# A Design for a Manufacturing-Constrained Off-Axis Four-Mirror Reflective System
## 4 Conclusions  
The introduction of manufacturing constraints in the optical design process can guarantee the optical performance of the system and facilitate the assembly and machining process, which is beneficial to further develop the applications of optical systems.  
In this paper, a novel method for finding the initial structure parameters of an off-axis four-mirror reflective system considering manufacturing constraints via GA is proposed. In this method, in order to find the appropriate initial structural parameters, the GA is employed to optimize the objective function, which is established according to aberration analyses and manufacturing constraints analyses. The GA is computationally efficient when finding the global minima in complex, highly nonlinear and high-dimensional parameter spaces. In addition, the reasonable definition of the objective function and the optimization strategy ensure the possibility of using GA successfully. This method helps solve the complex optical alignment problem in optical systems. The off-axis four mirror system designed according to this method can be put into use without optical alignment once the machining is completed. This method is also applicable to off-axis three-mirror systems, off-axis five-mirror systems and coaxial reflective systems.  
The feasibility of the proposed method is validated by designing an off-axis four-mirror afocal system with an entrance pupil diameter of 170 mm, a field of view of \(3^{\circ}\times 3^{\circ}\), an operating wave band of 650-900 nm, and a compression ratio of five times, in which all the mirrors are guaranteed to be distributed along a cylinder 337 mm in radius to facilitate the ultraprecise raster milling. The results show that the optimization method based on GA can provide a good starting point for the design of the reflective optical system with special requirements. In the future, by integrating this method with different optical structures, a synthetic method to deal with the design of manufacturing-constrained optical systems can be obtained.  
Conceptualization, Z.L. and F.F.; methodology, R.L.; software, R.L.; validation, R.L.; formal analysis, R.L. and Z.L.; investigation, R.L., Z.L. and Y.D.; resources, F.F.; data curation, R.L.; writing--original draft preparation, R.L.; writing--review and editing, R.L., Z.L., Y.D. and F.F.; visualization, R.L.; supervision, F.F.; project administration, F.F.; funding acquisition, F.F. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Key Research and Development Program of China (Grant No. 2017 YFA0701200), the Postdoctoral Innovative Talent Support Program of China (BX20190230), the National Natural Science Foundation (No. 61635008), and the \"111\" project conducted by the State Administration of Foreign Experts Affairs and the Ministry of Education of China (No. B07014)  
Figure 16: Distortion of the off-axis four-mirror system. (**a**) Grid distortion. (**b**) F-Tan (Theta) distortion.  
**Acknowledgments**: The authors express their sincere thanks to [PERSON] from Tianjin University for valuable discussions.  
**Conflicts of Interest**: The authors declare no conflict of interest.