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

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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.