Reconstructing the topology of optical polarization knots

Abstract

Knots are topological structures describing how a looped thread can be arranged in space. Though most familiar as knotted material filaments, it is also possible to create knots in singular structures within three-dimensional physical fields such as fluid vortices 1 and the nulls of optical fields 2-4. Here we produce, in the transverse polarization profile of optical beams, knotted lines of circular transverse polarization. We generate and observe both simple torus knots and links as well as the topologically more complicated figure-eight knot. The presence of these knotted polarization singularities endows a nontrivial topological structure on the entire three-dimensional propagating wavefield. In particular, the contours of constant polarization azimuth form Seifert surfaces of high genus 5 , which we are able to resolve experimentally in a process we call seifertometry. This analysis reveals a level of topological complexity, present in all experimentally generated polarization fields, that goes beyond the conventional reconstruction of polarization singularity lines.

Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects

“…The oriented areas on the Poincarana sphere determining the 3D geometric phase(30) and (31a), cf. 2D case in Figs.…”

Complete Control of Multichannel, Angle‐Multiplexed, and Arbitrary Spatially Varying Polarization Fields

“…The utilization of a vectorial optical field with complete control over the phase, amplitude, and polarization has recently become a topic of extensive interest . In particular, vectorial optical fields with an arbitrary spatially varying polarization distribution are playing an increasingly important role in the scientific discoveries of many physical systems and a variety of applications, ranging from optical encryption to communication . For example, vector beams have been considered to be a promising candidate for an optical mode‐division multiplexing system, optical storage, and quantum key distribution .…”

“…The applications of optical images with spatially varying polarization manipulation include dynamic displays and information encryption . An attempt has also been made to reconstruct a Seifert surface structure by analyzing the polarization ellipse orientations of knotted polarization singularities …”

Multi-twist polarization ribbon topologies in highly-confined optical fields

“…These beams are interesting both at the fundamental and applied levels. For instance, they can be used to enhance measurement sensitivity [9,10], to transport high-power in nonlinear media [11], or to generate exotic optical beams with peculiar topological structures [12,13]. In particular, spatially structured light fields with field components along the propagation direction were predicted by Freund to show so-called optical polarization Möbius strips and twisted ribbons [14,15], with the former recently confirmed experimentally in tightly focused fields [16] as well as in the originally proposed scheme of crossing beams [17], and numerically in the scattering from dielectric spheres [18].…”