A Novel Finite Element Method Toolbox for Interface Phenomena in Plasmonic Structures

该项目的研究将导致开发一个高效的计算平台，为光散射的物理问题提供有价值的见解。总体目标是提供一个工具箱，可供多个社区使用，以准确计算电磁场，补充实验测量。这项研究将通过填补奇异结构中电磁场理论和数学建模的空白来推进几个领域的知识。该项目旨在包括跨学科（计算机科学，物理学，数学），以团队为基础的方法与研究生培训。该团队将专注于创建直接影响当前和未来实验的应用数学研究成果。该项目包括在会议上组织小型专题讨论会，以促进与其他研究人员的讨论。该研究项目将被用作招聘工具，以进一步促进STEM领域的妇女和代表性不足的少数民族。该项目的目标是开发一个最先进的计算和数学平台，准确有效地捕获等离子体结构中电磁场的多尺度行为。等离子体结构通常由金属和陶瓷制成，并且在金属-电介质界面处表现出光学频率的表面电磁波，称为表面等离子体。在过去的几十年里，人们对在纳米光子器件中引导和限制表面等离子体激元产生了极大的兴趣，并将其应用于天线，隐身等。表面等离子体激元是亚波长的，在拐角附近是超奇异的，因此对几何形状高度敏感。科学家和工程师通常使用的商业软件是基于有限元法，并且很难近似这些奇异结构中的电磁近场。需要一种适应的数学框架来捕获多尺度，并且需要用于计算等离子体结构中的电磁场的数值方法来避免不准确的预测。该项目将开发一种基于有限元的方法来解决这一问题，该方法涉及界面附近的特定处理，精确捕获表面等离子体，以及提取电磁近场的高度振荡行为的功能框架。具体工作包括：（1）在经典框架中解决任何2D几何结构的等离子体问题，（2）当出现超奇异行为时解决任何2D几何结构的等离子体问题，以及（3）将框架扩展到3D等离子体问题和其他等离子体模型。该项目的成果将导致对基础物理的深入了解，开发准确的方法，并将其应用于现实问题。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The research in this project will lead to the development of an efficient computational platform that will provide valuable insight into this physical problem of light scattering. The overarching goal is to provide a toolbox that can be used by several communities to accurately compute the electromagnetic field, complementing experimental measurements. The research will advance knowledge in several fields by filling a gap in the theory and mathematical modeling of electromagnetic fields in exotic structures. The project is designed to include an interdisciplinary (computer science, physics, mathematics), team-based approach with graduate training. The team will focus on creating applied math research results that directly impact current and future experiments. The project includes activities to organize mini-symposia at conferences to foster discussions with other researchers. This research project will be used as a recruiting tool to further promote women and underrepresented minorities in STEM-fields. The goal of this project is to develop a state-of-the-art computational and mathematical platform that accurately and efficiently capture the multiscale behaviors of the electromagnetic field in plasmonic structures. Plasmonic structures are commonly made of metals and dielectrics, and exhibit at optical frequencies surface electromagnetic waves at the metal-dielectric interfaces, called surface plasmons. Over the past decades there has been a great interest to guide and confine surface plasmons in nanophotonic devices, with applications to antennas, cloaking, and others. Surface plasmons are sub-wavelength, hyper-singular near corners, and consequently highly sensitive to the geometry. Commercial software commonly used by scientists and engineers is based on the finite element method, and poorly approximates the electromagnetic near-field in these exotic structures. There is a need for an adapted mathematical framework to capture the multiple scales, and for numerical approaches for computing the electromagnetic field in plasmonic structures to avoid inaccurate predictions. This project will develop a finite element based approach to address this problem that involves specific treatments near the interface the accurately capture the surface plasmons, and a functional framework to extract the highly oscillatory behaviors of the electromagnetic near-field. Specific work includes: (1) solve plasmonic problems for any 2D geometries in a classical framework, (2) solve plasmonic problems for any 2D geometries when hyper-singular behaviors appear, and (3) extend the framework to 3D plasmonic problems and other plasmonic models. Results from this project will lead to insights into the underlying physics, develop accurate methods, and apply them to realistic problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.