Fig. 1: Optical magneto-electric effect.

Magneto-electric (ME) coupling leads to different electrodynamics within the bulk and on the surface. Inside the magneto-electric medium, two linearly polarized light propagate with different speeds because the wave equation is modified by origin-independent magneto-electric coupling T_ij and electric quadrupole susceptibility S_ijk (this effect is called gyrotropic birefringence²⁰). On the surface, axion magneto-electric coupling θ^(z) comes into play additionally, contributing to the surface Hall conductivity. This is most readily seen by writing the action for optical axion electrodynamics allowing for a spatially varying θ parameter, S_OA = (e²/2πh)∫θ(x)E⋅B. Integrating by parts in the presence of an interface along the z direction where the axion angle jumps at the interface δθ and identifying the coefficient of A_i with the surface current density we find \({{{{{{{\bf{J}}}}}}}}=-({e}^{2}/2\pi h)\delta \theta \hat{z}\times {{{{{{{\bf{E}}}}}}}}\). Thus an electric field parallel to the surface sets up a current in the orthogonal direction along the surface, indicative of a surface Hall effect. S_OA does not modify the propagation of electromagnetic fields within the bulk medium where θ does not vary spatially.