Nanophotonics 2019; 8(5): 685–701 Review article Yabin Jin*, Bahram Djafari-Rouhani* and Daniel Torrent* Gradient index phononic crystals and metamaterials https://doi.org/10.1515/nanoph-2018-0227 arrangements of scatterers in a given matrix, and they Received December 21, 2018; revised February 8, 2019; accepted were firstly proposed in 1993 [1, 2]. The attention that February 11, 2019 phononic crystals received originally was due to the existence of a phononic Bragg band gap where the prop- Abstract: Phononic crystals and acoustic metamaterials agation of acoustic waves was forbidden. In this regime are periodic structures whose effective properties can be the wavelength λ of the acoustic field is comparable to tailored at will to achieve extreme control on wave propa- the periodicity a of the lattice, λ ≈ a, and to be observ- gation. Their refractive index is obtained from the homog- able it is also required that the thickness D of the bulk enization of the infinite periodic system, but it is possible phononic crystals be at least four or five periodicities, to locally change the properties of a finite crystal in such D > > a. In the subwavelength range, λ > > a, it is pos- a way that it results in an effective gradient of the refrac- sible to find local resonances in the scatterers, and the tive index. In such case the propagation of waves can be designed structures exhibit hybridization band gaps accurately described by means of ray theory, and differ- which give rise to novel effects such as negative mass ent refractive devices can be designed in the framework of density or negative elastic modulus. In this exotic regime wave propagation in inhomogeneous media. In this paper the structure behaves as a special type of materials called we review the different devices that have been studied for “acoustic metamaterials”, which were firstly proposed in the control of both bulk and guided acoustic waves based the seminal work [3] in 2000. Over the past two decades, on graded phononic crystals. dramatically increasing efforts have been devoted to the Keywords: gradient index; phononic crystals; metamate- study of acoustic artificial structured materials driven by rials; lenses; homogenization. both fundamental scientific curiosities with properties not found previously and diverse potential applications with novel functionalities [4–16]. While interesting, most of the extraordinary proper- 1 Introduction ties of metamaterials are in general single-frequency or narrow-band, since outside the resonant regime meta- Phononic crystals and acoustic metamaterials enable materials behave as common composites. However, to achieve innovative properties for the propagation of non-resonant phononic crystals in the low-frequency mechanical waves (air-borne sound waves, water-borne regime behave as homogeneous non-dispersive materi- acoustic waves, water waves, elastic waves, surface als whose effective parameters can be easily tailored, acoustic waves and Lamb waves) inapproachable in and in this regime gradient index (GRIN) acoustic mate- natural materials. Phononic crystals consist of periodic rials, or GRIN devices, are easily doable. These devices were firstly proposed for acoustic waves by Torrent et al. in 2007 [17] and for elastic waves by Lin et al. in *Corresponding authors: Yabin Jin, School of Aerospace Engineering 2009 [18], and they allow to manipulate acoustic waves and Applied Mechanics, Tongji University, 200092, Shanghai, China, e-mail: [email protected]. to enforce them to follow curved trajectories. They are https://orcid.org/0000-0002-6991-8827; characterized by a spatial variation of acoustic refractive Bahram Djafari-Rouhani, Institut d’Electronique, de index, which is designed by locally changing the geom- Microélectonique et de Nanotechnologie, UMR CNRS 8520, etry of units. For instance, the effective index obviously Département de Physique, Université de Lille, 59650 Villeneuve depends on the filling ratio of scatterers, namely, the d’Ascq, France, e-mail: [email protected]; and Daniel Torrent, GROC-UJI, Institut de Noves Tecnologies de la size of scatterers, which is initially proposed for GRIN Imatge, Universitat Jaume I, 12071, Castello, Spain, control of acoustic waves [17–19]; instead, the variation e-mail: [email protected] in the lattice spacing while keeping the size of scatterers Open Access. © 2019 Yabin Jin, Bahram Djafari-Rouhani, Daniel Torrent et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. 686 Y. Jin et al.: GRIN phononic crystals and metamaterials is also achievable [20, 21]; for triangular shape of scat- since from the technological point of view low-dimen- terers, rotating the angles of the triangular shape can sional materials are more interesting than bulk materi- also affect the effective acoustic velocity [22]; for lead- als. The excellent compatibility of phononic crystals and rubber pillared metamaterial plate, by changing the metamaterials with the nanoelectromechanical systems height of lead layer in pillars, the effective mass density has been proven for applications in wireless telecom- is tuned, resulting in a change in effective phase velocity munications, sensing or thermal control, among others following a given law [23]; for phononic crystal plates, [76–81]. However, the propagation of elastic waves in the effective velocity of antisymmetric Lamb wave is plates is in general composed of three polarizations, directly related with the thickness of plates, which can which travel at different speeds and for which refractive also further affect the effective velocity of symmetric devices designed for one of these polarizations will not mode [24–26]; and gradient effective index can also be work for the other two. Recently, the designs of GRIN achieved from coiling-space structures [27–30]. In addi- devices based on phononic crystal plates have demon- tion to the geometric parameters, the effective refractive strated that the simultaneous control of all fundamental index can also be tuned in relation to the elastic proper- modes propagating in thin elastic plates in a broadband ties of the scatterers, for instance, the choice of materi- frequency region is possible [25, 26, 63]. als [18, 31], or to an external stimulus such as electric The objective of this review is to provide a compre- field [32] or temperature [33]. hensive picture of the evolution of the domain of GRIN The attained effective refractive index of the GRIN devices for mechanical waves, to demonstrate the wide device can be smaller or larger than that of the background variety of applications at the nanoscale that these medium, which corresponds to phase advance [20] or devices offer and to present the challenges to be accom- delay approaches in wave propagation. Nevertheless, plished to further develop this field. The paper is organ- the phase delay approach is mostly employed as higher ized as follows: after this introduction, we will review the refractive index enables to design more advanced fundamental ideas of the homogenization of sonic and functionalities and reduce the entire width of the devices. phononic crystals in Section 2 and of phononic crystal Comparing to the wavelength, the thinner the device, the plates in Section 3, which offers efficient tools to obtain larger will be the highest required index. If the width of the effective elastic properties and, consequently, their the whole device is downscaled to the subwavelength effective refractive index. Then we will review GRIN regime, the GRIN device becomes a metasurface [34, 35]. devices for bulk acoustic and elastic waves in Section 4 However, to find a high enough effective refractive index and for flexural waves in Section 5; the advanced full to design GRIN metasurfaces is a big challenge that needs control of polarizations in elastic plates with multi- new technologies in material science. Recently, it has modal GRIN devices will be reviewed in Section 6. The been reported that soft porous materials [36, 37] with soft- last section will present some conclusions and future matter techniques can achieve a relative refractive index challenges. higher than 20 [38], resulting in soft GRIN metasurface [39]. GRIN devices can also be designed with negative index of refraction at frequencies lying in the first nega- tive slope of the acoustic band structure [21]. However, 2 Homogenization of sonic and these devices would have a narrow band, and it may limit phononic crystals potential applications. GRIN phononic crystals and metamaterials can be Graded materials with specific variations of the refractive applied to various types of waves in a long frequency index are obviously not found in nature, and they have limit, such as surface water waves [40, 41] for a few to be artificially engineered. The inclusion of scatterers in hertz, air-borne sound waves [19, 42–51] for 103–10 5 Hz, a given matrix is an excellent way for the realization of water-borne acoustic waves [20, 31, 52–54] for 104–10 6 Hz, artificially graded materials, since the average behavior of Rayleigh waves [55–59] for 10–108 Hz and Lamb waves the scatterers is to modify the effective velocity of acous- [23–25, 32, 60–72] for 103–10 8 Hz, among others, with func- tic waves in the matrix, and this effective velocity can be tionalities like focusing [18, 19], waveguiding [70, 73], tuned by means of the size of the inclusions. When these mirage [74], beam splitting or deflection [41, 62, 63], inclusions