## 4 Verification of Compensation Correction Models  
### A Specific Compensation Correction Case  
In this section, the off-axis TMA telescope described in Section 2 will be used to verify the constructed compensation correction models. The specific steps of the simulation experiment using the SM as the compensator to compensate for TM misalignments and PM figure errors are as follows:  
1. In the optical design model of the off-axis three-mirror telescope (ZEMAX software is used for modeling in this paper), the perturbation parameters (TM misalignments and PM figure errors) shown in Table 1 are introduced randomly. 2. The dynamic data linking function of MATLAB and ZEMAX is used to calculate the full-field distributions of the Fringe Zernike astigmatism(C5/6), the Fringe Zernike coma(C7/8), and the Fringe Zernike spherical aberration(C9) before and after perturbation of the telescope, they are calculated with 12 \(\times\) 12 equally spaced FOV points in \(1^{\circ}\times 1^{\circ}\) and shown in Figure 2a,b, Figure 3a,b, Figure 4a,b, respectively. 3. By using the SM compensation correction model constructed in Section 3, the linear compensation adjustments, angle compensation adjustments, and axial compensation adjustments required by SM can be calculated, as shown in Table 2.
4. The calculated compensation adjustments are introduced into the perturbed telescope and the full-field distributions of the Fringe Zernike astigmatism(C5/6), the Fringe Zernike coma(C7/8), and the Fringe Zernike spherical aberration(C9) after compensation adjustment of the telescope are calculated, as shown in Figures 2c, 3c and 4c. 5. In addition, in order to evaluate the compensation correction accuracy more clearly, Figure 5a-c show the full-field distributions of C5/6, C7/8, and C9 differences before perturbation and after compensation adjustment of the telescope, respectively. By comparing Figure 2a and Figure 2b, Figure 3a and Figure 3b, and Figure 4a and Figure 4b, respectively, it can be seen that the full-field distribution of astigmatism, coma, and spherical aberration of the system changes obviously compared with the design state (it should be noted that the numerical scale of each sub-figure is different) after introducing the misalignments and figure errors. Specifically, the full-field distribution of astigmatism and coma changes not only in amplitude but also in shape; however, the change of the full-field distribution of spherical aberration is only reflected in the amplitude, and the shape of its full-field distribution does not change. As analyzed in Section 2, the main reason for this phenomenon is that the misalignments and PM astigmatic figure errors will introduce field-constant astigmatism, field-linear astigmatism, and field-quadratic astigmatism, and the coupling of these three types of aberration fields will change the shape of the full-field distribution of astigmatism. Likewise, the coupling effect of field-constant coma and field-linear coma induced by misalignments and PM coma figure errors causes the shape of the full-field distribution of the coma to change. However, the misalignments and PM spherical aberration figure errors only introduce field-constant spherical aberration, which makes the shape of the full field-of-view distribution of the spherical aberration unchanged. By comparing Figure 2a and Figure 2c, Figure 3a and Figure 3c, and Figure 4a and Figure 4c, respectively, it can be seen that the full-field distribution of astigmatism, coma, and spherical aberration of the telescope after compensation correction are close to the design state in terms of both amplitude and shape. It can be seen from Figure 5 that the correction accuracy of astigmatism and coma of the telescope in the full field-of-view  
Figure 4: Distribution of Fringe Zernike spherical aberration (C9) in the full-field of the off-axis TMA telescope: (**a**) before perturbation, (**b**) after perturbation, and (**c**) after compensation correction.