Effects of non-linear vacuum electrodynamics on the polarization plane of light

We consider the Pleba{\'n}ski class of nonlinear theories of vacuum electrodynamics, i.e., Lagrangian theories that are Lorentz invariant and gauge invariant. Our main goal is to derive the transport law of the polarization plane in such a theory, on an unspecified general-relativistic spacetime and with an unspecified electromagnetic background field. To that end we start out from an approximate-plane-harmonic-wave ansatz that takes the generation of higher harmonics into account. By this ansatz, the electromagnetic field is written as an asymptotic series with respect to a parameter $\alpha$, where the limit $\alpha \to 0$ corresponds to sending the frequency to infinity. We demonstrate that by solving the generalized Maxwell equations to zeroth and first order with respect to $\alpha$ one gets a unique transport law for the polarization plane along each light ray. We exemplify the general results with the Born-Infeld theory.