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A certified classification of first-order controlled coaxial telescopes

Audric Drogoul

TL;DR

The paper addresses the certified classification of on-axis three-mirror telescope configurations by modeling admissible first-order optics as a real semi-algebraic set defined via transfer-matrix equations. It adopts a real algebraic-geometry framework, leveraging elimination theory and a real-root classification algorithm to partition the parameter space into connected components, each associated with a precise topological invariant. The main contributions are a semi-algebraic description of the solution space, an exact invariant-based nomenclature that differentiates topologically distinct configurations, and validated results for codimensions 2 and 3 in focal and afocal cases. This framework provides a mathematically certified, topology-aware foundation for optical design exploration and paves the way for extending to four-mirror systems.

Abstract

This paper is devoted to an intrinsic geometrical classification of three-mirror telescopes. The problem is formulated as the study of the connected components of a semi-algebraic set. Under first order approximation, we give the general expression of the transfer matrix of a reflexive optical system. Thanks to this representation, we express the semi-algebraic set for focal telescopes and afocal telescopes as the set of non-degenerate real solutions of first order optical conditions. Then, in order to study the topology of these sets, we address the problem of counting and describe their connected components. In a same time, we introduce a topological invariant which encodes the topological features of the solutions. For systems composed of three mirrors, we give the semi-algebraic description of the connected components of the set and show that the topological invariant is exact.

A certified classification of first-order controlled coaxial telescopes

TL;DR

The paper addresses the certified classification of on-axis three-mirror telescope configurations by modeling admissible first-order optics as a real semi-algebraic set defined via transfer-matrix equations. It adopts a real algebraic-geometry framework, leveraging elimination theory and a real-root classification algorithm to partition the parameter space into connected components, each associated with a precise topological invariant. The main contributions are a semi-algebraic description of the solution space, an exact invariant-based nomenclature that differentiates topologically distinct configurations, and validated results for codimensions 2 and 3 in focal and afocal cases. This framework provides a mathematically certified, topology-aware foundation for optical design exploration and paves the way for extending to four-mirror systems.

Abstract

This paper is devoted to an intrinsic geometrical classification of three-mirror telescopes. The problem is formulated as the study of the connected components of a semi-algebraic set. Under first order approximation, we give the general expression of the transfer matrix of a reflexive optical system. Thanks to this representation, we express the semi-algebraic set for focal telescopes and afocal telescopes as the set of non-degenerate real solutions of first order optical conditions. Then, in order to study the topology of these sets, we address the problem of counting and describe their connected components. In a same time, we introduce a topological invariant which encodes the topological features of the solutions. For systems composed of three mirrors, we give the semi-algebraic description of the connected components of the set and show that the topological invariant is exact.

Paper Structure

This paper contains 42 sections, 15 theorems, 91 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Polynomial systems
  3. Problem statement
  4. Notations
  5. Transfer Matrix
  6. Polynomials systems for focal telescopes
  7. Image constraint
  8. Focal constraint
  9. Petzval's constraint
  10. Telecentricity constraint
  11. Unknowns
  12. Constraints
  13. Polynomials systems for afocal telescope
  14. Magnification constraint
  15. Petzval constraint
  16. ...and 27 more sections

Key Result

Proposition 2.1

Let $N\geq 2$, the transfer matrix $M_N$ writes as where with

Figures (7)

  • Figure 1: Geometrical illustration of \ref{['eq:courbures']}. First order formula: $\frac{1}{s_k}+\frac{1}{s_k'}=2c_k$, Change of coordinate system: $d_k=s_k'-s_{k+1}$, Magnification definition: $\Omega_k = \frac{\rho_{k+1}}{\rho_k}=\frac{s_{k+1}}{s_k'}$
  • Figure 2: Topological invariant features. (a) Positive and (b) Negative magnifications. (c) Convex and (d) Concave mirrors.
  • Figure 3: Focal codim 2, $f=1$
  • Figure 4: Focal codim 2, $f=-1$
  • Figure 5: Afocal codim 2
  • ...and 2 more figures

Theorems & Definitions (33)

  • Proposition 2.1
  • proof
  • Theorem 3.1: The Elimination Theorem [Cox, chapter 3, § 1 ]
  • Theorem 3.2: The Extension Theorem [Cox, chapter 3, § 1]
  • Theorem 3.3: The Closure Theorem [Cox, chapter 4, § 7]
  • Remark 3.1
  • Lemma 3.1: LE202225-Prop. 11
  • Lemma 3.2
  • proof
  • Remark 3.2
  • ...and 23 more