# A Design for a Manufacturing-Constrained Off-Axis Four-Mirror Reflective System
## 2 Working Principle
### Search for the Initial Structure via GA
The chromosome with a larger value of fitness has a higher selection probability. Step 4: Crossover. Crossover is the operator that allows selected chromosomes to exchange some genes with the crossover probability \(p_{c}\), which is an important means to obtain excellent individuals in GA. The \(p_{c}\) is set between 0.8 and 1.0. In binary coding, crossover methods include single point crossover, two-point crossover and multi-point crossover. Step 5: Mutation. Mutation is the operation of changing some genes of a chromosome to form new individuals with a certain probability (mutation probability). For the mutation in binary, the chromosomes are mutated at one point selected randomly. The mutation of bit is the inversion of bit: 0 becomes 1 or 1 becomes 0. In GA, mutation is employed to avoid converging to local minima. The mutation probability \(p_{m}\) is approximately 0.01. Step 6: After finishing the process of crossover and mutation operations, some new individuals are produced to join the surviving individuals so as to form a new generation. Step 7: Termination conditions. The termination condition is usually set as the maximum number of generations or no obvious changes in the value of the objective function. Step 8: Decode and calculate the initial structure parameters \(\alpha_{1}\), \(\alpha_{2}\), \(\beta_{1}\), \(\beta_{2}\), \(k_{1}\), \(k_{2}\), \(k_{4}\), \(d_{2}\), and \(\theta_{3}\). The flow chart of the whole design process is shown in Figure 7, which is described as follows:  
Figure 7: Flowchart of the design process. Step 1: Establish the objective function \(F=f(w_{i}\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},k_{1},k_{2},k_{4},d_{2},\theta_{ 3})\) by analyzing the aberrations and manufacturing constraints. Step 2: Set the weights and use GA to optimize the parameters (\(w_{i}\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},k_{1},k_{2},k_{4},d_{2},\theta_{ 3})\) to minimize the objective function. Step 3: Based on the optimized parameters in the previous step, calculate the configuration parameters \(R_{1}\), \(R_{2}\), \(R_{4}\), \(d_{1}\), \(d_{3}\), and \(d_{4}\). Step 4: The configuration parameters were imported for further analysis into the commercial software, and then, the final system is obtained. In this study, the objective function consists of two parts: the aberration constraints and the manufacturing constraints, in which the calculation of manufacturing constraints is very time-consuming because of the ray tracing. This phenomenon is more obvious when the population size and the number of iterations is large. In order to solve this problem, here we propose a strategy to speed up the calculation. The fitness calculation process of the GA is improved as shown in the flow chart in Figure 8. In this improved strategy, the aberration constraints \(f_{a}\) is first calculated instead of the objective function \(F\), and then the value of \(f_{a}\) is compared with the threshold. The inferior/poor individuals will directly use the \(f_{a}^{-1}\) as the fitness to implement the selection operation without calculating manufacturing constraints \(f_{c}\); As for the excellent individuals, \(f_{c}^{-1}\) will be calculated and added to \(f_{a}^{-1}\) to make them more competitive, which is more like an incentive process. As the iteration goes on, the proportion of individuals with small aberrations and good structural configurations will increase in the population.