# Non-Rotationally Symmetric Field Mapping for Back-Scanned Step/Stare Imaging System
## 4 Comparison and Analysis  
In order to explain the distortion characteristic differences of optical systems under different field mapping equations during back-scanning, another two medium-wave infrared optical systems are successfully designed in meeting the same design specifications in Table 1.  
The optical system designed by Zemax software according to the field mapping condition \(h=f^{\prime}\tan\theta\) is shown in Figure 8. The afocal telescope is composed of five lenses. From left to right, the lens 1, the lens 3, and the lens 4 are made of silicon and have a positive power. The lens 2 and the lens 5 are made of germanium and have a negative power. The entrance pupil is placed on the front surface of the lens 1. The rear surface of the lens 2 is a secondary aspheric surface, and the remaining surfaces are spherical. The lens 3 is a field lens, and the optical power is very small. The fast steering mirror is placed on the exit pupil of the afocal telescope, and the optical axis is turned by 90 degrees. The imager focuses the scene onto the FPA. The optical system have good imaging performance and low pupil aberrations. Figure 6: 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 20 times. Figure 7: Image points wander during back-scanning under field mapping condition \(h_{\mathrm{x}}\,=\,f^{\prime}\,g_{x},h_{y}\,=\,f^{\prime}\,g_{y}\): –0.4 deg to +0.4 deg back-scan in object space. The optimization commands of Zemax software are utilized to set the relationship among the image height, the focal length and the incident angle of the imaging system to meet Equation (2). For the above system, the MTF is close to the diffraction limit. The maximum distortion is less than 0.02% within the FPA. When FSM remain still, the distortion of the optical system is controlled well. Figure 9 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 displays symmetrical barrel distortion. In this moment, the distortion seemed to be controlled and the image quality seemed to be well. Further, the FSM placed behind the afocal telescope could be considered to have nothing to do with the image quality by optical designers. Actually, the movement required to track an image viewed through a rotating mirror is a nonlinear function of the mirror movement, but this fact is always neglected [20]. During the exposure of the FPA, the back-scan is over \(\pm\)0.4 deg in object space. The lines in Figure 10 indicate motion (in units of pixel IFOV) of the chief ray at nine image points during the 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  
Figure 8: Optical system according to the field mapping condition \(h=f^{\prime}\tan\theta\)  
Figure 9: Grid distortion of the optical system in Figure 8. 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. largest. Compared with the case when the fast mirror has no motion, the maximum distortion image shift is 3.13 pixels. This image wander will severely degrade the image quality. The optical system in Figure 8 is optimized according to the condition \(h=f^{\prime}\theta\). Only the lens radius, air distance, and aspheric coefficient are slightly different, and the rest remain the same. The optimization commands of Zemax software are utilized to set the relationship among the image height, the focal length and the incident angle of the imaging system to meet Equation (6). For the above system, the MTF is close to the diffraction limit. For this optical system, the maximum distortion is less than 0.012% within the FPA.