Tryfon Antonakakis

Asymptotics for metamaterials and photonic crystals

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.

High-frequency homogenization of zero-frequency stop band photonic and phononic crystals

We present an accurate methodology for representing the physics of waves, in periodic structures, through effective properties for a replacement bulk medium: this is valid even for media with zero-frequency stop bands and where high-frequency phenomena dominate. Since the work of Lord Rayleigh in 1892, low-frequency (or quasi-static) behaviour has been neatly encapsulated in effective anisotropic media; the various parameters come from asymptotic analysis relying upon the ratio of the array pitch to the wavelength being sufficiently small. However, such classical homogenization theories break down in the high-frequency or stop band regime whereby the wavelength to pitch ratio is of order one. Furthermore, arrays of inclusions with Dirichlet data lead to a zero-frequency stop band, with the salient consequence that classical homogenization is invalid. Higher-frequency phenomena are of significant importance in photonics (transverse magnetic waves propagating in infinite conducting parallel fibres), phononics (anti-plane shear waves propagating in isotropic elastic materials with inclusions) and platonics (flexural waves propagating in thin-elastic plates with holes). Fortunately, the recently proposed high-frequency homogenization (HFH) theory is only constrained by the knowledge of standing waves in order to asymptotically reconstruct dispersion curves and associated Floquet–Bloch eigenfields: it is capable of accurately representing zero-frequency stop band structures. The homogenized equations are partial differential equations with a dispersive anisotropic homogenized tensor that characterizes the effective medium. We apply HFH to metamaterials, exploiting the subtle features of Bloch dispersion curves such as Dirac-like cones, as well as zero and negative group velocity near stop bands in order to achieve exciting physical phenomena such as cloaking, lensing and endoscope effects. These are simulated numerically using finite elements and compared to predictions from HFH. An extension of HFH to periodic supercells enabling complete reconstruction of dispersion curves through an unfolding technique is also introduced.

Wave Mechanics in Media Pinned at Bravais Lattice Points

The propagation of waves through microstructured media with periodically arranged inclusions has applications in many areas of physics and engineering, stretching from photonic crystals through to seismic metamaterials. In the high-frequency regime, modeling such behavior is complicated by multiple scattering of the resulting short waves between the inclusions. Our aim is to develop an asymptotic theory for modeling systems with arbitrarily shaped inclusions located on general Bravais lattices. We then consider the limit of pointlike inclusions, the advantage being that exact solutions can be obtained using Fourier methods, and go on to derive effective medium equations using asymptotic analysis. This approach allows us to explore the underlying reasons for dynamic anisotropy, localization of waves, and other properties typical of such systems, and in particular their dependence upon geometry. Solutions of the effective medium equations are compared with the exact solutions, shedding further light on the underlying physics. We focus on examples that exhibit dynamic anisotropy as these demonstrate the capability of the asymptotic theory to pick up detailed qualitative and quantitative features.

Metamaterial-like transformed urbanism

Viewed from the sky, the urban fabric pattern appears similar to the geometry of structured devices called metamaterials; these were developed by Physicists to interact with waves that have wavelengths in the range from nanometers to meters (from electromagnetic to seismic metamaterials). Visionary research in the late 1980s based on the interaction of big cities with seismic signals and more recent studies on seismic metamaterials, made of holes or vertical inclusions in the soil, has generated interest in exploring the multiple interaction effects of seismic waves in the ground and the local resonances of both buried pillars and buildings. Here, we use techniques from transformational optics and theoretically validate, by numerical experiments, that a district of buildings could be considered as a set of above-ground resonators, purely elastic, interacting with an incident seismic signal. We hope that our proposal will contribute to all theoretical and experimental efforts in design of cities of the future, from a metamaterial standpoint.

Unveiling Extreme Anisotropy in Elastic Structured Media

Periodic structures can be engineered to exhibit unique properties observed at symmetry points, such as zero group velocity, Dirac cones, and saddle points; identifying these and the nature of the associated modes from a direct reading of the dispersion surfaces is not straightforward, especially in three dimensions or at high frequencies when several dispersion surfaces fold back in the Brillouin zone. A recently proposed asymptotic high-frequency homogenization theory is applied to a challenging time-domain experiment with elastic waves in a pinned metallic plate. The prediction of a narrow high-frequency spectral region where the effective medium tensor dramatically switches from positive definite to indefinite is confirmed experimentally; a small frequency shift of the pulse carrier results in two distinct types of highly anisotropic modes. The underlying effective equation mirrors this behavior with a change in form from elliptic to hyperbolic exemplifying the high degree of wave control available and the importance of a simple and effective predictive model.

An asymptotic theory for waves guided by diffraction gratings or along microstructured surfaces

An effective surface equation, that encapsulates the detail of a microstructure, is developed to model microstructured surfaces. The equations deduced accurately reproduce a key feature of surface wave phenomena, created by periodic geometry, that are commonly called Rayleigh–Bloch waves, but which also go under other names, for example, spoof surface plasmon polaritons in photonics. Several illustrative examples are considered and it is shown that the theory extends to similar waves that propagate along gratings. Line source excitation is considered, and an implicit long-scale wavelength is identified and compared with full numerical simulations. We also investigate non-periodic situations where a long-scale geometrical variation in the structure is introduced and show that localized defect states emerge which the asymptotic theory explains.