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Standard weak coupling methods are used to study collective modes in thesuperconducting state of a double-layer system with intralayer and interlayerinteraction, as well as a Josephson-type coupling and single particle hoppingbetween the layers by calculating the electronic polarization functionperpendicular to the layers. New analytical results are derived for the modefrequencies corresponding to fluctuations of the relative phase and amplitudeof the layer order parameters in the case of interlayer pairing and finitehopping $t$. A new effect is found for finite $k$-dependent hopping: then theamplitude and phase fluctuations are coupled. Therefore two collective modesmay appear in the dynamical c-axis conductivity below the threshold energy forbreaking Cooper pairs. With help of numerical calculations we investigate thetemperature dependence of the collective modes and show how a plasmoncorresponding to charge fluctuations between the layers evolves in the normalstate.Comment: 17 pages, latex, 8 ps figure

Collective modes in the electronic polarization of double-layer systems in the superconducting state