  Graphene-based hyperbolic metamaterials have been predicted to transport
evanescent fields with extraordinarily large vacuum wave-vectors. It is
particularly at much higher wave vector values that the commonly employed
descriptional models involving structure homogenization and assumptions of an
approximatively local graphene conductivity start breaking down. Here, we
combine a non-local quantum conductivity model of graphene with an exact
mathematical treatment of the periodic structure in order to develop a tool-set
for determining the hyperbolic behavior of these graphene-based hyperbolic
metamaterials. The quantum conductivity model of graphene facilitates us to
predict the plasmonic amplification in graphene sheets of the considered
structures. This allows us to reverse the problem of Ohmic and temperature
losses, making this simple yet powerful arrangement practically applicable. We
analyze the electric field distribution inside of the finite structures,
concluding that Bloch boundary solutions can be used to predict their behavior.
With the transfer matrix method we show that at finite temperature and
collision loss we can compensate for losses, restoring imaging qualities of the
finite structure via an introduction of chemical imbalance.
