Stability criterion for Gaussian pulse propagation through negative index materials

We analyze the dynamics of propagation of a Gaussian light pulse through a medium having a negative index of refraction employing the recently reported projection operator technique. The governing modified nonlinear Schrödinger equation, obtained by taking into account the Drude dispersive model, is expressed in terms of the parameters of Gaussian pulse, called collective variables, such as width, amplitude, chirp, and phase. This approach yields a system of ordinary differential equations for the evolution of all the pulse parameters. We demonstrate the dependence of stability of the fixed-point solutions of these ordinary differential equations on the linear and nonlinear dispersion parameters. In addition, we validate the analytical approach numerically utilizing the method of split-step Fourier transform.

Figure 1

Nonlinear dispersive parameters used for depicting pulse parameter dynamics through NIM at ${\omega }_{\mathit{pe}}/{\omega }_{\mathit{pm}}=0.8$.

Figure 2

Propagation of a smoothly evolving Gaussian pulse through NIM at a stable fixed point. (a) Variation of pulse parameters with respect to propagation distance with the solid line denoting that plotted by the projection operator method and the circled line corresponding to that plotted by the direct split-step Fourier transform method. (b) Intensity of a smoothly evolving Gaussian pulse propagating through NIM.

Figure 3

Propagation of an unstable Gaussian pulse through NIM at an unstable fixed point. (a) Variation of pulse parameters with respect to propagation distance with the solid line denoting that plotted by the projection operator method and the circled line corresponding to that plotted by the direct split-step Fourier transform method. (b) Intensity of unstable Gaussian pulse propagating through NIM.

Figure 4

Temporal position plotted as a function of propagation distance for NIM with various values of ${\sigma }_1$ and ${\sigma }_2$ at ${\omega }_{\mathit{pm}}/{\omega }_{\mathit{pe}}=0.8$.