  The normal and the lateral Casimir interactions between corrugated ideal
metallic plates in the presence of an amplifying or an absorptive dielectric
slab has been studied by the path-integral quantization technique. The effect
of the amplifying slab, which is located between corrugated conductors, is to
increase the normal and lateral Casimir interactions, while the presence of the
absorptive slab diminishes the interactions. These effects are more pronounced
if the thickness of the slab increases, and also if the slab comes closer to
one of the bounding conductors. When both bounding ideal conductors are flat,
the normal Casimir force is non-monotonic in the presence of the amplifying
slab and the system has a stable mechanical equilibrium state, while the force
is attractive and is weakened by intervening the absorptive dielectric slab in
the cavity. By replacing one of the flat conductors by a flat ideal permeable
plate the force becomes non-monotonic and the system has an unstable mechanical
equilibrium state in the presence of either an amplifying or an absorptive
slab. When the left side plate is conductor and the right one is permeable, the
force is non-monotonic in the presence of a double-layer DA
(dissipative-amplifying) dielectric slab with a stable mechanical equilibrium
state, while it is purely repulsive in the presence of a double-layer AD
(amplifying-dissipative) dielectric slab.
