25 research outputs found
High-frequency homogenization of zero frequency stop band photonic and phononic crystals
We present an accurate methodology for representing the physics of waves, for
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. However such classical homogenization theories break down in
the high-frequency or stop band regime.
Higher frequency phenomena are of significant importance in photonics
(transverse magnetic waves propagating in infinite conducting parallel fibers),
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
Theoretical extensions and applications of high frequency homogenization on photonics, phononics and platonics
In the context of acoustics, single polarization electromagnetism and elastic plates we consider microstructured media that have an
underlying periodic structure and we develop an asymptotic continuum
model that captures the essential microstructural behaviour entirely
in a macroscale setting. The asymptotics are based upon a two-scale
approach and are valid even at high frequencies when the wavelength
and microscale length are of the same order. The general theory is
illustrated via one- and two-dimensional model problems that can have
zero-frequency stop bands that preclude conventional averaging and
homogenization theories. Localized defect modes created by material or shape
variations are also modelled using the theory and compared to
numerical simulations.Open Acces
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, modelling such behaviour is complicated by multiple
scattering of the resulting short waves between the inclusions. Our aim is to
develop an asymptotic theory for modelling systems with arbitrarily-shaped
inclusions located on general Bravais lattices. We then consider the limit of
point-like 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, localisation 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
Asymptotic theory of microstructured surfaces: 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 such as 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 to full numerical simulations. We also investigate non-periodic
situations where a long-scale geometric variation in the structure is
introduced and show that localised defect states emerge which the asymptotic
theory explains
Homogenization Techniques for Periodic Structures
International audienceWe describe a selection of mathematical techniques and results that suggest interesting links between the theory of gratings and the theory of homogenization, including a brief introduction to the latter. We contrast the "classical" homogenization, which is well suited for the description of composites as we have known them since their advent until about a decade ago, and the "non-standard" approaches, high-frequency homogenization and high-contrast homogenization, developing in close relation to the study of photonic crystals and metamaterials, which exhibit properties unseen in conventional composite media, such as negative refraction allowing for super-lensing through a flat heterogeneous lens, and cloaking, which considerably reduces the scattering by finite size objects (invisibility) in certain frequency range. These novel electromagnetic paradigms have renewed the interest of physicists and applied mathematicians alike in the theory of gratings
Gratings: Theory and Numeric Applications
International audienceThe book containes 11 chapters written by an international team of specialist in electromagnetic theory, numerical methods for modelling of light diffraction by periodic structures having one-, two-, or three-dimensional periodicity, and aiming numerous applications in many classical domains like optical engineering, spectroscopy, and optical telecommunications, together with newly born fields such as photonics, plasmonics, photovoltaics, metamaterials studies, cloaking, negative refraction, and super-lensing. Each chapter presents in detail a specific theoretical method aiming to a direct numerical application by university and industrial researchers and engineers
Gratings: Theory and Numeric Applications, Second Revisited Edition
International audienceThe second Edition of the Book contains 13 chapters, written by an international team of specialist in electromagnetic theory, numerical methods for modelling of light diffraction by periodic structures having one-, two-, or three-dimensional periodicity, and aiming numerous applications in many classical domains like optical engineering, spectroscopy, and optical telecommunications, together with newly born fields such as photonics, plasmonics, photovoltaics, metamaterials studies, cloaking, negative refraction, and super-lensing. Each chapter presents in detail a specific theoretical method aiming to a direct numerical application by university and industrial researchers and engineers.In comparison with the First Edition, we have added two more chapters (ch.12 and ch.13), and revised four other chapters (ch.6, ch.7, ch.10, and ch.11
Rayleigh–Bloch waves along elastic diffraction gratings
International audienceRayleigh–Bloch (RB) waves in elasticity, in contrast to those in scalar wave systems, appear to have had little attention. Despite the importance of RB waves in applications, their connections to trapped modes and the ubiquitous nature of diffraction gratings, there has been no investigation of whether such waves occur within elastic diffraction gratings for the in-plane vector elastic system. We identify boundary conditions that support such waves and numerical simulations confirm their presence. An asymptotic technique is also developed to generate effective medium homogenized equations for the grating that allows us to replace the detailed microstructure by a continuum representation. Further numerical simulations confirm that the asymptotic scheme captures the essential features of these waves