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Inverse methods for freeform optical design

J. H. M. ten Thije Boonkkamp, K. Mitra, M. J. H. Anthonissen, L. Kusch, P. A. Braam, W. L. IJzerman

Abstract

We present a systematic derivation of three mathematical models of increasing complexity for optical design, based on Hamilton's characteristic functions and conservation of luminous flux, and briefly explain the connection with the mathematical theory of optimal transport. We outline several iterative least-squares solvers for our models and demonstrate their performance for a few challenging problems.

Inverse methods for freeform optical design

Abstract

We present a systematic derivation of three mathematical models of increasing complexity for optical design, based on Hamilton's characteristic functions and conservation of luminous flux, and briefly explain the connection with the mathematical theory of optimal transport. We outline several iterative least-squares solvers for our models and demonstrate their performance for a few challenging problems.

Paper Structure

This paper contains 14 sections, 80 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Hamilton's characteristic functions
  3. Geometrical description of optical systems: an optimal transport point of view
  4. Conservation of luminous flux
  5. Reflector and lens equations: a hierarchy of mathematical models
  6. Parallel-to-far-field reflector
  7. Point-to-far-field lens
  8. Parallel-to-near-field reflector
  9. Summary of mathematical models
  10. Iterative least-squares methods
  11. Numerical examples
  12. Point-to-far-field lens
  13. Parallel-to-near-field reflector
  14. Concluding remarks

Figures (7)

  • Figure 1: Interpretation of the mixed characteristics of the first and second kind.
  • Figure 2: Stereographic projections of the unit sphere $\mathrm{S}^{2}$.
  • Figure 3: Sketch of a parallel-to-far-field reflector.
  • Figure 4: Sketch of a point-to-far-field lens.
  • Figure 5: Sketch of a parallel-to-near-field reflector.
  • ...and 2 more figures