Building Paraxial Optical Skyrmions Using Rational Maps

Abstract

A simple mathematical expression based on rational maps to describe all optical paraxial skyrmion known to date, including Néel‐type and Bloch‐type skyrmions, bimerons, and anti‐skyrmions, is introduced. This expression is derived solely from topological considerations and outlines the rules that fully polarized paraxial light fields must obey to qualify as optical skyrmions. It is shown that rational maps can be implemented experimentally by superposing a pair of orthogonally polarized Laguerre–Gaussian modes. Novel optical skyrmion fields, called multi‐skyrmions, are obtained upon generalizing the proposed expression, laying the foundation for the exploration of skyrmion nucleation and annihilation mechanisms in optics.

Topological Approach of Characterizing Optical Skyrmions and Multi‐Skyrmions

“…[14][15][16][17][18][19] Unlike magnetic skyrmions, freely propagating paraxial optical skyrmions are only constrained by Maxwell's equations, and therefore offer a versatile platform for the investigation of exotic topological structures. [20] The generation of topological states of light opens up new avenues for the controlled interaction of DOI: 10.1002/lpor.202300155 photons with material quasi-particles such as plasmons, phonons, and excitons. [21] Experimentally, polarization textures and hence skyrmions can be assessed by measuring the spatially varying reduced Stokes vector S(x, y) across the light profile, mapping the local polarization states onto the Poincaré sphere, just like the local spin of magnetic skyrmions is mapped onto the Bloch sphere.…”

“…[ 14–19 ] Unlike magnetic skyrmions, freely propagating paraxial optical skyrmions are only constrained by Maxwell's equations, and therefore offer a versatile platform for the investigation of exotic topological structures. [ 20 ] The generation of topological states of light opens up new avenues for the controlled interaction of photons with material quasi‐particles such as plasmons, phonons, and excitons. [ 21 ]…”

“…Experimentally, polarization textures and hence skyrmions can be assessed by measuring the spatially varying reduced Stokes vector $$\mathbf {S}(x,y)$$ across the light profile, mapping the local polarization states onto the Poincaré sphere, just like the local spin of magnetic skyrmions is mapped onto the Bloch sphere. [ 22 ] The polarization texture itself can take on almost unlimited shapes, including Néel‐type (hedgehog) and Bloch‐type skyrmions, [ 12,15,20,23,24 ] but the underlying topology is characterized by a single invariant, the skyrmion number, n , which counts how many times $$\mathbf {S}$$ wraps around the Poincaré sphere. [Note that while every Skyrmion beam is a Poincaré beam, the reverse does not hold: The mapping must obey specific mapping rules.…”

“…In this research article, we derive and demonstrate an alternative, topological method to calculate skyrmion numbers that avoids products of polarization gradients and significantly increases precision (for low n ) and accuracy (in the presence of noise). We evaluate n for a variety of experimentally generated optical skyrmions and multi‐skyrmions, the latter being constructed by combining multiple skyrmions cores as discussed in [ 20 ] and. [ 26 ] Our method provides geometric insight that is missing from the surface integral representation: allowing us, for example, to interpret multi‐skyrmions as combination of individual skyrmion structures rather than just providing an overall skyrmion number.…”

Optical skyrmions

“…These beams are similar to the Poincaré beams, in which all possible polarisations appear at some point in the transverse plane [4], but they are subtly different. Skyrmions are topological features [19] that are characterised by an integer Skyrmion number which is a property of an underlying Skyrmion field. This field is constructed from the local (normalised) Stokes parameters and Skyrmion field lines have the physically appealing interpretation that they correspond to lines of constant polarisation [20].…”