Geometrodynamics of spinning light: Berry phase, spin Hall effect, anisotropyK. Y. Bliokh, A. Niv, V. Kleiner, E. HasmanMicro and Nanooptics Laboratory, Faculty of Mechanical Engineering, Technion, Haifa 32000, IsraelNonlinear Physics Centre, RSPhysSE, The Australian National University, Canberra ACT 0200, Australiaemail: k.bliokh@gmail.com Abstract. The semiclassical evolution of spinning particles

has recently been re-examined in condensed matter physics, high energy physics, and optics, resulting in the prediction of the intrinsic spin Hall effect associated with the Berry phase. We report a unified theory and a direct observation of two mutual phenomena: a spin-dependent deflection (the spin Hall effect) of photons and the precession of the Stokes vector along the coiled light trajectory. Our measurements verify the dynamical action of the topological Berry-phase monopole in the evolution of light.

Spin-orbit interaction of lightGeometrical optics describes propagation of light in

a smooth inhomogeneous medium in the short-wavelength limit. Polarization and trajectory are uncoupled in the zero-order approximation. The first-order corrections account for the coupling between the trajectory and polarization, i.e., the spin-orbit interaction of light. It yields the two mutual effects:(A) The Berry phase which describes the influence of

the trajectory on the spin, resulting in the parallel-transport geometric evolution of polarization.

(B) The spin Hall effect which represents the reaction of spin on the trajectory. Due to this effect, right-hand and left-hand circularly polarized beams are deflected in the opposite directions even in isotropic but inhomogeneous medium.

Topological monopoleThe effects of the spin-orbit coupling of light are

described by the Berry-phase topological monopole in the momentum space. (A) Geometrically, the monopole field provides the

curvature underlying the parallel transport of the polarization.

(B) Dynamically, it produces a Lorentz-force-type term in the ray equations, which describes the spin Hall effect of light:

Here S3 is the spin helicity (3rd component of the Stokes vector), which perturbs the trajectory through the topological term.

ExperimentTo observe the topological dynamics of light (Berry

phase and spin Hall effect) we aimed to use a helical light trajectory. Such trajectory was produced by launching a polarized laser beam at a grazing angle inside a glass cylinder. As a result of total internal reflection, the grazing beam propagates along the smooth cylinder surface. The output polarization and position of the beam were measured as dependent on the input polarization and a number of coils in the helical trajectory.

Equations of motion and resultsThe propagation along a totally-reflecting surface

produces an extra birefringence between p and s polarizations. The medium becomes effectively anisotropic. The Berry phase and anisotropy are entangled in a complex polarization evolution, which is described by a precession of the Stokes vector on the Poincaré sphere. As a result, the light evolution in the experiment is described by coupled equations of motion for the polarization (the Stokes vector) and trajectory:

Here is the Stokes vector, R-1 and T-1 are the curvature and torsion of the ray trajectory, and b is the binormal to the ray.

Direct measurements of the output Stokes parameters and spin-dependent shift of the light beam are in perfect agreement with the above theory. This verifies the geometrodynamics of light including simultaneous action of the three phenomena:(i) Berry phase, (ii) spin Hall effect, and (iii) anisotropy.

,S S

13S Rk

-- bkr

2 11 ,1 2,0n R T- - - --

33 kS

k -

kr kk monopole

spin

Lorentz

spin Hall effectBerry phaseanisotropy

S

Published: K.Y. Bliokh et al., Nature Photon. (December 2008).