Search

The stacking order of two-dimensional transition metal dichalcogenides (TMDs) is attracting tremendous interest as an essential component of van der Waals heterostructures. A common and fast approach to distinguish between the AAprime (2H) and AB (3R) configurations uses the relative edge orientation of each triangular layer (theta) from optical images. Here, we highlight that this method alone is not sufficient to fully identify the stacking order. Instead we propose a model and methodology to accurately determine the bilayer configuration of WSe2 using second harmonic generation (SHG) and Raman spectroscopy. We demonstrate that the SHG response of the AB phase (theta = 0) deg layers is more intense than the signal from the single layer structure. However, the SHG totally vanishes in the AAprime and ABprime phases (theta = 60 deg) and 0 deg respectively) of homo-bilayer WSe2. Also, several optical features of homo-bilayer WSe2 are found to depend on the details of the stacking order, with the difference being the clearest in the low frequency (LF) Raman frequencies, as confirmed by DFT simulation. This allows unambiguous, high-throughput, nondestructive identification of stacking order in TMDs, which is not robustly addressed in this emerging research area.

Resonances, also known as quasinormal modes (QNM) in the non-Hermitian case, play a ubiquitous role in all domains of physics ruled by wave phenomena, notably in continuum mechanics, acoustics, electrodynamics, and quantum theory. The non-Hermiticity arises from the system losses, whether they are material (Joule losses in electromagnetism) or linked to the openness of the problem (radiation losses). In this paper, we focus on the latter delicate matter when considering bounded computational domains mandatory when using e.g. Finite Elements. Dispersive perfectly matched layers and absorbing boundary conditions are studied.

Light emission of europium (Eu3+) ions placed in the vicinity of optically resonant nanoantennas is usually controlled by tailoring the local density of photon states (LDOS). We show that the polarization and shape of the excitation beam can also be used to manipulate light emission, as azimuthally or radially polarized cylindrical vector beam offers to spatially shape the electric and magnetic fields, in addition to the effect of silicon nanorings (Si-NRs) used as nanoantennas. The photoluminescence mappings of the Eu3+ transitions and the Si phonon mappings are strongly dependent of both the excitation beam and the Si-NR dimensions. The experimental results of Raman scattering and photoluminescence are confirmed by numerical simulations of the near-field intensity in the Si nanoantenna and in the Eu3+-doped film, respectively. The branching ratios obtained from the experimental PL maps also reveal a redistribution of the electric and magnetic emission channels. Our results show that it is possible to spatially control both electric and magnetic dipolar emission of Eu3+ ions by switching the laser beam polarization, hence the near-field at the excitation wavelength, and the electric and magnetic LDOS at the emission wavelength. This paves the way for optimized geometries taking advantage of both excitation and emission processes., Comment: 8 pages, 4 figures

We study the dynamic surface response of neolithic stone settlements obtained with seismic ambient noise techniques near the city of Carnac in French Brittany. Surprisingly, we find that menhirs (neolithic human size standing alone granite stones) with an aspect ratio between 1 and 2 periodically arranged atop a thin layer of sandy soil laid on a granite bedrock, exhibit fundamental resonances in the range of 10 to 25 Hz. We propose an analogic Kelvin-Voigt viscoelastic model that explains the origin of such low frequency resonances. We further explore low frequency filtering effect with full wave finite element simulations. Our numerical results confirm the bending nature of fundamental resonances of the menhirs and further suggest additional resonances of rotational and longitudinal nature in the frequency range 25 to 50 Hz. Our study thus paves the way for large scale seismic metasurfaces consisting of granite stones periodically arranged atop a thin layer of regolith over a bedrock, for ground vibration mitigation in earthquake engineering., Comment: 6 pages, 7 figures

We present an approach to analyzing the safety of asynchronous, independent, non-deterministic, turn-to-bearing horizontal maneuvers for two vehicles. Future turn rates, final bearings, and continuously varying ground speeds throughout the encounter are unknown but restricted to known ranges. We develop a library of formal proofs about turning kinematics, and apply the library to create a formally verified timing computation. Additionally, we create a technique that evaluates future collision possibilities that is based on waves of position possibilities and relies on the timing computation. The result either determines that the encounter will be collision-free, or computes a safe overapproximation for when and where collisions may occur., Comment: Coq proofs are at https://bitbucket.org/ykouskoulas/ottb-foundation-proofs; Accepted 08 Mar 2022 (International Journal on Software Tools for Technology Transfer)

We design planar silicon antennas for controlling the emission rate of magnetic or electric dipolar emitters. Evolutionary algorithms coupled to the Green Dyadic Method lead to different optimized geometries which depend on the nature and orientation of the dipoles. We discuss the physical origin of the obtained configurations thanks to modal analysis but also emphasize the role of nanoscale design of the LDOS. We complete our study using finite element method and demonstrate an enhancement up to 2000 of the magnetic Purcell factor in europium ions. Our work brings together random optimizations to explore geometric parameters without constraint, a first order deterministic approach to understand the optimized designs and a modal analysis which clarifies the physical origin of the exaltation of the magnetic Purcell effect., Comment: 18 pages, 11 figures, research article

pyGDM is a python toolkit for electro-dynamical simulations of individual nano-structures, based on the Green Dyadic Method (GDM). pyGDM uses the concept of a generalized propagator, which allows to solve cost-efficiently monochromatic problems with a large number of varying illumination conditions such as incident angle scans or focused beam raster-scan simulations. We provide an overview of new features added since the initial publication [Wiecha, Computer Physics Communications 233, pp.167-192 (2018)]. The updated version of pyGDM is implemented in pure python, removing the former dependency on fortran-based binaries. In the course of this re-write, the toolkit's internal architecture has been completely redesigned to offer a much wider range of possibilities to the user such as the choice of the dyadic Green's functions describing the environment. A new class of dyads allows to perform 2D simulations of infinitely long nanostructures. While the Green's dyads in pyGDM are based on a quasistatic description for interfaces, we also provide as new external python package pyGDM2_retard a module with retarded Green's tensors for an environment with two interfaces. We have furthermore added functionalities for simulations using fast-electron excitation, namely electron energy loss spectroscopy and cathodoluminescence. Along with several further new tools and improvements, the update includes also the possibility to calculate the magnetic field and the magnetic LDOS inside nanostructures, field-gradients in- and outside a nanoparticle, optical forces or the chirality of nearfields. All new functionalities remain compatible with the evolutionary optimization module of pyGDM for nano-photonics inverse design., Comment: 29 pages, 20 figures

In atomically thin transition metal dichalcogenide semiconductors, there is a crossover from indirect to direct bandgap as the thickness drops to one monolayer, which comes with a fast increase of the photoluminescence signal. Here, we show that for different alloy compositions of WS2(1-x)Se2x this trend may be significantly affected by the alloy content and we demonstrate that the sample with the highest Se ratio presents a strongly reduced effect. The highest micro-PL intensity is found for bilayer WS2(1-x)Se2x (x = 0.8) with a decrease of its maximum value by only a factor of 2 when passing from mono- to bi-layer. To better understand this factor and explore the layer-dependent band structure evolution of WS2(1-x)Se2x, we performed a nano-angle resolved photoemission spectroscopy study coupled with first-principles calculations. We find that the high micro-PL value for bilayer WS2(1-x)Se2x (x = 0.8) is due to the overlay of direct and indirect optical transitions. This peculiar high PL intensity in WS2(1-x)Se2x opens the way for spectrally tunable light-emitting devices.

There is great interest in using formal methods to guarantee the reliability of deep neural networks. However, these techniques may also be used to implant carefully selected input-output pairs. We present initial results on a novel technique for using SMT solvers to fine tune the weights of a ReLU neural network to guarantee outcomes on a finite set of particular examples. This procedure can be used to ensure performance on key examples, but it could also be used to insert difficult-to-find incorrect examples that trigger unexpected performance. We demonstrate this approach by fine tuning an MNIST network to incorrectly classify a particular image and discuss the potential for the approach to compromise reliability of freely-shared machine learning models., Comment: Part of 3rd Workshop on Formal Methods for ML-Enabled Autonomous Systems (FoMLAS 2020)

Diverting, and controlling, elastic vibrations impacting upon infrastructure is a major challenge for seismic hazard mitigation, and for the reduction of machine noise and vehicle vibration in the urban environment. Seismic metamaterials (SMs), with their inherent ability to manipulate wave propagation, provide a key route for overcoming the technological hurdles involved in this challenge. Engineering the structure of the SM serves as a basis to tune and enhance its functionality, and inspired by split rings, swiss-rolls, notch-shaped and labyrinthine designs of elementary cells in electromagnetic and mechanical metamaterials, we investigate altering the structure geometries of SMs with the aim of creating large bandgaps \textcolor{black}{in a subwavelength regime}. We show that square stiff inclusions, perform better in comparison to circular ones, whilst keeping the same filling fraction. En route to enhancing the bandgap, we have also studied the performance of SMs with different constituent materials; we find that steel columns, as inclusions, show large bandgaps, however, the columns are too large for steel to be a feasible material in practical or financial terms. Non-reinforced concrete would be preferable for industry level scaling up of the technology because, concrete is cost-effective, easy to cast directly at the construction site and easy to provide arbitrary geometry of the structure. As a part of this study, we show that concrete columns can also be designed to exhibit bandgaps if we cast them within a soft soil coating surrounding the protected area for various civil structures like a bridge, building, oil pipelines etc.

Liquid wave studies have shown that under certain constructive interference conditions, an abnormal size wave could be generated at a specific point. This type of wave, described long ago in the literature, was named 'Draupner Wave' in 1995. It was the first rogue wave, more than ten meters high, to be detected by a measuring instrument, occurring at the Draupner platform in the North Sea off the coast of Norway. For very soft and thick soils in continental sedimentary basin, with high water content, low shear wave velocity and wave bouncing effects on the edges of the basin, one can question the theoretical existence of such a type of freak waves that may have come unnoticed amongst surface seismic waves. This short communication presents recent observations and the conditions and limits of the analogy between gravity and seismic waves., Comment: 13 pages, 7 figures

Some properties of electromagnetic metamaterials have been translated, using some wave analogies, to surface seismic wave control in sedimentary soils structured at the meter scale. Two large scale experiments performed in 2012 near the French cities of Grenoble [Brule et al., PRL 112, 133901, 2014] and Lyon [Brule et al., Sci. Rep. 7, 18066, 2017] have confirmed the usefulness of this methodology and its potential influence on soil-structure interaction. We present here a new perspective on the in-situ experiment near Lyon, which unveils energy corridors in the seismic lens. We further introduce a concept of time-modulated seismic metamaterial underpined by an effective model based on Willis's equations. As a first application, we propose that ambient seismic noise time-modulates structured soils that can be viewed as moving media. In the same spirit, a design of an analogous seismic computer is proposed making use of ambient seismic noise. Seismic signals transmitted between remote seismic computers can help forming an Internet of Things. We further recall that ancient Roman theaters and forests of trees are examples of large scale structures that behave in a way similar to electromagnetic metamaterials: invisibility cloaks and rainbows, respectively. Seismic metamaterials can thus not only be implemented for shielding, lensing and cloaking of potentially deleterious Rayleigh waves, but they also have potential applications in energy harvesting and analogous computations using ambient seismic noise and this opens new vistas in seismic energy harvesting via natural or artificial soil structuring., Comment: 27 pages, 8 figures

Metre size inertial resonators located in the ground have been theoretically shown to interact with a seismic wave (attenuation, band gaps) to enable protection of surface structures such as buildings. The challenge for Civil Engineering is to both reduce the size of these resonators and to increase their efficiency. Here we explore steel spheres, connected to a concrete bulk medium, either by a coating of rubber, or rubber and steel ligaments, or air and steel ligaments. We show that for a cubic lattice periodicity of 1 metre, we achieve stop bands in the frequency range 14 to 20 Hz; by splitting spheres in 2 and 8 pieces, we tune down the stop bands frequencies and further increase their bandwidth. We thus demonstrate we are able to provide a variety of inertial resonators with stop bands below 10 Hz i.e., in the frequency range of interest for earthquake engineering.

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

In this paper, we describe Fourier-based Wave Front Sensors (WFS) as linear integral operators, characterized by their Kernel. In a first part, we derive the dependency of this quantity with respect to the WFS's optical parameters: pupil geometry, filtering mask, tip/tilt modulation. In a second part we focus the study on the special case of convolutional Kernels. The assumptions required to be in such a regime are described. We then show that these convolutional kernels allow to drastically simplify the WFS's model by summarizing its behavior in a concise and comprehensive quantity called the WFS's Impulse Response. We explain in particular how it allows to compute the sensor's sensitivity with respect to the spatial frequencies. Such an approach therefore provides a fast diagnostic tool to compare and optimize Fourier-based WFSs. In a third part, we develop the impact of the residual phases on the sensor's impulse response, and show that the convolutional model remains valid. Finally, a section dedicated to the Pyramid WFS concludes this work, and illustrates how the slopes maps are easily handled by the convolutional model., Comment: 21 pages, 7 figures

This paper introduces Whittemore, a language for causal programming. Causal programming is based on the theory of structural causal models and consists of two primary operations: identification, which finds formulas that compute causal queries, and estimation, which applies formulas to transform probability distributions to other probability distribution. Causal programming provides abstractions to declare models, queries, and distributions with syntax similar to standard mathematical notation, and conducts rigorous causal inference, without requiring detailed knowledge of the underlying algorithms. Examples of causal inference with real data are provided, along with discussion of the implementation and possibilities for future extension., Comment: 7 pages, 6 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

This paper proposes a causal inference relation and causal programming as general frameworks for causal inference with structural causal models. A tuple, $\langle M, I, Q, F \rangle$, is an instance of the relation if a formula, $F$, computes a causal query, $Q$, as a function of known population probabilities, $I$, in every model entailed by a set of model assumptions, $M$. Many problems in causal inference can be viewed as the problem of enumerating instances of the relation that satisfy given criteria. This unifies a number of previously studied problems, including causal effect identification, causal discovery and recovery from selection bias. In addition, the relation supports formalizing new problems in causal inference with structural causal models, such as the problem of research design. Causal programming is proposed as a further generalization of causal inference as the problem of finding optimal instances of the relation, with respect to a cost function., Comment: 30 pages, 9 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

This paper introduces a causation coefficient which is defined in terms of probabilistic causal models. This coefficient is suggested as the natural causal analogue of the Pearson correlation coefficient and permits comparing causation and correlation to each other in a simple, yet rigorous manner. Together, these coefficients provide a natural way to classify the possible correlation/causation relationships that can occur in practice and examples of each relationship are provided. In addition, the typical relationship between correlation and causation is analyzed to provide insight into why correlation and causation are often conflated. Finally, example calculations of the causation coefficient are shown on a real data set., Comment: 31 pages, 6 figures

The spirally arranged stems of the spikemoss Selaginella lepidophylla, an ancient resurrection plant, compactly curl into a nest-ball shape upon dehydration. Due to its spiral phyllotaxy, older outer stems on the plant interlace and envelope the younger inner stems forming the plant centre. Stem curling is a morphological mechanism that limits photoinhibitory and thermal damages the plant might experience in arid environments. Here, we investigate the distinct conformational changes of outer and inner stems of S. lepidophylla triggered by dehydration. Outer stems bend into circular rings in a relatively short period of desiccation, whereas inner stems curl slowly into spirals due to hydro-actuated strain gradient along their length. This arrangement eases both the tight packing of the plant during desiccation and its fast opening upon rehydration. The insights gained from this work shed light on the hydro-responsive movements in plants and might contribute to the development of deployable structures with remarkable shape transformations in response to environmental stimuli.

The regularity of earthquakes, their destructive power, and the nuisance of ground vibration in urban environments, all motivate designs of defence structures to lessen the impact of seismic and ground vibration waves on buildings. Low frequency waves, in the range $1$ to $10$ Hz for earthquakes and up to a few tens of Hz for vibrations generated by human activities, cause a large amount of damage, or inconvenience, depending on the geological conditions they can travel considerable distances and may match the resonant fundamental frequency of buildings. The ultimate aim of any seismic metamaterial, or any other seismic shield, is to protect over this entire range of frequencies, the long wavelengths involved, and low frequency, have meant this has been unachievable to date. Elastic flexural waves, applicable in the mechanical vibrations of thin elastic plates, can be designed to have a broad zero-frequency stop-band using a periodic array of very small clamped circles. Inspired by this experimental and theoretical observation, all be it in a situation far removed from seismic waves, we demonstrate that it is possible to achieve elastic surface (Rayleigh) and body (pressure P and shear S) wave reflectors at very large wavelengths in structured soils modelled as a fully elastic layer periodically clamped to bedrock. We identify zero frequency stop-bands that only exist in the limit of columns of concrete clamped at their base to the bedrock. In a realistic configuration of a sedimentary basin 15 meters deep we observe a zero frequency stop-band covering a broad frequency range of $0$ to $30$ Hz.

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

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

We apply genetic programming techniques to the `shepherding' problem, in which a group of one type of animal (sheep dogs) attempts to control the movements of a second group of animals (sheep) obeying flocking behavior. Our genetic programming algorithm evolves an expression tree that governs the movements of each dog. The operands of the tree are hand-selected features of the simulation environment that may allow the dogs to herd the sheep effectively. The algorithm uses tournament-style selection, crossover reproduction, and a point mutation. We find that the evolved solutions generalize well and outperform a (naive) human-designed algorithm.

This paper considers the computational power of constant size, dynamic Bayesian networks. Although discrete dynamic Bayesian networks are no more powerful than hidden Markov models, dynamic Bayesian networks with continuous random variables and discrete children of continuous parents are capable of performing Turing-complete computation. With modified versions of existing algorithms for belief propagation, such a simulation can be carried out in real time. This result suggests that dynamic Bayesian networks may be more powerful than previously considered. Relationships to causal models and recurrent neural networks are also discussed.

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.