Knotted optical vortices in exact solutions to Maxwell's equations

TL;DR

This paper presents a method to construct exact solutions to Maxwell's equations featuring knotted optical vortices with complex topologies, including algebraic links and cable knots, which are preserved over time.

Contribution

It introduces a novel class of exact electromagnetic solutions with knotted vortex lines based on algebraic links and complex polynomials, expanding the known topological structures in optics.

Findings

Exact solutions with knotted vortex lines are constructed.

Vortex topology remains invariant over time.

Includes complex algebraic links beyond simple knots.

Abstract

We construct a family of exact solutions to Maxwell's equations in which the points of zero intensity form knotted lines topologically equivalent to a given but arbitrary algebraic link. These lines of zero intensity, more commonly referred to as optical vortices, and their topology are preserved as time evolves and the fields have finite energy. To derive explicit expressions for these new electromagnetic fields that satisfy the nullness property, we make use of the Bateman variables for the Hopf field as well as complex polynomials in two variables whose zero sets give rise to algebraic links. The class of algebraic links includes not only all torus knots and links thereof, but also more intricate cable knots. While the unknot has been considered before, the solutions presented here show that more general knotted structures can also arise as optical vortices in exact solutions to Maxwell's equations.