High-Order Numerical Simulation of Focusing Nonlinear Waves in the Non-Paraxial Regime

该项目的主要目标是建立对聚焦非线性电磁波的定量预测能力。描述这类问题的标准数学模型是非线性薛定谔方程(NLS)。到目前为止，这个方程已经得到了比较好的研究。然而，有迹象表明，NLS模型可能过于简单化。另一种选择是由更全面的非线性Helmholtz方程(NLH)提供的，实际上，NLS是通过采用所谓的近轴近似并忽略重要的后向散射现象而得到的。与NLS方程不同的是，关于NLH解的可解性和解的唯一性的研究相对较少，而且这个方程对数值逼近提出了相当大的挑战。非线性是一个主要的障碍，因为它意味着撞击波和(后)散射波不能分开。另一个关键的困难是与前向传播波相比，后向散射的幅度很小。在这个项目的过程中，Pi和他的同事们将开发、实施和测试一个有效的数值程序来积分NLH。它将涉及对先前提出的方法进行重大修改和改进，这些方法已经被证明是成功的，而且实际上是文献中无与伦比的。该方法采用高阶有限差分近似。它的中心元素是一种特殊的双向非局部人工边界条件，使外边界对所有出射的波都是透明的，同时能够准确地规定给定的入射信号。期望借助这种方法，对非线性光学中的一些关键悬而未决的问题，特别是非傍轴和后向散射是否可以阻止聚焦的非线性波的崩塌(爆破)，以及NLH是否能够维持宽度仅为几个波长量级的所谓窄空间孤子，获得有价值的新见解。在该项目的过程中，将建立一种数值方法来模拟强激光在各种介质和材料中的传输。这种方法有坚实的数学基础，有望帮助解决理论非线性光学中的一些具有挑战性的问题。除了其潜在的理论优点外，该方法从应用的角度来看也将是有用的。事实上，激光在材料中的传输通常伴随着非线性自聚焦和后向散射现象。定量分析和预测这些关键现象的能力对于现代科学和工程中的许多应用都是极其重要的。后者的范围从遥感大气传感(地面强大的激光向天空发送脉冲，后向散射辐射占探测到的信号的很大一部分)到激光手术(激光光束在组织中的传播)，再到沿着光纤传输信息。还存在其他可能的应用，其涉及例如同向传播或反向传播的激光之间的相互作用。它们可以为下一代光电子电路设计所谓的全光开关提供一种工具。拟议的数值方法将产生一个强大的工具，用于对上述应用以及许多其他应用进行准确和稳健的分析。

The key objective of the project is to build a quantitative predictive capability for the focusing nonlinear electromagnetic waves. A standard mathematical model for describing this type of problems is the nonlinear Schroedinger equation (NLS). By now, this equation is relatively well studied. There are, however, indications that the NLS model may be oversimplified. An alternative is provided by a more comprehensive nonlinear Helmholtz equation (NLH), from which the NLS is, in fact, derived by employing the so-called paraxial approximation and neglecting the important phenomenon of backscattering. In contradistinction to the NLS, relatively little is known about the solvability of the NLH and uniqueness of its solutions.Moreover, this equation presents a considerable challenge for the numerical approachas well. Nonlinearity is a major hurdle, as it implies that the impinging and(back)scattered waves cannot be separated. Another key difficulty is the small magnitude of backscattering compared to that of the forward propagating wave.In the course of the project, the PI and his colleagues will develop, implement,and test an efficient numerical procedure for integrating the NLH. It will involvemajor modifications and improvements to the previously proposed methodology thathas already proven successful and, in fact, unparalled in the literature. The methodology employs a high-order finite-difference approximation. Its central element is a special two-way nonlocal artificial boundary condition that makes the outer boundary transparent for all the outgoing waves and at the same time is capable of accurately prescribing the given impinging signal. It is expected that with the help of this methodology, a valuable new insight will be gained into a number of key outstanding questions in nonlinear optics, in particular, whether the nonparaxiality and backscattering may arrest the collapse (blow-up) of focusing nonlinear waves, and whether the NLH is capable of sustaining the so-called narrow spatial solitons, with the width on the order of only several wavelengths.In the course of the project, a numerical methodology will be built to simulate thepropagation of intense laser light through a variety of media and materials. This methodology has a solid mathematical foundation, and is expected to help addressa number of challenging issues in the theoretical nonlinear optics. In additionto its potential theoretical merits, the methodology will be useful from the standpoint of applications as well. Indeed, the propagation of laser beams in materials is typically accompanied by the phenomena of nonlinear self-focusing and backscattering. The capability to quantitatively analyze and predict these key phenomena is extremely important for many of applications in modern science and engineering. The latter range from remote atmosphere sensing (when an earth-based powerful laser sends pulses to the sky, and backscattered radiation accounts for a substantial part of the detected signal), to laser surgery (propagation of laser beams in tissues), to transmitting information along optical fibers. There are other possible applications that involve, e.g., interactions between the co-propagating or counter-propagating laser beams. They may provide a vehicle for designing the so-called all-optical switches for the next generation of opto-electronic circuits. The proposed numerical methodology will yield a powerful tool for the accurate and robust analysis of the foregoing applications, along with many others.