Parametric amplification and instability in time-periodic dielectric slabs.

We study the phenomenon of parametric amplification in the context of time-periodic dielectric slabs. These structures show particular promise inasmuch as they are capable of very large amplifications when illuminated by an electromagnetic wave of half the modulation frequency. Successive studies have corroborated this finding but none have yet been able to ascertain the nature of amplification in such devices. On top of that, some studies have raised speculations regarding the instability of a time-periodic slab which are off the mark. The problem lies in the poor understanding (or lack thereof) of the mathematical devices necessary to tackle such problems. We successfully carry out the tasks by tapping into the rich mathematical theory of Hill's equation. Specifically, we make use of the Folquet's theorem in its complete form which brings to light novel physical phenomena that the more prevalent simplified form fails to account for. Also, useful mathematical concepts such as coexistence are employed which to the best of our knowledge have not yet been applied in the field of time-varying optics. Our analytical method proves an effective means of assessing the amplifier's performance, e.g., estimating how long it takes for the device to reach steady state. We further delineate the link between amplification and instability and correct the misconceptions surrounding the subject by presenting a rigorous analysis of the instability problem in such structures.