SGER: Stimulated Raman Scattering in the Transient Limit

在瞬变极限下描述受激拉曼散射的方程是非线性的和可积的。然而，该系统在长距离传播后的物理行为与光纤中的光脉冲等其他可积系统有很大的不同。在光纤中，光脉冲分解成若干个孤子和一个色散波，而在受激拉曼散射的瞬变区域，解趋向于一个依赖于(距离)x(时间)的相似解。这个最后的结果虽然不能从以前的数值结果中推断出来，但直到最近才被证明，它的许多后果仍然有待阐明。第一个关注的问题是第二代Stokes和有限T2对可能的自相似解的实验观测施加的限制。我们将对这一现象进行调查。我们还将探索瞬时拉曼效应对限制与钛宝石激光器结合使用的拉曼频率下移器转换效率的影响。我们还将探索类似技术可应用于光纤光栅、二能级系统、光折变材料、以及其他系统，它们要么已经被证明具有自相似性，要么因为拥有长期记忆而成为有前途的候选系统。最后，在数学方面，我们将与Tom Seidman合作研究这些解的稳定性，并与其他数学家合作，获得频率失谐时自相似情况的渐近公式。

The equations which describe stimulated Raman scattering in the transient limit are nonlinear and integrable. However, the physical behavior of this system after propagating, long distances is quite different from other integrable systems such as light pulses in optical fibers. In optical fibers, the light pulses break up into a number of solitons and a dispersive wave contribution, while in the transient regime of stimulated Raman scattering, the solutions tend toward a similarity solution which depends on (distance) x (time). This last result, while inferrable from previous numerical results, was only recently proved, and many of its consequences remain to be elucidated. The first issue of concern is the limitation which second Stokes generation and finite T2 impose on possible experimental observation of the self- similar solution. We will investigate this phenomenon. We will also explore the impact of the transient Raman effect on limiting the conversion efficiency of Raman frequency downshifters which are used in conjunction with Ti:Sapphire lasers. We will also explore the extent to which similar techniques can be applied to fiber gratings, two-level systems, photorefractive materials, and other systems which have either already been shown to exhibit self- similarity or are promising candidates because they have long-term memory. Finally, on the mathematical front, we will collaborate with Tom Seidman in studying the stability of these solutions and with other mathematicians in obtaining asymptotic formulae for the self-similar case when the frequencies are detuned.