  We consider the response of etalons created by a combination of a
conventional mirror and a metasurface, composed of a periodic lattice of metal
scatterers with a resonant response. This geometry has been used previously for
perfect absorption, in so-called Salisbury screens, and for hybridization of
localized plasmons with Fabry-Perot resonances. The particular aspect we
address is if one can assume an environment-independent reflectivity for the
metasurface when calculating the reflectivity of the composite system, as in a
standard Fabry-Perot analysis, or whether the fact that the metasurface
interacts with its own mirror image renormalizes its response. Using lattice
sum theory, we take into account all possible retarded dipole-dipole
interactions of scatterers in the metasurface amongst each other, and through
the mirror. We show that while a layer-by-layer Fabry-Perot formalism captures
the main qualitative features of metasurface etalons, in fact the mirror
modifies both the polarizability and reflectivity of the metasurface in a
fashion that is akin to Drexhage's modification of the radiative properties of
a single dipole.
