# Optical Pupil Shift Correction Method for Large Ground-Based Optical Telescopes
## 4 Simulation Analysis and Experimental Verification
### Compensation of Optical-Axis Deviation under the Influence of Gravity  
The best-fit aspheric surface of the secondary mirror was calculated from the deformation results of several discrete nodes on the secondary mirror. The position and angular deviations of this mirror with respect to the optical axis of the primary mirror were obtained. The optical aberration was compensated by adjusting the position and angle of the Hexapod platform. Before and after the compensation, the variation in the angular deviation of the secondary mirror relative to the main mirror's optical axis with respect to the elevation angle is depicted in Figure 12. The variation in the position deviation with respect to the elevation angle is displayed in Figure 13.  
The deviations in position and angle were used as inputs for analyzing the optical-axis deviation of the main system in the simplified optical model. The variations in the optical-axis deviation of the main system with the elevation angle before and after compensation are depicted in Figure 14.  
Figure 12: The angle deviations between primary and secondary mirror vs. the telescope elevation angle. The data results were obtained through the deformation analysis of the telescope. (**a**) Before compensation; (**b**) after compensation. Figure 13: The position deviations between primary and secondary mirror vs. the telescope elevation angle. The data results were obtained through the deformation analysis of the telescope. (**a**) Before compensation; (**b**) after compensation. Figure 12 leads to the conclusion that the optical axis still deviates significantly after compensation. Based on the coma-free theory of the primary and secondary mirrors of the Cassegrain system, the optical axis was compensated by rotating the secondary mirror around the coma-free point, with respect to the target offset of the calibration camera. The optical-axis deviation after compensation is illustrated in Figure 15.  
Subsequently, since the Hartmann detector is the conjugate surface of the primary mirror, the effectiveness of the compensation method was verified by the pupil shift of effective sub-aperture on the Hartmann detector during the actual calibration of the adaptive optical imaging system. The pupil shift of effective detection sub-apertures of the Hartmann detector before and after compensation is shown in Figure 16.  
Figure 14: The position deviations between primary and secondary mirror vs. the telescope elevation angle. The data results were obtained through the overall optical model of the telescope. (**a**) Before compensation; (**b**) after compensation. Figure 15: The optical-axis deviation of the main system after compensation by rotating the secondary mirror around the coma-free point vs. the telescope elevation angle. The data results were obtained through the overall optical model of the telescope. Figure 16: The pupil shift of effective sub-aperture on the Hartmann detector. The bigger red circle refers to the pupil of the primary mirror. The smaller red circle refers to the cover of the second mirror. The green square grid is the effective sub-aperture on the Hartmann detector and one of the green square grid is equivalent to the pupil shift of the conjugate surface by 10 mm. (**a**) Before compensation (elevation angle of the telescope is 25\({}^{\circ}\), 45\({}^{\circ}\), 60\({}^{\circ}\), 80\({}^{\circ}\)); (**b**) after compensation (elevation angle of the telescope is 25\({}^{\circ}\), 45\({}^{\circ}\), 60\({}^{\circ}\), 80\({}^{\circ}\)). The following conclusions can be drawn from the calculated results of the secondary mirror deviations shown in Figures 12 and 13 and the optical-axis deviations of the main system in Figures 14 and 15. The maximum deviations in the position and angle of both the primary and secondary mirrors before compensation occurred when the optical axis was orientated in the horizontal direction. The maximum angular deviation was close to 38\({}^{\prime\prime}\), and the maximum position deviation 0.5 mm.