## 2 Theoretical Basis
### The NAT of Misaligned System  
When the secondary mirror of the catadioptric optical system is tilted, the optical system can be regarded as a misaligned optical system. Based on NAT, the total aberration of misaligned system is equal to the sum of the aberration contribution of each surface in the system, but the center of the aberration field is offset, and the offset is represented by the aberration field decenter vector \(\overrightarrow{\sigma_{j}}\)[20]. Based on the third-order NAT, the system wave aberration equation when the secondary mirror is tilted can be expressed as follows [21]:  
\[\begin{array}{l}W\left(\overrightarrow{H},\overrightarrow{\rho}\right) =\sum\limits_{j}W_{040j}\left(\overrightarrow{\rho}\cdot\overrightarrow{ \rho}\right)^{2}\cdot\\ +\sum\limits_{j}W_{131j}\left[\left(\overrightarrow{H  
with the zernike polynomial. Third-order coma and astigmatism coefficients of the zernike polynomial are expressed in the same matrix, as follows [22]:  
\[ \begin{bmatrix}\mathbb{C}_{5}\\ \mathbb{C}_{6}\\ \mathbb{C}_{7}\\ \mathbb{C}_{8}\end{bmatrix} = \begin{bmatrix}H_{\mathrm{x}}^{2}-H_{\mathrm{y}}^{ 2}&H_{\mathrm{x}}&-H_{\mathrm{y}}&1&0&0&0&0\\ 2H_{x}H_{\mathrm{y}}&H_{\mathrm{y}}&H_{\mathrm{x}}&0&1&0&0&0\\ 0&0&0&0&0&H_{\mathrm{x}}&1&0\\ 0&0&0&0&0&H_{\mathrm{y}}&0&1\end{bmatrix}\begin{bmatrix}\alpha_{0}\\ \alpha_{1}\\ \alpha_{2}\\ \alpha_{3}\\ \alpha_{4}\\ \beta_{0}\\ \beta_{1}\\ \beta_{2}\end{bmatrix}  \tag{2}\]  
\[ \begin{cases}\alpha_{0}=\frac{1}{2}\sum\limits_{j}W_{222j}\\ \alpha_{1}=-\sum\limits_{j}W_{222j}\sigma_{\mathrm{jx}}\\ \alpha_{2}=-\sum\limits_{j}W_{222j}\sigma_{\mathrm{jy}}\\ \alpha_{3}=\frac{1}{2}\sum\limits_{j}W_{222j}\Big{(}\sigma_{\mathrm{jx}}^{2}- \sigma_{\mathrm{jy}}^{2}\Big{)}\\ \alpha_{4}=\sum\limits_{j}W_{222j}\sigma_{\mathrm{jx}}\sigma_{\mathrm{jy}} \end{cases}\begin{cases}\beta_{0}=\frac{1}{3}\sum\limits_{j}W_{131j}\\ \int\limits_{j}W_{131j}\sigma_{\mathrm{jx}}\\ \beta_{2}=-\frac{1}{3}\sum\limits_{j}W_{131j}\sigma_{\mathrm{jy}}\\ \end{cases} . \tag{3}\]  
Among them, \(\sigma_{\mathrm{jx}}\) and \(\sigma_{\mathrm{jy}}\) are the x and y components of the aberration field decenter vector, respectively. \(\alpha_{0}\) represents the astigmatism coefficient when the optical system is coaxial, \(\alpha_{1}\) and \(\alpha_{2}\) represent the astigmatism coefficients introduced by the first-order term of misalignment, \(\alpha_{3}\) and \(\alpha_{4}\) represent the astigmatism coefficients introduced by the square term of misalignment, \(\beta_{0}\) represents the coma coefficient when the optical system is coaxial. \(\beta_{1}\) and \(\beta_{2}\) represent the coma coefficients introduced by the first-order term of misalignment. The derivation method of the aberration coefficient related to the misalignment has been relatively mature, and the specific steps are shown in the Appendix A. Therefore, the tilt sensitivity is the changes of coma and astigmatism coefficients of the zernike polynomial before and after tilt, namely \(C_{5 ij}-C_{5j}\), \(C_{6 ij}-C_{6j}\), \(C_{7 ij}-C_{7j}\), \(C_{8 ij}-C_{8j}\): \(i\) represents the different tilt directions of the secondary mirror, respectively \(+\)x, \(-\)x, \(+\)y and \(-\)y, j represents different fields of view. \(C_{5j}\), \(C_{6j}\), \(C_{7j}\) and \(C_{8j}\) represent the coma and astigmatism coefficients of the zernike polynomial before tilt (secondary mirror on-axis). \(C_{5 ij}\), \(C_{6 ij}\), \(C_{7 ij}\) and \(C_{8 ij}\) represent the coma and astigmatism coefficients of the zernike polynomial after tilt. Since the tilt angle of the secondary mirror is usually very small, the third-order aberration is the main aberration of the misaligned system, ignoring the effect of higher-order aberration. Therefore, the contributions of the third-order coma and astigmatism to the zernike polynomial coefficients should be considered when the secondary mirror is tilted.