# Optical Design of the Goaly3 Multi-Mirror Telescope System with a Wide Field of View
## 3 Evaluation of the Performance of the Golay3 MMT  
Figure 8 shows the configuration of the Golay3 MMT with the optimized corrector group. The parameters of the final system are listed in Table 4. The semi-diameter of the sub-mirror is 43.75 mm and the fill factor is 25.5%. The aberration tolerances of the Golay3 MMT are evaluated by the wavefront error. According to the [PERSON] criteria [13], the performance is almost indistinguishable from perfect if the wavefront error is less than or equal to one-quarter of a wavelength. The tolerances of the geometrical aberrations corresponding to the wavefront error of one-quarter wavelength are calculated and their values are represented by the red bars in Figure 9.  
Acronyms LSA, RSA, ACA, SPC, SES and OSC represent the longitudinal spherical aberration (\(\lambda\)/\(n^{\prime}\)sin\({}^{2}u^{\prime}\)), residual spherical aberration (6/\(\lambda\)/\(n^{\prime}\)sin\({}^{2}u^{\prime}\)), axial chromatic aberration (\(\lambda\)/\(n^{\prime}\)sin\({}^{2}u^{\prime}\)), spherochromatian (4/\(n^{\prime}\)sin\({}^{2}u^{\prime}\)), secondary spectrum (\(\lambda\)/\(n^{\prime}\)sin\({}^{2}u^{\prime}\)) and off-face against sine condition (usually <0.0025 mm), respectively. The parameter \(n^{\prime}\) refers to the refraction index in the image space, and \(u_{m^{\prime}}\) is the angle of the marginal ray in the image plane. These geometrical aberrations can be calculated by ray tracing and their values are represented by the blue bars in Figure 9. As can be seen from Figure 10, the values of the five geometrical aberrations are far less than their corresponding aberration tolerances. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Surface** & **Radius (mm)** & **Thickness (mm)** & **Semi-Diameter (mm)** & **Conic** \\ \hline Sub-mirror & \(-\)2100.0 & \(-\)674.224 & 43.75 & 0 \\ Secondary mirror & \(-\)1291.344 & 869.686 & 62.500 & 11.714 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The parameters of the final Golay3 MMT. Figure 8: The configuration of the Golay3 MMT with the corrector group. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Corrector Lens** & **1** & **2** & **3** \\ \hline Material & N-BK7 & N-LAF35 & N-F2 \\ Radius of curvature (front surface) (mm) & \(-\)756.425 & 329.505 & 330.970 \\ Radius of curvature (back surface) (mm) & \(-\)140.597 & 172.986 & 243.969 \\ Thickness (mm) & 12.000 & 14.499 & 15.000 \\ Semi-Diameter (mm) & 40.987 & 39.017 & 36.088 \\ Conic (back surface) & 0 & 0 & 0 \\
4 th-order term (back surface) & \(-7.508\times 10^{-7}\) & \(3.239\times 10^{-10}\) & \(-3.566\times 10^{-14}\) \\
6 th-order term (back surface) & \(7.176\times 10^{-7}\) & \(8.302\times 10^{-11}\) & \(-1.573\times 10^{-13}\) \\
8 th-order term (back surface) & \(1.418\times 10^{-7}\) & \(-3.899\times 10^{-10}\) & \(2.199\times 10^{-13}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The parameters of the three-element corrector group. The MTF curves of the Golay3 MMT in Figure 10 indicate that the first cutoff frequency \(\rho_{r}\) in each field of view is around 80 lp/mm. The MTF values of different orientation in the mid and high frequency region (above 80 lp/mm) are different due to the asymmetry of the sub-mirrors' distribution. A slight degradation in the sagittal MTF is observed in the mid and high frequency region when going to the outermost fields. The point spread function (PSF) for the most off-axis field is shown in Figure 11 and still delivers a nearly diffraction-limited performance. Though the final system satisfies all the design specifications, it should be noted that the image size is 56.65 mm in diameter, which means that a detector with several hundreds of millions of elements or an array of detectors is needed. From the field curvature curves in Figure 12, the field curvature in the tangential plane is more severe than its sagittal counterpart for the outer FOV (from 0.3\({}^{\circ}\) to 0.65\({}^{\circ}\)), with a maximum tangential shift of 0.255 mm. The maximum distortion of the Golay3 MMT is 3% for the outermost FOV.