Analytical Studies of Nonlinear Optics in Periodic Media

本文的目的是分析研究非线性光在新型周期介质中的传播新现象。具体地说，孤立波在组合线性和非线性晶格、一般奇偶时间对称晶格和一般二维晶格中的分岔和线性稳定性将通过发展一种复杂的指数渐近技术来解析确定。孤立波的位置和它们的线性稳定性特征值都将被明确地计算出来；由此可以解析地得到周期介质中孤波的存在性和线性稳定性。从更广泛的角度来看，周期介质中的非线性光学是目前光学和应用数学研究的前沿。通过适当的微尺度周期结构工程，可以以各种不同的方式控制和引导光的传播，这为微尺度光学数据处理设备，低失真图像传输以及超高速计算机的潜在光学芯片提供了前景。研究这些令人兴奋但又困难的物理问题需要先进的数学技术。在这个项目中，我们将开发一种强大的指数渐近方法，它可以得到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，例如微观周期结构的数据处理。此外，该项目将促进研究生在这一重要领域的跨学科培训。

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.