RIA: Investigating the Exact Behavior of Helmholtz Equationin a Collinear Degenerate Four-Wave Mixing Arrangement

一维矢量亥姆霍兹方程 非线性各向同性Kerr介质是最重要的 非线性光学的基本方程 通过 自然界中，这个方程能够描述任何一个- X（3）维光场位形 材料. 因此，它在一个 一些非常重要和潜在有用的 非线性相互作用，如共线 简并四波混频和光学 双稳态 然而，到目前为止，非线性亥姆霍兹 方程在这样的配置已被处理 大概吧 现阶段， 这些近似技术提供了一个 足够准确的图片（甚至是整个图片） 与这个问题有关。 的精确解 这个问题很重要，首先是 原则，然后作为一种测试已经存在的 理论，并最终可能导致新的重大 结果 本研究提案的目的是 研究Helmholtz方程的精确性态 在各向同性非线性克尔介质中， 计划，这将涉及分析和 计算机模拟 该研究启动奖的具体目标是： a）确定该方程的可积情况，使用 哈密顿技术，并反过来使用它们来测试 以前得到的近似结果。 B）至 发现全新的现象，如空间混沌 在各向同性克尔介质中， 被以前的近似程序忽略了。 c）、 来提高我们对波在 X（3）媒体，这将反过来有利于其他相关的 研究领域。

The one-dimensional vectorial Helmholtz equation in nonlinear isotropic Kerr media is one of the most fundamental equations in nonlinear optics. By nature, this equation is capable of describing any one- dimensional optical field configuration in X(3) materials. As a result, it plays a central role in a number of very important and potentially useful nonlinear interactions, such as that of collinear degenerate four-wave mixing and that of optical bistability. However so far, the nonlinear Helmholtz equation in such configurations has been treated approximately. At this stage, it is by no means clear that these approximate techniques provide an accurate enough picture (or even the whole picture) associated with this problem. The exact solution of this problem is very important, first as a matter of principle, then as a way to test already existing theories and finally it may lead to new significant results. It is the purpose of this research proposal to investigate the exact behavior of Helmholtz equation in isotropic nonlinear Kerr media in a comprehensive program, that will involve both analytical and computer simulations. Specific goals in this Research Initiation Award are: a) to identify integrable cases of this equation using Hamiltonian techniques and in turn use them to test previously obtained approximate results. b) to discover totally new phenomena, such as spatial chaos in isotropic Kerr media that may have been overlooked by previous approximate procedures. c) to improve our understanding of wave propagation in X(3) media, which will in turn benefit other associated areas of research.