Classification of obscuration-free reflective polygonal light beams

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.