Classification of obscuration-free reflective polygonal light beams

TL;DR

This paper explores the topological structure of obscuration-free reflective polygonal light beams, establishing a correspondence between their connected components and those of reflective polygonal chains, aiding optical system design.

Contribution

It introduces a topological invariant linking optical configurations to polygonal chains, providing a new method for classifying light beam spaces in optical systems.

Findings

Connected components correspond to polygonal chain configurations.

Homotopy construction links light beam space to polygonal chains.

Topological invariant aids in optical configuration classification.

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.