# Optimization Design for Sparse Planar Array in Satellite Communications
## 2 Sparse Planar Array Optimization Design Model
### Array Antenna Mathematical Underpinnings  
The electrical performance of space-borne phased array antennas is primarily evaluated on the basis of several key metrics, including antenna gain, antenna directivity, antenna pattern, beam scanning range, and antenna polarization. Antenna directivity \(D(\theta,\varphi)\) is a critical parameter that quantitatively quantifies the strength of antenna directivity. The directivity \(D(\theta,\varphi)\) is defined as the ratio of the far-field power density \(S(\theta,\varphi)\) at a distance \(r\) from the antenna to the power density \(S_{0}\) of an ideal non-directivity antenna at the same position with the same radiant power \(P_{r}\). \[D(\theta,\varphi)=\frac{S(\theta,\varphi)}{S_{0}}=\frac{|E(\theta,\varphi)|^{2 }}{|E_{0}|^{2}}. \tag{1}\]  
The physical parameter denoting the ability of an antenna to convert energy effectively is the antenna efficiency. The total antenna efficiency \(e_{0}\) is influenced by the reflection loss at the input of the antenna, conductor loss at the output of the antenna, and medium losses. The expression for the total antenna efficiency is given as  
\[e_{0}=e_{r}e_{c}e_{d}=\Big{(}1-|\Gamma|^{2}\Big{)}e_{cd}, \tag{2}\]  
where \(e_{r}\) represents the corresponding efficiency in the circumstance that energy cannot be fully transmitted to the antenna due to the mismatch between the antenna and the transmission line, which is related to the reflection coefficient \(\Gamma\). \(e_{c}\) represents the efficiency attributed to the energy dissipation caused by the heating of the antenna conductor during its operation. \(e_{d}\) represents the efficiency attributed to the energy dissipation caused by the propagation of electromagnetic waves in a dielectric medium. Typically, the product of these two efficiency factors \(e_{c}\) and \(e_{d}\) is utilized to denote the radiation efficiency \(e_{cd}\) of the antenna. The gain of an antenna is determined through the multiplication of its efficiency with its directivity. \[G(\theta,\phi)=D(\theta,\phi)\cdot e_{0}. \tag{3}\]  
In spaceborne phased array antennas, a variation in the steering angle \(\theta_{m}\) during beam scanning results in a corresponding change in the antenna gain. When \(\theta_{m}\) is not significantly large, \(G(\theta_{m},\varphi)=G(\theta_{0},\varphi)\cdot\cos\theta_{m}\). \(G(\theta_{0},\varphi_{0})\) represents the gain in the normal direction of the antenna. However, when the phased array antenna is scanned at large angle, the gain of the antenna decreases correspondingly. The concern of the antenna pattern for spaceborne phased array antennas primarily focuses on parameters such as sidelobe level, beam width, and beam scanning range. The antenna beams can be electrically steered in both azimuth and elevation directions, with the spacing between array elements being a critical factor affecting the antenna's scanning range. In practical engineering, there exists a constraint relationship between the antennaelement spacing and the maximum achievable scanning angle of the antenna, which can be expressed as follows:  
\[d\leq\frac{\lambda}{1+\sin(\theta_{m})}. \tag{4}\]  
In satellite communications, the utilization of sparse arrays facilitates the reconfiguration of the array element layout in the antenna pattern as compared to traditional phased arrays. The use of a smaller number of antenna elements can achieve comparable wide-angle scanning performance. Antenna polarization refers to the polarization of the electromagnetic waves emitted by an antenna. According to the reciprocity theorem, the polarization characteristics of the same antenna acting as a transmitting antenna and a receiving antenna are identical. As the electromagnetic waves radiated by an antenna differ in various directions, different polarizations are observed in different directions. Linear polarization, circular polarization, and elliptical polarization are three commonly used polarization methods that are determined on the basis of the amplitude and phase difference of the electric field vector.