# A Design for a Manufacturing-Constrained Off-Axis Four-Mirror Reflective System
## 3 Design Approach  
In this section, an off-axis four-mirror afocal reflective system, which is satisfying manufacturing constraints, is presented as an example. Table 1 shows the specification of the system. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Specification** \\ \hline Entrance pupil diameter/mm & 170 \\ Exit pupil diameter/mm & 34 \\ Field of view/(\({}^{\circ}\)) & \(3\times 3\) \\ Wavelength/nm & 650–900 \\ Compression ratio & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specification of the off-axis four-mirror optical system. Figure 8: Improvement of the fitness calculation. According to the method proposed in Section 2, the GA program is written to calculate the initial structural parameters of the off-axis four-mirror system. The main parameters of GA are the population size \(n=50\), \(p_{c}=0.9\), \(p_{m}=0.01\), and evolutionary generations are 200. The objective function is composed of primary aberration coefficients and structural layout constraints with weights  
\[ \begin{array}{l}F=f(w_{i},\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},k_{1},k_{ 2},k_{4},d_{2},\theta_{3})\\ =w_{1}|S_{1}|+w_{2}|S_{2}|+w_{3}|S_{3}|+w_{4}|S_{4}|+w_{5}|S_{5}|+w_{6}|\sigma_{ \text{ADBS}}|+w_{7}|C|+w_{j}|O|.\end{array}  \tag{13}\]  
Certain conditions are set to obtain reasonable initial structural parameters: \(\alpha_{1}>0\), \(\alpha_{2}<0\), \(\beta_{1}<0\) and \(\beta_{2}>0\). In this case, we know that there is an intermediate image plane in this structure. In addition, these parameters need to satisfy some boundary conditions, which are shown in Table 2.  
According to the 9 parameters with ranges, GA is used to optimize the objective function \(F\) mentioned before. Figure 9 shows the convergence curve of the objective function. The data point on the curve indicates the elitist individual of each generation. It can be seen that the value of the objective function decreases as iteration times increase, and the objective function is convergent. Multiple groups of initial structure parameters have been obtained by changing the weight. Five groups of data with the lowest convergent \(F\) values are listed in Table 3. The optimization results of each group are close, which proves the stability of the GA in dealing with optical optimization with special requirements. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Parameter** & \(\alpha_{1}\) & \(\alpha_{2}\) & \(\beta_{1}\) & \(\beta_{2}\) & \(k_{1}\) & \(k_{2}\) & \(k_{4}\) & \(d_{2}\)\(\times\)10 & \(\theta_{3}\) \\ \hline Range & (0, 0.5) & (\(-1, 0\)) & (\(-5, 0\)) & (\(0, 5\)) & (\(0, 5\)) & (\(0, 5\)) & (\(0, 5\)) & (\(42, 45\)) & (\(1.6, 2\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters range. Figure 9: Convergence curve of the objective function. Figure 11: Optimized structure layout of the off-axis four-mirror system. Figure 12: Initial structure modulation transfer function (MTF) of the off-axis four-mirror system. Figure 13: Optimized structure MTF of the off-axis four-mirror system. The configuration parameters of the optimized structure are presented in Table 6. After further optimization, the structural parameters are still close to the initial structure, which proves that it is feasible to use GA to find the initial structure with special requirements. In the next step, the freeform surface is introduced to further improve the imaging quality [30]. The freeform surface is a category of non-rotational symmetric surfaces. Compared with the traditional optical surface such as spherical and aspheric surfaces, it has a stronger ability to correct aberrations. With the development of manufacturing technology, freeform surfaces are more and more used in illumination and imaging systems. In this structure, we choose the Zernike polynomial surface to optimize the mirrors [31]. Zernike polynomials can correspond to Seidel aberration coefficients in optical design.