Modulation instability in second harmonic generation in metamaterials with quadratic nonlinearity

Abstract

Modulation instability (MI) in second harmonic generation in metamaterials (MMs) with quadratic nonlinearity is studied. The analytical expressions for MI gain spectra are obtained, from which the generation condition of MI and the respective roles of dispersion, group velocity mismatch and phase mismatch in MI are identified. It is found that the region of phase mismatch where the MI can take place in MMs is in contrast with its counterpart in conventional materials due to the negative wave vector direction. The MI gains mainly rely on the phase mismatch. Since magnetic permeability is dominant for negative refraction, so the refractive index difference of the fundamental frequency (FF) and second harmonic (SH) wave can fluctuate in a large range in MMs. While the frequency is near the critical point, the refractive index difference of the FF and SH wave is small, so the phase mismatch is small and the MI gain is low. However, while the frequency is far away from the critical point, the refractive index difference of the FF and SH wave becomes large, this leads to a large phase mismatch, and the giant gain with more sidelobes can be obtained for the large phase mismatch occurring in MMs with quadratic nonlinearity.