Acoustic Metamaterials

This chapter introduces the field of acoustic metamaterials in light of correspondences with related phenomena in electromagnetics. The semantic frontier between phononic/photonic crystals (PCs) and metamaterials is underpinned by low-frequency high-contrast and high-frequency homogenization models for periodic structures, the former being well suited for metamaterials, while the latter unveils the band structure and associated anomalous dispersion of PCs. We find it therefore worthwhile to outline the corresponding asymptotic models for waves propagating in such structured media. The mathematics behind the physical scene are illustrated by numerical simulations including cloaking, lensing and confinement effects via artificial anisotropy (motivated by transformational optics and acoustics), negative refraction and slow waves.

In this chapter we investigate the propagation of acoustic waves in a two-dimensional array of cylindrical pillars on the surface of a semi-infinite substrate. Through the computation of the acoustic band diagram and transmission spectra of periodic pillars arranged in different symmetries, we show that these structures possess acoustic metamaterial features for surface acoustic waves. The pillars on the top of the surface introduce new guided modes in the non-radiative region of the substrate outside the sound cone. The modal shape and polarization of these guided modes are more complex than those of classical surface waves propagating on a homogeneous surface. Significantly, an in-plane polarized wave and a transverse wave with sagittal polarization appear that are not supported by the free surface. In addition, the band diagram of the guided modes defines band gaps that appear at frequencies markedly lower than those expected from the Bragg mechanism. We identify them as originating from local resonances of the individual cylindrical pillar and we show their dependence on the geometrical parameters, in particular with the height of the pillars. The frequency positions of these band gaps are invariant with the symmetry, and thereby the period, of the lattices, which is unexpected in band gaps based on Bragg mechanism. However, the role of the period remains important for defining the non-radiative region limited by the slowest bulk modes and influencing the existence of new surface modes of the structures. The surface acoustic wave transmission across a finite array of pillars corroborates the signature of the locally resonant band gaps for surface modes and their link with the symmetry of the source and its polarization. Numerical simulations based on an efficient finite element method and considering Lithium Niobate pillars on a Lithium Niobate substrate are used to illustrate the theory.

The design of periodic and quasiperiodic structures possessing innovative filtering properties for elastic waves opens the way to the realization of elastic metamaterials. In these structures prestress has a strong influence, ‘shifting’ in frequency, but also ‘annihilating’ or ‘nucleating’ band gaps. The effects of prestress are demonstrated with examples involving flexural waves in periodic and quasiperiodic beams and periodic plates. Results highlight that prestress can be employed as a ‘tuning parameter’ for continuously changing vibrational properties of elastic metamaterials.

We study sound transmission through plates perforated with subwavelength holes. Experimental results are analyzed in the light of both a rigid solid model as well as a full elasto-acoustic theory. A discussion comparing sound and optics is given based upon an analytical framework. We show that, unlike light, sound is transmitted through individual subwavelength holes, in a perfectly rigid thin film approximately in proportion to their area. Moreover, hole arrays in perfectly rigid thin films do not exhibit full sound transmission due to the absence of lattice resonances. Therefore, the resonant full transmission observed in hole arrays is not extraordinary in the case of sound. However extraordinary sound screening well beyond that predicted by the mass law is observed. Finally, we find a strong interplay between Wood anomaly minima and intrinsic plate modes (Lamb modes), which results in fundamentally unique behavior of sound as compared to light.

Routine applications of ultrasound imaging combine array technology and beamforming (BF) algorithms for image formation. Although BF is very robust, it discards a significant proportion of the information encoded in ultrasonic signals. Therefore, BF can reconstruct some of the geometrical features of an object but with limited resolution due to the diffraction limit. Inverse scattering theory offers an alternative approach to BF imaging that has the potential to break the diffraction limit and extract quantitative information about the mechanical properties of the object. High-resolution, quantitative imaging is central to modern diagnostic technology to achieve cost-effective detection through high sensitivity and limited false positive rate. This chapter lays out a framework encompassing theoretical and experimental results, and in which inverse scattering and modern array technology can be combined together to achieve super-resolution, quantitative imaging.

Time reversal is a physical concept that allows focussing of waves both spatially and temporally regardless of the complexity of the propagation medium. Time reversal mirrors have been demonstrated first in acoustics, then with electromagnetic waves, and are being intensively studied in many fields ranging from underwater communications to sensing.

In this chapter we review the principles of time reversal and in particular its ability to focus waves in complex media. We show that this focussing effect depends on the complexity of the propagation medium rather than on the time reversal mirror itself. A modal approach is utilized to explain the results and grasp the physical mechanisms underlying the concept.

A particular focus is given to the possibility of overcoming the diffraction barrier from the far field using time reversal. With this aim, we return to the first proof of concept of this original approach. Those results are explained in terms of the coherent excitation of subwavelength modes. In particular, we show that a finite size medium consisting of coupled subwavelength resonators, which we call a resonant metalens, supports modes which radiate spatial information of the near field of a source efficiently in the far field. We show that such a process, due to reversibility, enables us to beat the diffraction limit using far field time reversal, and especially that this result occurs due to the inherent broadband nature of time reversal. We then generalize the concept to other types of media, and finally show experimentally that it is also valid for acoustic waves, demonstrating deep subwavelength focal spots obtained within an array of soda cans.

Metamaterials are becoming a prominent class of artificial materials that allow us to have very precise and specific optical properties. Its associated engineering flexibility opens up a wide range of applications and provides an effective route in molding the flow of energy. By drawing analogies between electromagnetic and acoustic wave frameworks, many concepts like invisibility cloaking and subwavelength imaging can be transplanted from electromagnetic to acoustic waves easily. However, we need quite different ways of constructing the artificial materials and devices for acoustics. Here, we show how anisotropic metamaterials can be constructed to control the constitutive parameters of the effective medium through positioning hard plates in different preferred directions. We will then use them to construct an acoustic carpet cloak, an acoustic hyperlens and a superlens as examples.

In this chapter we review the development of the concept of transformation acoustics, through which sound fields can be arbitrarily manipulated by complex acoustic materials. We describe the theory and the design equations in several different forms, and we present several explicit design examples using transformation acoustics. After briefly describing some theoretical offshoots from the original idea, we conclude with a summary of approaches for engineering composite materials with the smoothly inhomogeneous and anisotropic properties needed for many transformation acoustics devices.

Acoustic cloaking is the mechanism representing the ideal acoustic stealth. We introduce and discuss the acoustic cloak, a material shell that renders an object acoustically ‘invisible’ thanks to its presence surrounding the object. It has been shown that cloaking shells require very complex parameters to be realized. This complexity comes from the fact that their acoustic parameters must be anisotropic, inhomogeneous and divergent near the cloaked object. This chapter explains how to engineer artificial structures, which have been called acoustic metamaterials or metafluids, that respond dynamically as anisotropic and inhomogeneous materials. The metafluids are made from arrays of isotropic and homogeneous elastic cylinders or by metallic plates cylindrically corrugated. We also propose solutions to remove the divergences appearing in the design of cloaking shells. It is therefore predicted that, although difficult to realize, cloaking shells are not impossible by using metafluids based on the homogenization of periodic structures.

This chapter presents an overview of the scattering cancellation approach applied to acoustic waves, inspired by the use of plasmonic cloaks for electromagnetic waves. Using an analogous analytical approach, we show here that isotropic and homogeneous acoustic metamaterial covers may provide strong scattering reduction over moderately broad bandwidths of operation in a variety of acoustic scenarios of interest. This chapter outlines the basic physics of this approach, along with numerical examples for moderately sized elastic and fluid objects, thereby providing insights into the anomalous suppression of acoustic scattering produced by this cloaking technique.

In this chapter, we use analogies between the governing equations for linear surface liquid waves (LSWs) and surface plasmon polaritons (SPPs) in order to control their trajectories using geometric transforms. These two routes towards cloaking are emerging areas of physics known as transformational acoustics and plasmonics. We first analyze cloaking of LSWs propagating through a circular cloak, which consists of concentric layers cut into a large number of small sectors with rigid pillars. This water wave cloak behaves as an effective anisotropic fluid. We experimentally observe the decreased backscattering around the frequency 10 Hertz of a fluid with low viscosity and finite density (Methoxynonafluorobutane) from a cylindrical rigid obstacle surrounded by the cloak when it is located a couple of wavelengths away from the acoustic source. We then study theoretically, and numerically, the cloaking of SPPs propagating on a structured metal surface, and we manufacture and experimentally validate a plasmonic carpet working in the visible spectrum using dielectric plots of TiO₂ arranged along a quasi-conformal grid on a metal plate.

This chapter consists of three parts. In the first part we recall the elastodynamic equations under coordinate transformations. The idea is to use coordinate transformations to manipulate waves propagating in an elastic material. Then we study the effect of transformations on a mass-spring network model. The transformed networks can be realized with “torque springs”, which are introduced here and are springs with a force proportional to the displacement in a direction other than the direction of the spring terminals. Possible homogenizations of the transformed networks are presented, with potential applications to cloaking. In the second and third parts we present cloaking methods that are based on cancelling an incident field using active devices which are exterior to the cloaked region and that do not generate significant fields far away from the devices. In the second part, the exterior cloaking problem for the Laplace equation is reformulated as the problem of polynomial approximation of functions. An explicit solution is given that allows cloaking of larger objects at a fixed distance from the cloaking device, compared to previous explicit solutions. In the third part we consider the active exterior cloaking problem for the Helmholtz equation in 3D. Our method uses the Green’s formula and an addition theorem for spherical outgoing waves to design devices that mimic the effect of the single and double layer potentials in Green’s formula.