Copropagation of two pulses of different frequencies and modulation instabilities induced by cross-phase modulation in metamaterials

Nonlinear coupled equations describing the copropagation of two light pulses of different central frequencies in metamaterials (MMs) are derived. Based on the derived equations and the Drude model for dispersive MMs, the modulation instabilities (MIs) induced by cross-phase modulation (XPM) for different combination of the signs of refractive index experienced by the two optical pulses, respectively, are analyzed. It is shown that, in the absence of group-velocity mismatch, the property of MI only depends on the sign of group-velocity dispersion the two pulses experience, irrespective of the sign of refractive index. The group-velocity mismatch plays an important role in the occurrence of MI, especially when the central frequencies of the two pulses near the electric or magnetic plasma frequency of the MM. The theoretical results are confirmed by numerical simulations.