Complex optical vortex knots

TL;DR

This paper introduces a mathematical method to generate complex optical fields with vortex knots of any shape, expanding beyond previously limited types like torus and lemniscate knots, and demonstrates this for all knots up to 8 crossings.

Contribution

The authors develop a general construction for creating optical vortex knots of any topology, significantly broadening the range of knotted optical vortices achievable in experiments.

Findings

Successfully generated complex fields with all knots up to 8 crossings

Extended the class of known optical vortex knots beyond traditional types

Provided a mathematical framework for designing arbitrary knotted optical vortices

Abstract

The curves of zero intensity of a complex optical field can form knots and links: optical vortex knots. Both theoretical constructions and experiments have so far been restricted to the very small families of torus knots or lemniscate knots. Here we describe a mathematical construction that presumably allows us to generate optical vortices in the shape of any given knot or link. We support this claim by producing for every knot $K$ in the knot table up to 8 crossings a complex field $\Psi:\mathbb{R}^3\to\mathbb{C}$ that satisfies the paraxial wave equation and whose zeros have a connected component in the shape of $K$. These fields thus describe optical beams in the paraxial regime with knotted optical vortices that go far beyond previously known examples.