Mathematics and Computation of Nonlinear Problems in Diffractive Optics Modeling

点击翻译按钮获取中文摘要

Our proposed mathematical modeling techniques and computational methods will address key scientific challenges in applied mathematics including numerical solution of nonlinear Maxwell's equations in second harmonic generation with strong nonlinearities; multiscale modeling of second harmonic generation via Density Functional Theory, numerical solution of the inverse diffraction problems; and optimal design of guided mode grating resonance filters. We will initiate a mathematical modeling study of second harmonic generation for metal and diectric interfaces. A significant challenge is to explore novel multiscale, multi-physics models involving to capture the useful microscopic effects. Novel computational methods and mathematical modeling techniques will be developed to solve the underlying PDE problems. When nonlinear optical effects become significantly strong, numerical methods based on the linearization of the essentially nonlinear problem are inadequate. Novel techniques must be explored to develop stable numerical schemes. New robust solution methods for the associated optimal design and inverse problems will also be developed to investigate the critical ill-posedness of the model problems.The goal of this project is to develop new mathematical models and computational algorithms that meet the basic research needs and provide the necessary modeling tools for scientists in the areas of diffractive optics and nonlinear optics. The proposed project has the potential not only to evolve new science, but also to lead to novel mathematics and computational methods. The capabilities for controlling and manipulating light are of paramount significance for many areas of our society, and have applications in critical areas such as energy, sensing, imaging, and information technology. Nonlinear diffractive optics is a fundamental and vigorously growing technology with diverse applications including fast optical switches, plasmonic materials, optical computing, optical microscopy, spectroscopy, and optical metamaterials. The recent enabling high-performance computing facilities and modern lithographic techniques have led to a substantial surge of applications of subwavelength structures, establishing diffractive optics and nonlinear diffractive optics as two of the most rapidly advancing areas of modern optical science. The continuous technology developments have given rise to exciting innovation such as "invisibility cloaks" and "superlenses"; near-field or super-resolution optical microscopy, and other resonance phenomena. The future development and analysis of modern optics devices and theory will benefit from the availability of efficient computational modeling tools and mathematical analysis techniques. Our computational models and optimal design tools will provide an inexpensive and easily controllable virtual prototype of the structures in the design and fabrication of optical devices, potentially leading to faster information processing devices that consume less power, sensors with higher sensitivity, and nonlinear optical devices with high conversion efficiency.