Inverse Diffraction Problems in Optics

本项目将支持解决以下几类工业应用驱动的直接和反问题的数学问题和计算方法的研究：衍射光学中的反问题，结构增强非线性光学效应的建模，非线性光学中的优化设计问题。提出的工作的主要主题包括：唯一性，稳定性和正则性的反问题的光学。衍射光学逆设计与优化设计问题的数值计算。周期结构非线性二次谐波产生的建模。结构增强非线性光学效应。非线性光学中偏微分方程模型的适定性。界面最小二乘有限元法求解模型问题。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元件的建模和设计中自然出现。微衍射光学是一种基本的新兴技术，其中光学器件是具有非常小的结构特征的微机械加工。利用衍射效应，衍射结构实现了传统光学无法实现的功能。当前和潜在的应用领域包括矫正透镜、电子显示器、微传感器、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推动了对严格的数学模型、数学分析和数值算法的需求，以准确描述复杂衍射结构的衍射，计算电磁矢量场，从而预测给定衍射结构在线性和非线性光学介质中的性能，并进行新结构的优化设计。该研究对发展新科学和指导工业设计和制造新型光学器件具有重要的潜力。

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.