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Magnetoelectric phenomena are intimately linked to relativistic effects andalso require the material to break spatial inversion symmetry and time reversalinvariance. Magnetoelectric coupling can substantially affect light-matterinteraction and lead to non-reciprocal light propagation. Here, we confirm on afully experimental basis, without invoking either symmetry-based ormaterial-specific assumptions, that the optical magnetoelectric effect inmaterials with non-parallel magnetization ($\boldsymbol{M}$) and electricpolarization ($\boldsymbol{P}$) generates a trilinear term in the refractiveindex, $\deltan\propto\boldsymbol{k}\cdot(\boldsymbol{P}\times\boldsymbol{M})$, where$\boldsymbol{k}$ is the propagation vector of light. Its sharp magnetoelectricresonances, that are simultaneously electric and magnetic dipole activeexcitations, make Co$_{2}$Mo$_3$O$_8$ an ideal compound to demonstrate thisfundamental relation via independent variation of $\boldsymbol{M}$,$\boldsymbol{P}$ and $\boldsymbol{k}$. Remarkably, the material shows almostperfect one-way transparency in moderate magnetic fields at some of themagnetoelectric resonances.

Confirming the trilinear form of the optical magnetoelectric effect in the polar honeycomb antiferromagnet Co$_{2}$Mo$_3$O$_8$