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，包括非线性麦克斯韦方程的数值解，在二次谐波产生强非线性;通过密度泛函理论，逆衍射问题的数值解，二次谐波产生的多尺度建模;和导模光栅谐振滤波器的优化设计。我们将开始一个金属和电介质界面二次谐波产生的数学模型研究。一个重大的挑战是探索新的多尺度，多物理模型，涉及捕捉有用的微观效应。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时，基于本质上非线性问题的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方案。本项目的目标是发展新的数学模型和计算算法，以满足基本研究的需要，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。该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于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中具有应用。非线性衍射光学是一种基础的和蓬勃发展的技术，具有多种应用，包括快速光学开关，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，使亚波长结构的应用激增，使衍射光学和非线性衍射光学成为现代光学科学发展最快的两个领域。技术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”;近场或超分辨率光学显微镜，以及其他共振现象。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在光学器件的设计和制造中提供廉价且易于控制的结构虚拟原型，可能导致更快的信息处理设备，消耗更少的功率，具有更高灵敏度的传感器，以及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.