Knotting fractional-order knots with the polarization state of light

TL;DR

This paper introduces a novel class of light beams with polarization singularities exhibiting torus-knot topology, achieved through bicircular superpositions, expanding the understanding of symmetry and topology in optical fields.

Contribution

The work constructs polarization-invariant light beams for arbitrary symmetry parameters using bicircular superpositions, revealing their torus-knot topology and experimentally characterizing their singularities.

Findings

Beams exhibit invariance under generalized coordinated rotations.

Polarization singularities form torus-knot topologies.

Experimental observation confirms theoretical predictions.

Abstract

The fundamental polarization singularities of monochromatic light are normally associated with invariance under coordinated rotations: symmetry operations that rotate the spatial dependence of an electromagnetic field by an angle $\theta$ and its polarization by a multiple $\gamma\theta$ of that angle. These symmetries are generated by mixed angular momenta of the form $J_\gamma = L + \gamma S$ and they generally induce M\"obius-strip topologies, with the coordination parameter $\gamma$ restricted to integer and half-integer values. In this work we construct beams of light that are invariant under coordinated rotations for arbitrary $\gamma$, by exploiting the higher internal symmetry of 'bicircular' superpositions of counter-rotating circularly polarized beams at different frequencies. We show that these beams have the topology of a torus knot, which reflects the subgroup generated by the torus-knot angular momentum $J_\gamma$, and we characterize the resulting optical polarization singularity using third-and higher-order field moment tensors, which we experimentally observe using nonlinear polarization tomography.