Asymptotics for metamaterials and photonic crystals

Abstract

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.

Elastic metamaterials and dynamic homogenization: a review

“…Asymptotic homogenization was also limited in application to describing only the behavior of the fundamental Bloch mode at low frequencies [72,73]. Recently, however, it has been extended to higher frequencies and higher Bloch modes [74][75][76][77] (see also [78]). Additionally, there have been other efforts to bridge the scales for the study of dispersive systems based upon variational formulations [79], micromechanical techniques [80], Fourier transform of the elastodynamic equations [81], and strain projection methods [82].…”

‘Parabolic’ trapped modes and steered Dirac cones in platonic crystals

“…The area of high-frequency homogenization for waves in structured media has been recently developed in the papers by Craster and co-workers [9][10][11][12][13][14]. These studies include a wide range of formulations for waves in lattices as well as structured continuous media.…”

“…In the concluding §7, we also discuss a concept of effective mass (as a scalar and tensorial quantity) and its link to the band structure for waves in the structured material. As a long established and useful idea in solid-state physics [19][20][21], it has now received novel mathematical treatments, mainly owing to the pioneering studies in high frequency homogenization by Craster and colleagues [9][10][11]14] among others [22,23]. Electronic supplementary material to the paper includes animations showing the evolutions of band surfaces with varying aspect ratios of the rectangular lattice.…”

Front Matter

“…The elastic waves carry the heat (phonons). It is worth remarking that, subsequent to pioneering work by Bensoussan, Lions, and Papanicolaou in Chapter 4 of their book [14], there has been a resurgence of interest in Foreword xvii high-frequency homogenization at stationary points in the dispersion diagram, which may be local minima or maxima, or even saddle points [15][16][17][18][19][20][21]. The wave is a modulated Bloch wave and modulation satisfies appropriate effective equations.…”