Analytical and Numerical Studies of Nonlinear Light Propagation in Two-dimensional Photonic Lattices

本计画的目的是研究二维光子晶格中非线性光传播的新现象。特别地，一个特殊的重要的解类（自困非线性局域状态，通常称为孤子）的基础数学模型将分析和数值研究。本文将研究三个相互关联的问题：(i)二维光子晶格所允许的所有可能类型的孤子态的确定和分类；（ii）确定这些孤子的线性稳定性；（iii）发展更有效的数值方法来计算这些孤子及其线性稳定性特征值。适合这些物理问题的数学模型将是具有周期或准周期势的二维非线性薛定谔方程。周期和准周期介质中的非线性光学是近年来光学和应用数学的一个令人兴奋的前沿领域。通过适当的微尺度周期结构工程，可以以各种方式控制和引导光的传播。这种可控转向指向了有前途的技术应用，如微型光学数据处理设备和用于超快计算机的潜在光学芯片。从物理学的角度来看，该项目将对微尺度周期介质中的非线性波动现象有更深入的认识。从数学的角度来看，这些研究将促进周期性介质中非线性波的分析和数值方法的发展。此外，本研究与非线性光学以外的研究密切相关，如玻色-爱因斯坦凝聚在光学晶格中的负载。该项目将促进研究生在这些重要研究领域的跨学科培训。

The objective of this project is to investigate novel phenomena of nonlinear light propagation in two-dimensional photonic lattices. In particular, a special important class of solutions (self-trapped nonlinear localized states, often called solitons) of the underlying mathematical models will be studied both analytically and numerically. Three inter-related issues will be investigated: (i) determination and classification of all possible types of soliton states admitted by two-dimensional photonic lattices; (ii) determination of linear stability properties of these solitons; (iii) development of more efficient numerical methods for computing these solitons and their linear-stability eigenvalues. The mathematical models appropriate for these physical problems will be two-dimensional nonlinear Schroedinger equations with periodic or quasi-periodic potentials. Nonlinear optics in periodic and quasi-periodic media is an exciting frontier of optics and applied mathematics these days. With appropriate engineering of micro-scale periodic structures, light propagation can be controlled and steered in various ways. This controlled steering then points to promising technological applications such as micro-scale optical data-processing devices and potential optical chips for ultrafast computers. From a physical viewpoint, this project will lead to a deeper understanding on nonlinear wave phenomena in micro-scale periodic media. From a mathematical viewpoint, these studies will advance the analytical and numerical methodologies for the treatment of nonlinear waves in periodic media. In addition, this research is intimately related to research in areas beyond nonlinear optics, such as Bose-Einstein condensates loaded in optical lattices. The project will facilitate interdisciplinary training of a graduate student in these important areas of research.