Fractional vortex Hilbert's Hotel

TL;DR

This paper explores how fractional vortex plates can realize the infinite and transfinite mathematics of Hilbert's Hotel paradox, enabling the creation of arbitrary topological charges in light propagation.

Contribution

It introduces the concept of fractional vortex plates as a physical realization of transfinite mathematics, demonstrating their ability to generate arbitrary topological charges simultaneously.

Findings

Fractional vortex plates can create multiple topological charges simultaneously.

They realize a physical analogy of Hilbert's Hotel paradox in optics.

The vortex state becomes completely undefined and arbitrary at the singularity.

Abstract

We demonstrate how the unusual mathematics of transfinite numbers, in particular a nearly perfect realization of Hilbert's famous hotel paradox, manifests in the propagation of light through fractional vortex plates. It is shown how a fractional vortex plate can be used, in principle, to create any number of "open rooms," i.e. topological charges, simultaneously. Fractional vortex plates are therefore demonstrated to create a singularity of topological charge, in which the vortex state is completely undefined and in fact arbitrary.