Direct and Inverse Problems in Diffractive Optics Modeling

Michigan State University, East Lansing MI

Investigators

Abstract

Over the next five years, the Principal Investigator proposes to examine the mathematical issues and develop computational methods for solving the following classes of direct and inverse problems motivated 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 harmonic generation. An interface least-squares finite element method for solving the 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 facilities and microlithographic fabrication techniques have led to an explosion of applications 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 equation models, mathematical analysis, and numerical algorithms to 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.