Knotting fractional-order knots with the polarization state of light

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 Möbius-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.

arXiv commentSubmitted Manuscript, including a subset of the figures from the published Supplementary Information

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