  We report on results of our theoretical study of the c-axis infrared
conductivity of bilayer high-Tc cuprate superconductors using a microscopic
model involving the bilayer-split (bonding and antibonding) bands. An emphasis
is on the gauge-invariance of the theory, which turns out to be essential for
the physical understanding of the electrodynamics of these compounds. The
description of the optical response involves local (intra-bilayer and
inter-bilayer) current densities and local conductivities. The local
conductivities are obtained using a microscopic theory, where the
quasiparticles of the two bands are coupled to spin fluctuations. The coupling
leads to superconductivity and is described at the level of generalized
Eliashberg theory. Also addressed is the simpler case of quasiparticles coupled
by a separable and nonretarded interaction. The gauge invariance of the theory
is achieved by including a suitable class of vertex corrections. The resulting
response of the model is studied in detail and an interpretation of two
superconductivity-induced peaks in the experimental data of the real part of
the c-axis conductivity is proposed. The peak around 400/cm is attributed to a
collective mode of the intra-bilayer regions, that is an analogue of the
Bogolyubov-Anderson mode playing a crucial role in the theory of the
longitudinal response of superconductors. For small values of the bilayer
splitting, its nature is similar to that of the transverse plasmon of the
phenomenological Josephson superlattice model. The peak around 1000/cm is
interpreted as a pair breaking-feature that is related to the electronic
coupling through the spacing layers separating the bilayers.
