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A general process is proposed to experimentally design anisotropic inhomogeneous metamaterials obtained through a change of coordinate in the Helmholtz equation. The method is applied to the case of a cylindrical transformation that allows to perform cloaking. To approximate such complex metamaterials we apply results of the theory of homogenization and combine them with a genetic algorithm. To illustrate the power of our approach, we design three types of cloaks composed of isotropic concentric layers structured with three types of perforations: curved rectangles, split rings and crosses. These cloaks have parameters compatible with existing technology and they mimic the behavior of the transformed material. Numerical simulations have been performed to qualitatively and quantitatively study the cloaking efficiency of these metamaterials., Comment: 34 pages, 12 figures, 5 tables

The discovery of photonic crystals thirty years ago, in conjunction with research advances in plasmonics and metamaterials, has inspired the concept of decameter scale metasurfaces, coined seismic metamaterials, for an enhanced control of surface (Love and Rayleigh) and bulk (shear and pressure) elastodynamic waves. The powerful mathematical tools of coordinate transforms, effective medium and Floquet-Bloch theories, which have revolutionized nanophotonics, can be translated in the language of civil engineering and geophysics. Experiments on seismic metamaterials made of buried elements in the soil demonstrate that the fore mentioned tools make possible a novel description of complex phenomena of soil-structure interaction during a seismic disturbance. But the concepts are already moving to more futuristic concepts and the same notions developed for structured soils are now used to examine the effects of buildings viewed as above surface resonators in megastructures such as metacities. But this perspective of the future should not make us forget the heritage of the ancient peoples. Indeed, we finally point out the striking similarity between an invisibility cloak design and the architecture of some ancient megastructures as antique Gallo-Roman theaters and amphitheatres., Comment: 13 pages, 16 figures

We outline some recent research advances on the control of elastic waves in thin and thick plates, that have occurred since the large scale experiment [Phys. Rev. Lett. 112, 133901, 2014] that demonstrated significant interaction of surface seismic waves with holes structuring sedimentary soils at the meter scale. We further investigate the seismic wave trajectories in soils structured with buildings. A significant substitution of soils by inclusions, acting as foundations, raises the question of the effective dynamic properties of these structured soils. Buildings, in the case of perfect elastic conditions for both soil and buildings, are shown to interact and strongly influence elastic surface waves; such site-city seismic interactions were pointed out in [Bulletin of Seismological Society of America 92, 794-811, 2002], and we investigate a variety of scenarios to illustrate the variety of behaviours possible., Comment: 15 pages, 9 figures

We create a passive wave splitter, created purely by geometry, to engineer three-way beam splitting in electromagnetism in transverse electric polarisation. We do so by considering arrangements of Indium Phosphide dielectric pillars in air, in particular we place several inclusions within a cell that is then extended periodically upon a square lattice. Hexagonal lattice structures more commonly used in topological valleytronics but, as we discuss, three-way splitting is only possible using a square, or rectangular, lattice. To achieve splitting and transport around a sharp bend we use accidental, and not symmetry-induced, Dirac cones. Within each cell pillars are either arranged around a triangle or square; we demonstrate the mechanism of splitting and why it does not occur for one of the cases. The theory is developed and full scattering simulations demonstrate the effectiveness of the proposed designs.

We combine theories of scattering for linearized water waves and flexural waves in thin plates to characterize and achieve control of water wave scattering using floating plates. This requires manipulating a sixth-order partial differential equation with appropriate boundary conditions of the velocity potential. Making use of multipole expansions, we reduce the scattering problem to a linear algebraic system. The response of a floating plate in the quasistatic limit simplifies, considering a distinct behavior for water and flexural waves. Unlike similar studies in electromagnetics and acoustics, scattering of gravity-flexural waves is dominated by the zeroth-order multipole term and this results in non-vanishing scattering cross-section also in the zero-frequency limit. Potential applications lie in floating structures manipulating ocean waves., Comment: 19 pages, 4 figures

Recent static experiments on twist effects in chiral three-dimensional mechanical metamaterials have been discussed in the context of micropolar Eringen continuum mechanics, which is a generalization of Cauchy elasticity. For cubic symmetry, Eringen elasticity comprises nine additional parameters with respect to Cauchy elasticity, of which three directly influence chiral effects. Here, we discuss the behavior of the static case of an alternative generalization of Cauchy elasticity, the Milton-Briane-Willis equations. We show that in the homogeneous static cubic case only one additional parameter with respect to Cauchy elasticity results, which directly influences chiral effects. We show that the Milton-Briane-Willis equations qualitatively describe the experimentally observed chiral twist effects, too. We connect the behavior to a characteristic length scale.

We propose an alternative approach called backward transformation for the design of platonic cloaks, without resorting to in-plane body forces and pre-stresses, which can lead to unphysical features. It is shown that the Kirchhoff-Love equation for an anisotropic heterogeneous plate is actually form invariant for a class of transformations with a vanishing Hessian. This formalism is detailed and numerically validated with three-dimensional simulations in the time domain. Results published in the literature fall in the class of transformations proposed here., Comment: 5 pages, 5 figures

Structured soils with regular meshes of metric size holes implemented in first ten meters of the ground have been theoretically and experimentally tested under seismic disturbance this last decade. Structured soils with rigid inclusions embedded in a substratum have been also recently developed. The influence of these inclusions in the ground can be characterized in different ways: redistribution of energy within the network with focusing effects for seismic metamaterials, wave reflection, frequency filtering, reduction of the amplitude of seismic signal energy, etc. Here we first provide some time-domain analysis of the flat lens effect in conjunction with some form of external cloaking of Rayleigh waves and then we experimentally show the effect of a finite mesh of cylindrical holes on the ellipticity of the surface Rayleigh waves at the level of the Earth's surface. Orbital diagrams in time domain are drawn for the surface particle's velocity in vertical (x,z) and horizontal (x,y) planes. These results enable us to observe that the mesh of holes locally creates a tilt of the axes of the ellipse and changes the direction of particle movement. Interestingly, changes of Rayleigh waves ellipticity can be interpreted as changes of an effective Poisson ratio. However, the proximity of the source is also important to explain the shape of the ellipses. We analyze these observations in terms of wave mode conversions inside the mesh and we propose to broaden the discussion on the complexity of seismic wave phenomena in structured soils such as soils foundations and on the coupling effects specific to the soil-structure interaction., Comment: 7 pages, 5 figures

Photonic crystals are periodic structures which prevent light propagation along one or more directions in certain frequency intervals. Their band spectrum is usually analyzed using Floquet-Bloch decomposition. This spectrum is located on the real axis, and it enters the complex plane when absorption and dispersion is considered in the dielectric permittivity of material constituents. Here, we review fundamental definition and properties of dispersion law and group velocity in photonic crystals and we illustrate them with numerical examples., Comment: book chapter

We make precise some results on the cloaking of displacement fields in linear elasticity. In the spirit of transformation media theory, the transformed governing equations in Cosserat and Willis frameworks are shown to be equivalent to certain high contrast small defect problems for the usual Navier equations. We discuss near-cloaking for elasticity systems via a regularized transform and perform numerical experiments to illustrate our near-cloaking results. We also study the sharpness of the estimates from [H. Ammari, H. Kang, K. Kim and H. Lee, J. Diff. Eq. 254, 4446-4464 (2013)], wherein the convergence of the solutions to the transmission problems is investigated, when the Lam\'e parameters in the inclusion tend to extreme values. Both soft and hard inclusion limits are studied and we also touch upon the finite frequency case. Finally, we propose an approximate isotropic cloak algorithm for a symmetrized Cosserat cloak., Comment: 7 figures, 7 tables; Note that the earlier version of this preprint was titled 'Some results in near-cloaking for elasticity systems'. This new version of the manuscript has also seen some major upgrade. We have added a new section on 'Cloaking parameters and isotropic approximation'. In there, we propose an approximate isotropic cloak algorithm for a symmetrized Cosserat cloak

We analyse the propagation of pressure waves within a fluid filled with a three-dimensional array of rigid coated spheres (shells). We first draw band diagrams for corresponding Floquet-Bloch waves. We then dig a channel terminated by a cavity within each rigid shell and observe the appearance of a low frequency stop band. The underlying mechanism is that each holey shell now acts as a Helmholtz resonator supporting a low frequency localized mode: Upon resonance, pressure waves propagate with fast oscillations in the thin water channel drilled in each shell and are localized in each fluid filled inner cavity. The array of fluid filled shells is approximated by a simple mechanical model of springs and masses allowing for asymptotic estimates of the low frequency stop band. We finally propose a realistic design of periodic macrocell with a large defect surrounded by 26 resonators connected by thin straight rigid wires, which supports a localized mode in the low frequency stop band., Comment: 7 pages, 12 figures

Following the advent of electromagnetic metamaterials at the turn of the century, researchers working in other areas of wave physics have translated concepts of electromagnetic metamaterials to acoustics, elastodynamics, as well as to heat, mass and light diffusion processes. In elastodynamics, seismic metamaterials have emerged in the last decade for soft soils structured at the meter scale, and have been tested thanks to full-scale experiments on holey soils five years ago. Born in the soil, seismic metamaterials grow simultaneously on the field of tuned-resonators buried in the soil, around building's foundations or near the soil-structure's interface, and on the field of above-surface resonators. In this perspective article, we quickly recall some research advances made in all these types of seismic metamaterials and we further dress an inventory of which material parameters can be achieved and which cannot, notably from the effective medium theory perspective. We finally envision perspectives on future developments of large scale auxetic metamaterials for building's foundations, forests of trees for seismic protection and metamaterial-like transformed urbanism at the city scale., Comment: 12 pages, 11 figures

We inspect the propagation of shear polarized surface waves akin to Love waves through a forest of trees of same height atop a guiding layer on a soil substrate. We discover that the foliage of trees { brings a radical change in} the nature of the dispersion relation of these surface waves, which behave like spoof plasmons in the limit of a vanishing guiding layer, and like Love waves in the limit of trees with a vanishing height. When we consider a forest with trees of increasing or decreasing height, this hybrid "Spoof Love" wave is either reflected backwards or converted into a downward propagating bulk wave. An asymptotic analysis shows the forest behaves like an anisotropic wedge with effective boundary conditions., Comment: 5 pages, 6 figures

This paper explores the concept of near-cloaking in the context of time-dependent heat propagation. We show that after the lapse of a certain threshold time instance, the boundary measurements for the homogeneous heat equation are close to the cloaked heat problem in a certain Sobolev space norm irrespective of the density-conductivity pair in the cloaked region. A regularised transformation media theory is employed to arrive at our results. Our proof relies on the study of the long time behaviour of solutions to the parabolic problems with high contrast in density and conductivity coefficients. It further relies on the study of boundary measurement estimates in the presence of small defects in the context of steady conduction problem. We then present some numerical examples to illustrate our theoretical results., Comment: 31 pages, 7 figures, 1 table

In [AIP Advances 6, 121707 (2016)], a soil structured with concrete columns distributed within two specially designed seismic cloaks thanks to a combination of transformational elastodynamics and effective medium theory was shown to detour Rayleigh waves of frequencies lower than 10 Hz around a cylindrical region. The aforementioned studies motivate our exploration of interactions of surface elastic waves propagating in a thick plate (with soil parameters) structured with concrete pillars above it. Pillars are 40 m in height and the plate is 100 m in thickness, so that typical frequencies under study are below 1 Hz, a frequency range of particular interest in earthquake engineering. We demonstrate that three seismic cloaks allow for an unprecedented flow of elastodynamic energy. These designs are achieved by first computing ideal cloaks' parameters deduced from a geometric transform in the Navier equations that leads to almost isotropic and symmetric elasticity (4th order) and density (2nd order) tensors. To do this we extend the theory of Non-Euclidean cloaking for light as proposed by the theoretical physicists Leonhardt and Tyc. In a second step, ideal heterogeneous nearly isotropic cloak's parameters are approximated by averaging elastic properties of sets of pillars placed at the nodes of a bipolar coordinate grid, which is an essential ingredient in our Non-Euclidean cloaking theory for elastodynamic waves. Cloaking effects are studied for a clamped obstacle (reduction of the disturbance of the wave wavefront and its amplitude behind a clamped obstacle). Protection is achieved through reduction of the wave amplitude within the center of the cloak.These results represent a first step towards designs of Non-Euclidean seismic cloaks for surface (Rayleigh and Love) waves propagating in semi-infinite elastic media structured with pillars., Comment: Latex, 10pages, 4 figures. Pacs: 41.20.Jb,42.25.Bs,42.70.Qs,43.20.Bi,43.25.Gf

It is well known in metamaterials that local resonance and hybridization phenomena dramatically influence the shape of dispersion curves; the metasurface created by a cluster of resonators, subwavelength rods, atop an elastic surface being an exemplar with these features. On this metasurface, band-gaps, slow or fast waves, negative refraction and dynamic anisotropy can all be observed by exploring frequencies and wavenumbers from the Floquet-Bloch problem and by using the Brillouin zone. These extreme characteristics, when appropriately engineered, can be used to design and control the propagation of elastic waves along the metasurface. For the exemplar we consider, two parameters are easily tuned: rod height and cluster periodicity. The height is directly related to the band-gap frequency, and hence to the slow and fast waves, while the periodicity is related to the appearance of dynamic anisotropy. Playing with these two parameters generates a gallery of metasurface designs to control the propagation of both flexural waves in plates and surface Rayleigh waves for half-spaces. Scalability with respect to the frequency and wavelength of the governing physical laws allows the application of these concepts in very different fields and over a wide range of lengthscales., Comment: Paper currently submitted to Frontiers in Mechanics of Materials

The present study investigates a way to design dykes which can filter the wavelengths of ocean surface waves. This offers the possibility to achieve a structure that can attenuate waves associated with storm swell, without affecting coastline in other conditions. Our approach is based on low frequency resonances in metamaterials combined with Bragg frequencies for which waves cannot propagate in periodic lattices.

Recent years have heralded the introduction of metasurfaces that advantageously combine the vision of sub- wavelength wave manipulation, with the design, fabrication and size advantages associated with surface excitation. An important topic within metasurfaces is the tailored rainbow trapping and selective spatial frequency separation of electromagnetic and acoustic waves using graded metasurfaces. This frequency dependent trapping and spatial frequency segregation has implications for energy concentrators and associated energy harvesting, sensing and wave filtering techniques. Different demonstrations of acoustic and electromagnetic rainbow devices have been performed, however not for deep elastic substrates that support both shear and compressional waves, together with surface Rayleigh waves; these allow not only for Rayleigh wave rainbow effects to exist but also for mode conversion from surface into bulk waves. We design, and test experimentally, a device creating not only elastic Rayleigh wave rainbow trapping, but also selective mode conversion of surface Rayleigh waves to shear waves., Comment: Article currently submitted for review. Comments are welcome!

We first review classical results on cloaking and mirage effects for electromagnetic waves. We then show that transformation optics allows the masking of objects or produces mirages in diffusive regimes. In order to achieve this, we consider the equation for diffusive photon density in transformed coordinates, which is valid for diffusive light in scattering media. More precisely, generalizing transformations for star domains introduced in [Diatta and Guenneau, J. Opt. 13, 024012, 2011] for matter waves, we numerically demonstrate that infinite conducting objects of different shapes scatter diffusive light in exactly the same way. We also propose a design of external light-diffusion cloak with spatially varying sign-shifting parameters that hides a finite size scatterer outside the cloak. We next analyse non-physical parameter in the transformed Fick's equation derived in [Guenneau and Puvirajesinghe, R. Soc. Interface 10, 20130106, 2013], and propose to use a non-linear transform that overcomes this problem. We finally investigate other form invariant transformed diffusion-like equations in the time domain, and touch upon conformal mappings and non-Euclidean cloaking applied to diffusion processes., Comment: 42 pages, Latex, 14 figures. V2: Major changes : some formulas corrected, some extra cases added, overall length extended from 21 pages (V1) to 42 pages (present version V2). The last version will appear at Journal of Optics

We introduce the concept of transformation crystallography which consists of the application of geometric transforms to periodic structures. We consider motifs with three-fold, four-fold and six-fold symmetries according to the crystallographic restriction theorem. Furthermore, we define motifs with five-fold symmetry such as quasi-crystals generated by a cut-and-projection method. We analyze elastic wave propagation in the transformed crystals and (Penrose-type) quasi-crystals with the finite difference time domain freeware SimSonic. We consider geometric transforms underpinning the design of seismic cloaks with square, circular, elliptical and peanut shapes in the context of triangular, square and honeycomb crystals. Interestingly, the use of morphing techniques leads to the design of cloaks with interpolated geometries reminiscent of Victor Vasarely's artwork. Employing the case of transformed graphene-like (honeycomb) structures allows one to draw useful analogies between large-scale seismic metamaterials such as soils structured with columns of concrete or grout with soil and nanoscale biochemical metamaterials. We further point out similarities between cloaks for elastodynamic and hydrodynamic waves and cloaks for diffusion (heat or mass) diffusion processes, notably with respect to invisibility and protection. Experimental data extracted from field test analysis of soil structured with boreholes demonstrates the application of bio-inspired seismic metamaterials. We conclude that these novel materials hold strong applications in biophysics and geophysics., Comment: 25 pages, 14 figures

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 homogenisation 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 behaviour 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., Comment: 5 pages, 4 figures

We explore interactions of elastic waves propagating in plates (with soil parameters) structured with concrete pillars buried in the soil. Pillars are 2 m in diameter, 30 m in depth and the plate is 50 m in thickness. We study the frequency range 5 to 10 Hz, for which Rayleigh wave wavelengths are smaller than the plate thickness. This frequency range is compatible with frequency ranges of particular interest in earthquake engineering. It is demonstrated in this paper that two seismic cloaks' configurations allow for an unprecedented flow of elastodynamic energy associated with Rayleigh surface waves. The first cloak design is inspired by some approximation of ideal cloaks' parameters within the framework of thin plate theory. The second, more accomplished but more involved, cloak design is deduced from a geometric transform in the full Navier equations that preserves the symmetry of the elasticity tensor but leads to Willis' equations, well approximated by a homogenization procedure, as corroborated by numerical simulations. The two cloaks's designs are strikingly different, and the superior efficiency of the second type of cloak emphasizes the necessity for rigor in transposition of existing cloaks's designs in thin plates to the geophysics setting. Importantly, we focus our attention on geometric transforms applied to thick plates, which is an intermediate case between thin plates and semi-infinite media, not studied previously. Cloaking efficiency (reduction of the disturbance of the wave wavefront and its amplitude behind an obstacle) and protection (reduction of the wave amplitude within the center of the cloak) are studied for ideal and approximated cloaks' parameters. These results represent a preliminary step towards designs of seismic cloaks for surface Rayleigh waves propagating in sedimentary soils structured with concrete pillars., Comment: V2: Major changes. Quantitative study carried out. New figures. Investigation of elastic trapped modes and resonances. Latex, 13 pages, 8 pages. Pacs: 41.20.Jb,42.25.Bs,42.70.Qs,43.20.Bi,43.25.Gf. The last version will appear at AIP Advances

In electromagnetism, acoustics, and quantum mechanics, scattering problems can routinely be solved numerically by virtue of perfectly matched layers (PMLs) at simulation domain boundaries. Unfortunately, the same has not been possible for general elastodynamic wave problems in continuum mechanics. In this paper, we introduce a corresponding scattered-field formulation for the Navier equation. We derive PMLs based on complex-valued coordinate transformations leading to Cosserat elasticity-tensor distributions not obeying the minor symmetries. These layers are shown to work in two dimensions, for all polarizations, and all directions. By adaptative choice of the decay length, the deep subwavelength PMLs can be used all the way to the quasi-static regime. As demanding examples, we study the effectiveness of cylindrical elastodynamic cloaks of the Cosserat type and approximations thereof.

We report on a new concept of cloaking objects in diffusive light regime using the paradigm of the scattering cancellation and mantle cloaking techniques. We show numerically that an object can be made completely invisible to diffusive photon density waves, by tailoring the diffusivity constant of the spherical shell enclosing the object. This means that photons' flow outside the object and the cloak made of these spherical shells behaves as if the object were not present. Diffusive light invisibility may open new vistas in hiding hot spots in infrared thermography or tissue imaging., Comment: 16 pages, 5 figures

A prerequisite for achieving seismic invisibility is to demonstrate the ability of civil engineers to control seismic waves with artificially structured soils. We carry out large-scale field tests with a structured soil made of a grid consisting of cylindrical and vertical holes in the ground and a low frequency artificial source (< 10 Hz). This allows the identification of a distribution of energy inside the grid, which can be interpreted as the consequence of an effective negative refraction index. Such a flat lens reminiscent of what Veselago and Pendry envisioned for light opens avenues in seismic metamaterials to counteract the most devastating components of seismic signals.

We investigate the elastic stop band properties of a theoretical cubic array of iron spheres connected to a bulk of concrete via iron or rubber ligaments. Each sphere can move freely within a surrounding air cavity, but ligaments couple it to the bulk and further facilitate bending and rotational motions. Associated low frequency local resonances are well predicted by an asymptotic formula. We find complete stop bands (for all wave-polarizations) in the frequency range $[16,21]$ Hertz (resp. $[6,11]$ Hertz) for $7.4$-meter (resp. $0.74$-meter) diameter iron spheres with a $10$-meter (resp. $1$-meter) center-to-center spacing, when they are connected to concrete via steel (resp. rubber) ligaments. The scattering problem shows that only bending modes are responsible for damping and that rotational modes are totally overwritten by bending modes. Regarding seismic applications, we further consider soil as a bulk medium, in which case the relative bandwidth of the low frequency stop band can be enlarged through ligaments of different sizes that allow for well separated bending and rotational modes. We finally achieve some damping of elastodynamic waves from $8$ to $49$ Hertz (relative stop band of $143$ percent) for iron spheres $0.74$-meter in diameter that are connected to soil with six rubber ligaments of optimized shapes. These results represent a preliminary step in the design of seismic shields placed around, or underneath, the foundations of large civil infrastructures.

We analyse wave propagation in two-dimensional bianisotropic media with the Finite Element Method (FEM). We start from the Maxwell-Tellegen's equations in bianisotropic media, and derive some system of coupled Partial Difference Equations (PDEs) for longitudinal electric and magnetic field components. Perfectly Matched Layers (PMLs) are discussed to model such unbounded media. We implement these PDEs and PMLs in a finite element software. We apply transformation optics in order to design some bianisotropic media with interesting functionalities, such as cloaks, concentrators and rotators. We propose a design of metamaterial with concentric layers made of homogeneous media with isotropic permittivity, permeability and magneto-electric parameters that mimic the required effective anisotropic tensors of a bianisotropic cloak in the long wavelength limit (homogenization approach). Our numerical results show that well-known metamaterials can be transposed to bianisotropic media., Comment: 26 pages, 8 figures

We propose that wave propagation through a class of mechanical metamaterials opens unprecedented avenues in seismic wave protection based on spectral properties of auxetic-like metamaterials. The elastic parameters of these metamaterials like the bulk and shear moduli, the mass density, and even the Poisson ratio, can exhibit negative values in elastic stop bands. We show here that the propagation of seismic waves with frequencies ranging from 1Hz to 40Hz can be influenced by a decameter scale version of auxetic-like metamaterials buried in the soil, with the combined effects of impedance mismatch, local resonances and Bragg stop bands. More precisely, we numerically examine and illustrate the markedly different behaviors between the propagation of seismic waves through a homogeneous isotropic elastic medium (concrete) and an auxetic-like metamaterial plate consisting of 64 cells (40mx40mx40m), utilized here as a foundation of a building one would like to protect from seismic site effects. This novel class of seismic metamaterials opens band gaps at frequencies compatible with seismic waves when they are designed appropriately, what makes them interesting candidates for seismic isolation structures.

We propose solar-shaped thermal concentrators designed with orthoradial layers and obtained in practice through the homogenization of an ideal thermal concentrator. Considering the spectral regime of the heat equation, we quantitatively evaluate at different pulsations the effectiveness of the homogenized concentrators by comparing the thermal flux existing in an ideal concentrator and the thermal flux in an homogenized concentrator. Dependence on the pulsation is shown to be negligible and plotting the effectiveness of the homogenized concentrators as a function of the number of orthoradial layers $N$, we determine the number of layers needed to achieve a certain effectiveness. Significantly high numbers $N$(ranging from a hundred to tens of thousands layers) are found highlighting the fact that achieving high effectiveness demands a high level of engineering of the homogenized concentrator., Comment: 12 pages, 8 figures

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.

Numerical simulations shed light on control of shear elastic wave propagation in plates structured with inertial resonators. The structural element is composed of a heavy core connected to the main freestanding plate through tiny ligaments. It is shown that such a configuration exhibits a complete band gap in the low frequency regime. As a byproduct, we further describe the asymmetric twisting vibration of a single scatterer via modal analysis, dispersion and transmission loss. This might pave the way to new functionalities such as focusing and self-collimation in elastic plates.

A new analogy between optical propagation and heat diffusion in heterogeneous anisotropic media has been proposed recently [S. Guenneau, C. Amra, and D. Veynante, Optics Express Vol. 20, 8207-8218 (2012)]. A detailed derivation of this unconventional correspondence is presented and developed. In time harmonic regime, all thermal parameters are related to optical ones in artificial metallic media, thus making possible to use numerical codes developed for optics. Then the optical admittance formalism is extended to heat conduction in multilayered structures. The concepts of planar micro-cavities, diffraction gratings, and planar transformation optics for heat conduction are addressed. Results and limitations of the analogy are emphasized., Comment: 20 pages, 11 figures

In this paper, we bring to the awareness of the scientific community and civil engineers, an important fact: the possible lack of wave protection of transformational elastic cloaks. To do so, we propose spherical cloaks described by a non-singular asymmetric elasticity tensor depending upon a small parameter $\eta,$ that defines the softness of a region one would like to conceal from elastodynamic waves. By varying $\eta$, we generate a class of soft spheres dressed by elastodynamic cloaks, which are shown to considerably reduce the soft spheres' scattering. Importantly, such cloaks also provide some wave protection except for a countable set of frequencies, for which some large elastic field enhancement (resonance peaks) can be observed within the cloaked soft spheres, hence entailing a possible lack of wave protection. We further present an investigation of trapped modes in elasticity via which we supply a good approximation of such Mie-type resonances by some transcendental equation. Next, after a detailed presentation of spherical elastodynamic PML of Cosserat type, we introduce a novel generation of cloaks, the mixed cloaks, as a solution to the lack of wave protection in elastodynamic cloaking. Indeed, mixed cloaks achieve both invisibility cloaking and protection throughout a large range of frequencies. We think, mixed cloaks will soon be generalized to other areas of physics and engineering and will in particular foster studies in experiments., Comment: V2: major changes. More details on the study of trapped modes in elasticity. Mixed cloaks introduced. Latex files, 27 pages, 14 figures. The last version will appear at Journal of Physics D: Applied Physics. Pacs:41.20.Jb,42.25.Bs,42.70.Qs,43.20.Bi,43.25.Gf. arXiv admin note: text overlap with arXiv:1403.1847

We analyse basic thermal cloaks designed via different geometric transforms applied to thermal cloaking. We evaluate quantitatively the efficiency of these heterogeneous anisotropic thermal cloaks through the calculation of the standard deviation of the isotherms. The study addresses the frequency regime and we point out the cloak's spectral efficiencies. We find that all these cloaks have comparable efficiency irrespective of whether or not they have singular conductivity at their inner boundary. However, approximate cloaking with multi-layered cloak critically depends upon the homogenization algorithm and a large number of thin layers (at least fifty) is required to reduce substantially the standard deviation of the isotherms., Comment: 13 pages, 7 figures. The results from this article will be presented on a poster at the Metamaterials 2014 Copenhague conference

We propose to use morphing algorithms to deduce some approximate wave pictures of scattering by cylindrical invisibility cloaks of various shapes deduced from the exact computation (e.g. using a finite element method) of scattering by cloaks of two given shapes, say circular and elliptic ones, thereafter called the source and destination images. The error in L2 norm between the exact and approximate solutions deduced via morphing from the source and destination images is typically less than 1 percent if control points are judiciously chosen. Our approach works equally well for rotators and concentrators, and also unveils some device which we call rotacon since it both rotates and concentrates electromagnetic fields. However, our approach is shown to break down for superscatterers (i.e. when the geometric transform underpinning the metamaterial is non-monotonic): In this case, the error in L2 norm is about 25 percent. We stress that our approach might greatly accelerate numerical studies of 2D and 3D cloaks (e.g. it takes less than 1 minute to deduce 50 images from the exact computations of the source and destination images with any morphing algorithm in 2D). The only price to pay is a human intervention since the accuracy of morphing highly depends upon control points., Comment: Last version with four more references, Fig. 6. more legible, and improvement of some parts of the text like "concluding remarks" and "acknowledgements"

We propose a cloak for coupled shear and pressure waves in solids. Its elastic properties are deduced from a geometric transform that retains the form of Navier equations. The spherical shell is made of an anisotropic and heterogeneous medium described by an elasticity tensor C' (without the minor symmetries) which has 21 non-zero spatially varying coefficients in spherical coordinates. Although some entries of C, e.g. some with a radial subscript, and the density (a scalar radial function) vanish on the inner boundary of the cloak, this metamaterial exhibits less singularities than its cylindrical counterpart studied in [M. Brun, S. Guenneau, A.B. Movchan, Appl. Phys. Lett. 94, 061903 (2009).] In the latter work, C' suffered some infinite entries, unlike in our case. Finite element computations confirm that elastic waves are smoothly detoured around a spherical void without reflection., Comment: Version 3: minor typos corrected. Figures captions improved. 5 figures. Key words: 3D elastic cloaking, seismic metamaterials. This paper is the cover of the 14 July 2014 issue of Applied Physics Letters

Negative indices are revisited through the thin-film admittance formalism. Effective indices and phase delay associated with wave propagation through negative index layers are carefully defined and computational rules easily implementable in standard thin-film software are derived from this approach. This admittance formalism is then used to recover the main features of the perfect lens and to highlight the benefit of such negative index materials to improve the performances of quarter-wavelength Bragg mirrors and Fabry-Perot band-pass filters., Comment: 9 pages, 6 Figures

The use of negative index materials is highly efficient for tayloring the spectral dispersion properties of a quarter-wavelength Bragg mirror and for obtaining a resonant behavior of a multilayer Fabry-Perot cavity over a very large spectral range. An optimization method is proposed and validated on some first promising devices., Comment: 4 pages, 3 figures

We investigate the twofold functionality of a cylindrical shell consisting of a negatively refracting heterogeneous bianisotropic (NRHB) medium deduced from geometric transforms. The numerical simulations indicate that the shell enhances their scattering by a perfect electric conducting (PEC) core, whereas it considerably reduces the scattering of electromagnetic waves by closely located dipoles when the shell surrounds a bianisotropic core. The former can be attributed to a homeopathic effect, whereby a small PEC object scatters like a large one as confirmed by numerics, while the latter can be attributed to space cancelation of complementary bianisotropic media underpinning anomalous resonances counteracting the field emitted by small objects (external cloaking). Space cancellation is further used to cloak a NRHB finite size object located nearby a slab of NRHB with a hole of same shape and opposite refracting index. Such a finite frequency external cloaking is also achieved with a NRHB cylindrical lens. Finally, we investigate an ostrich effect whereby the scattering of NRHB slab and cylindrical lenses with simplified parameters hide the presence of dipoles in the quasi-static limit., Comment: 16 pages, 15 figures

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.

We discuss the concept of cloaking for elastic impedance tomography, in which, we seek information on the elasticity tensor of an elastic medium from the knowledge of measurements on its boundary. We derive some theoretical results illustrated by some numerical simulations., Comment: latex, 2 figures, 11 pages, submitted

In this chapter we 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. By no means do we purport to imply that homogenization theory is an exclusive method for studying gratings, neither do we hope to be exhaustive in our choice of topics within the subject of homogenization. Our preferences here are motivated most of all by our own latest research, and by our outlook to the future interactions between these two subjects. We have also attempted, in what follows, to contrast the "classical" homogenization (Section 11.1.2), 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 (Section 11.2) and high-contrast homogenization (Section 11.3), which have been 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.

We investigate a high-order homogenization (HOH) algorithm for periodic multilayered stacks. The mathematical tool of choice is a transfer matrix method. Expressions for effective permeability, permittivity and magnetoelectric coupling are explored by frequency power expansions. On the physical side, this high-order homogenization uncovers a magnetoelectric coupling effect (odd order approximation) and artificial magnetism (even order approximation) in moderate contrast photonic crystals. Comparing the effective parameters' expressions of a stack with three layers against that of a stack with two layers, we note that the magnetoelectric coupling effect vanishes while the artificial magnetism can still be achieved in a center symmetric periodic structure. Furthermore, we numerically check the effective parameters through the dispersion law and transmission curves of a stack with two dielectric layers against that of an effective bianisotropic medium: they present a good agreement in the low frequency (acoustic) band until the first stop band, where the analyticity of the logarithm function of transfer matrix ($\log\{T\}$) breaks down., Comment: 29 pages, 9 figures