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Physical Review A

Similarity reduction and perturbation solution of the stimulated-Raman-scattering equations in the presence of dissipation

Decio Levi B, Curtis R. Menyuk, Pavel Winternitz A

Published Apr 1, 1994• 14 citations
Publisher

Abstract

Similarity and other group-invariant solutions of stimulated Raman scattering (SRS) in the presence of dissipation are studied. The group-theoretical analysis reduces the SRS equations to ordinary differential equations which in the most interesting cases, i.e., in the case of the self-similar and soliton solutions, are studied perturbatively to derive the effect of dissipation.

DOI
10.1103/physreva.49.2844

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Similarity and other group-invariant solutions of stimulated Raman scattering (SRS) in the presence of dissipation are studied. The group-theoretical analysis reduces the SRS equations to ordinary differential equations which in the most interesting cases, i.e., in the case of the self-similar and soliton solutions, are studied perturbatively to derive the effect of dissipation.
The equations that govern stimulated Raman scattering in the transient limit have long been known to have soliton solutions, a Lax pair, and other properties that are normally associated with integrable systems. Yet the behavior of this system is known to be quite different from the usual behavior associated with integrable systems. Solitons are transient, and the behavior of the system at long distances is dominated by self-similarity. This behavior was first discovered numerically.1 In this presentation, we show that the self-similar solution can be derived by using symmetry reduction. We then show for fairly general initial conditions precisely which self-similar solution the system tends toward at long distances. These results are verified numerically. We argue that this behavior in which self-similar solutions dominate the longdistance evolution should often appear in nonlinear systems with memory. Finally, we outline an experiment that could observe the self-similar solutions.
Systems that exhibit stimulated Raman scattering in the transient limit have long been the subject of theoretical and experimental study, and recent experimental developments have made possible the observation of the long-distance evolution of these systems. These systems are integrable using the inverse scattering method; however, in contrast to the usual behavior in integrable systems, solitons do not dominate the long-distance behavior. Instead, solitons are transient, and self-similarity appears in the long-distance limit. In this paper, the similarity solutions of the equations which describe stimulated Raman scattering are first reviewed from a physical standpoint. Inverse scattering theory is then used to show that given physically reasonable initial data in which the pump and Stokes waves have no linear frequency mismatch, the solution will converge toward a self-similar solution. These results are confirmed by a numerical comparison of the solution of the transient, stimulated Raman-scattering equations to the appropriate self-similar solutions. The length scale over which convergence occurs and the transient generation of solitons are observed. From a physical standpoint, the appearance of self-similarity rather than solitons in the long-distance limit can be attributed to the presence of memory in the Raman-active medium. Other systems with memory also exhibit self-similarity, and the results which are described here may have applications to a wide array of optical systems with memory. An experiment which could observe self-similarity is described.
Abstract The semiclassical equations of SRS are derived by the following method: The many‐level molecule is described by the Schrödinger equation and subsequently all amplitudes but those of the essential two levels are eliminated by approximate integration. Especially the case of equal group velocities for pump and Stokes wave is considered. Terms describing a frequency shift due to dynamical Stark effect and a change of refraction index by vibrational excitation can be removed by a suitable transformation. Under various, physically different, assumptions the system of molecular and field equations is reduced to the equations describing a quasimonochromatic pulse propagating in a resonant two‐level system, and in this way we can find a rich manifold of solutions. E.g., the 2π‐pulse of self‐induced transparency can be transferred to SRS. By other assumptions we are led to equations defining the Bäcklund transformation for the sine‐Gordon equation. For this case also soliton type solutions can be constructed.
A system of Maxwell-Bloch type equations (MBE's), describing stimulated Raman scattering with both Stokes and anti-Stokes waves taken into account, is investigated. We introduce variables S3{\mathit{S}}_{3}S3​ and ${\mathit{S}}_{\ifmmode\pm\else\textpm\fi{}}$, which are bilinear in the electromagnetic fields, and prove that the corresponding equations possess Lax representation. This fact is used to obtain additional solutions for S3{\mathit{S}}_{3}S3​, ${\mathit{S}}_{\ifmmode\pm\else\textpm\fi{}}$ and for the MBE's which include solitons, periodical waves, and self-similarity solutions. The transient and bright threshold solitons are also analyzed. \textcopyright{} 1996 The American Physical Society.