  We study the influence of different kinds of gaps in a quasiparticle spectrum
on longitudinal and transverse optical conductivities of bilayer graphene. An
exact analytical expression for magneto-optical conductivity is derived using a
low-energy two-band Hamiltonian. We consider how the layer asymmetry gap caused
by a bias electric field and a time-reversal symmetry breaking gap affect the
absorption lines. The limit of zero magnetic field is then analyzed for an
arbitrary carrier density in the two-band model. For a neutral bilayer
graphene, the optical Hall and longitudinal conductivities are calculated
exactly in the four-band model with four different gaps and zero magnetic
field. It is shown that two different time-reversal symmetry breaking states
can be distinguished by analyzing the dependence of the optical Hall
conductivity on the energy of photon. These time-reversal symmetry breaking
states are expected to be observed experimentally via optical polarization
rotation either in the Faraday or Kerr effects. We analyze a possibility of
such an experiment for a free-standing graphene, graphene on a thick substrate,
and graphene on a double-layer substrate.
