Antonakakis T, Craster R V, Guenneau S
Department of Mathematics , Imperial College London , London SW7 2AZ, UK ; European Organization for Nuclear Research , CERN 1211 , Geneva 23, Switzerland.
Proc Math Phys Eng Sci. 2013 Apr 8;469(2152):20120533. doi: 10.1098/rspa.2012.0533.
Metamaterial and photonic crystal structures are central to modern optics and are typically created from multiple elementary repeating cells. We demonstrate how one replaces such structures asymptotically by a continuum, and therefore by a set of equations, that captures the behaviour of potentially high-frequency waves propagating through a periodic medium. The high-frequency homogenization that we use recovers the classical homogenization coefficients in the low-frequency long-wavelength limit. The theory is specifically developed in electromagnetics for two-dimensional square lattices where every cell contains an arbitrary hole with Neumann boundary conditions at its surface and implemented numerically for cylinders and split-ring resonators. Illustrative numerical examples include lensing via all-angle negative refraction, as well as omni-directive antenna, endoscope and cloaking effects. We also highlight the importance of choosing the correct Brillouin zone and the potential of missing interesting physical effects depending upon the path chosen.
超材料和光子晶体结构是现代光学的核心，通常由多个基本重复单元构成。我们展示了如何通过一个连续体，进而通过一组方程，渐近地替代此类结构，该连续体和方程组能够捕捉通过周期性介质传播的潜在高频波的行为。我们所采用的高频均匀化方法在低频长波长极限下恢复了经典的均匀化系数。该理论是专门针对二维方形晶格在电磁学中发展起来的，其中每个单元在其表面都包含一个具有诺伊曼边界条件的任意孔洞，并针对圆柱体和分裂环谐振器进行了数值实现。具有代表性的数值示例包括全角度负折射透镜效应，以及全向天线、内窥镜和隐身效应。我们还强调了选择正确布里渊区的重要性，以及根据所选路径可能错过有趣物理效应的可能性。
