Classification of obscuration-free reflective polygonal light beams
Pierre Franck, Audric Drogoul
TL;DR
The paper addresses the problem of classifying obscuration-free planar optical configurations modeled by reflective polygonal surfaces by studying the connected components of the configuration space $E$. It builds a sequence of polygonal models—$LO^N(\mathbb{R}^2)$, $LR^N(\mathbb{R}^2)$, and $Faisc^N(\mathbb{R}^2)$—and introduces exact topological invariants $N$, $N^{R}$, Car, Car$^{F}$, yielding exact invariants $I^{F}=(N^{F},Car^{F})$ (and variants $I^{S}$) that classify components; symmetry is handled via a $\mathbb{Z}/2\mathbb{Z}$ quotient. The framework is applied to three-mirror telescopes to define paired on-off axis nomenclature, with numerical evidence suggesting at least 160 connected components in the paired model, aligning with prior onaxis24 results. Overall, the invariants provide a practical labeling scheme that reduces brute-force exploration to topology-guided classification, aiding optical design and exhaustive sampling of admissible configurations.
Abstract
In this paper, we study the connected components of an obscuration-free planar polygonal light beam space modeling light propagation in optical systems composed of reflective surfaces and a focal plane. Through homotopy construction, we demonstrate that the connected components of this space are in bijection with the connected components of the reflective polygonal chains space, whose elements are the polygonal chains with their respective mirrors' orientations taken into account. In order to prove this, we introduce a topological invariant that provides an intelligible way for opticians to name homotopy-equivalent obscuration-free optical configurations thanks to previous work with polygonal chains.