Photonic Time Crystals and Parametric Amplification: Similarity and Distinction

Photonic time crystals (PTC) arise in time-modulated media. PTCs are manifested by the generation and amplification of so-called "time reversed" waves propagating in the direction opposite the incoming light. Superficially, the observed phenomenon bears a 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 exhibits 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, rather unexpectedly, modulating permittivity in the ultrafast (few femtoseconds) rate is not a necessity. Also, one can emulate all the PTC features using materials with a few picoseconds response time commensurate with the propagation time through the medium.