Analytical Studies of Nonlinear Optics in Periodic Media

这一建议的目的是解析地研究非线性光在新型周期介质中传播的新现象。具体地说，通过发展复杂的指数渐近技术，将解析地确定线性和非线性组合格子、一般宇称时间对称格子和一般二维格子中孤立波的分叉和线性稳定性。我们将显式计算孤立波的位置及其线性稳定性特征值，从而解析地得到周期介质中这些孤立波的存在性和线性稳定性性质。从更广泛的角度来看，周期介质中的非线性光学目前处于光学和应用数学研究的前沿。随着微尺度周期性结构的适当设计，光的传播可以以各种不同的方式进行控制和操纵，这表明在微尺度光学数据处理设备、低失真图像传输以及用于超高速计算机的潜在光学芯片方面具有很好的应用前景。研究这些令人兴奋但困难的物理问题需要先进的数学技术。在这个项目中，我们将开发一个强大的指数渐近方法，它可以给出关于周期介质中孤立波的存在性和稳定性的非常详细的信息。这些信息将使我们能够评估光孤子在各种物理应用中的潜力，例如在微观周期结构上的数据处理。此外，该项目还将促进这一重要领域的研究生的跨学科培训。

The objective of this proposal is to analytically investigate novel phenomena of nonlinear light propagation in new types of periodic media. Specifically, bifurcation and linear stability of solitary waves in combined linear and nonlinear lattices, general parity-time symmetric lattices and general two-dimensional lattices will be analytically determined through the development of a sophisticated exponential asymptotics technique. Both the positions of solitary waves and their linear-stability eigenvalues will be explicitly calculated; hence the existence and linear-stability properties of these solitary waves in periodic media will be analytically obtained. From a broader perspective, nonlinear optics in periodic media is currently at the forefront of research in optics and applied mathematics. With appropriate engineering of microscale periodic structures, light propagation can be controlled and steered in various different ways, which point to promising applications for micro-scale optical data-processing devices, low-distortion image transmission, and potential optical chips for ultrafast computers. The study of these exciting but difficult physical problems calls for advanced mathematical techniques. In this project, we will develop a powerful exponential asymptotics method, which can yield very detailed information on the existence and stability of solitary waves in periodic media. This information will allow us to assess the potential of optical solitons for various physical applications such as data processing on microscopic periodic structures. In addition, this project will facilitate interdisciplinary training of a graduate student in this important area.