Parametric resonances in a photonic time crystal with periodic square modulation of its permittivity ɛ(t)

We study parametric resonances in a ``temporal photonic crystal'' or ``photonic time crystal'' (PTC) slab whose permittivity is assumed to be modulated periodically in time in a stepwise manner (with abrupt transitions). These resonances in the light reflected and transmitted by the PTC slab occur when the modulation frequency assumes a series of eigenvalues that are inversely proportional to the slab thickness, in addition to having a value that is twice that of the frequency of incidence. We compare the reflection and transmission spectra with those for a harmonically modulated slab and find marked, qualitative differences. These depend crucially on the modulation strength and on the impedance contrast at the slab's interfaces. In the special case of weak modulation, the differences in behavior between the two forms of modulation can be traced to a simple scaling factor. In principle, these findings could be corroborated with infrared light incident on PTC slabs tens of micrometers thick.