jith reflecting unit is  
\[U_{f}(\theta,\varphi)=\cos^{2\eta_{f}}\theta\;(0\leq\theta\leq\frac{\pi}{2}) \tag{2}\]2  
After the above analysis and design, the array is first modeled in HFSS software 21.1 according to the design scheme, and the structure of the array is shown in Figure 8a. The xoz and yoz planes are selected as the beam scanning plane of the antenna. Combined with the spherical coordinate system analysis, the angle of the main beam pointing in the scanning plane is expressed as \(\theta_{m}\), and the array is arranged at 3.5 GHz. The compensated phase required by the reflecting unit at each position of the reflectarray is calculated according to Formula (10), and then the calculated compensated phase is quantized by the 3-bit compensated phase quantization-processing results in Table 1 to obtain the working state of each unit. When \(\theta_{m}\) is \(0^{\circ}\) (the state distribution of array units is shown in Figure 8b), the state distribution diagram of units under each beam scanning angle can be calculated.