From Non-Asymptotic to Nonlocal Homogenization of Electromagnetic Metamaterials

在过去的二十年里，人们对超材料（MM）的兴趣激增，超材料是一种精心设计的人工结构，用于控制波的传播，并产生天然材料无法提供的物理效果。这些不寻常的物理现象包括完美透镜、负折射和隐形。根据谷歌Scholar的数据，已有超过9万篇研究论文、书籍章节和书籍致力于MM的科学和应用。MM的分析和设计依赖于它们的大尺度（“宏观”）参数和获得这些参数所需的有效介质理论（“均质化”）。典型MM的行为是非局部的：也就是说，它们在空间中给定点的响应可能取决于不同点的激励。然而，非局部状态的数学和数值均匀化方法是稀缺和不足的。该项目将消除这一阻碍MM进一步发展的关键障碍。现有的均质化理论，其中一些可以追溯到19世纪的经典物理学，将因此得到显著扩展。广泛的技术影响将在MM领域，不一定是电磁。非局部均质化将使我们能够明确地确定哪些超材料的应用实际上是可行的，并优化超材料器件的性能。更广泛的影响也将出现在非局部介质科学中，例如溶剂中生物分子的非局部静电学，这在蛋白质建模和药物发现中有应用。本研究的主要目的是通过建立非局域双尺度均匀化理论来推进电磁超材料的分析、模拟和应用。该理论涉及电磁场在粗尺度和细尺度上的Trefftz近似，以及一组明智选择的自由度。这不仅是对传统理论的充实和必要的延伸，也是对近年来发展起来的非渐近局域理论的延伸。对于典型的例子，如层状介质或光子晶体，PI将在透射/反射系数方面显示出一致的数量级精度提高。本研究的智力价值在于发展了非局部均匀化的新范式，提出了与之相关的新计算方法，并将新方法应用于电磁超材料。

The last two decades have witnessed an explosion of interest in metamaterials (MM) -- artificial structures judiciously designed to control wave propagation and to produce physical effects not available in natural materials. These unusual physical phenomena include perfect lensing, negative refraction, and cloaking. According to Google Scholar, over 90,000 research papers, book chapters and books have been devoted to the science and applications of MM. Analysis and design of MM relies on their large-scale ("macroscopic") parameters and on effective medium theories ("homogenization") needed to obtain these parameters. The behavior of typical MM is nonlocal: that is, their response at a given point in space may depend on the excitation at a different point. Yet mathematical and numerical homogenization methods for nonlocal regimes are scarce and insufficient. This project will remove this critical impediment to further progress in MM. Existing homogenization theories, some of which date back to classical physics of the 19th century, will thereby be significantly extended. The broad technical impact will be in the field of MM, not necessarily electromagnetic. Nonlocal homogenization will allow us to establish unambiguously which applications of metamaterials are practically feasible and to optimize the performance of metamaterial devices. The broader impact will also be in the science of nonlocal media, for example nonlocal electrostatics of biomolecules in solvents, which has applications in protein modeling and drug discovery.The chief objective of this research is to advance the analysis, simulation and applications of electromagnetic metamaterials by developing a nonlocal two-scale homogenization theory. This theory involves Trefftz approximations of the electromagnetic fields on both coarse and fine scales, and a judiciously chosen set of degrees of freedom. This is a substantial, and necessary, extension not only of traditional theories, but also of the non-asymptotic but local theory the PI has developed in recent years. For canonical examples such as layered media or photonic crystals, the PI will demonstrate a consistent order-of-magnitude accuracy improvement in the transmission/reflection coefficients. The intellectual merit of the this research is in the development of a new paradigm of nonlocal homogenization, of new computational methods related to it, and in the application of new methodology to electromagnetic metamaterials.