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Optical Caustics as Lagrangian Singularities: Classification and Geometric Structure

Rongqi Shang, Donglin Ma

TL;DR

The paper develops a rigorous symplectic-contact framework for optical phase space, proving that 3D light rays correspond to Reeb orbits in a 5D contact manifold and project to a 4D symplectic phase space. It provides intrinsic, wavefront-based definitions of caustic surfaces as projections of Lagrangian/Legendrian data and delivers explicit expressions for convex-lens caustics. Employing catastrophe theory, it classifies stable caustics in 3D and connects them to Seidel aberrations, establishing precise correspondences with A2, A3, A4, and D4± types. Building on this, the authors propose Topological Optical Correction (TOC), a topology-aware optimization strategy that navigates bifurcation sets in Zernike control space to robustly reduce high-order aberrations and manage caustic structures, with implications for high-precision optics and adaptive systems.

Abstract

This paper develops a rigorous mathematical framework for light propagation by constructing the optical phase space with its symplectic structure and the extended phase space with its contact structure. We prove that light rays in three-dimensional Euclidean space correspond to Reeb orbits in a five-dimensional contact manifold, which are then projected onto a four-dimensional symplectic manifold via symplectic reduction. Leveraging the advantages of phase space, we provide a rigorous definition of caustic surfaces as singularities of the Lagrangian submanifold projection and derive explicit expressions for caustic surfaces in convex lens systems. Furthermore, based on singularity theory, we present a complete classification of stable caustic surfaces and establish a correspondence with classical Seidel aberration theory. Building upon this theory, we propose a method of \emph{topological optical correction} that overcomes the limitations of traditional optimization algorithms in dealing with complex caustic structures. This work provides a new mathematical paradigm for the design and correction of high-precision optical systems.

Optical Caustics as Lagrangian Singularities: Classification and Geometric Structure

TL;DR

The paper develops a rigorous symplectic-contact framework for optical phase space, proving that 3D light rays correspond to Reeb orbits in a 5D contact manifold and project to a 4D symplectic phase space. It provides intrinsic, wavefront-based definitions of caustic surfaces as projections of Lagrangian/Legendrian data and delivers explicit expressions for convex-lens caustics. Employing catastrophe theory, it classifies stable caustics in 3D and connects them to Seidel aberrations, establishing precise correspondences with A2, A3, A4, and D4± types. Building on this, the authors propose Topological Optical Correction (TOC), a topology-aware optimization strategy that navigates bifurcation sets in Zernike control space to robustly reduce high-order aberrations and manage caustic structures, with implications for high-precision optics and adaptive systems.

Abstract

This paper develops a rigorous mathematical framework for light propagation by constructing the optical phase space with its symplectic structure and the extended phase space with its contact structure. We prove that light rays in three-dimensional Euclidean space correspond to Reeb orbits in a five-dimensional contact manifold, which are then projected onto a four-dimensional symplectic manifold via symplectic reduction. Leveraging the advantages of phase space, we provide a rigorous definition of caustic surfaces as singularities of the Lagrangian submanifold projection and derive explicit expressions for caustic surfaces in convex lens systems. Furthermore, based on singularity theory, we present a complete classification of stable caustic surfaces and establish a correspondence with classical Seidel aberration theory. Building upon this theory, we propose a method of \emph{topological optical correction} that overcomes the limitations of traditional optimization algorithms in dealing with complex caustic structures. This work provides a new mathematical paradigm for the design and correction of high-precision optical systems.

Paper Structure

This paper contains 56 sections, 206 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Motivation and Structure
  4. Symplectic Geometry of Optical Phase Space
  5. Contact Geometry of Extended Phase Space
  6. Three-Dimensional Phase Space Optics
  7. Basic Construction
  8. From 6D Symplectic Space to 5D Contact Space
  9. Equivalence of Rays and Reeb Vector Field
  10. Correspondence Between Rays and Reeb Vector Field
  11. Second Reduction: From 5D Contact Space to 4D Symplectic Space
  12. Existence
  13. Uniqueness
  14. Closedness
  15. Non-degeneracy
  16. ...and 41 more sections

Figures (4)

  • Figure 1: Schematic diagram of caustic surface for a convex lens.
  • Figure 2: Schematic diagrams of different caustic surfaces.
  • Figure 3: Schematic of correspondence between $D_4^-$ and trefoil aberration.
  • Figure 4: Schematic diagrams of catastrophe functions corresponding to different caustic surfaces.

Theorems & Definitions (33)

  • proof
  • proof
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  • proof
  • proof
  • proof
  • proof
  • ...and 23 more