  Standard weak coupling methods are used to study collective modes in the
superconducting state of a double-layer system with intralayer and interlayer
interaction, as well as a Josephson-type coupling and single particle hopping
between the layers by calculating the electronic polarization function
perpendicular to the layers. New analytical results are derived for the mode
frequencies corresponding to fluctuations of the relative phase and amplitude
of the layer order parameters in the case of interlayer pairing and finite
hopping $t$. A new effect is found for finite $k$-dependent hopping: then the
amplitude and phase fluctuations are coupled. Therefore two collective modes
may appear in the dynamical c-axis conductivity below the threshold energy for
breaking Cooper pairs. With help of numerical calculations we investigate the
temperature dependence of the collective modes and show how a plasmon
corresponding to charge fluctuations between the layers evolves in the normal
state.
