# Analysis and Design of an X-Band Reflectarray Antenna for Remote Sensing Satellite System
## 2 Reflectarray Antenna  
The proposed reflectarray antenna is composed of an array of reflecting elements arranged on planar circular disk in front of a Yagi-Uda feeding antenna, as shown in Figure 1. It is designed to be operating at a center frequency of 8.5 GHz, which is suitable for the down-link for remote sensing satellite systems. The center of the array is placed at the origin. An \(x\)-polarized Yagi-Uda feeder is centered at \(\left(x_{f},\,y_{f},\,z_{f}\right)=\left(0,\,0,\,F\right)\), where \(F\) is the focal distance of the proposed reflectarray. The focal to diameter ratio of the proposed reflectarray antenna is \(F/D\)=1. The diameter of the proposed reflectarray antenna is 352.9 mm which corresponds to 10\(\lambda_{0}\) at the center frequency. The array elements of the reflectarray are distributed periodically on a square grid of length 17.6 mm, which corresponds to \(\lambda_{0}/2\) at the center frequency. The incident field on each reflecting elements at a certain angle can be locally considered as a plane wave with a phase proportional to the distance from the phase center of the feeder to each element. In order to produce a focused beam, the field must be reflected from each unit cell with an appropriate phase shift. This phase shift is adjusted independently for each element to produce a progressive phase shift distribution of the reflected field that produces a focused beam in the required direction. For a planar array having \(M\times N\) elements arranged on a rectangular grid on the \(x-y\) plane with a uniform separation, the array factor \(AF(\theta,\varphi)\) can be written as [15]:  
\[AF(\theta,\varphi)=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}\mathrm{A}_{m,n}e^{jk(md_{x }u+nd_{y}v)} \tag{1}\]  
where \(\mathrm{A}_{m,n}\) is the complex excitation of the element \(\left(m,n\right)\), \(k\) is the free space wavenumber, \(u=\sin\theta\cos\varphi+\beta_{x}\) and \(v=\sin\theta\sin\varphi+\beta_{y}\), and \(\beta_{x}\) and \(\beta_{y}\) are the progressive phase shift between array elements in the \(x\) and \(y\) directions, respectively. For the case of a pencil beam oriented in the direction \(\left(\theta_{o},\varphi_{o}\right)\), the values of these progressive phase shifts can be expressed as:  
\[\beta_{x}=-\sin\theta_{o}\cos\varphi_{o} \tag{2}\]  
\[\beta_{y}=-\sin\theta_{o}\sin\varphi_{o} \tag{3}\]  
where \(d_{x}\) and \(d_{y}\) are the spacing between each two successive elements in the \(x\) and \(y\) directions, respectively. Thus, the phase shift on the \(mm\)th element \(\Delta\varphi_{m,n}\) is obtained as  
Figure 1: A reflectarray antenna fed by a Yagi-Uda antenna. \(\Delta\varphi_{m,n}=\beta_{x}+\beta_{y}\). For the case of a reflectarray, the magnitude of the excitation at the elements \(|A_{m,n}|\) is determined by the amplitude distribution of the fields due to the feeding element on the aperture of the reflectarray. On the other hand, the total phase at the \(mm\)th element is the summation of the phase of the field distribution of the feeding element plus the phase distribution of the reflection coefficient on the aperture of the reflectarray. The key point in the design of the reflectarray antenna is to determine the required total phase distribution for the obtained amplitude distribution by the feeding element to obtain the required radiation pattern. The next step is the implementation of the reflecting elements of the reflectarray antenna to verify this total phase distribution. For simple radiation patterns such as a broadside pencil beam or a tilted pencil beam, this phase distribution can be obtained analytically in a closed form as follows:  
\[\Delta\varphi_{m,n}=2\pi N^{\prime}-k\bigg{(}r_{m,n}-\overset{\rightarrow}{R \cdot}\overset{\cdot}{r_{o}}\bigg{)} \tag{4}\]  
where \(N^{\prime}=1,2,3,\ldots\), \(r_{m,n}\) is the distance from the feed to each array element, \(\hat{R}\) is the position vector from each element to the array center \((0,0,0)\) and \(\hat{r_{o}}\) is the position vector in the direction of the main beam of the reflectarray.