  Light scattering by a periodic atomic array is studied when the atoms couple
with the mode of a high-finesse optical resonator and are driven by a laser.
When the von-Laue condition is not satified, there is no coherent emission into
the cavity mode, and the latter is pumped via inelastic scattering processes.
We consider this situation and identify conditions for which different
non-linear optical processes can occur. We show that these processes can be
controlled by suitably tuning the strength of laser and cavity coupling, the
angle between laser and cavity axis, and the array periodicity. We characterize
the coherence properties of the light when the system can either operate as
degenerate parametric amplifier or as a source of antibunched-light. Our study
permits us to identify the individual multi-photon components of the nonlinear
optical response of the atomic array and the corresponding parameter regimes,
thereby in principle allowing one for controlling the nonlinear optical
response of the medium.
