Some Mathematical Problems in Design of Free-Form Mirrors and Lenses Público
Palta, Hasan (2012)
Published
Abstract
In this dissertation, we investigate several optics-related problems. The problems discussed in Chapters 1, 2 and 3 are concerned with the determination of surfaces reshaping collimated beams of light to obtain a priori given intensities on prescribed target sets. In optics, such transformations are performed by lenses and/or mirrors whose shapes need to be determined in order to satisfy the application requirements. These are inverse problems, which in analytical formulations lead to nonlinear partial differential equations of Monge-Ampère type. In Chapter 4, we present several different designs of radiant energy concentrators. Our goal in these designs is to obtain a device that can capture solar rays with maximal efficiency.
Table of Contents
Contents
1. The Design of a Free-Form Lens 7
1.1. Introduction to the Problem . . . . . . . . . . . . . . . . . . . 7
1.2. The PDE Describing the Two-Lens Problem . . . . . . . . . . 11
1.3. Geometry of Refractors . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1. Refracting Hyperboloids . . . . . . . . . . . . . . . . . 13
1.4. General Refracting Surfaces . . . . . . . . . . . . . . . . . . . 17
1.5. The Existence of a Solution . . . . . . . . . . . . . . . . . . . . 19
1.5.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.2. The Existence Theorem . . . . . . . . . . . . . . . . . 23
1.5.3. Refractors Defined by a Finite Number of Hyperboloids. . . . . . . . . . . . 27
1.5.4. A Uniqueness Theorem . . . . . . . . . . . . . . . . . 31
1.6. The Method of Supporting Hyperboloids and the Algorithm 32
1.7. An Estimate of the Parameters ηi . . . . . . . . . . . . . . . . 35
1.8. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2. A Light Beam Reshaping XR System 44
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2. Description of the System . . . . . . . . . . . . . . . . . . . . 45
2.3. Derivation of the PDE . . . . . . . . . . . . . . . . . . . . . . 46
2.4. A Geometric Approach to the XR Problem . . . . . . . . . . . 51
2.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4.2. The XR Surfaces . . . . . . . . . . . . . . . . . . . . . . 51
2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3. A Collimated Source Problem 59
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2. Description of the Problem . . . . . . . . . . . . . . . . . . . . 59
3.3. The Derivation of the PDE . . . . . . . . . . . . . . . . . . . . 61
3.4. The Rotationally Symmetric Case . . . . . . . . . . . . . . . . 65
3.5. An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4. A Problem in Non-Imaging Optics 76
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2. Description of the Problem . . . . . . . . . . . . . . . . . . . . 78
4.3. A Particular Design . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3.1. Example . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4. A Mixed Design . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5. A Refractive Surface . . . . . . . . . . . . . . . . . . . . . . . 88
Bibliography 93
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