## 3 Compensation Correction Models for the Off-Axis TMA Telescope
\Bigg{\}}, \tag{11}\]\[C_{7,\mathcal{L}}^{SM} =-\Big{[}C_{7,\mathcal{C}}^{TM}+(n^{\prime}-n)_{(FIGURE)}C_{7,\mathcal{C }}\Big{]}\] \[C_{8,\mathcal{C}}^{SM} =-\Big{[}C_{8,\mathcal{C}}^{TM}+(n^{\prime}-n)_{(FIGURE)}C_{8, \mathcal{C}}\Big{]}\Bigg{\}}, \tag{12}\] \[C_{8,\mathcal{L}}^{SM} =-C_{8,\mathcal{L}}^{TM}\] \[C_{9,\mathcal{C}}^{SM} =-\Big{[}C_{9,\mathcal{C}}^{TM}+\big{(}n^{\prime}-n\big{)}_{(FIGURE) }C_{9,\mathcal{C}}\Big{]}. \tag{13}\]  
Based on Equations (11)-(13), some aberration coefficients of the \(q\)-th FOV points related to the compensation adjustments of SM can be obtained (hereinafter referred to as correlation coefficients), as shown in the following equations:  
\[C_{5,\mathcal{C}}^{SM} =f_{1}^{(q)}(x_{SM},y_{SM},\theta_{SM},\phi_{SM},z_{SM}), \tag{14}\] \[C_{5,\mathcal{L}}^{SM} =f_{2}^{(q)}(x_{SM},y_{SM},\theta_{SM},\phi_{SM},z_{SM}),\] (15) \[C_{5,\mathcal{Q}}^{SM} =f_{3}^{(q)}(x_{SM},y_{SM},\theta_{SM},\phi_{SM},z_{SM}),\] (16) \[C_{6,\mathcal{C}}^{SM} =f_{4}^{(q)}(x_{SM},y_{SM},\theta_{SM},\phi_{SM},z_{SM}),\] (17) \[C_{6,\mathcal{L}}^{SM} =f_{5}^{(q)}(x_{SM},y_{SM},\theta_{SM},\phi_{SM},z_{SM}),\] (18) \[C_{6,\mathcal{Q}}^{SM} =f_{6}^{(q)}(x_{SM},y_{SM},\theta_{SM},\phi_{SM},z_{SM}),\] (19) \[C_{8,\mathcal{C}}^{SM} =-\Big{[}C_{8,\mathcal{C}}^{TM}+(n^{\prime}-n)_{(FIGURE)}C_{8, \mathcal{C}}\Big{]}\Bigg{\}}, \tag{12}\]0 (20) \[C_{8,\mathcal{C}}^{SM} =-\Big{[}C_{8,\mathcal{C}}^{TM}+(n^{\prime}-n)_{(FIGURE)}C_{8, \mathcal{C}}\Big{]}\Bigg{\}}, \tag{12}\]1 (21) \[C_{8,\mathcal{C}}^{SM} =-\Big{[}C_{8,\mathcal{C}}^{TM}+(n^{\prime}-n)_{(FIGURE)}C_{8, \mathcal{C}}\Big{]}\Bigg{\}}, \tag{12}\]2 (23) \[C_{8,\mathcal{C}}^{SM} =-\Big{[}C_{8,\mathcal{C}}^{TM}+(n^{\prime}-n)_{(FIGURE)}C_{8, \mathcal{C}}\Big{]}\Bigg{\}}, \tag{12}\]3  
In Equations (14)-(24), \(x_{SM}\) and \(y_{SM}\) represent the linear compensation adjustments of SM in the x- and y-axis, respectively; \(\theta_{SM}\) and \(\phi_{SM}\) represent the angular compensation adjustments of SM about the x- and y-axis, respectively; \(z_{SM}\) represent the axial compensation adjustments of SM in z-axis; \(f_{i}^{(q)}(i=1,2\cdots,11)\) represents the functional relationship between the correlation coefficients and the compensation adjustments of SM, which is a complex nonlinear equation system. Considering that the values of misalignments in active optics are very small (Generally, linear misalignments are on the level of micrometers, and angular misalignments are on the level of arcseconds [30]). Therefore, using the power series expansion and selecting only the first-order term of the power series, \(f_{i}^{(q)}(i=1,2\cdots,11)\) can be approximately replaced by a linear system of equations:  
\[C_{8,\mathcal{C}}^{SM} =-\Big{[}C_{8,\mathcal{C}}^{TM}+(n^{\prime}-n)_{(FIGURE)}C_{8, \mathcal{C}}\Big{]}\Bigg{\}}, \tag{12}\]4  
where \(f_{0i}^{(q)}\) is a constant related to the structural parameters and Gaussian parameters of the system; \(\frac{\partial f_{i}^{(q)}}{\partial(i)}\) represents the first-order partial derivative of the correlation coefficients to the adjustments of each dimension of SM. By selecting some FOV points that can cover the full FOV, the overdetermined linear equation system formed by Equation (25) can be used to calculate the compensation adjustments of SM. It should be noted that \(f_{0i}^{(q)}\) and \(\frac{\partial f_{i}^{(i)}}{\partial(\rangle}\) are obtained by fitting based on the principle of multiple linear regression [11]. The specific fitting method is as follows:  
1. [label=0.] 2. Making each dimension of SM randomly generate N groups of perturbations according to a standard uniform distribution within a certain range. Meanwhile, bringing the generated N groups of perturbations into Equations (2)-(4), so that N groups of sample data can be obtained: \[C_{8,\mathcal{C}}^{SM} =-\Big{[}C_{8,\mathcal{C}}^{TM}+(n^{\prime}-n)_{(FIGURE)}C_{8, \mathcal{C}}\Big{]}\Bigg{\}}, \tag{12}\]5 (26)
3. Integrating this batch of sample data into the structural form shown in Equation (25), so the following equations could be obtained: \[