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

非线性各向同性克尔介质中的一维矢量Helmholtz方程是非线性光学中最基本的方程之一。从本质上讲，该方程能够描述X(3)材料中的任意一维光场组态。因此，它在许多非常重要和潜在有用的非线性相互作用中扮演着中心角色，例如共线简并四波混频和光学双稳相互作用。然而，到目前为止，这种构型下的非线性Helmholtz方程仍被近似处理。在这个阶段，还不清楚这些近似技术是否提供了与这个问题相关的足够准确的图景(甚至是整个图景)。这个问题的准确解决是非常重要的，首先是作为一个原则问题，然后是作为检验现有理论的一种方式，最后它可能会导致新的重大结果。这项研究的目的是研究亥姆霍兹方程在各向同性非线性克尔介质中的精确行为，这是一个综合的程序，包括分析和计算机模拟。这一研究启蒙奖的具体目标是：a)使用哈密顿技术确定该方程的可积情况，并反过来用它们来检验先前获得的近似结果。B)发现全新的现象，例如各向同性克尔介质中的空间混沌，这可能是以前的近似过程所忽略的。C)提高我们对波在X(3)介质中传播的理解，这反过来将有益于其他相关研究领域。

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.