Ray propagation of circularly polarized light in an inhomogeneous medium: the optical Magnus effect from Maxwell's equations

Abstract

We derive the ray propagation of circularly polarized light in an inhomogeneous, isotropic and lossless medium, including the first correction to geometrical optics, the optical Magnus (spin Hall) effect. Starting from Maxwell's equations and the eikonal expansion to first order in $1/k_0$, we show that the polarization is parallel transported along the ray, identify the Berry connection and curvature of the helicity states, and obtain a ray action in which the Berry phase appears as a correction to the optical path length. This action leads to a Hamiltonian in which the refractive index is evaluated at a position shifted by the Berry connection, $\mathcal H_σ(\mathbf r,\mathbf p)=|\mathbf p|-n(\mathbf r+\mathbf A_σ(\mathbf p)/k_0) \simeq|\mathbf p|-n(\mathbf r)-k_0^{-1}\boldsymbol\nabla n\cdot\mathbf A_σ(\mathbf p)$. In terms of the physical position, the equations of motion contain the Berry curvature as an anomalous velocity. We give the equivalent non-canonical bracket form, the gauge transformation, and the equivalent time-parametrized formulation with the Hamiltonian $\hbarω=\hbar c|\mathbf k|/n$, and we state the approximations involved. The resulting ray equations are compared with analytic and numerical results. A planar interface, the helical ray in a cylindrical medium and the exact conservation of angular momentum provide checks. The formulation is set up for numerical propagation of rays in a medium with a spatially varying refractive index.