Control of Space-Time Trajectories of Noise-Driven Optical Extreme Events in Metamaterial Waveguides

Abstract

Metamaterials offer the potential to precisely manipulate electromagnetic wave propagation in ways that cannot be achieved with materials found in nature. The formation and propagation of optical spatial solitons in metamaterials has been already investigated [1]. Here we report the theoretical and numerical investigations on temporal-spectral dynamics of nonlinear extreme events arising from the initial noise-perturbed plane wave in metamaterial waveguides. A typical waveguide structure used here is a planar structure with a metamaterial core and a part of the structure, in the form of the substrate, is replaced with a magnetooptic material. We assume that the core material is isotropic and it has a negative permittivity and negative permeability thus the form of the metamaterials considered here is transparently double-negative [1]. Based on numerical simulations of an appropriate extension to the nonlinear Schrodinger equation (NLSE) including self-steepening and magnetooptic effects [2], Fig. 1(a) illustrates the trajectories of the spontaneous modulation instability (MI) patterns, such as Akhmediev breathers (AB), Kuznetsov-Ma (KM) breathers and Peregrine solitons (PS) [3]. More interestingly, the presence of the self-steepening term in the NLSE leads to a characteristic group velocity vg0 of the trajectory of all localized wave structures. This can be also retrieved through typical asymmetric spectra of these localized structures. If one calculates the weight-average frequency of the power spectrum, the evolution of the group velocity vg can be estimated by using the group-velocity dispersion relation of the waveguide. Some of the relevant results are shown in Fig. 1(b), where we show that vg grows from nearly zero up to reach a maximum value when the self-steepening term begins to play a role on the spectrum symmetry breaking of the MI patterns, vg then evolves with small oscillations and leads to the eventual saturation. Note that the final mean value of vg is close to the measured group velocity vg0 obtained from Fig. 1(a) when considering a single KM breather. Next, by taking advantage of the magnetooptic property of the metamaterial waveguides [2], one can precisely control the trajectory of the MI patterns by carefully adding an external magnetic field. More particularly it induces a change in group velocity of the MI patterns. Our numerical simulations reveal that the impact of the self-steepening term of the NLSE and the magnetic-optic effect can cancel each other by choosing appropriate parameters. Fig. 1(c) shows the value of magnetic field required to minimize vg as a function of the self-steepening coefficient and they exhibit a linear behaviour. However, higher-order linear or nonlinear effects, e.g. higher order dispersions and stimulated Raman scattering, may induce the acceleration of the MI patterns and breather waves. In such conditions, the control of their trajectories still needs more advanced techniques.