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

自2004年首次通过实验分离出来以来，石墨烯（蜂窝晶格中的单层碳原子）已经改变了等离子体和光子学领域。石墨烯具有卓越的机械，化学和电子特性，并且正在寻找从通信和军事能力到医学科学和生物传感的工程感兴趣的设备。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其电学特性。工程界对石墨烯进行了广泛的研究并不奇怪，然而，在应用数学文献中几乎没有报道。有机会做出宝贵的贡献。该项目的目标是为光栅结构中石墨烯的研究提供严格的数学框架，并描述用于高度精确模拟这些模型的计算框架。有了这些，该项目的目的是指导工程师的设计，提供感兴趣的设备在一个大大加快fashion.The家庭的高阶扰动表面（HOPS）的方法，其中首席研究员在过去十年中开发的，是一个理想的选择，为手头的模型。然而，需要进一步的算法增强和扩展，以产生对从业人员仍然有用的工具。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、数值模拟和设计的最新技术水平，其目标如下：（1）虽然石墨烯等材料已经被纳入二维标量配置的现有模型中，但这些努力必须扩展到三维矢量电磁模拟的情况;（2）变换场展开法（TFE）是一种稳定的HOPS算法，它不仅能得到严格的存在性、唯一性和解析性结果，而且能提供高精度的数值逼近，这些递推式必须推广到三维矢量电磁场的情形;（3）PI所提倡的曲面公式受到与Dirichlet和Neumann迹相关的曲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，这些算子没有困扰直接Dirichlet的“Dirichlet特征值”，（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作为项目的一部分纳入该框架;以及（5）除了使用这些HOPS方法来识别以石墨烯为特征的光栅结构之外，这个奖项反映了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，被认为是值得支持的。

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.