Geometrical phase and Hall effect associated with the transverse spin of light

By analyzing the vectorial Helmholtz equation within the thin-layer approach, we find that light acquires another geometrical phase, in addition to the usual one (the optical Berry phase), during the propagation along a curved path. Unlike the optical Berry phase, the additional geometrical phase is induced by the curvature of the curve and associated with the transverse spin of light. Furthermore, we show an additional Hall effect of light induced by the torsion of the curve and associated with the transverse spin of light, which is different from the usual spin Hall effect of light. Finally, we demonstrate that the usual and transverse-spin-dependent geometrical phase phenomena are described by different geometry-induced U(1) gauge fields in different adiabatic approximations. In the general case, these gauge fields are united in an effective SO(3) gauge field, and the optical Berry phase and transverse-spin-dependent geometrical phase are united in a general geometrical phase of light.