## 2 Principle  
### Optical Path Structure and Design Parameters of an Unobscured Off-Axis Three-Mirror Telescope  
Taking an unobscured off-axis three-mirror anastigmatic (TMA) telescope, with an aperture stop (located on the PM) of 0.5 m and an F-number of 12 used as an example, this paper aims to study its compensation correction method. The telescope is designed for the verification of key technology of space-based optical surveys. This type of off-axis optical system can be obtained by shifting the entrance pupil of an on-axis parent system with the same design parameters (e.g., radius, conic constants) and larger aperture. The optical path structure of the off-axis TMA telescope and the pupil transformation relationship with its on-axis parent system are both shown in Figure 1. In addition, the SM and TM of the optical system introduce tilts and decenters in the design process for the purpose of maximizing the balance of off-axis FOV aberrations and increasing the effective FOV of the telescope. The specific optical design parameters are given in Appendix A. The effective FOV of the telescope is \(1^{\circ}\times 1^{\circ}\) (the central FOV is offset by \(-0.5^{\circ}\) in the y direction), and the average RMS wavefront error of full FOV is only 0.0417 \(\lambda\) (\(\lambda\) = 632.8 nm). The spectral range of the telescope is 255-1000 nm. One of its operating bands at 632.8 nm is selected as the evaluation unit of image quality in this paper. Evidently, it is a diffraction-limited system with competent imaging performance. According to NAT, when an optical system is perturbed, new aberrations will not be produced but the field dependence of aberrations (such as the linear dependence of the field, and the quadratic dependence of the field) will be changed [23; 33]. However, different perturbation parameters often induce the same type of net aberration fields, that is, there are coupling effects among different perturbation parameters. They either compensate and offset each other to make the image quality better or accumulate and superimpose each other to make the image quality worse. This section aims to determine the compensation correction strategy for the off-axis TMA telescope described in the previous section by analyzing the coupling effect and compensation relationship between the net aberration fields induced by perturbation parameters. The wave aberration function of perturbed  
Figure 1: The optical path structure of the off-axis TMA telescope and pupil transformation relationship with its on-axis parent system. pupil-offset off-axis optical systems with designed tilts and decenters can be expressed as [11]  
\[ \begin{array}{c}W_{off-axis}=\sum\limits_{j}\sum\limits_{p=0}^{\infty}\sum \limits_{n=0}^{\infty}\sum\limits_{m=0}^{\infty}\left(W_{klm}+\Delta W_{klm} \right)_{j}\left(\overset{\rightarrow}{H}_{Aj}\cdot\overset{\rightarrow}{H}_ {Aj}\right)^{p}\left(\overset{\rightarrow}{\rho}\cdot\overset{\rightarrow}{ \rho}\right)^{n}\left(\overset{\rightarrow}{H}_{Aj}\cdot\overset{ \rightarrow}{\rho}\right)^{m}\\ \overset{\rightarrow}{H}_{Aj}=\overset{\rightarrow}{H}-\overset{ \rightarrow}{\sigma}_{j}^{\ast}-\overset{\rightarrow}{\sigma}_{j}^{\ast}\\ \overset{\rightarrow}{\rho}=\overset{\rightarrow}{R}\overset{ \rightarrow}{\rho}^{\ast}+h\end{array}  \tag{1}\]  
where \(k=2p+m\), \(l=2n+m\), \(j\) is the optical surface number; \(W_{klm}\) represents the aberration coefficient for a particular aberration type; \(\Delta W_{klm}\) represents the net change of \(W_{klm}\) induced by axial misalignments of optical surfaces; \(\overset{\rightarrow}{\sigma}^{\ast}\) denotes the aberration field decenter vector induced by lateral misalignments of optical surfaces; \(\overset{\rightarrow}{\sigma}^{\ast}\) denotes the aberration field decenter vector introduced in the design process of optical systems. Both \(\overset{\rightarrow}{\sigma}^{\prime}\) and \(\overset{\rightarrow}{\sigma}^{\ast\#}\) can be obtained by ray tracing using the method proposed in [34].