  Discrete dimension is introduced via an extended Dirac operator to include
the interlayer interactions in a geometric framework of generic
$d+1$-dimensional bilayer systems. The photon and its Kaluza-Klein partners in
this extended space-time will involve a pair of vector fields together with a
scalar field, which describes a pair of local jumping between the layers. The
quartic potential of the scalar field triggers an abelian Higgs mechanism,
which gives a mass to one vector field, which describes a short-range
interaction. The discrete derivative along the extra dimension generates a
non-diagonal mass matrix, whose eigenstates mix the fermions on two layers with
a mixing angle. The couplings of the Kaluza-Klein siblings of photon including
massless, massive, and scalar ones with the mass eigenstates of fermions will
also depend on the mixing angle. Thus interlayer interactions in bilayer
systems are manifested elegantly by these new features via the extra discrete
dimension. In specific cases, the short-range force, weak and strong coupling
regimes of the fermions with different masses can lead to new physical
consequences to be discovered in different realistic $2+1$ and
$3+1$-dimensional bilayer systems.
