Direct and Inverse Problems in Diffractive Optics Modeling

在接下来的五年里，首席研究员建议研究数学问题并开发计算方法来解决以下由工业应用驱动的直接和逆问题：周期结构（光栅）的散射和衍射，非线性衍射光学中的二次谐波产生，以及衍射光学中的逆和优化设计问题。本文的主要工作内容如下：周期性手性结构的衍射和微扰衍射结构的散射的建模、分析和计算。电磁共振的建模与设计。——二阶谐波非线性Maxwell方程的适定性。求解三维模型问题的界面最小二乘有限元法。结构（光栅）增强非线性光学效应。涂层对非线性衍射光学的影响。——衍射结构优化设计。最优设计问题的松弛技术（均质化）和局部方法。反衍射问题的唯一性、稳定性和规律性研究。衍射光学逆设计与优化设计问题的数值解。近年来，高性能计算设备和微光刻制造技术的发展使得衍射技术在光学领域的应用爆炸式增长，使衍射光学成为当前光学工程研究中发展最快的领域之一。实际应用和科学发展推动了对严格的偏微分方程模型、数学分析和数值算法的需求，以描述复杂光栅结构的衍射，计算电磁场，从而预测给定衍射结构在线性、手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。所提出的研究对于发展新的数学和科学以及为工业设计和制造新的光学器件提供指导具有重要的潜力。

Over the next five years, the Principal Investigator proposes to examine the mathematical issues and develop computational methodsfor solving the following classes of direct and inverse problemsmotivated by industrial applications: scattering and diffraction by periodic structures (gratings), second harmonic generation in nonlinear diffractive optics, and inverse and optimal design problems in diffractive optics. The main topics of the proposed work are as follows:-- Modeling, analysis, and computation of the diffraction by periodic chiral structures and the scattering by a perturbed diffractive structure. Modeling and design of electromagnetic resonances.-- Well-posedness of the nonlinear Maxwell equations in second harmonicgeneration. An interface least-squares finite element method for solvingthe three-dimensional model problem. Structure (grating) enhanced nonlinear optical effects. Effects of coatings on nonlinear diffractive optics.-- Optimal design of diffractive structures. A relaxation technique (homogenization) and a local approach for the optimal design problem. Uniqueness, stability, and regularity studies of the inverse diffraction problem. Numerical solution of inverse and optimal design problems in diffractive optics.The recent enabling technologies of high-performance computing facilitiesand microlithographic fabrication techniques have led to an explosion ofapplications of diffraction in optics, establishing diffractive optics as one of the most rapidly advancing areas of current research in optical engineering. The practical applications and scientific developments have driven the need for rigorous partial differential equationmodels, mathematical analysis, and numerical algorithmsto describe the diffraction of complicated grating structures, to compute electromagnetic fields and thus to predict the performance of a given diffractive structure in linear, chiral, and nonlinear optical media, as well as to carry out optimal design of new structures. The proposed research has significant potential for evolving new mathematics and science and providing industry with guidance to design and fabricate new optical devices.