# Optical Design of the Goaly3 Multi-Mirror Telescope System with a Wide Field of View
## 4 Tolerance Analysis of the Golay3 MMT  
Tolerance analysis plays an important role in optical design because it is directly related to the optical fabrication, alignment and cost. The tolerance analysis of the Golay3 MMT system is carried out using the Monte Carlo method. The criterion of the tolerance analysis is the sagittal MTF value decreasing less than 0.1 at 150 lp/mm (nominal 0.174). The thickness of the rear surface of lens 3 (i.e., the position of the image plane) is defined as the compensator. The analysis is repeated with tightening the tolerances with higher sensitivities and loosening the tolerances with lower sensitivities.  
The results of the tolerance analysis are listed in Table 5. It was found that most parameters are flexible and insensitive. The surface and element decenter tolerances of corrector len1 and len2 are relatively more sensitive than other tolerances due to the aspheric nature of their back surfaces. However, these tolerances are still within a reasonable and acceptable range. With this tolerance budget, a Monte Carlo analysis was performed, and the results indicate that the probability of the sagittal MTF exceeding 0.11 is more than 90% at the frequency of 150 lp/mm. The maximum change in the back focus is 0.485 mm. The configuration of the sub-mirrors in the pupil plane determines that the telescope MTF has different values in different directions (sagittal and tangential). In the case of the Golay3 structure in Figure 1 in the paper, the contribution of sub-mirror 1 to the sagittal and tangential MTF is obviously different from that of sub-mirror 2 and sub-mirror 3. Therefore, when the criterion of tolerance analysis is determined, the tolerance value of each sub-mirror should be different. To ensure that Golay3 MMT achieves good thermal-mechanical stability, the mirrors are made of Silicon Carbide, and the mirror barrel and support structure are manufactured from invar.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Tolerance Type**} & \multicolumn{2}{c}{**Sub-**} & \multicolumn{2}{c}{**Secondary**} & \multicolumn{2}{c}{**Lens 1**} & \multicolumn{2}{c}{**Lens 2**} & \multicolumn{2}{c}{**Lens 3**} \\ \cline{3-10}  & **Mirrors** & **Mirrors** & **Front** & **Back** & **Front** & **Back** & **Front** & **Back** \\ \hline Radius (mm) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.05\) \\ Thickness (mm) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.05\) & — \\ Element decenter X (mm) & \(\pm 0.1\) & \(\pm 0.1\) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.1\) & \\ Element decenter Y (mm) & \(\pm 0.1\) & \(\pm 0.1\) & \(\pm 0.05\) & \(\pm 0.05\) & \(\pm 0.1\) & \\ Element tilt X (degree) & \(\pm 0.02\) & \(\pm 0.02\) & \(\pm 0.02\) & \(\pm 0.02\) & \(\pm 0.02\) & \(\pm 0.02\) \\ Element tilt Y (degree) & \(\pm 0.02\) & \(\pm 0.02\) & \(\pm 0.02\) & \(\pm 0.02\) & \(\pm 0.02\) & \(\pm 0.02\) \\ Surface decenter X (mm) & — & — & \(\pm 0.1\) & \(\pm 0.05\) & \(\pm 0.1\) & \(\pm 0.05\) & \(\pm 0.1\) & \(\pm 0.1\) \\ Surface decenter Y (mm) & — & — & \(\pm 0.1\) & \(\pm 0.05\) & \(\pm 0.1\) & \(\pm 0.04\) & \(\pm 0.1\) & \(\pm 0.1\) \\ Surface tilt X (degree) & — & — & \(\pm 0.02\) & \(\pm 0.014\) & \(\pm 0.02\) & \(\pm 0.016\) & \(\pm 0.02\) & \(\pm 0.016\) \\ Surface tilt Y (degree) & — & — & \(\pm 0.02\) & \(\pm 0.014\) & \(\pm 0.02\) & \(\pm 0.016\) & \(\pm 0.02\) & \(\pm 0.016\) \\ Refraction index & — & — & \(\pm 0.001\) & \(\pm 0.001\) & \(\pm 0.001\) & \(\pm 0.001\) \\ Abbe number & — & — & \(\pm 0.0642\) & \(\pm 0.0493\) & \(\pm 0.0364\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: The tolerance values of the Golay3 MMT.  
Figure 12: The field curvature and distortion of the Golay3 MMT.