Influence of nonlinear dispersion on modulation instability of coherent and partially coherent ultrashort pulses in metamaterials

The combination of dispersive magnetic permeability with nonlinear polarization leads to a series of nonlinear dispersion terms in the propagation equations for ultrashort pulses in metamaterials. Here we present an investigation of modulation instability (MI) of both coherent and partially coherent ultrashort pulses in metamaterials to identify the role of nonlinear dispersion in pulse propagation. The Wigner–Moyal equation for partially coherent ultrashort pulses and the nonlinear dispersion relation for MI in metamaterials are derived. Combining the standard MI theory with the unique properties of the metamaterial, the influence of the controllable first-order nonlinear dispersion, namely self-steepening, and the second-order nonlinear dispersion on both coherent and partially coherent MI, in both negative-index and positive-index regions of the metamaterial for all physically possible cases is analyzed in detail. For the first time to our knowledge, we demonstrate that the role of the second-order nonlinear dispersion in MI is equivalent to that of group-velocity dispersion (GVD) to some extent, and thus due to the role of the second-order nonlinear dispersion, MI may appear in the otherwise impossible cases, such as in the normal GVD regime. PACS 42.25.Kb; 42.65.Sf; 78.20.Ci