OP: High Accuracy Modeling of Graphene Plasmonics in Three Dimensional Grating Structures

自从2004年首次通过实验分离出石墨烯（蜂窝晶格中的单层碳原子）以来，它已经改变了等离子体和光子学领域。石墨烯具有卓越的机械、化学和电子特性，并正在进入通信、军事、医学和生物传感等工程领域。特别值得注意的是石墨烯的半金属特性，它允许人们调整其电性能。在工程界对石墨烯进行了广泛的研究，这并不奇怪，然而，在应用数学文献中却很少有报道。这是一个做出有价值贡献的机会。该项目的目标是为研究光栅结构中的石墨烯提供一个严格的数学框架，并描述一个高度精确模拟这些模型的计算框架。有了这些，该项目旨在指导工程师设计，以大大加快的方式交付感兴趣的设备。在过去十年中，首席研究员开发的高阶表面摄动（HOPS）方法家族是手头模型的理想选择。然而，需要进一步的算法增强和扩展来产生对从业者有用的工具。特别是，该项目将推进以二维材料为特征的光栅结构的建模、数值模拟和设计方面的最新技术，其目标如下：(1)虽然石墨烯等材料已被纳入现有的二维、标量配置模型，但这些努力必须扩展到三维矢量电磁模拟的情况；(2)变换场展开法（TFE）是一种稳定的hop算法，它不仅可以得到严格的存在唯一性和解析性结果，而且可以提供高精度的数值近似，这些递推必须推广到三维矢量电磁情况；(3) PI提倡的表面公式受到与狄利克雷和诺伊曼轨迹相关的表面积分算子的选择的影响，该项目将研究阻抗-阻抗算子的性能，这些算子没有“狄利克雷特征值”，而“狄利克雷特征值”困扰着以前实现中使用的直接狄利克雷-诺伊曼算子；(4) PI和合作者最近研究了随机粗糙光栅散射的统计特性，并将三维结构中的二维材料纳入该框架，作为项目的一部分；(5)除了使用这些HOPS方法识别石墨烯光栅结构外，该项目还旨在设计其最佳性能。该奖项反映了美国国家科学基金会的法定使命，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。

Since it was first isolated experimentally in 2004, graphene (a single layer of carbon atoms in a honeycomb lattice) has transformed the fields of plasmonics and photonics. Graphene has remarkable mechanical, chemical, and electronic properties, and is finding its way into devices of engineering interest from communications and military capabilities to medical sciences and biological sensing. Of particular note is graphene's semi-metallic character which permits one to tune its electrical properties. It is not surprising that extensive investigation of graphene has been conducted in the engineering community, however, there is very little to report in the applied mathematics literature. There is an opportunity to make valuable contributions. The goal of this project is to provide a rigorous mathematical framework for the study of graphene in grating structures and to describe a computational framework for highly accurate simulation of these models. With these, the project aims to guide the design of engineers to deliver devices of interest in a greatly expedited fashion.The family of High-Order Perturbation of Surfaces (HOPS) methods, which the Principal Investigator has developed over the past decade, are an ideal choice for the model at hand. However, further algorithmic enhancements and extensions are required to produce tools which will continue to be useful to practitioners. In particular, the project will advance the state of the art in the modeling, numerical simulation, and design of grating structures featuring two-dimensional materials with the following objectives: (1) While materials such as graphene have been incorporated into the existing models in two-dimensional, scalar configurations, these efforts must be extended to the case of three-dimensional vector electromagnetic simulations; (2) The Method of Transformed Field Expansions (TFE) is a stabilized HOPS algorithm with which one can not only derive rigorous existence, uniqueness, and analyticity results, but also provide highly accurate numerical approximations, and these recursions must be extended to the three-dimensional vector electromagnetic case; (3) The surface formulations advocated by the PI are affected by the choice of surface integral operator relating Dirichlet and Neumann traces, and the project will investigate the performance of Impedance-Impedance Operators which are free of the "Dirichlet Eigenvalues" which plague the straightforward Dirichlet-Neumann Operators used in previous implementations; (4) the PI and collaborators recently studied the statistical properties of scattering by random rough gratings and will include two-dimensional materials in three-dimensional structures into this framework as part of the project; and (5) In addition to using these HOPS methods to identify grating structures featuring graphene, the project also aims to design them for optimal performance.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.