  We develop a quantum optical formalism to treat a two-dimensional array of
atoms placed in an optical cavity. Importantly, and in contrast to typical
treatments, we account for cooperative dipole-dipole effects mediated by the
interaction of the atoms with the outside, non-cavity-confined modes. Based on
the observation that scattering to these modes is largely inhibited due to
these cooperative effects, we construct a generic formalism, independent of the
specific cavity structure, and apply it to an array of non-saturated atoms. By
further considering the atomic motion, we show that the inhibited damping can
lead to a favorable scaling of the optomechanical parameters of an atom-array
membrane placed within a cavity. The developed formalism lays the basis for
further investigation of many-body QED with atom arrays in transversely
confined geometries.
