  Despite its great technological importance, the magnetoelectric (ME)
couplings in \BF are barely understood. By using a first-principles approach,
we uncover the {\it dynamic} ME couplings of the long-range spin-cycloid in
BiFeO$_3$. Based on a microscopic Hamiltonian, our first-principles approach
disentangles the hidden ME couplings due to spin-current and
exchange-striction. Beyond the spin-current polarization governed by the
inverse Dzyaloshinskii-Moriya interaction \cite{iDM}, various spin-current
polarizations derived from both ferroelectric and antiferrodistortive
distortions cooperatively produce the strong non-reciprocal directional
dichroism or the asymmetry in the absorption of counter-propagating light in
BiFeO$_3$. Our systematic approach can be generally applied to any multiferroic
material, laying the foundation for revealing hidden ME couplings on an atomic
scale and for exploiting optical ME effects in the next generation of
technological devices such as optical diodes.
