## 2 Theoretical Basis
### The NAT of Misaligned System
When the secondary mirror is located on the axis, there is no quantity related to the the aberration field decenter vector, that is, \(\alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha_{4}=\beta_{1}=\beta_{2}=0\), we can get the coma and astigmatism coefficients of the zernike polynomial before tilt as follows:  
\[ \begin{bmatrix}\mathbb{C}_{5j}\\ \mathbb{C}_{6j}\\ \mathbb{C}_{7j}\\ \mathbb{C}_{8j}\end{bmatrix} = \begin{bmatrix}\alpha_{0}\Big{(}H_{\mathrm{x}}^ {2}-H_{\mathrm{y}}^{2}\Big{)}\\ \alpha_{0}\cdot 2H_{x}H_{\mathrm{y}}\\ \beta_{0}H_{\mathrm{x}}\\ \beta_{0}H_{\mathrm{y}}\end{bmatrix} . \tag{4}\]  
The variation of the zernike polynomial coefficients of the third-order coma and astigmatism with the field of view for the different tilt directions of the secondary mirror will be derived below. The difference between the zernike polynomial coefficients when the secondary mirror is tilted and the secondary mirror on the axis is the contribution of the aberration to the sensitivity. 3.1.1. Sensitivity When the Secondary Mirror Is Tilted around the X-Axis  
Referring to (3), when the secondary mirror is tilted around the x-axis, \(\sigma_{yz}=0\), then \(\alpha_{1}=\alpha_{4}=\beta_{1}=0\), we can get the coma and astigmatism coefficients of the zernike polynomial when the secondary mirror is tilted about the x-axis, as follows:  
\[ \begin{bmatrix}C_{5 ij}\\ C_{6 ij}\\ C_{7 ij}\\ C_{8 ij}\end{bmatrix} = \begin{bmatrix}H_{x}^{2}-H_{y}^{2}&H_{x}&-H_{y}&1&0&0&0&0 \\ 2H_{x}H_{y}&H_{y}&H_{x}&0&1&0&0&0\\ 0&0&0&0&H_{x}&1&0\\ 0&0&0&0&0&H_{y}&0&1\end{bmatrix}\begin{bmatrix}\alpha_{0}\\ 0\\ \alpha_{2}\\ \alpha_{3}\\ 0\\ \beta_{0}\\ \beta_{2}\end{bmatrix} = \begin{bmatrix}\alpha_{0}\left(H_{x}^{2}-H_{y}^{2} \right)-\alpha_{2}H_{y}+\alpha_{3}\\ \alpha_{0}\cdot 2H_{x}H_{y}+\alpha_{2}H_{x}\\ \beta_{0}H_{x}\\ \beta_{0}H_{y}+\beta_{2}\end{bmatrix} . \tag{5}\]  
The sensitivity matrix of the secondary mirror tilt around the x-axis can be obtained by subtracting (4) from (5), as follows:  
\[ \begin{bmatrix}C_{5 ij}-C_{5j}\\ C_{6 ij}-C_{6j}\\ C_{7 ij}-C_{7j}\\ C_{8 ij}-C_{8j}\end{bmatrix} = \begin{bmatrix}-\alpha_{2}H_{y}+\alpha_{3}\\ \alpha_{2}H_{x}\\ 0\\ \beta_{2}\end{bmatrix} . \tag{6}\]  
3.1.2. Sensitivity When the Secondary Mirror Is Tilted around the Y-Axis  
Referring to (3), when the secondary mirror is tilted around the y-axis, \(\sigma_{yz}=0\), then \(\alpha_{2}=\alpha_{4}=\beta_{2}=0\), we can get the coma and astigmatism coefficients of the zernike polynomial when the secondary mirror is tilted about the y-axis, as follows:  
\[ \begin{bmatrix}C_{5 ij}\\ C_{6 ij}\\ C_{7 ij}\\ C_{8 ij}\end{bmatrix} = \begin{bmatrix}H_{x}^{2}-H_{y}^{2}&H_{x}&-H_{y}&1&0&0&0 &0\\ 2H_{x}H_{y}&H_{x}&0&1&0&0&0\\ 0&0&0&0&0&H_{x}&1&0\\ 0&0&0&0&0&H_{y}&0&1\end{bmatrix}\begin{bmatrix}\alpha_{0}\\ \alpha_{1}\\ 0\\ \alpha_{3}\\ 0\\ \beta_{0}\\ \beta_{1}\\ 0\end{bmatrix} = \begin{bmatrix}\alpha_{0}\left(H_{x}^{2}-H_{y}^{2}\right)+ \alpha_{1}H_{x}+\alpha_{3}\\ \alpha_{0}\cdot 2H_{x}H_{y}+\alpha_{1}H_{y}\\ \beta_{0}H_{x}+\beta_{1}\\ \beta_{0}H_{y}\end{bmatrix} . \tag{7}\]  
The sensitivity matrix of the secondary mirror tilt around the y-axis an be obtained by subtracting (4) from (7), as follows:  
\[ \begin{bmatrix}C_{5 ij}-C_{5j}\\ C_{6 ij}-C_{6j}\\ C_{7 ij}-C_{7j}\\ C_{8 ij}-C_{8j}\end{bmatrix} = \begin{bmatrix}\alpha_{1}H_{x}+\alpha_{3}\\ \alpha_{1}H_{y}\\ \beta_{1}\\ 0\end{bmatrix} . \tag{8}\]  
Referring to (6) and (8), the change of the zernike polynomial coefficients with the field of view when the secondary mirror is tilted can be obtained. The zernike polynomial coefficients are functions of the aberration coefficients related to misalignment and the field of view. \(C_{5 ij}-C_{5j}\) and \(C_{6 ij}-C_{6j}\) represent the sensitivity of the astigmatism terms, \(C_{7 ij}-C_{7j}\) and \(C_{8 ij}-C_{8j}\) represent the sensitivity of the coma terms.