  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.
