Knotting fractional-order knots with the polarization state of light

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 Mobius-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 rational 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.