On the properties of two pulses propagating simultaneously in different dispersion regimes in a nonlinear planar waveguide

Abstract

Properties of two pulses propagating simultaneously in different dispersion regimes, anomalous and normal, in a Kerr-type planar waveguide are studied in the framework of the nonlinear Schrödinger equation. It is found that the presence of the pulse propagating in a normal dispersion regime can cause termination of catastrophic self-focusing of the pulse with anomalous dispersion. It is also shown that the coupling between pulses can give rise to spatiotemporal splitting of the pulse propagating in anomalous dispersion regime, but it does not necessarily lead to catastrophic self-focusing of the pulse with normal dispersion. For the limiting case when the dispersive term of the pulse propagating in normal dispersion regime can be neglected an indication (based on the variational estimation) of a possibility of a stable self-trapped propagation of both pulses is obtained. This stabilization is similar to the one which was found earlier in media with saturation-type nonlinearity.

Collapse of optical wave arrested by cross-phase modulation in nonlinear metamaterials

“…For OMs, some previous authors reported that nonlinearity necessary for catastrophic self-focusing in a configuration of two beams is approximately three times smaller than in the case when they propagate as single beam [20][21][22][23]. Therefore, under the action of XPM, Figure 2(b) shows that peak intensity of both waves will monotonically increase with increasing propagation until collapse occurs, agreeing with the results obtained in [36][37][38][39]. However, for MMs, the situation becomes qualitatively different, as can be seen from Figure 2(a).…”

“…Case (i) is similar to the conventional materials, while case (ii) is simple since optical wave always spread even for very large strengths of self-focusing nonlinearity [36][37][38][39]. Therefore, in the following, we will only in detail discuss the last case (iii) unseen in OMs, paying attention to collapse of probe wave arrested by XPM.…”

“…Meanwhile, up to now, the nature and the fundamental character of the interplay between coupled beam/pulse have been well understood previously in nonlinear dispersive OMs [36][37][38][39][40][41][42][43][44][45]. For example, in focusing medium, catastrophic self-focusing of the second beam also occurs if it occurs for one of coupled beams for two cylindrically symmetric spatially overlapped beams [36][37][38][39][40]. In self-defocusing medium, Agrawal theoretically demonstrated that the induced focusing of the probe beam occurs when it copropagates with the pump beam whose intensity peak at a place different from that of the probe wave [40].…”

“…In our case, the variational approach amounts to a system of ordinary differential equations for the evolution of the width, amplitude, and relative phase of coupled waves. By using The variational method is based on defining a Lagrangian function for the system from which the equations of motions can be derived [30,39,46]. It is known that the Lagrangian density that corresponds to the set of CNLSEs is given by…”

“…Obviously, note that we do not specify the sign of n j in deriving Equation (7), so they are formally applied to OMs [39,46]. Further, it is worth to note that in the case of without XPM, the coupled equations in Equation (7) will be decoupled; thus, they are reduced to the results obtained by M. Desaix previously in OMs [46].…”