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Annual Meeting Optical Society of America

Self-similarity in stimulated Raman scattering

Curtis R. Menyuk

Published Jan 1, 1992• 5 citations
Publisher

Abstract

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.

DOI
10.1364/oam.1992.thn3

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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.
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.
The high pulse repetition rates (~100 MHz) of radio-frequency free-electron lasers make possible the investigation of molecular response to a rapid pulse train.1 When the molecular decay time T2 is much larger than the time between pulses T, molecular memory is carried from pulse to pulse. For stimulated Raman scattering (SRS) the memory effect can lead to an increase in the size of the Raman dipole resulting in enhanced scattering. We analyze the 1-D SRS equations in the condition that the scattering under an individual pulse occurs in the transient regime (~10 ps). We obtain a general expression for the behavior of SRS which reduces to the following long-time limits: (1) When T2 ≪ T, the SRS from each pulse in the train develops independently from the others; (2) When T2 ≫ T, the transient SRS response is equivalent to the steady-state response to a constant pump at the time-averaged pump intensity. We also perform this analysis for a medium such as the atmosphere where the density of scatterers and Raman bandwidth decrease exponentially with distance.
We use the dressed-state representation to generate the full quantum-mechanical treatment of stimulated Raman scattering (SRS). Equations of motion and propagation are obtained from the Heisenberg equations of motion in the plane-wave limit and under conditions such that anti-Stokes and higher-order Stokes processes can be neglected. Equations for a stochastic model, based upon antinormal ordering and a quantum-classical correspondence are developed from them, suitable for numerical ensemble calculations for the study of amplified quantum fluctuations and spontaneously generated phase waves and solitons. Several alternative approaches, which have more general validity but present greater difficulty in numerical implementation, are also developed by introduction of quasiprobability functions and their associated Fokker-Planck equations from which corresponding stochastic equations of motion and propagation are derived. The Q representation, whose moments correspond to expectation values of products of antinormally ordered fields, and normally ordered atomic operators yield stochastic equations identical to the equations for the stochastic model developed directly from the Heisenberg equations for the same initial and boundary conditions, giving strong credence to the stochastic model. The Wigner representation, whose moments correspond to expectation values of symmetrically ordered operators, leads to a set of equations that are valid where the stochastic model fails, namely, where conditions are such that quantum fluctuations and nonlinearity are simultaneously important. The stochastic model, which is shown to be, by far, the simplest to implement in numerical calculations, is shown also to be valid for the conditions of SRS.