  We introduce a new theoretical approach for analyzing pump and probe
experiments in non-linear acousto-optic systems. In our approach, the effect of
coherently pumped polaritons is modeled as providing time-periodic modulation
of the system parameters. Within this framework, propagation of the probe pulse
is described by the Floquet version of Maxwell's equations and leads to such
phenomena as frequency mixing and resonant parametric production of polariton
pairs. We analyze light reflection from a slab of insulating material with a
strongly excited phonon-polariton mode and obtain analytic expressions for the
frequency-dependent reflection coefficient for the probe pulse. Our results are
in agreement with recent experiments by Cartella et al. which demonstrated
light amplification in resonantly excited SiC insulator. We show that, beyond a
critical pumping strength, such systems should exhibit Floquet parametric
instability, which corresponds to resonant scattering of the pump polaritons
into pairs of finite momentum polaritons. We find that the parametric
instability should be achievable in SiC using current experimental techniques
and discuss its signatures, including the non-analytic frequency dependence of
the reflection coefficient and the probe pulse afterglow. We discuss possible
applications of the parametric instability phenomenon and suggest that similar
types of instabilities can be present in other photoexcited non-linear systems.
