Iterative freeform lens design for prescribed irradiance on curved target

Introduction

Manipulating the irradiance distributions of artificial light sources are very crucial for lighting and laser applications. For example, a Gaussian laser beam needs to be converted into a flattop one for improving laser material processing abilities. Compared with traditional spherical and aspherical optics, freeform optics has much more freedom, which can produce very complex irradiance distributions that are previously unimaginable. In fact, diffractive optical elements (DOEs), metasurfaces or graphene oxide lenses, could also realize the same goal while remaining flat^1-3. However, freeform optics is still an energy efficient and cost-effective choice especially for macro dimensions. Freeform optics design for irradiance tailoring on a given target is a very difficult inverse problem. Komissarov, Boldyrev and later, Schruben showed that the design of a freeform reflector for a point source could be formulated as a second order nonlinear partial differential equation (PDE) of Monge-Ampère (MA) type^{4, 5}. In Schruben's formulation, the MA equation is derived by mainly merging two types of equations. The first type is the energy conservation between the source intensity and the target irradiance. The second type is the ray tracing equations that describe the coordinate relationships from source to target. In addition, the reflector surface is constrained to have continuous second derivatives. Unfortunately, Schruben did not present the final expression of the MA equation and gave no hint on the numerical calculation. This is probably because that the derivation process is too complicated and the final MA equation is very difficult to solve. Wu et al. made a great effort to formulate a freeform refractive surface using the direct determination and employed Newton's method to solve the final MA equation⁶. Ries and Muschaweck created a different formulation process and solved a set of equivalent nonlinear PDEs, but they kept silent on the numerical techniques⁷.

Many other numerical methods have been developed for designing freeform reflectors and lenses. A common way is to approximate the freeform surface with sufficient quadric surfaces and to optimize their geometry^8-12, but the computations may become slow for high-resolution irradiance tailoring. Ray mapping methods are also commonly used, and they simplify the design with two separate steps: i) ray map computation and ii) surface construction following the ray...