# Non-Rotationally Symmetric Field Mapping for Back-Scanned Step/Stare Imaging System
## 2 Mathematical Model for Back-Scanned Step/Stare Imaging System  
Figure 1 is a principle schematic diagram of utilizing the plane mirror to achieve the line of sight stability. It is assumed that the optical axis and the mirror are in the state A at the beginning of the exposure of the FPA. At this time, the light beam from the infinity object point on the optical axis is turned 90 degrees by the plane mirror, and then focused on the center of the FPA by the imager. During the exposure process, with the movement of the platform, the optical axis changes to the state B. The angle between the original optical axis A and the current optical axis B is \(\alpha\). By turning the mirror to the state B, the angle between the mirror A and the mirror B is \(\alpha\)/2. The light beam from the infinity object point on the optical axis is turned by the mirror, and then still focused on the center of the FPA by the imager. The change of the optical axis is compensated by the change of the angle of the mirror. During the integration time of the FPA, the light beam from the infinity object point on the optical axis is always focused on the center of the FPA. During the rotation of the mirror, projection distortion is produced in the image plane due to the change of the optical axis. This projection distortion makes off-axis field points wander during the exposure time, which reduces the SNR of the target. In the following, we want to find optimal field mapping between object space and image space, which would make all field points stable on the FPA during back-scanning and the image blurring could be mitigated. It is assumed that the relationship among the image height, the focal length of the imager and the incident angle of the imaging system in Figure 1 can be expressed by Equation (1):  
\[h=f^{\prime}\tan\theta \tag{1}\]  
where \(h\) is the image height, \(f^{\prime}\) is the focal length of the imager, \(\theta\) is the angle between the incident light and the optical axis. In Figure 2, the FOVs in object space are conventionally defined for optics, based on the x-y-z coordinate system. When the optical axis is in the state A, the image point coordinates of the chief ray OI on the FPA are shown in Equation (2),  
\[\left\{ \begin{array}{l}h_{x}=f^{\prime}\tan(\theta_{x})\\ h_{y}=f^{\prime}\tan(\theta_{y})\end{array} \right. \tag{2}\]  
where \(h_{x}\) is the image height in x direction, \(h_{y}\) is the image height in y direction. Figure 1: A schematic diagram of utilizing the plane mirror to achieve the line of sight stability. Considering that y-direction of the FOV is the scanning direction. When the change angle of the optical axis is \(\alpha\) and the mirror rotates \(\alpha/2\), the image point coordinates \(\left(h_{x}{{}^{\prime}},h_{y}{{}^{\prime}}\right)\) of the chief ray OI on the FPA will come to be expressed by Equation (3):  
\[\left\{ \begin{array}{c}h_{x}{{}^{\prime}}=f^{\prime}\tan(\theta_{x})\frac{ \cos(\alpha)\left[1+\tan^{2}(\alpha)\right]}{1+\tan(\theta_{y})\tan(\alpha)}\\ h_{y}{{}^{\prime}}=f^{\prime}\left[\tan(\theta_{y}-\alpha)+\tan(\alpha)\right] \end{array} \right. \tag{3}\]  
Calculating the difference between image points \(\left(h_{x}{{}^{\prime}},h_{y}{{}^{\prime}}\right)\) and \(\left(h_{x},h_{y}\right)\) in Equation (4),  
\[\left\{ \begin{array}{c}h_{x}{{}^{\prime}}-h_{x}=f^{\prime}\tan(\theta_{x}) \left[\frac{1}{\cos(\alpha)+\tan(\theta_{y})\sin(\alpha)}-1\right]\\ h_{y}{{}^{\prime}}-h_{y}=f^{\prime}\left[\tan(\theta_{y}-\alpha)+\tan(\alpha) -\tan(\theta_{y})\right]\end{array} \right. \tag{4}\]  
it can be seen that the position of the image point of the chief ray OI changes during back-scanning. It is assumed that the relationship among the image height, the focal length of the imager and the incident angle of the imaging system in Figure 1 is shown in Equation (5):  
\[h=f^{\prime}\theta \tag{5}\]  
When the optical axis is in the state A, the image point coordinates of the chief ray OI on the FPA are expressed by Equation (6):  
\[\left\{