# Non-Rotationally Symmetric Field Mapping for Back-Scanned Step/Stare Imaging System
## 4 Comparison and Analysis
When fast steering mirror does not move, the distortion of the optical system is controlled well. Figure 11 shows the grid distortion map. In order to visualize the distortion characteristics, the distortion deviation is enlarged by 200 times. As can be seen from the figure, this optical system still displays symmetrical barrel distortion. The field mapping of Equation (6) minimizes image wander for field points along a single axis. Line-scan systems can use this mapping and achieve adequate results. However, this optimal one-dimensional field mapping is not ideal for two-dimensional imaging systems. During the exposure of the FPA, the back-scan is over \(\pm 0.4\) deg in object space. The lines in Figure 12 indicate motion (in units of pixel IFOV) of the chief ray at nine image points during the  
Figure 11: Grid distortion. The X point is the actual field point, and the grid is the theoretical field point. In order to visually display the distortion characteristics, the distortion deviation is enlarged by 200 times. Figure 10: Image points wander during backscanning under field mapping condition \(h=f^{\prime}\tan\theta\): \(-0.4\) deg to \(+0.4\) deg back-scan in object space. back-scan operation. As can be seen from the figure, except for the central image point, the other off-axis image points have image wander. The image wander at the four edge fields of view is the largest. Compared with the case when the fast mirror has no motion, the maximum distortion image shift is 3.18 pixels. The image wander is similar to that in Figure 12. This image wander will also severely degrade the image quality. Additionally, we compared the optical system based on the proposed model \(h_{x}=f^{\prime}\delta_{x},h_{y}=f^{\prime}\theta_{y}\) to the optical systems based on the model \(h=f^{\prime}\tan\theta\) and \(h=f^{\prime}\theta\) in our experiments, and the distortion and image wander are selected as the evaluation parameters. Table 2 lists the maximum distortion when the FSM does not move and the maximum image point wander during back-scanning in the previous three optical systems. The optical systems that meet \(h=f^{\prime}\tan\theta\) and \(h=f^{\prime}\theta\) have a distortion of less than 0.02%, which is approximately no distortion. At this time, the movement of the image points are caused by the projection distortion during the movement of the FSM. For the optical system that meets the conditions of non-rotational projection, the maximum distortion is less than 0.46%. When the FSM does not move, a large distortion remains in the optical system, which is more than 23 times that of the conventional optical systems. During the rotation of the FSM, the maximum wander of the image points on the FPA is only 0.22 pixels, and the remaining distortion just compensates for the projection distortion, which verifies the effectiveness of the proposed method, as \"one cannot make an omelet without breaking eggs\". However, in order to achieve no image point wander during the back-scanning, the optical system is more complicated than the traditional optical systems, and non-rotationally symmetric free-form surfaces are required to achieve a non-rotational symmetry field mapping.