  We consider the propagation of light in arbitrarily curved step-index optical fibers. Using a multiple-scales approximation scheme, set-up in Fermi normal coordinates, the full vectorial Maxwell equations are solved in a perturbative manner. At leading order, this provides a rigorous derivation of Rytov's law. At next order, we obtain non-trivial dynamics of the electromagnetic field, characterized by two coupling constants, the phase and the polarization curvature moments, which describe the curvature response of the light's phase and its polarization vector, respectively. The latter can be viewed as an inverse spin Hall effect of light, where the direction of propagation is constrained along the optical fiber and the polarization evolves in a frequency-dependent way. 
Polarization transport in optical fibers beyond Rytov's law