Simulation of the Wave-Matter Interactions: Geometrical Optics and Nano Optics

更广泛的意义和重要性：拟议的项目致力于提供可靠的模型和最先进的数值方法来研究波-物质相互作用，如几何光学和纳米光学中的实际应用，产生于地震成像、医学成像、纳米技术、纳米科学等。所提出的模型和数值方法基于几何光学、物理光学、纳米光学、计算化学和科学计算的知识，旨在为应用提供数值工具和指导。将开展跨学科合作，以扩大拟议方法的实际应用。该项目还涉及计算数学方面的研究和教育的整合。将鼓励研究生和本科生以及来自代表性不足群体的成员参加该项目，以增进他们的知识和研究。拟议的项目将进一步加强对学生的培训和教育，并鼓励他们在未来的科学、技术、工程和数学(STEM)领域追求职业生涯。PI将开发可靠的模型和有效的数值方法来模拟几何光学和纳米光学中的波-物质相互作用。当物质的大小远远大于波长时，相互作用等价于模拟非均匀介质中的高频波，物质的平均和连续嵌入的影响作为介质常数。PI将结合几何光学和物理光学以及科学计算来设计和分析有效的数值方法。当物质的尺寸达到纳米尺度时，相互作用需要同时模拟物质的运动和波的演化，就像在纳米光学中研究纳米结构的光学响应一样，其中必须考虑物质的量子效应。PI将以半经典理论为指导，推导出描述相互作用的简单模型，并提出多尺度方法来模拟相互作用。所提出的模型和方法包括以下核心思想：(1)对于模拟高频波，格林函数的惠更斯-基尔霍夫积分作为推进波传播的公式和机制，其中几何光学提供了格林函数的渐近近似。用精确的方法求解哈密顿-雅可比方程，可以有效地计算出近似格林函数的相位和振幅。基于低阶矩阵分解的多级算法可以有效地计算振荡惠更斯-基尔霍夫积分。而压缩相位和幅度的数据压缩技术确保了在二维和三维空间中构建波的可行性。(2)为了研究纳米结构的光学响应，用麦克斯韦方程经典地处理波并保留物质的量子力学描述的半经典理论提供了光-物质相互作用的表征。Born-Oppenheimer近似和从头算分子动力学可以用来解决用量子力学处理多体系统的困难，这导致了简单的半经典模型，这些模型可以用所提出的多尺度格式进行数值跟踪。

Broader Significance and Importance: The proposed project is devoted to providing reliable models and the state-of-the-art numerical methods for studying the wave-matter interactions as in geometrical optics and nano optics with practical applications arising from seismic imaging, medical imaging, nanotechnology, nanosciences, and so on. The proposed models and numerical methods are based on knowledge of geometrical optics, physical optics, nano optics, computational chemistry and scientific computing, aiming to provide numerical tools and guidance for applications. Interdisciplinary collaborations will be pursued to extend the practical applications of the proposed methods. The project also involves the integration of research and education in computational mathematics. Graduate and undergraduate students, and members from underrepresented groups will be encouraged to participate in the project to enhance their knowledge and research. The proposed project will further the training and education of students and encourage them to pursue future career in science, technology, engineering and mathematics (STEM). The PI will develop reliable models and efficient numerical methods to simulate the wave-matter interactions as in geometrical optics and nano optics. When the size of the matter is much larger than the wavelength, the interactions are equivalent to simulating high frequency waves in nonhomogeneous media with the effect of matter averaged and embedded continuously as medium constants. The PI will combine geometrical optics and physical optics along with scientific computing to design and analyze efficient numerical methods. When the size of the matter reaches nanoscale, the interactions require to simulate the motion of the matter and the evolution of the waves simultaneously as in the study of the optical responses of nanostructures in nano optics, where the quantum effects of the matter must be considered. The PI will use the semi-classical theories as the guideline to derive simple models to characterize the interactions and propose multiscale methods to simulate the interactions. The proposed models and methods consist of the following core ideas: (1) for simulating high frequency waves, the Huygens-Kirchhoff integral with the Green functions serves as the formulation and mechanism for advancing the wave propagation, where geometrical optics provides asymptotic approximations of the Green functions. The phase and amplitude to approximate the Green functions can be computed efficiently with accurate methods for Hamilton-Jacobi equations. The oscillatory Huygens-Kirchhoff integral can be evaluated efficiently with multilevel algorithms based on low-rank matrix decompositions. And data compression techniques to compress the phase and amplitude ensure the feasibility of building the waves in both two and three-dimensional spaces. And (2) for studying the optical responses of nanostructures, the semi-classical theories that treat the waves classically with the Maxwell equations and retain the quantum mechanical description of the matter provide characterizations of the light-matter interactions. The Born-Oppenheimer approximation and the ab initio molecular dynamics can be utilized to resolve the difficulty of dealing with the many-body system quantum mechanically, which results in simple semi-classical models that are numerically trackable with proposed multiscale schemes.