Knotted nodal lines in superpositions of Bessel-Gaussian light beams

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".