Efficient and robust algorithms for the design of manufacturable optical freeform surfaces and their experimental validation

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Refractive and reflective optical components, whose surfaces offer a much higher number of degrees of freedom in comparison with conventional lenses and reflectors (axially symmetric and cylindric spheres and aspheres), are subsumed under the term "freeform optics". For their computer-aided design, algorithms are used that compute the optical surfaces from the given source and target light densities and other boundary conditions. This project addresses the first interdisciplinary advancement of these algorithms on the mathematical as well as on the technical and optical side in order to make them available for engineering applications in complex optical systems.State of the art design algorithms do not offer satisfactory solutions to this problem even for a point light source, because neither do they meet production requirements, in particular, concerning the regularity of the surfaces, nor do they offer the possibility for the design of multiple surfaces (two lens surfaces, lens and reflector, etc.). Given the source intensity distribution none of the known methods can take a nonzero extent of the light source into account without employing an iterative correction.In preliminary work, the applicants have developed a method that generates for given point light sources smooth surfaces by rigorously modeling the underlying problem in terms of a Monge-Ampère equation. Furthermore, the method is capable of generating surfaces that produce highly complex target illuminations. Similar to optimal transport problems, the inverse reflector problem can be formulated as a second boundary value problem for the Monge-Ampère equation which is a strongly nonlinear second order partial differential equation subject to additional boundary conditions. In order to solve this problem numerically, a B-spline collocation method and a finite-difference method are used for mutual validation. Nested iteration methods are applied to ensure the convergence of the solver and to accelerate the calculations. The particularly challenging boundary condition is realized through a Picard-type iteration. Until now, no other solution algorithms are known for this boundary value problem.As part of the submitted project, the algorithms developed by the applicants are generalized to spatially extended light sources which are essential for real-life applications. Moreover, methods are developed that handle the design of optical systems with multiple surfaces. By combining both of these steps, a method shall be constructed that simultaneously handles arbitrarily bounded optical surfaces, multiple optical surfaces and spatially extended light sources. Besides correctness, further essential goals of the project are robustness, numerical efficiency and manufacturability of the calculated solutions.