# A Design for a Manufacturing-Constrained Off-Axis Four-Mirror Reflective System
## 2 Working Principle
### Search for the Initial Structure via GA  
GA is a global optimization algorithm that does not depend on the initial parameters. It uses the principle of nature to \"evolve\" toward an optimal solution. For highly nonlinear and high-dimensional parameter optimization problems, GA can often find the optimal solution effectively [28]. In our work, GA is used to optimize the objective function to obtain the initial optical system parameters. The algorithm iteratively optimizes the objective function by relying on biologically inspired operators such as mutation, crossover and selection. Figure 5: The layout of the four-mirror system. Based on the analyses of the manufacturing constraints, aberrations and the structure configuration in the previous section, the objective function was established. The objective function of aberrations is composed of weighted aberrations which can be expressed as \(f_{a}\),  
\[f_{a}=f_{a}(w_{i},\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},k_{1},k_{2},k_{4})=w _{1}|S_{1}|+w_{2}|S_{2}|+w_{3}|S_{3}|+w_{4}|S_{4}|+w_{5}|S_{5}|, \tag{10}\]  
where \(w_{i}(i=1,2\ldots 5)\) is the weight of the corresponding term. The weight given to each term depends on its importance in the system. The higher weight values are set to the aberration coefficients with high requirements and vice versa. According to Equation (10), the objective function \(f_{a}\) is a comprehensive reflection of the aberration in the optical system. The smaller \(f_{a}\) value indicates a better imaging quality of the optical system. The objective function of manufacturing and structural constraints \(f_{c}\) consists of the ADRS, the tool spindle path \(C\) and some other structural constraints \(O\), such as the obscuration elimination and telecentric in the image space, are as follows:  
\[f_{c}=f_{c}(w_{i},\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},k_{1},k_{2},k_{4}, d_{2},\theta_{3})=w_{6}|\sigma_{\text{ADRS}}|+w_{7}|C|+w_{i}|O|, \tag{11}\]  
where \(d_{2}\) and \(\theta_{3}\) are the additional variables added for structure control. The manufacturing and structural constraints function \(f_{c}\) and imaging quality constraints function \(f_{a}\) constitute the objective function \(F\):  
\[F=f_{a}+f_{c}. \tag{12}\]  
By establishing the objective function, the problem of solving the initial structural parameters of the four-mirror optical system is transformed into the optimization of the objective function \(F\). The small value of aberration coefficients, ADRS and tool spindle path \(C\) will be obtained by minimizing \(F\). In this paper, GA is introduced to optimize the objective function, that is, GA is introduced to solve the parameters of the initial structure. The optimization process of GA is shown in Figure 6, which is briefly described as follows:  
Figure 6: Flowchart of the genetic algorithm (GA) process. Step 1: Encode the parameters and initialize the population. Encoding is the basis of GA, and the encoding mechanism has an essential influence on the performance and efficiency of the algorithm. In this paper, binary is used to encode the parameters [29]. First, convert parameters \(\alpha_{1}\), \(\alpha_{2}\), \(\beta_{1}\), \(\beta_{2}\), \(k_{1}\), \(k_{2}\), \(k_{4}\), \(d_{2}\), and \(\theta_{3}\) from decimal numbers to binary numbers. The sequence of 9 parameters represents a chromosome, which is a solution of the objective function. Each generation consists of a certain amount of chromosomes, and the population size is set empirically. Then, the initial population of GA is formed by randomly generating multiple chromosomes. Step 2: Evaluate the fitness. The reciprocal of the objective function is used as the fitness function to calculate the fitness of each chromosome in the population. The value of fitness is the main performance index to describe the performance of an individual in GA. The larger fitness value indicates a good individual's performance and vice versa. Step 3: Selection. Once the fitness is calculated, several pairs of chromosomes were selected as parents for breeding.