Inverse Diffraction Problems in Optics

9803604这个项目将支持对数学问题和计算方法的研究，以解决以下几类工业应用中的正反问题：衍射光学中的反问题，结构增强的非线性光学效应的建模，以及非线性光学中的优化设计问题。主要工作包括：光学反问题的唯一性、稳定性和正则性。衍射光学中逆问题和优化设计问题的数值计算。周期结构中非线性二次谐波产生的建模。结构增强的非线性光学效应。非线性光学中一类偏微分方程模型的适定性。求解模型问题的界面最小二乘有限元方法。非线性光学设计与优化的计算方法。这些问题在微衍射光学系统的建模和设计中自然会出现。微衍射光学是一种将光学器件微细加工成微小结构特征的基础新兴技术。利用衍射效应，衍射结构实现了传统光学无法实现的功能。当前和潜在应用领域的例子包括矫正透镜、电子显示器、微型传感器、激光器、光存储系统、光通信组件和集成光电半导体器件。实际应用和科学发展需要严格的数学模型、数学分析和数值算法来精确描述复杂衍射结构的绕射，计算电磁矢量场，从而预测给定的衍射结构在线性和非线性光学介质中的性能，以及进行新结构的优化设计。这项研究对发展新的科学具有重要的潜力，并为工业设计和制造新的光学器件提供指导。

9803604 Bao This project will support research on the mathematical issues and computational methods for solving the following classes of direct and inverse problems motivated by industrial applications: inverse problems in diffractive optics, modeling of structure enhanced nonlinear optical effects, and optimal design problems in nonlinear optics. The main topics of the proposed work include: Uniqueness, stability, and regularity properties for inverse problems in optics. Numerical computation of inverse and optimal design problems in diffractive optics. Modeling of nonlinear second harmonic generation in periodic structures. Structure enhanced nonlinear optical effects. Well-posedness of a partial differential equation model in nonlinear optics. Interface least-squares finite element methods for solving the model problem. Computational methods for design and optimization in nonlinear optics. These problems arise naturally in modeling and design of micro diffractive optics. Micro diffractive optics is a fundamental, emerging technology in which optical devices are micromachined with very small structural features. Exploiting diffraction effects, diffractive structures perform functions unattainable with conventional optics. Examples of current and potential application areas include corrective lenses, electronic displays, microsensors, lasers, optical storage systems, optical communication components, and integrated opto-electronic semiconductor devices. The practical applications and scientific developments have driven the need for rigorous mathematical models, mathematical analysis, and numerical algorithms to accurately describe the diffraction of complicated diffractive structures, to compute electromagnetic vector fields and thus to predict the performance of a given diffractive structure in linear and nonlinear optical media, as well as to carry out optimal design of new structures. The research has significant potential for evo lving new science and providing industry with guidance to design and fabricate new optical devices.