# The Development of a Spaceborne SAR Based on a Reflector Antenna
## 2 Principles and Classification of Reflector Antennas  
### Basic Principles of Reflector Antennas  
Spaceborne SAR systems utilize various types of antennas, primarily including planar phased array antennas and reflector antennas. The selection of an appropriate antenna type depends on mission requirements, satellite platform capabilities, and SAR system performance. Due to its high gain and low sidelobe characteristics, reflector antennas are advantageous for achieving high-resolution imaging in spaceborne SAR applications. Among various reflector antenna types, the rotationally symmetric parabolic reflector antenna is the most common.  
The simplest form of a reflector antenna typically consists of two main components: a reflector surface significantly larger than the operating wavelength and a relatively small feed. Figure 1 illustrates the structural schematic of a parabolic reflector antenna.  
The vertex of the paraboloid is denoted as \(O^{\prime}\), and the focal point is located at \(F\). A symmetric parabolic reflector is fully defined by its diameter \(d\) and focal length \(f\), and it is typically characterized by its \(f/d\) ratio. The diameter \(d\) represents the physical size of the reflector, while the \(f/d\) ratio describes its curvature. As \(f/d\) approaches infinity, the reflector surface approximates a planar surface. A rotationally symmetric paraboloid is generated by rotating a parabolic curve about its axis of symmetry. Any point on its surface satisfies the following equations [20]:  
In cylindrical coordinates (origin at \(O^{\prime}\)):  
\[\rho^{2}=4 fz \tag{1}\]  
where \(f\) is the focal length of the paraboloid, \(\rho\) is the radial distance from point \(M\) on the paraboloid to the \(z\)-axis, and \(z\) is the axial coordinate along the paraboloid's symmetry axis.  
In rectangular coordinates (origin at \(O^{\prime}\)):  
\[x^{2}+y^{2}=4 fz \tag{2}\]  
In comparison, a more convenient coordinate system for application is the spherical coordinate system with the focus \(F\) as the origin. The spherical coordinates of an arbitrary point \(M\) on the parabolic surface are denoted as \((r^{\prime},\theta^{\prime},\varphi^{\prime})\); thus,  
\[r^{\prime}\sin\theta^{\prime}=\rho \tag{3}\]  
\[r^{\prime}\cos\theta^{\prime}=f-z \tag{4}\]  
where \(r^{\prime}\) is the distance from the focus \(F\) to an arbitrary point \(M\) on the reflector surface, and \(\theta^{\prime}\) is the angle between \(r^{\prime}\) and the \(z\)-axis.  
The sum of the squares of the above two equations is obtained as follows:  
\[r^{\prime}=\sqrt{\rho^{2}+(f-z)^{2}}=\sqrt{4 fz+f^{2}-2 fz+z^{2}}=f+z=2f-r^{ \prime}\cos\theta^{\prime} \tag{5}\]  
Therefore,  
Figure 1: Structural schematic of a parabolic reflector antenna (feed placed at the focus).  
\[r^{\prime}=\frac{2f}{1+\cos\theta^{\prime}}=\frac{f}{\cos^{2}\frac{\theta^{\prime} }{2}} \tag{6}\]  
This is the spherical coordinate equation of a point on the parabolic surface with point \(F\) as the origin. The geometric parameters of the paraboloid include the diameter \(d\), focal length \(f\), and the half opening angle \(\theta_{m}\). Since  
\[\rho=r^{\prime}\sin\theta^{\prime}=\frac{f}{\cos^{2}(\theta^{\prime}/2)}2\sin \frac{\theta^{\prime}}{2}\cos\frac{\theta^{\prime}}{2}=2f\tan\frac{\theta^{ \prime}}{2} \tag{7}\]  
we have  
\[\frac{d}{2}=2f\tan\frac{\theta_{m}}{2} \tag{8}\]  
or  
\[\tan\frac{\theta_{m}}{2}=\frac{d}{4f} \tag{9}\]  
where \(\theta_{m}\) is the half opening angle of the paraboloid.  
It follows that among these three parameters, only two are independent. Specifying any two uniquely determines the shape of the paraboloid. Similarly, based on the principle of geometrical optics, when a point source is positioned at the focal point \(F\), the reflected waves from the parabolic surface propagate parallel to the axis, forming a collimated beam (plane wave).