Photonic Time Crystals and Parametric Amplification: similarity and distinction

Photonic Time crystals (PTC) arise in time-modulated media when the frequency of modulation of permittivity is on the order of twice the frequency of light and are manifested by the generation and amplification of so-called time reversed waves propagating in the direction opposite to the incoming light. Superficially, the observed phenomenon bears resemblance to the widely known phenomena of optical parametric generation (OPG) and amplification (OPA) using second or third order optical nonlinearities. I show that while indeed the same physical mechanism underpins both PTC and OPA , the difference arises from the boundary conditions. Thus , while dispersion for both PTC and OPA exhibit the same bandgap in momentum space, only in the case of PTC can one have propagation in that bandgap with exponential amplification. I also show that PTC can be engineered with both second and third order nonlinearities, and that rather unexpectedly, modulating permittivity on the ultrafast (few fs) rate is not a necessity, and that one can emulate all the PTC features using materials with a few picoseconds response time commensurate with the propagation time through the medium.