Confirming the trilinear form of the optical magnetoelectric effect in the polar honeycomb antiferromagnet Co${}_2$Mo${}_3$O${}_8$

Magnetoelectric phenomena are intimately linked to relativistic effects and also require the material to break spatial inversion symmetry and time reversal invariance. Magnetoelectric coupling can substantially affect light-matter interaction and lead to non-reciprocal light propagation. Here, we confirm on a fully experimental basis, without invoking either symmetry-based or material-specific assumptions, that the optical magnetoelectric effect in materials with non-parallel magnetization ($\mathit{M}$) and electric polarization ($\mathit{P}$) generates a trilinear term in the refractive index, $\delta n\propto\boldsymbol{k}\cdot(\boldsymbol{P}\times\boldsymbol{M})$,where$\boldsymbol{k}$ is the propagation vector of light. Its sharp magnetoelectric resonances, that are simultaneously electric and magnetic dipole active excitations, make Co${}_2$Mo${}_3$O${}_8$ an ideal compound to demonstrate this fundamental relation via independent variation of $\mathit{M}$, $\mathit{P}$ and $\mathit{k}$. Remarkably, the material shows almost perfect one-way transparency in moderate magnetic fields at some of the magnetoelectric resonances.