Monge-Ampère equations with applications in optic design

  • Monge–Ampère-Gleichungen mit Anwendungen in der Optik

Kremer, Elisa; Müller, Siegfried (Thesis advisor); Dahmen, Wolfgang (Thesis advisor)

Aachen : RWTH Aachen University (2020, 2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2020

Abstract

Since antiquity humans are interested in the design of mirrors and lenses to create light distributions. Mathematically the problem is strongly connected to the optimal transport problem, where for an efficient transport between two different distributions is asked for. In both, illumination optics and optimal transport, the problem can be transferred to solving a fully nonlinear partial differential equation of Monge–Ampère type equipped with non-standard transport boundary conditions. In this thesis we develop a versatile, robust numerical method to solve partial differential equations of such a type. Versatile means that the method works for most illumination problems of practical interest, especially the difficult near field case, and robust means that the algorithm contains as few parameters as possible and does not have to be tweaked in a long trial and error process for every new configuration. The method is based on a C^1 finite element projection method designed to solve fully nonlinear partial differential equations with Dirichlet conditions. We discuss the key elements implementing such complicated elements and how to incorporate the nonlinear transport boundary conditions in the finite element method. In order to solve practically relevant problems, we identify the critical parts of the algorithm and introduce stabilisation techniques and inevitable regularisation within a nested iteration process. We confirm with a wide range of benchmark problems that the resulting algorithm yields numerical accuracy, geometric flexibility and high order of approximation for smooth solutions in both, classical optimal transport problems and optic design problems. In a second part we discuss potential exploitation in the context of extended light sources. Although mathematically the hereto considered illumination system already provides major challenges, during the design of compact optics the size of the source is not negligible and, hence, these systems cannot be modelled by a partial differential equation of Monge–Ampère type. Current design techniques like the phase space method are based on least-squares methods and often yield suboptimal results. Typically these methods are initialised with surfaces designed assuming a point source approximation. We develop a merging process that takes several different point source approximations into account and could thereby improve the overall performance of the phase space method. The guiding idea for the merging procedure is a convex combination of optical surfaces based on a unified parametrisation.

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