Effective skyrmion number for mixed polarization states of light
Phys. Rev. A 114, 043503 – Published 6 October, 2026
DOI: https://doi.org/10.1103/q9k3-yzn9
Abstract
Polarization skyrmions are usually characterized through a unit Stokes vector field, which defines a map from the transverse plane to the Poincaré (Bloch) sphere and allows one to assign a skyrmion number under appropriate boundary conditions. This description assumes complete polarization. In many optical situations, however, the local polarization state is partially polarized and is therefore described by a spectral polarization matrix rather than by a single Jones vector. We formulate a density-matrix-based effective extension of the skyrmion-number construction for statistically stationary, quasimonochromatic paraxial fields in terms of the normalized spectral polarization matrix. Its eigenvectors define two orthogonal polarization eigenstates, corresponding to antipodal points on the sphere, while the eigenvalue imbalance is fixed by the local degree of polarization (DoP) . Averaging the opposite skyrmion-density contributions of these two eigenpolarizations leads to an effective skyrmion number in which the usual pure-state density is weighted by . The resulting quantity reduces to the conventional skyrmion number for fully polarized fields and assigns zero weight to locally unpolarized regions. In general, it is not an integer topological invariant; rather, it is an integrated effective measure, constructed from local Stokes parameters, which can be interpreted geometrically as a DoP-weighted signed solid-angle integral of the polarization eigentexture on the Poincaré sphere. We illustrate the construction with a regular analytic family of polarization eigentextures with the winding order and fixed radial DoP profiles, showing how the effective number depends jointly on the winding order and on the spatial overlap between the DoP and the skyrmion density.
