# A Design for a Manufacturing-Constrained Off-Axis Four-Mirror Reflective System
## 3 Design Approach
The Zernike polynomial surface is composed of the conical surface and additional aspheric terms defined by the Zernike polynomial coefficients. The Fringe ordering of the Zernike polynomials is chosen, with sag described mathematically by  
\[z=\frac{cr^{2}}{1+\sqrt{1-(1+k)c^{2}r^{2}}}+\sum_{i=1}^{N}C_{i}Z_{i}(\rho,\rho), \tag{14}\]  
where \(c\) is the curvature of the base sphere, \(r\) is the radial coordinate of the surface, \(k\) is the conic constant, \(\rho\) is the radial coordinate of the surface normalized by \(R_{\text{norm}}\), (that is, \(\rho=r/R_{\text{norm}}\)), \(\varphi\) is the azimuthal component of the surface aperture, and \(C_{i}\) is the weight factor of the \(i^{th}\) Zernike term,\(Z_{i}\). After setting the tertiary mirror to the Zernike surface, the Aspheric surface high-order coefficients and Zernike coefficients of the tertiary mirror are shown in Table 7, and the configuration parameters of the final off-axis four-mirror system are shown in Table 8.  
The final structure layout of the off-axis four-mirror system is shown in Figure 14, in which the rays tracing of all FOVs are plotted with different colors. All mirrors in the final structure are distributed approximately along a cylinder. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mirror** & **R (mm)** & **d (mm)** & **Conic** \\ \hline Primary & \(-1947.38\) & \(-636.00\) & \(1.020\) \\ Secondary & \(-865.21\) & \(438.27\) & \(4.297\) \\ Quaternary & \(964.89\) & \(820.00\) & \(0.942\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Configuration parameters of the optimized off-axis four-mirror system. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mirror** & **Fourth Order** & **Sixth Order** & **Zernike Third** & **Zernike Fifth** \\  & **Coefficient** & **Coefficient** & **Coefficient** & **Coefficient** \\ \hline Tertiary & \(-1.810\times 10^{-10}\) & \(-7.727\times 10^{-15}\) & \(6.6\times 10^{-4}\) & \(-1.506\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Aspheric surface high-order coefficients and Zernike coefficients of the tertiary mirror. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mirror** & **R (mm)** & **d (mm)** & **Conic** & **Fourth Order Coefficient** & **Sixth Order Coefficient** \\ \hline Primary & \(-1682.54\) & \(-636.00\) & \(1.003\) & \(-1.039\times 10^{-13}\) & \(-5.334\times 10^{-20}\) \\ Secondary & \(-615.69\) & \(438.27\) & \(4.039\) & \(-2.681\times 10^{-11}\) & \(-1.514\times 10^{-16}\) \\ Quaternary & \(1009.27\) & \(820.00\) & \(0.986\) & \(-4.901\times 10^{-12}\) & \(-1.796\times 10^{-17}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Configuration parameters of the final off-axis four-mirror system. The ADRS and the maximum deviation from the reference surface of each mirror in the final structure are listed in Table 9. The ADRS values show that mirrors in the system are close to the reference surface. The modulation transfer function (MTF) of the system is shown in Figure 15. The MTFs at 13 cycles/mr are above 0.6 for all FOVs, which indicates that the system has a good optical performance. Figure 16a shows the gird distortion of the final off-axis four-mirror system. The paraxial FOV grid coincides with the actual FOV grid approximately, which indicates that there is no obvious distortion in the system. Figure 16b shows the F-Tan (Theta) distortion of the system. The distortion increases gently as the FOV increases and the maximum distortion is approximately 0.4%. Figure 14: Final structure layout of the off-axis four-mirror system. Figure 15: Final structure MTF of the off-axis four-mirror system.