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

该项目的研究将导致开发一种高效的计算平台，这将为光散射的这个物理问题提供有价值的见解。首要目标是提供一个工具箱，可供几个社区使用，以准确计算电磁场，补充实验测量。这项研究将填补奇异结构中电磁场理论和数学模型的空白，从而促进几个领域的知识进步。该项目旨在包括跨学科(计算机科学、物理、数学)、以团队为基础的方法和研究生培训。该团队将专注于创造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研讨会的活动，以促进与其他研究人员的讨论。这一研究项目将被用作招聘工具，进一步促进妇女和在STEM领域任职人数不足的少数群体。该项目的目标是开发一个最先进的计算和数学平台，准确和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电介质制成，并在光学频率下在金属-介质界面上表现出表面电磁波，称为表面等离子体。在过去的几十年里，在纳米光子器件中引导和限制表面等离子体激元一直是人们非常感兴趣的事情，并将其应用于天线、隐形和其他领域。表面等离子体是亚波长的，在角落附近是超奇异的，因此对几何高度敏感。科学家和工程师常用的商业软件是基于有限元方法的，不能很好地逼近这些奇异结构中的电磁近场。需要一个适当的数学框架来捕捉多个尺度，并需要数值方法来计算等离子体结构中的电磁场，以避免不准确的预测。该项目将开发一种基于有限元的方法来解决这个问题，其中包括在界面附近进行特定处理，以精确捕获表面等离子激元，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)在经典框架下求解任意二维几何的等离子体激元问题，(2)当超奇异行为出现时，求解任意二维几何的等离子体激元问题，以及(3)将该框架扩展到三维等离子体激元问题和其他等离子体激元模型。这个项目的结果将导致对潜在物理学的洞察，开发准确的方法，并将它们应用于现实问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.