Second harmonic generation and nonlinear frequency conversion in photonic time-crystals

We study the nonlinear process of second harmonic generation in photonic time-crystals, materials with refractive index that varies abruptly and periodically in time, and obtain the phase matching condition for this process. We find conditions for which the second harmonic generation is highly enhanced even in the absence of phase matching, governed by the exponential growth of the modes residing in the momentum gap of the photonic time crystal. Additionally, under these conditions, a cascade of higher-order harmonics is generated at growing exponential rates. The process is robust, with no requirement for phase-matching, the presence of a resonance or a threshold, drawing energy from the modulation.