Modeling of Micro-Optics and Electromagnetics

PI建议对两名数学科学行业研究生研究助理的专业和学术发展进行培训。培训活动的第一个目标是为研究生提供技能和经验，使他们在学术/工业环境中成为强有力的竞争者。 另一个目标是培养具有资格的研究生，使他们成为学术/工业环境中科学和技术发展的重要贡献者。 拟议的研究重点是线性和非线性介质中光学微结构建模和设计的计算方法和数值分析，以及求解麦克斯韦方程组的逆源问题。 这项研究将扩大我们与工业界的现有合作，并开发新的工业合作。本论文的主要研究内容如下：（1）微衍射光学的计算建模：TM偏振衍射问题的界面最小二乘有限元方法的不断发展。 双周期结构绕射问题的公式化及数值分析。(2)非线性衍射光学的建模与设计。偏微分方程模型的适定性研究。非线性光栅的优化设计。 反衍射问题：非线性材料薄涂层对二次谐波产生的影响。(3)脑磁图（MEG）中麦克斯韦方程的逆源问题。癫痫灶在活体人脑中的定位。 模型问题的建模、计算和渐近分析，对于发展新的科学和指导工业界设计和制造新的光学器件以及解决电磁学中的逆源问题具有重要的潜力。 该GOALI项目由MPS多学科活动办公室（OMA）和数学科学部（DMS）联合支持。

The PI proposes training on the professional and academic development of two Mathematical Sciences Industry-Based Graduate Research Assistants. The first objective of the training activitiesis to prepare the graduate students with skills and experiences that will make them strong competitors in an academic/industrial environment. Another objective is to produce the graduate students who have qualifications that will make them important contributors to science and technology developments in an academic/industrial environment. The proposed research focuses on computational methods and numerical analysis for modeling and design of optical micro-structures in linear and nonlinear media, and solving inverse source problems for Maxwell's equations. The research will extend our existing collaboration with industry and develop a new industrial collaboration. It will provide opportunities for the graduate assistants to engage in University-Industry joint research programs and make important contributions to an exciting area of applied and industrial mathematics.The main topics of the proposed research are as follows:(1) Computational modeling of micro diffractive optics.Continuous developments of interface least-squares finite element methods for diffraction problems in TM polarization. Formulation and numerical analysis for the diffraction problem bybiperiodic structures. (2) Modeling and design in nonlinear diffractive optics.Well-posedness study of a PDE model. Optimal design of nonlinear gratings. Inverse diffraction problems.The effect of thin coatings of nonlinear material on second harmonic generation. (3) Inverse source problems for Maxwell's equations in magnetoencephalography (MEG). Determination of locations of epileptic foci in the living human brain. Modeling, computation, and asymptotic analysis of the model problem.The research has significant potential for evolving new science and providing industry with guidance to design and fabricate new optical devices and to solve the inverse source problems in electromagnetics. This GOALI project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).