Geometrical spinoptics & symplectic geometry

The object of this talk is to illustrate the usefulness of symplectic geometry in understanding new features of geometrical optics for spinning light in refractive media, e.g., the subtle transverse shift of light beams across an interface, which has been coined the Spin Hall Effect of Light [1]. We reformulate Fermat’s principle to describe spinning light rays in Riemannian three-manifolds, (M, g). The procedure relies on the generic, spinning and colored, coadjoint orbits of the Euclidean group, E(3). Our presymplectic model for spinoptics [2] yields, via its characteristic foliation, equations of motion obtained by the prescription of minimal coupling—borrowed from General Relativity. Spin-induced geodesic deviation is analyzed; it is shown to provide a refinement of the Snell-Descartes laws, viz., the Spin Hall Effect of Light, derived using the symplectic scattering of Souriau between “in” and “out” photonic coadjoint orbits. Geometrical spinoptics is then generalized to the case of Finsler three-manifolds, (M,F ), providing an effective model for anisotropic media [3]. The original Euclidean symmetry forces us to single out a Finsler-Cartan structure. Again, minimal coupling to the Cartan connection generates a foliation on the indicatrix-bundle, F−1(1), that significantly deviates from the direction of the classical Finsler geodesic spray. This might provide new insights into the theory of geometrical optics in the presence of anisotropy, with applications to, e.g., Faraday-active optical media. Further extension of the formalism to arbitrarily polarized light is achieved by means of the complexification of the tangent “wave-plane”. Our previous choice of an E(3)-coadjoint orbit yields a new, exact, presymplectic “evolution space” (V, σ) featuring, apart from color and spin terms, the Berry and Pancharatnam connections at a single stroke. Moreover, the characteristic foliation of σ completely specifies the geodesic deviation of light rays, and the evolution of the polarization in an optical medium modelled by a Riemannian three-manifold. Linearization of these equations of motion in a conformally flat background metric, g, (defined in terms of a slowly variable refractive index) helps us recover a set of differential equations [1] which have been recently and spectacularly confirmed on experimental grounds [4].