三维介质异向介质材料共振附近的均匀化和人工磁性
Homogenization near resonances and artificial magnetism in 3D dielectric metamaterials
摘要：
现在已经确定，具有适当缩放的高介电常数的平行电介质纤维的周期性阵列的均匀化可以导致（可能）负的频率相关的有效磁导率。然而，基于二维方法的这一结果仅在线性极化磁场的情况下成立，从而减少了其对无限圆柱形障碍物的应用。在本文中，我们考虑放置在$\mathbb{R}^3$的有界域的介质结构，并执行完整的3D渐近分析。主要成分是一种新的平均方法，用于表征消失期极限内的体有效磁场。我们证明了一个矢量谱问题的周期性细胞，确定微共振和编码的振荡行为的磁场，人工磁性的产生。在宏观层面上，我们推导出一个有效的磁导率张量，我们可以明确作为频率的函数。至于改变符号的磁导率后，我们可以预见，周期性的大块介电夹杂物可能是一个有效的替代非常流行的金属分裂环结构提出的彭德里。
英文摘要：
It is now well established that the homogenization of a periodic array of parallel dielectric fibers with suitably scaled high permittivity can lead to a (possibly) negative frequency-dependent effective permeability. However this result based on a two-dimensional approach holds merely in the case of linearly polarized magnetic fields, reducing thus its applications to infinite cylindrical obstacles. In this paper we consider a dielectric structure placed in a bounded domain of $\mathbb{R}^3$ and perform a full 3D asymptotic analysis. The main ingredient is a new averaging method for characterizing the bulk effective magnetic field in the vanishing-period limit. We evidence a vectorial spectral problem on the periodic cell which determines micro-resonances and encodes the oscillating behavior of the magnetic field from which artificial magnetism arises. At a macroscopic level we deduce an effective permeability tensor that we can be make explicit as a function of the frequency. As far as sign-changing permeability are sought after, we may foresee that periodic bulk dielectric inclusions could be an efficient alternative to the very popular metallic split-ring structure proposed by Pendry.
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