Knotting of optical vortices

Abstract

Optical vortices form three-dimensional lines of darkness in scalar light. They are places where the phase becomes undefined and hence singular in value. We study the ability of optical vortices to form knots and links of darkness in scalar optical fields. We describe a construction to create complex scalar fields that contain a fibred knot or link as its zero set. This procedure starts by constructing braids with strands that follow a lemniscate trajectory as they increase in height. These braids are closed using Milnor maps to form a function with a knotted or linked zero set. This braid contains a minimal amount of information rather than the minimum number of crossings, taking advantage of symmetries in the construction. The knots and links we construct exhibit patterns in their Alexander and Jones polynomial coefficients, as well as in their Conway notation as parameters in the construction are varied. We use these patterns to propose a tabulation of the knots and links we can construct. The knots and links we can construct are examined as solutions of the paraxial equation using polynomial solutions. We show that a wide range of vortex topologies are possible and report an experimental implementation of the technique. We also consider the Helmholtz and Schr¨odinger equations and attempt to construct solutions to these equations with knotted phase singularities. We conclude with a geometric approach to optical vortex control. This is used to study the initial value problem of paraxial propagation and attempts to construct a function that describes the optical vortex paths on propagation

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