## 4 Verification of Compensation Correction Models
### Monte Carlo Analyses and Comparison for Different Compensation Correction Models  
In this section, Monte Carlo analyses will be conducted on SMCM and TMCM within the perturbation ranges of the misalignments, figure errors are shown in Table 3. The simulation analyses are divided into four cases, including the first case, the second case, the third case, which has increases in turn in terms of the perturbation ranges, and the fourth case, which has the same perturbation range as the first but contains a 5% measurement error. For the Monte Carlo analyses of SMCM, within the given perturbation ranges, 500 sets of perturbation parameters (TM misalignments and PM figure errors) were randomly generated according to a standard uniform distribution, representing 500 random perturbation systems to be corrected by SMCM. For the Monte Carlo analyses of TMCM, also within the given perturbation ranges, 500 sets of perturbation parameters (SM misalignments and PM figure errors) were randomly generated according to the standard uniform distribution, representing 500 random perturbation systems to be corrected by TMCM. Here we choose the average RMS wavefront error (WFE) of the full FOV to evaluate the compensation correction effects of these two models (ignoring the piston, tilt, and defocus terms). After simulation, the compensation correction results of the SMCM and TMCM are shown in Figures 6 and 7, respectively. Furthermore, Table 4 shows the root-mean-square deviation of the average RMS WFE under different cases, and the definition of the root-mean-square deviation is shown in the following equation:  
\[RMSD=\sqrt{\frac{1}{n}{\sum\limits_{l=1}^{n}}\Big{[}\big{(}WFE^{Average}\big{)} _{l}^{correction}-\big{(}WFE^{Average}\big{)}_{l}^{design}\Big{]}^{2}} \tag{36}\]  
where \(n\) is the number of perturbed samples (\(n=500\)); \(\big{(}WFE^{Average}\big{)}_{l}^{correction}\) is the average RMS WFE after compensation correction for perturbation sample \(i\); \(\big{(}WFE^{Average}\big{)}_{i}^{design}\) is the average RMS WFE in nominal design (0.0417 \(\lambda\)). Equation (36) can be used to evaluate the correction accuracy of the compensation correction models.  
Figure 6: Average RMS WFE before and after compensation of the optical system based on SMCM: (**a**) Case 1, (**b**) Case 2, (**c**) Case 3, and (**d**) Case 4. It could be noted that the black and red spots represent the average RMS WFE before and after compensation, respectively.  
Figure 7: Average RMS WFE before and after compensation of the optical system based on TMCM: (**a**) Case 1, (**b**) Case 2, (**c**) Case 3, and (**d**) Case 4. It could be noted that the black and red spots represent the average RMS WFE before and after compensation, respectively.  
As can be seen from Figure 6, SMCM can obtain convergent correction results for different cases. Anyway, it can be seen from Table 4 that the correction accuracy of the SMCM is basically maintained at the order of \(10^{-3}\) under different cases. In addition, as can be seen from Figure 7--although TMCM also achieved convergent correction results--by comparing Figures 6 and 7, it can be found that the correction performance of TMCM is significantly worse than that of SMCM. This is also verified by the correction accuracy results of TMCM in Table 4, especially in Case 2, Case 3, and Case 4, the correction accuracy of TMCM decreases by one order of magnitude compared with that of SMCM. The reason for this phenomenon may be that in the off-axis TMA telescope, the wave aberrations of the system are more sensitive to the SM misalignments than the TM misalignments (it can be seen by comparing the numerical scale of the ordinate in Figures 6 and 7), that is to say, under the same perturbation range, the contribution of SM misalignments to the system aberrations is greater than the TM misalignments. This means that the SM is more capable of compensating for wave aberrations, that is, the perturbation errors (the TM misalignments and the PM figure errors) that can be tolerated is greater.