# Non-Rotationally Symmetric Field Mapping for Back-Scanned Step/Stare Imaging System
## 2 Mathematical Model for Back-Scanned Step/Stare Imaging System
\begin{array}{c}h_{x}=f^{\prime}\theta_{x}\\ h_{y}=f^{\prime}\theta_{y}\end{array} \right. \tag{6}\]  
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 (7):  
\[\left\{ \begin{array}{c}h_{x}{{}^{\prime}}=f^{\prime}\tan^{-1}[\tan(\theta_{x })\cos(\alpha)]\\ h_{y}{{}^{\prime}}=f^{\prime}\theta_{y}\end{array} \right. \tag{7}\]  
Figure 2: The field of views (FOVs) in object space are conventionally defined for optics, based on the x-y-z coordinate system. The optical axis is Oz. OI is the chief ray of a certain FOV. (**a**) The projection of OI on the xOz plane is \(\text{OI}_{\text{x}}\). The angle between \(\text{OI}_{\text{x}}\) and Oz is \(\theta_{x}\), which is defined as the FOV in x direction of the chief ray OI. (**b**) The projection of OI on the yOz plane is \(\text{OI}_{y}\). The angle between \(\text{OI}_{\text{y}}\) and Oz is \(\theta_{y}\), which is defined as the FOV in y direction of the chief ray OI. Calculating the difference between image points \(\left(h_{x}\text{'},h_{y}\text{'}\right)\) and \(\left(h_{x},h_{y}\right)\) in Equation (8),  
\[\left\{ \begin{array}{c}h_{x}\text{'}-h_{x}=f\text{'}\left[\tan^{-1}[\tan( \theta_{x})\cos(\alpha)]-\theta_{x}\right]\\ h_{y}\text{'}-h_{y}=0\end{array} \right. \tag{8}\]  
it can be seen that for the line field of view \(\theta_{x}=0\), the image wander is zero at different angles \(\theta_{y}\). For the condition of \(\theta_{x}\
eq 0\) and \(\alpha\
eq 0\), the image wander always exists during scanning. The existence of projection distortion is because the optical axis of the imaging system is changed by the angle \(\alpha\), but \(\theta_{x}\) is still based on the original optical axis. Figure 3 shows a schematic diagram of the new projection relationship. Where Oz is the original optical axis. When the optical axis is in the state B, \(\theta_{y}=\alpha\), the new optical axis changes to Oz'. The projection point of OIy on the original optical axis Oz is \(C\). The intersection of IyC and the new optical axis is \(C\text{'}\). IxC' is parallel to IxC. The angle between OIx' and the new optical axis Oz' is \(\delta_{x}\), and we define \(\delta_{x}\) as the field angle in the x direction. \(L,M,N\) are the direction cosines of OI in the original optical axis coordinate system. \[\tan(\theta_{x})=\frac{L}{N},\cos(\alpha)=\frac{N}{\text{OC}^{\prime}},\tan( \delta_{x})=\frac{L}{\text{OC}^{\prime}} \tag{9}\]  
Using Equation (9), we can get the Equation (10):  
\[\tan(\delta_{x})=\tan(\theta_{x})\cos(\alpha) \tag{10}\]  
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 (11):  
\[\left\{ \begin{array}{c}h_{x}=f\text{'}\delta_{x}\\ h_{y}=f\text{'}\theta_{y}\end{array} \right. \tag{11}\]  
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 \(\left(h_{x}\text{'},h_{y}\text{'}\right)\) of the chief ray OI on the FPA will come to be expressed by Equation (12):  
\[\left\{ \begin{array}{c}h_{x}\text{'}=f\text{'}\tan^{-1}[\tan(\theta_{x}) \cos(\alpha)]=f\text{'}\delta_{x}\\ h_{y}\text{'}=f\text{'}\theta_{y}\end{array} \right. \tag{12}\]  
Figure 3: A schematic diagram of the new projection relationship. Oz is the original optical axis. When the optical axis is in the state B, \(\theta_{y}=\alpha\), the new optical axis changes to Oz’. The projection point of OIy on the original optical axis Oz is \(C\). The intersection of IyC and the new optical axis is \(C\text{'}\). IxC’ is parallel to IxC. The angle between OIx’ and the new optical axis is \(\delta_{x}\), which can be defined the field angle in the x direction.