  We consider the optical properties of finite systems composed of a series of
graphene sheets separated by thin dielectric layers. Because these systems
respond as conductors to electric fields in the plane of the graphene sheets
and as insulators to perpendicular electric fields, they can be expected to
have properties similar to those of hyperbolic metamaterials. We show that
under typical experimental conditions graphene/dielectric multilayers have
enhanced Purcell factors, and enhanced photonic densities-of-states in both the
THz and mid-IR frequency range. These behaviors can be traced to the coupled
plasmon modes of the multi-layer graphene system. We show that these results
can be obtained with just a few layers of graphene.
