Identifying the Magnetoelectric Modes of Multiferroic BiFeO$_3$

We have identified three of the four magnetoelectric modes of multiferroic BiFeO$_3$ measured using THz spectroscopy. Excellent agreement with the observed peaks is obtained by including the effects of easy-axis anisotropy along the direction of the electric polarization. By distorting the cycloidal spin state, anisotropy splits the $\Psi_{\pm 1}$ mode into peaks at 20 and 21.5 cm$^{-1}$ and activates the lower $\Phi_{\pm 2}$ mode at 27 cm$^{-1}$ (T=200 K). An electromagnon is identified with the upper $\Psi_{\pm 1}$ mode at 21.5 cm$^{-1}$. Our results also explain recent Raman and inelastic neutron-scattering measurements.