Knotted nodal lines in superpositions of Bessel-Gaussian light beams

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 Schrodinger equation for a free particle. The 2D Schrodinger propagator is expressed in terms of Bessel functions, which allows to obtain directly superpositions of beams with the desired topology of nodal lines. Four types of knots are constructed in an explicit way: the unknot, the Hopf link, the (3,3)-torus knot, and the trefoil. It is also shown, using the example of the figure-eight knot, that more complex structures require a larger number of constituent beams as well as high precision both from the numerical and the experimental sides. A tiny change of beam's intensity can lead to the knot switching. The figures refer to structures that can be called nanoknots. However, they equally apply to larger knots of up to several tens of the wavelength, if the scaling possibility provided by the presence of the parameter gamma is used.

Automated summary

A simple analytical method is proposed to create superpositions of Bessel-Gaussian light beams with knotted nodal lines, based on the equivalence between the paraxial wave equation and the two-dimensional Schrodinger equation. Four types of knots are explicitly constructed and it is shown that more complex structures require a larger number of constituent beams as well as high precision.