# Analysis and Design of an X-Band Reflectarray Antenna for Remote Sensing Satellite System
## 4 Field Distribution of the Feeding Antenna on the Aperture of the Reflectarray  
In order to obtain the required beam pattern of the reflectarray, the amplitudes and phase distribution on the plane of the reflectarray should be determined. In previous studies of reflectarray antennas, the field distribution is presented as a simple analytical approximation based on \((\cos\theta)^{q}\), where the value of \(q\) is chosen to obtain the corresponding approximate radiation pattern of the feeding antenna. In addition, the phase distribution of the field on the aperture of the reflectarray is calculated in terms of the distance from the center of the feeding point to each point on the aperture of the reflectarray. In this paper, this approximation is replaced by directly calculating this field distribution numerically using the commercially EM simulation software HFSS. The advantage of this method is that it does not require any assumptions. Figure 3a shows the 2D distribution of the magnitude of the complex total field on an aperture located at a distance \(F=352.9\) mm from the center of the fed element of the Yagi-Uda antenna. It should be noted that this is only a calculation plane and it does not represent any additional boundary to the simulation problem. This plane and the feeding antenna are included inside a common radiation boundary in the simulation process. This distribution can be presented as a radial function of the magnitude around the \(z\)-axis. Figure 3b shows the 1D distribution of the normalized amplitudes along the \(x\)-axis at the plane of the reflectarray. In addition, one can also obtain the corresponding phase distribution using the argument for this complex field, as shown in Figure 3b. These amplitude and phase distributions are discretized along the proposed grid of the reflectarray to obtain the amplitude and phase of the incident field on each element of the reflectarray.  
The next step is to use this amplitude distribution to find the required phase on each element to obtain the required radiation pattern. Then, it is required to design each element to introduce a phase reflection added to the phase distribution of the feeding element, such that the total phase on this element equals the required phase, to obtain the required radiation pattern.  
Figure 3: Complex field distribution of the feeding element at the plane of the reflectarray: (a) 2D representation, (b) normalized amplitude and phase along the \(x\)-axis.  
For the case of a pencil beam radiation pattern, the required total phase distribution on the aperture of the reflectarray antenna can be obtained analytically, as shown in Equation (4). Thus, the reflection phase of each reflecting element can be obtained by subtracting the phase of the feeding element from the required phase distribution of the aperture of the reflectarray. For the present reflectarray structure mentioned in Section 2 and the proposed Yagi-Uda feeding element, the required reflection phases on the aperture of the reflectarray for both broadside and tilted pencil beams with a tilting angle \(15^{\circ}\) are shown in Figure 4a,b, respectively. It should be noted that the total number of reflecting elements is equal to 316 unit cells, arranged uniformly in a planar grid with \(M\times N=20\times 20\) elements.  
By comparison, for the case of a shaped beam, such as a flat-top beam, it is required to determine the required phase distribution using an optimization algorithm because it cannot be determined directly using Equations (1)-(4), as in the case of broadside or tilted pencil beams. However, the phase distribution of the broadside pencil beam can be considered as a good starting point for the proposed optimization process to obtain the corresponding phase distribution for the flat-top beam.