Knotted optical vortices in exact solutions to Maxwell's equations

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.

Finsler geometries from topological electromagnetism

“…Another problem is to study the Finsler geometries corresponding to the polynomial action which implies including the constraints in the fundamental Finsler function. New generalizations of the Rañada fields have been presented recently in the literature, see e. g. [5][6][7][8] for topological solutions in the presence of the gravitational field and [9][10][11][12][13][14] for generalization to the non-linear electrodynamics. It should be interesting to generalize the construction presented in this paper to determine the Finsler geometries associated to these systems.…”

“…The study of the topological solutions to Maxwell's equations in vacuum, firstly proposed by Trautman and Rañada in [1][2][3], has revealed so far a rich interplay between physical systems and mathematical structures which was previously unexpected in the realm of classical electrodynamics and classical field theory [4]. Since then, the subject of the topological electromagnetic fields has gain momentum with very interesting problems investigated recently, such as the existence of topological solutions of the Einstein-Maxwell theory [5][6][7][8] and of the non-linear electrodynamics [9][10][11][12][13][14]. Also, it has been shown that there are interesting mathematical structures that can be associated to the physical systems a e-mail: adina.crisan@mep.utcluj.ro b e-mail: ionvancea@ufrrj.br (corresponding author) with topological electromagnetic fields and play an important role in their dynamics, such as twistors [15], fibrations [16] and rational functions [17,18] (see for recent reviews [19][20][21]).…”

Optical vortex knots and links via holographic metasurfaces

“…Besides, there also are exotic time-dependent solutions of Maxwell's equations, whose electric and magnetic field lines preserve topological configurations [34,41,69,70]. For example, Irvine and Bouwmeester studied the exact knotted solutions in free space based on a topological construction known as a Hopf fibration, in which all electric and magnetic field lines were closed loops and any two electric or magnetic field lines encode into knots and links [47].…”

The double copy: from optics to quantum gravity

“…Gravitational counterparts also exist, and have been studied in refs. [46][47][48][49][50][51], where the latter reference comments that the EM and gravity solutions can be related by the Weyl double copy mentioned above. The authors use twistor theory [52][53][54] -an elegant set of mathematical ideas relating algebraic geometry and complex analysis -to classify knotted radiation solutions in a compact way.…”