From Non-Asymptotic to Nonlocal Homogenization of Electromagnetic Metamaterials

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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.