Phenomena of Periodic Layered Media

本项目研究多层材料的物理性质，以指导高效和新颖的光电器件的设计，如激光器、天线和滤光片。这种材料的一个常见例子是堆叠的六角碳石墨烯薄片。其目的是从多种方式中发展设计的灵活性，在这些方式中，层可以相对于彼此放置，以实现所需的光学和电磁波相互作用，最终提供利用光的实用选择。分层过程自然地将层的几何结构与基础物理的数学方程缠绕在一起。这个项目关注于这些数学对象之间的相互作用，目的是建立层状介质的基础理论。由于这项调查针对的是潜在的物理现象，其结果将适用于依赖电磁设备的社会的各个方面。研究生和本科生将被整合到一个专门的研究小组，他们来自不同的背景，并将有机会在科学会议上展示他们的工作，以帮助他们拓宽科学视野并建立专业联系。有一个物理和数学对象位于该项目研究的中心，称为费米面，它与相关的布洛赫面一起包含关于晶体或周期材料中电磁波的核心信息，通过编码两个基本物理量之间的关系，能量(或频率)和动量(或波数)。根据周期介质的细节，这些对象是解析的或代数的变体，它们的结构编码关于分层方法的信息。层状材料可以是二维的，例如多层石墨烯薄片，也可以是三维的，其中层位于主体的内部，并提供一种隐藏的自由度。具体目标是建立分层方法与费米和布洛赫变种之间的联系，特别是它们的可约性和奇异性；分析这些变种的可约性的共振结果；对导致可约性和对称性破缺下可还原的持久性的内部对称性进行分类；通过层状图形模型的均化来创建新的块状介质；以及将这些研究应用于多层石墨烯和其他物理上可行的层状材料。这项研究将引用复数分析、偏微分方程、代数几何和微分算子的频谱理论等数学领域。这一奖项反映了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。

This project investigates the physical properties of multi-layered materials for the purpose of guiding the efficient and novel design of opto-electronic devices, such as lasers, antennas, and filters. A popular example of such a material is stacked sheets of hexagonal carbon graphene. The aim is to develop flexibility in design from the multitude of ways in which the layers can be placed relative to each other to achieve a desired interaction of optical and electromagnetic waves, ultimately delivering practical options for harnessing light. The layering process naturally intertwines the geometrical structure of the layers with the mathematical equations for the underlying physics. This project focuses on the interaction of these mathematical objects with the intent of building a foundational theory of layered media. Because the investigation targets the underlying physical phenomena, its results will be applicable across the diverse aspects of society that rely on electromagnetic devices. Students at the graduate and undergraduate levels will be integrated into a dedicated group of researchers from diverse backgrounds and will be given the opportunity to present their work at scientific conferences to help them broaden their scientific horizons and forge professional connections.There is a physical and mathematical object that lies at the center of the investigations in this project, called the Fermi surface, which together with the associated Bloch surface contains core information about electromagnetic waves in crystalline or periodic materials, by encoding a relation between two fundamental physical quantities, energy (or frequency) and momentum (or wavenumber). These objects are analytic or algebraic varieties, depending on the details of the periodic medium, and their structure encodes information about the layering method. The layered materials can be either two-dimensional, such as multi-layer graphene sheets, or three-dimensional, where the layers are internal to the bulk and provide a sort of hidden degree of freedom. Specific objectives are to establish connections between the layering method and the Fermi and Bloch varieties, particularly their reducibility and singularities; to analyze the resonant consequences of reducibility of these varieties; to classify internal symmetries that result in reducibility and the persistence of reducibility under symmetry breaking; to create novel bulk media by homogenization of layered graph models; and to apply these investigations to multi-layer graphene and other physically viable layered materials. The study will invoke the mathematical areas of complex analysis, partial differential equations, algebraic geometry, and spectral theory of differential operators.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.