## 2 Principle
### Optical Path Structure and Design Parameters of an Unobscured Off-Axis Three-Mirror Telescope
According to the relationship between optical path difference and figure error, Equations (2)-(4) can be modified as  
\[C_{5}=C_{5,C}+(n^{\prime}-n)_{(FIGURE)}C_{5,C}+C_{5,L}+C_{5,Q}\Bigg{\}}, \tag{5}\]  
\[C_{6}=C_{6,C}+(n^{\prime}-n)_{(FIGURE)}C_{6,C}+C_{6,L}+C_{6,Q}\Bigg{\}},\]  
\[C_{7}=C_{7,C}+(n^{\prime}-n)_{(FIGURE)}C_{7,C}+C_{7,L}\Bigg{\}},\]  
\[C_{8}=C_{8,C}+(n^{\prime}-n)_{(FIGURE)}C_{8,C}+C_{8,L}\Bigg{\}},\]  
\[C_{9}=C_{9,C}+\big{(}n^{\prime}-n\big{)}_{(FIGURE)}C_{9,C},\]  
where \(n^{\prime}\) and \(n\), respectively, represent the refractive index of the space where the incident light and the reflected light are located (it could be noted that \(n=1\), \(n^{\prime}=-1\) for odd reflections and \(n=-1\), \(n^{\prime}=1\) for even reflections); \({}_{(FIGURE)}C_{5,C}\) and \({}_{(FIGURE)}C_{6,C}\) represent field-constant astigmatism induced by PM astigmatic figure errors; \({}_{(FIGURE)}C_{7,C}\) and \({}_{(FIGURE)}C_{8,C}\) represent field-constant coma induced by PM coma figure errors; \({}_{(FIGURE)}C_{9,C}\) represents the field-constant spherical aberration induced by PM spherical aberration figure errors. In principle, only the same type of aberration fields can compensate for each other [33]. It can be seen from Equations (5)-(7) that misalignments and PM figure errors of the off-axis optical systems simultaneously introduce field-constant astigmatism, field-constant coma, and field-constant spherical aberration; this indicates that there are coupling effects between the aberration fields induced by misalignments and aberration fields induced by PM figure errors in the off-axis optical systems, and the possibility of mutual compensation exists. However, aberration compensation in the true sense includes not only the aberration fields induced by the target perturbation parameters (referring to PM figure errors) that need to be compensated for but also other types of aberration fields induced by compensation perturbation parameters (referring to misalignments), which need to be compensated at the same time. For instance, while using the field-constant aberrations induced by misalignments to compensate for the field-constant aberrations induced by PM figure errors, it can be seen from Equations (5)-(7) that misalignments will inevitably introduce additional field-linear aberrations and field-quadratic aberrations; these two types of aberration fields cannot be ignored in wide-field off-axis telescopes. To solve this problem, compensation perturbation parameters can be decomposed into different components, so that the different components can compensate for each other, thereby hopefully compensating for the extra-induced aberration fields. In fact, it is possible and relatively easy to decompose compensation perturbation parameters into different compensators according to different mirrors, that is, compensation between misalignments of different mirrors. This is mainly because misalignments of different mirrors will introduce the same type of aberration fields, and no additional type of aberration fields will be introduced after compensation is done between different mirrors. Taking the off-axis three-mirror telescope as an example, when PM is used as the coordinate reference, the compensation parameters can be decomposed into two components, one of which is the misalignments of SM and the other is the misalignments of TM. The misalignments of SM will introduce field-constant aberrations, field-linear aberrations, and field-quadratic aberrations, and so does the misalignments of TM. Among them, field-constant aberrations induced by the misalignments of SM and TM can be used to compensate for field-constant aberrations induced by PM figure errors, while the additional field-linear aberrations and field-quadratic aberrations induced by the misalignments of SM and TM can compensate each other. Based on this analysis, we propose the following mechanism of aberration field compensation for an off-axis three-mirror telescope.