Mathematical and Numerical Study of Electromagnetic Waves Interacting with Metamaterials

本计画主要研究波与负折射率超材料相互作用之数学分析与设计。这些负折射率的超材料具有一些奇异的特性数值NIM分析对于设计具有复杂几何形状的纳米结构，建立NIM具有重要意义。由于NIM模型的色散性，它远比自由空间的麦克斯韦方程复杂，以及引入的电和磁极化电流。精确有效地求解它们是相当具有挑战性的，在坚实的数学分析和建模方面做的工作很少。本项目的最终目标是发展一套有效的时域有限元方法，用于模拟波与异向介质的相互作用。异向介质的研究是自2000年以来众多学科的热门课题之一，它将给天线、波导和雷达的设计带来潜在的革命，亚波长分辨率的纳米光刻和成像（用于更好地理解从非侵入性地球物理探测和肿瘤检测获得的图像）、近场控制和操作（用于探测低水平的化学和生物制剂，操纵分子）和隐形斗篷（用于隐形技术）。为负折射率超材料开发强大而有效的算法将使电气工程、材料、光学、物理、纳米技术和生物医学技术等更广泛的领域受益。此外，拟议的项目将帮助PI招募和培训研究生（该项目将支持目前由PI监督的一名女博士生），以追求计算数学的职业生涯。

This project is concerned with the mathematical analysis and design of robust and efficient computational algorithms for modeling wave interactions with negative index metamaterials (NIMs). These negative index metamaterials have some exotic properties (such as near fieldrefocusing) never seen before in those natural electromagnetic materials.The numerical NIM analysis plays an important role for the design of the nano-structures with complicated geometries that establish a NIM.these NIM models are far more complicated than the well-studied Maxwell's equations in free space due to its dispersiveness, and the introduced electric and magnetic polarization currents.Solving them accurately and efficiently is quite challenging and very few work has been done in regards of solid mathematical analysis and modeling. The ultimate goal of this project is to develop an efficient set of time domain finite element methods that are mathematically sound, accurate and fast convergent for simulating wave interactions with metamaterials.Study of metamaterials is one of the hottest topics in many disciplinaries since 2000, with potential revolution in design of antenna, waveguides and radar, nanolithography and imaging at subwavelength resolution (used for better understanding of the images obtained from noninvasive geophysical probing and tumor detection), near field control and manipulation (used for detecting low levels of chemical and biological agents, manipulation of molecules), and invisibility cloak (used for stealth technology). Developing robust and efficient algorithms for negative index metamaterials will benefit broader areas such as electrical engineering, materials, optics, physics, nano-technology, and biomedical technology. Furthermore, the proposed project will help the PI recruit and train graduate students (this project will support a female Ph.D student currently supervised by the PI) to pursue careers in computational mathematics.