Ultra short pulse propagation in presence of nonlinear dispersions in metamaterial

The derivation of the general formula of the propagation in metamaterial in the presence of the second, third, and fourth orders of the linear dispersion is presented. The influence of the first, second and third order of non-linear dispersion have also been introduced. All linear and nonlinear dispersions parameters are formulated as functions of the normalized frequency. The effects of linear and nonlinear dispersions induce many temporally and spectrally variations that affect the pulse shape and spectrum. The effects of the fourth order linear dispersion and the third order nonlinear dispersion have been added here. The results showed that they have both temporal and spectral effects that cannot be neglected under certain circumstances.