Knotted nodal lines in superpositions of Bessel-Gaussian light beams

TL;DR

This paper presents an analytical method to generate superpositions of Bessel-Gaussian light beams with knotted nodal lines, enabling the creation of complex topological structures in light fields.

Contribution

It introduces a novel approach linking the paraxial wave equation to the 2D Schrödinger equation, allowing explicit construction of knotted nodal lines in light beams.

Findings

Successfully constructed four types of knots in light beams.

More complex knots require more beams and higher precision.

Small intensity changes can switch the knot topology.

Abstract

A simple analytical way of creating superpositions of Bessel-Gaussian light beams with knotted nodal lines is proposed. It is based on the equivalence between the paraxial wave equation and the two-dimensional Schr\"odinger equation for a free particle. The $2D$ Schr\"odinger propagator is expressed in terms of Bessel functions, which allows to obtain directly superpositions of beams with a desired topology of nodal lines. Four types of knots are constructed in the explicit way: the unknot, the Hopf link, the Borromean rings and the trefoil. It is also shown, using the example of the figure-eight knot, that more complex structures require larger number of constituent beams as well as high precision both from the numerical and the experimental side. A tiny change of beam's intensity can lead to the knot "switching".