On the Near-Field Reflector Problem and Optimal Transport Open Access

Abstract

In the near-field reflector problem, one is given a point source of light with some radiation intensity and a target set at a finite distance. The design problem consists of constructing a reflector that reflects the rays emitted from the source such that a given irradiance distribution is produced on the target. In recent years, the optimal transport framework has been applied successfully to various problems in the design of free-form lenses and reflectors. In this dissertation, the near-field problem is investigated in this context. In particular, it is shown that the notion of a weak solution to the near-field problem as an envelope of ellipsoids of revolution leads to a generalized Legendre-Fenchel transform. Aside from some interesting properties of this transform, it also gives rise to a variational problem that is naturally associated with the near-field reflector problem. Furthermore, the resulting variational problem resembles a generalized optimal transport problem and exhibits interesting analogies to other optimal transport problems arising in optical design and geometry, particularly to the far-field reflector problem and the methods developed by Glimm, Oliker, and Wang. However, for the near-field problem the solutions to the associated variational problem do not solve the reflector problem in general. This situation is illustrated by a number of examples and numerical experiments and is in sharp contrast to the problems that have been studied previously in the optimal transport framework. Interestingly, a connection between the solutions to the near-field problem and the variational problem can still be established. In particular, for discrete target sets an approximation result is presented, which shows that under a suitable choice of the admissible set the variational solution produces an irradiance distribution arbitrarily close to the prescribed irradiance distribution from the design problem. The variational functional is also compared to various functionals motivated by the geometric approach to the near-field problem that was developed by Kochengin and Oliker.

Table of Contents

1 Introduction 1
1.1 The Near-Field Reflector Problem . . . . . . . . . . . . . . . . 1
1.2 Reflectors Defined by Families of Ellipsoids . . . . . . . . . . . 7
1.3 Weak Formulation of the Near-Field Reflector Problem . . . . 23
2 A Generalized Legendre-Fenchel Transform 24
2.1 A Generalized Legendre-Fenchel Transform and the Associated
Variational Problem . . . . . . . . . . . . . . . . . . . . . 24
2.2 E-Polytopes and Irradiance Distributions Defined by Atomic
Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Examples and Numerical Experiments 43
3.1 Solutions of the Variational Problem May Not Solve the Reflector
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.1 First Example . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.2 Second Example . . . . . . . . . . . . . . . . . . . . . 50
3.2 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . 51
3.2.1 First Experiment . . . . . . . . . . . . . . . . . . . . . 51
3.2.2 Second Experiment . . . . . . . . . . . . . . . . . . . . 59
3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Large Variational Solutions as Approximate Solutions to the
Near-Field Single Reflector Problem 67
4.1 An Approximation Theorem . . . . . . . . . . . . . . . . . . . 67
4.2 Lipschitz Property of the Measure G . . . . . . . . . . . . . . 69
4.3 Large Maximizers Are Interior Points . . . . . . . . . . . . . . 74
4.4 Large Maximizers Illuminate the Whole Target Set . . . . . . 82
4.5 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . 83
5 Alternative Functionals: Weak Solutions as Extrema of Vari-
ational Problems 85
5.1 The Functional of Kochengin and Oliker . . . . . . . . . . . . 86
5.2 The Sum of the Logarithmic Focal Functions . . . . . . . . . . 87
5.3 The Integral of the (Logarithmic) Radial Function . . . . . . . 88
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Appendix 90
6.1 The Hausdorff Metric . . . . . . . . . . . . . . . . . . . . . . . 90
6.2 Blaschke's Selection Theorem . . . . . . . . . . . . . . . . . . 91
Bibliography 92

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	Legendre-Fenchel Transform Reflector Design Optimal Transport Near-Field Problem
Committee Chair / Thesis Advisor	Oliker, Vladimir, Emory University
Committee Members	Borthwick, David, Emory University Yang, Shanshuang, Emory University

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