# Optimal F-Number of Ritchey-Chretien Telescope Based on Tolerance Analysis of Mirror Components
## 3 Tolerance of Mirror Elements Versus the F-Number  
In the mounting of Ritchey-Chretien telescope, an on-axis collimated beam was shone on the assembly of the primary and secondary mirror, and both mirrors were tilted and laterally shifted for alignment until the zero-coma condition was achieved, which was verified with an interferometer at the exit pupil of the telescope. A similar process was repeated for inserting corrector lens module. At the final stage, the FPA was put on and the MTF of the system at a different field angle was verified by tilting the whole telescope with respect to the collimated beam. At this stage, the secondary mirror was the only compensator for adjustment. There were at least two key tolerance items in the manufacturing and assembly process of the optical telescope, namely optical surface tilt/shift errors, and the RMS random surface WFE. The tolerance analysis was also performed with CODE V(r) [2, 8, 15, 16, 17, 18, 19]. In the analysis, the minimum MTF of 0.25 at the spatial frequency of 71.429 lp/mm was set as the criteria. The results of the inverse tolerance analysis for the optical surface de-center error, optical surface tilt error and system RMS random surface WFE of M1 and M2 surface with respect to each f-number in different design cases is shown in Table 4. The analysis did not include any compensator, i.e., the position of all the components locate at the position as designed, including FPA. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **M1 \& M2 RMS** & \multicolumn{3}{c}{**Primary Mirror, M1 Surface**} & \multicolumn{3}{c}{**Secondary Mirror, M2 Surface**} \\ \cline{3-8} \(f_{\mathbf{n}}\) & **Surface WFE** & **Tilt Error \(\Delta\alpha\)**, \(\Delta\beta\) (arcmin) & **De-Center Ax**, \(\Delta\beta\) (mm) & **Tilt Error \(\Delta\alpha\)**, \(\Delta\beta\) (arcmin) & **De-Center Ax**, \(\Delta\beta\) (mm) \({}^{*}\) \\ \hline
7.5 & 27 & 0.16 & 58 & 0.67 & 55 \\
7.8 & 27 & 0.18 & 64 & 0.71 & 62 \\
8. & 25 & 0.19 & 68 & 0.74 & 65 \\
8.5 & 25 & 0.21 & 77 & 0.81 & 74 \\
8.8 & 24 & 0.23 & 84 & 0.85 & 80 \\
9.2 & 23 & 0.26 & 93 & 0.90 & 90 \\
9.7 & 22 & 0.27 & 101 & 0.90 & 98 \\
10.2 & 21 & 0.31 & 114 & 0.98 & 108 \\
10.7 & 20 & 0.34 & 126 & 1.04 & 119 \\ \hline \hline \end{tabular} \({}^{*}\Delta\alpha\), \(\Delta\beta\); displacement (de-center) in direction of x, y, z coordinate system expressed in Figure 2. **\(\Delta\alpha\)**, \(\Delta\beta\); tilt angles of rotation \(\alpha\), \(\beta\) are left–handed about the +x, +y axis respectively.  
\end{table} 
Table 4: Tolerance of the optical surface de-center error, optical surface tilt error and system RMS random surface wavefront error (WFE) of the M1 and M2 surface versus the f-number. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Design Case** & **(1)** & **(2)** & **(3)** & **(4)** & **(5)** & **(6)** & **(7)** & **(8)** & **(9)** \\ \hline Pupil Diameter/\(f_{\mathbf{n}}\) & 800/7.5 & 770/7.8 & 740/8.1 & 710/8.5 & 680/8.8 & 650/9.2 & 620/9.7 & 590/10.2 & 560/10.7 \\ \hline min. MTF & 0.39 & 0.37 & 0.36 & 0.36 & 0.34 & 0.33 & 0.32 & 0.31 & 0.30 \\ Distortion (\%) & 0.149 & 0.149 & 0.149 & 0.148 & 0.149 & 0.149 & 0.149 & 0.150 \\ [PERSON] [PERSON] (\({}^{\circ}\)) & 2.63 & 2.63 & 2.63 & 2.60 & 2.63 & 2.63 & 2.62 & 2.62 \\ Obstruction Ratio & 0.325 & 0.338 & 0.341 & 0.344 & 0.347 & 0.348 & 0.355 & 0.356 & 0.357 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optical performance of 9 design cases with optimization. It indicates from Table 4 that the surface de-center error and tilt error increased with the increase of the f-number, while the tolerance of RMS random surface WFE decreased with the increase of the f-number [20]. The relationship between all the tolerance items versus the f-number is plotted in Figure 3a,b for M1 and M2 respectively. Figure 3 can be viewed from two different ways. One is to determine the f-number first, and the budget of all the tolerance items of M1 and M2 can be drawn from the diagram.