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--这种人造结构经过精心设计，可以控制波的传播，并产生天然材料所不具备的物理效果。这些不寻常的物理现象包括完美透镜、负折射和隐身。根据Google 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规模(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的行为是非局域的：也就是说，他们在空间中给定点的响应可能取决于在不同点的激发。然而，针对非局部制度的数学和数值均化方法很少，也不够充分。这个项目将消除这一阻碍MM进一步发展的关键障碍。现有的均匀化理论，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。广泛的技术影响将出现在MM领域，而不一定是电磁领域。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是实际可行的，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的科学上，例如生物分子在溶剂中的非局部静电学，这在蛋白质建模和药物发现中都有应用。本研究的主要目的是通过发展非局部两尺度均匀化理论来促进电磁超材料的分析、模拟和应用。这一理论包括电磁场在粗略和精细尺度上的特雷夫兹近似，以及一组明智地选择的自由度。这不仅是对传统理论的实质性和必要的扩展，而且也是对PI近年来发展的非渐近但局部理论的扩展。对于典型的例子，如层状介质或光子晶体，PI将在传输/反射系数方面显示出一致的数量级精度改进。这项研究的智力价值在于发展了一种新的非局部均匀化范式，发展了与之相关的新的计算方法，并将新的方法学应用于电磁超材料。

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.