Mathematics and Computation of Nonlinear Problems in Diffractive Optics Modeling

我们提出的数学建模技术和计算方法将解决应用数学中的关键科学挑战，包括具有强非线性的二次谐波产生非线性麦克斯韦方程组的数值解；二次谐波产生的密度泛函多尺度模拟，反衍射问题的数值解；以及导模光栅谐振滤波器的优化设计。我们将开始对金属和介电界面的二次谐波产生进行数学建模研究。一个重大的挑战是探索新的多尺度，多物理模型，包括捕获有用的微观效应。将开发新的计算方法和数学建模技术来解决潜在的PDE问题。当非线性光学效应变得非常强烈时，基于非线性问题线性化的数值方法是不够的。必须探索新的技术来开发稳定的数值格式。对于相关的优化设计和逆问题，也将开发新的鲁棒解方法来研究模型问题的临界病态性。该项目的目标是开发新的数学模型和计算算法，以满足基本的研究需求，并为衍射光学和非线性光学领域的科学家提供必要的建模工具。拟议的项目不仅有可能发展新的科学，而且还可能导致新的数学和计算方法。控制和操纵光的能力对我们社会的许多领域都具有至关重要的意义，并且在能源，传感，成像和信息技术等关键领域都有应用。非线性衍射光学是一门基础的、正在蓬勃发展的技术，有着广泛的应用，包括快速光开关、等离子体材料、光学计算、光学显微镜、光谱学和光学超材料。近年来，高性能计算设备和现代光刻技术的发展使得亚波长结构的应用激增，从而使衍射光学和非线性衍射光学成为现代光学科学中发展最快的两个领域。科技的不断发展带来了令人兴奋的创新，如“隐形斗篷”和“超级透镜”；近场或超分辨光学显微镜，及其他共振现象。现代光学器件和理论的未来发展和分析将受益于有效的计算建模工具和数学分析技术的可用性。我们的计算模型和优化设计工具将为光学器件的设计和制造提供一种廉价且易于控制的结构虚拟原型，有可能导致消耗更少功率的更快的信息处理器件，具有更高灵敏度的传感器和具有高转换效率的非线性光学器件。

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.