  We discuss terahertz pump-probe experiments in hBN from the perspective of
Floquet optical materials and photonic time crystals. Anisotropic nature of
this material results in a large separation of frequencies of in-plane and
out-of-plane optical phonons and leads to several branches of phonon-polariton
excitations. We consider a slab of finite thickness pumped at the frequency of
middle polaritons, around 25 THz. We theoretically analyze properties of the
pump induced state and identify several interesting features, including
possible light amplification and parametric instability at 2 THz. We relate
these features to resonant pair scattering processes of polaritons excited by
the pump pulse. We show that amplification/instability frequency ranges depend
on both the pump intensity and the slab thickness. We point out that this setup
can be used utilized to strongly enhance optical nonlinearities at terahertz
frequencies. Richness and tunability of this system suggests that it holds
promise for practical applications in terahertz technology.
