Polarized Spinoptics and Symplectic Physics

TL;DR

This paper develops a geometric model of polarized spinoptics using symplectic and Riemannian geometry, explaining phenomena like the Spin Hall Effect of Light through differential equations derived from minimal coupling.

Contribution

It extends geometrical optics to include elliptically polarized light rays within a symplectic framework, connecting classical and wave optics phenomena.

Findings

Derived differential equations for light trajectories and polarization evolution.

Reproduced and justified equations related to the Spin Hall Effect of Light.

Automatically incorporated Berry and Pancharatnam connections in the geometric model.

Abstract

We recall the groundwork of spinoptics based on the coadjoint orbits, of given color and spin, of the group of isometries of Euclidean three-space; this model has originally been put forward by Souriau in his treatise "Structure des Syst\'emes Dynamiques", whose manuscript was initially entitled "Physique symplectique". We then set up a model of polarized spinoptics, namely an extension of geometrical optics accounting for elliptically polarized light rays in terms of a certain fibre bundle associated with the bundle of Euclidean frames of a given Riemannian three-manifold. The characteristic foliation of a natural presymplectic two-form introduced on this bundle via the Ansatz of minimal coupling is determined, yielding a set of differential equations governing the trajectory of light, as well as the evolution of polarization in this Riemannian manifold. Those equations, when specialized to the Fermat metric (for a slowly varying refractive index), enable us to recover, and justify, a set of differential equations earlier proposed in the literature, in another context, namely in terms of a semi-classical limit of wave optics. They feature a specific anomalous velocity responsible for the recently observed Spin Hall Effect of Light, namely a tiny spatial deflection of polarized light rays, transversally to the gradient of the refractive index. Our model, constructed from the start on purely geometric grounds, turns out to encode automatically the Berry as well as the Pancharatnam connections that usually appear in the framework of wave optics.