Muller matrix ellipsometry of dynamic magnetoelectric effects in multiferroics*

Far-IR spectra of magneto-electric (ME) and multiferroic materials are in the focus of modern experimental and theoretical studies. Bi-anisotropic optical properties of these materials require consideration of not only dielectric susceptibility tensor $\hat{{\varepsilon }}(\omega )$ but also magnetic permeability $\hat{{\mu }}(\omega )$ and ME $\hat{{\alpha }}(\omega)$ tensors that cannot be distinguished from a single transmission or reflection spectrum. We report on the application of Mueller matrix spectroscopic ellipsometry (MM-SE) for studies of elementary excitations in multiferroic materials such as TbMnO$_{3}$, TbMn$_{2}$O$_{5}$, and TbFe$_{3}$(BO$_{3})_{4}$ single crystals. We show that magnetic, electric, and ME dipole excitations, such as magnons, phonons, and electromagnons can be distinguished from each other using a single MM measurement without introducing any modeling arguments. The fit of MM spectra based on the Berreman's $4\times 4$ propagation matrix formalism allowed us to determine parameters of electromagnon excitations separating the electric $\hat{{\varepsilon }}(\omega )$ and ME $\hat{{\alpha }}(\omega )$ tensors components.