# Optical Pupil Shift Correction Method for Large Ground-Based Optical Telescopes
## 3 Calculation of Optical Pupil Shift and Active Compensation
### Effect of Secondary Mirror Position Error  
The effect of wavefront aberration caused by the change in the secondary mirror's position according to the aberration theory is comprehensively analyzed in this section. The change in position of the secondary mirror affects mainly the wavefront of astigmatism and coma for the main system. If the central vertex of the secondary mirror is considered the coordinate origin, the optical axis direction is the \(Z\)-axis, and the vertical optical-axis direction is established along the \(x\) and \(y\)-axes. The deviation of the secondary mirror can be defined as node displacement \(d_{z}\) in the z direction, node displacement \(d_{x}\) and \(d_{y}\) in the \(x\) and \(y\) directions, and node rotations \(t_{x}\) and \(t_{y}\) around the \(x\)- and \(y\)-axes in a total of five variations. Figure 7 illustrates the diagram of \(d_{y}\), \(t_{x}\), and \(d_{z}\) deviations. According to the third-order vector wave aberration theory, for the telescope system with a card structure, the wavefront of coma and astigmatism when the secondary mirror is in the ideal position are expressed as  
\[W_{coma}=(\sum_{j}W_{131j}H\cdot\rho)(\rho\cdot\rho), \tag{17}\]  
\[W_{ast}=\frac{1}{2}(\sum_{j}W_{222j}H^{2})\rho^{2}, \tag{18}\]  
where \(j\) is the number of surfaces of the optical elements in the optical system and \(W_{131j}\) and \(W_{222j}\) are the third-order coma and astigmatism coefficients of the wave aberration of the \(j\)th surface system. The vector schematic is shown in Figure 8.  
As the node displacement \(dx\) and \(dy\) or node rotation \(Tx\) and \(Ty\) of the \(j\)th plane would offset the axis of symmetry of the aberration of this plane by \(\sigma_{j}\), the equations of coma aberration and image dispersion would become  
\[W^{\prime}_{coma}=[(\sum_{j}W_{131j}H-\sum_{j}W_{131j}\sigma_{j})\cdot\rho]( \rho\cdot\rho), \tag{19}\]  
and  
\[W^{\prime}_{ast}=\frac{1}{2}[\sum_{j}W_{222j}H^{2}-2H(W_{222j}\sigma_{j})+\sum _{j}W_{222j}\sigma_{j}]\cdot\rho^{2}, \tag{20}\]  
Figure 7: Schematic diagram of secondary mirror position error. The \(d_{z}\) corresponds to out-of-focus, the \(d_{x}\) corresponds to displacement, and the \(t_{x}\) corresponds to tilt. respectively. For the coma, an additional term \(\left(\sum\limits_{j}W_{131j}\sigma_{j}\right)\cdot\rho\cdot(\rho\cdot\rho)\) is added to Equation (19). For astigmatism, \(H(W_{222j}\sigma_{j})\cdot\rho^{2}\) and \(W^{\prime}{}_{ast}=\frac{1}{2}[\sum\limits_{j}W_{222j}\sigma_{j}]\cdot\rho^{2}\) are added to Equation (20). If the deviation is small, the on-axis field-of-view (FOV) image dispersion increment term will be negligible relative to the coma increment term. For the telescope system with a cassette structure, when the displacement occurred along the optical axis direction, the aberration was defocused and spherical, which can be corrected by moving the secondary mirror along the axial direction. When the vertical optical axis direction has shift and tilt, a constant coma is produced in the FOV, which can be corrected by translating or tilting the secondary mirror along the vertical optical axis. The telescope system error mainly comprises the relative position errors of the primary and secondary mirrors. For the on-axis FOV, if the aberration is corrected, the central axis of the primary and secondary mirrors will intersect at a coma-free point (Figure 9). The distance between the coma-free point and the secondary mirror center vertex \(Z_{cfp}\) satisfies [32]:  
\[Z_{cfp}=\frac{2L(m_{2}^{2}-1)}{(m_{2}+1)^{2}[(m_{2}-1)-(m_{2}+1)b_{s2}]}, \tag{21}\]  
where \(L\) is the distance between the central vertex of the secondary mirror and the center of the focal plane in the ideal case and \(m_{2}\) is the secondary mirror magnification, i.e., the ratio of the focal length of the system to the focal length of the primary mirror.