  The analytic expressions for the maximum and minimum reflectance of optical
films coated with gapped graphene are derived in the application region of the
Dirac model with account of multiple reflections. The respective film
thicknesses are also found. In so doing the film material is described by the
frequency-dependent index of refraction and graphene by the polarization tensor
defined along the real frequency axis. The developed formalism is illustrated
by an example of the graphene-coated film made of amorphous silica. Numerical
computations of the maximum and minimum reflectances and respective film
thicknesses are performed at room temperature in two frequency regions
belonging to the near-infrared and far-infrared domains. It is shown that in
the far-infrared domain the graphene coating makes a profound effect on the
values of maximum reflectance and respective film thickness leading to a
relative increase of their values by up to 65% and 50%, respectively. The
maximum transmittance of a graphene-coated film of appropriately chosen
thickness is shown to exceed 90%. Possible applications of the obtained results
are discussed.
