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

自从2004年首次实验分离以来，石墨烯(蜂窝状晶格中的单层碳原子)已经改变了等离子体和光子学领域。石墨烯具有显著的机械、化学和电子性能，正在被用于从通信和军事能力到医学和生物传感等具有工程意义的设备。特别值得注意的是石墨烯的半金属特性，这使得人们可以调整它的电学性质。工程界对石墨烯进行了广泛的研究，这并不令人惊讶，然而，在应用数学文献中几乎没有报道。现在有机会做出有价值的贡献。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，并描述一个高精度模拟这些模型的计算框架。通过这些，该项目旨在指导工程师的设计，以大大加快交付感兴趣的设备的方式。首席调查员在过去十年中开发的高阶曲面摄动(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将继续对从业者有用的工具。具体地说，该项目将推进以二维材料为特征的光栅结构的建模、数值模拟和设计的技术水平，其目标如下：(1)尽管石墨烯等材料已经以二维标量构型被纳入现有的模型中，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(TFE)方法是一种稳定的跳跃算法，利用它不仅可以得到严格的存在、唯一性和解析性结果，而且还可以提供高精度的数值近似，并且这些递推必须扩展到三维矢量电磁情况；(3)PI倡导的表面公式受到与Dirichlet和Neumann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，这些算子没有困扰以前实现中的直接Dirichlet-Neumann算子的“Dirichlet-Neumann算子”；(4)PI及其合作者最近研究了随机栅格散射的统计特性，并将三维结构中的二维材料作为项目的一部分纳入该框架；和(5)除了使用这些跃点方法来识别具有石墨烯的栅格结构外，该项目还旨在将它们设计为最佳性能。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。

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.