Designing Freeform Optical Surfaces by the Monge-Ampère Equations: A Review

Background: The equations of Monge-Ampère type which arise in geometric optics are used to design illumination lenses and mirrors. The optical design problem can be formulated as an inverse problem: determine an optical system consisting of a reflector and/or refractor that converts a given light distribution of the source into a desired target light distribution. For two decades, the development of fast and reliable numerical design algorithms for the calculation of freeform surfaces for irradiance control in the geometrical optics limit is of great interest in current research. Objective: The objective of this paper is to summarize the types, algorithms, and applications of Monge-Ampère equations. It helps scholars better grasp the research status of Monge-Ampère equations and further explore the theory of Monge-Ampère equations. Methods: This paper reviews the theory and applications of Monge–Ampère equations from four aspects. We first discuss the concept and development of Monge–Ampère equations. Then we derive two different cases of Monge–Ampère equations. We also list the numerical methods of Monge–Ampère equation in actual scenes. Finally, the paper gives a brief summary and an expectation. Results: This paper reviews the theory and applications of Monge-Ampère equations from four aspects. We first discuss the concept and development of Monge-Ampère equations. Then we derive two different cases of Monge-Ampère equations. We also list the numerical methods of the Monge-Ampère equation in actual scenes. Finally, the paper gives a brief summary and an expectation. Conclusion: Monge-Ampère equation has been widely applied in the geometric optics field since the predetermined energy distribution and the boundary condition creation can be well satisfied. Although the freeform surfaces designed by the Monge-Ampère equations is developing rapidly, there is still plenty of room for development in the design of the algorithms.