Closed Light Paths in Equiangular Spiral Disks

TL;DR

This paper investigates the geometry of light paths in equiangular spiral disks, deriving equations for closed paths, classifying them, and discovering a stable asymmetric bow-tie trajectory.

Contribution

It introduces a new geometric analysis of light paths in equiangular spirals, deriving equations for closed paths and identifying a novel stable asymmetric trajectory.

Findings

Two types of closed light paths identified: degenerate and nondegenerate.

Nondegenerate paths include a stable asymmetric bow-tie shape.

Closed paths exist over specific deformation parameter intervals.

Abstract

A new type of deformation for microscopic laser disks, the \textit{equiangular spiral deformation} is proposed. First a short review of the geometry of light paths in equiangular spirals in the language of real two-dimensional geometric calculus is given. Second, the constituting equations for \textit{closed paths} inside equiangular spirals are derived. Third, their numerical solution is performed and found to yield two generic types of closed light paths. \textit{Degenerate} closed paths that exist over large intervals of the deformation parameter, and \textit{nondegenerate} closed paths which only exist over relatively small deformation parameter intervals spanning less than 1% of the nondegenerate intervals. Fourth, amongst the nondegenerate paths a \textit{stable asymmetric bow-tie} shaped light trajectory was found.