# Acoustic wave science realized by metamaterials

- Dongwoo Lee
^{1, 2}, - Duc Minh Nguyen
^{2}and - Junsuk Rho
^{2, 3}Email author

**4**:3

**DOI: **10.1186/s40580-017-0097-y

© Korea Nano Technology Research Society 2017

**Received: **8 December 2016

**Accepted: **18 January 2017

**Published: **7 February 2017

## Abstract

Artificially structured materials with unit cells at sub-wavelength scale, known as metamaterials, have been widely used to precisely control and manipulate waves thanks to their unconventional properties which cannot be found in nature. In fact, the field of acoustic metamaterials has been much developed over the past 15 years and still keeps developing. Here, we present a topical review of metamaterials in acoustic wave science. Particular attention is given to fundamental principles of acoustic metamaterials for realizing the extraordinary acoustic properties such as negative, near-zero and approaching-infinity parameters. Realization of acoustic cloaking phenomenon which is invisible from incident sound waves is also introduced by various approaches. Finally, acoustic lenses are discussed not only for sub-diffraction imaging but also for applications based on gradient index (GRIN) lens.

### Keywords

Metamaterials Sub-wavelength Cloaking Sub-diffraction imaging Gradient index## 1 Introduction

Metamaterials made of periodic or random artificial structures, defined as “meta-atoms” with size that is larger than the conventional atom and much smaller than the radiated wavelength, are used for deeply control and manipulation of waves. Since the properties of the metamaterials are governed by the meta-atom structures rather than their base materials, by careful designing and engineering the parameters of the meta-atom structures such as shape, geometry, size or orientation, fascinating functionalities beyond the capability of conventional materials can be realized. The concept of metamaterials was first proposed by Veselago [1] in 1968 for electromagnetic waves, but it needed to wait for around 30 years for the next step when Pendry reported artificial designs with effectively negative permeability and permittivity in 1999 [2, 3]. Metamaterials were then experimentally demonstrated by Smith and Shelby [4, 5] for negative refractive index structures and have since been a subject of numerous studies in a wide variety of wave-matter interaction, including not only photonics but also acoustic wave science.

Acoustic wave science studies the propagation of matter oscillation through an elastic medium such as air or water and therefore explains energy transfer through the medium. While the movement of oscillating materials is limited through its equilibrium position, vibrational waves can propagate in a long distance and can be reflected, refracted, attenuated or, more generally, manipulated by the medium. According to the oscillation frequency, acoustic waves have been classified to different fields that cover the audio, ultrasonic and infrasonic frequency range, or seismic waves at much larger scale which are waves of energy travelling through the Earth’s layer.

The advent of fabrication technology [6–8] together with development of simulation techniques such as finite element method (FEM) and finite difference time domain method (FDTD) have led to a revolution of metamaterials in controlling and manipulating acoustic waves in new ways not previously imagined [9, 10]. For instance, in acoustics, it is now possible to design acoustic lenses for sub-diffraction imaging [9–12] or design acoustic cloaking which is able to make an object acoustically invisible by bending the waves [13–16]. Also, an assembly of rubber-coated spheres into a bulk metamaterial can exhibit locally designed resonant structures [17].

Our objective here is to present a unified discussion of the advances of metamaterials in acoustic wave science. The review is organized as follows. We focus on the acoustic metamaterials in Sect. 2 with discussions in theory about acoustic parameters such as mass density and bulk modulus. The section is then followed by our review of metamaterial designs for controlling these two parameters to achieve unusual negative or near-zero values that cannot be found in nature. As a next part, acoustic cloaking is discussed in detail with different approaches. Lastly, superlens and hyperlens for sub-diffraction imaging are organized then, Luneburg and Eaton lens which are based on the concept of (GRIN) lens are introduced.

## 2 Acoustic metamaterials

*P*is the pressure and \(\rho ,B\) are the mass density and bulk modulus of materials, respectively. Physically, the mass density is defined as mass per unit volume and the bulk modulus reflects the medium’s resistance to external uniform compression. These two parameters are analogous to the electromagnetic parameters, permittivity \(\varepsilon\) and permeability \(\mu ,\) as can be seen in the following expression of the refractive index

*n*and the impedance

*Z*.

### 2.1 Negative mass density

*M*

_{2}positioning inside the cavity of mass

*M*

_{1}and coupling with the mass

*M*

_{1}through a spring of strength

*K*is shown in Fig. 2a. If we assume that the masses vibrate without friction under an external force \(F(\omega )\) with an angular frequency \(\omega ,\) moving equations given by Newton’s second law are described as

*M*

_{1},

*M*

_{2}, respectively and \(\omega_{0} = \sqrt {K/M_{2} }\) is the local resonance frequency. By assuming

*x*

_{1},

*x*

_{2}and

*F*are time-variant values and solving these differential equations for the external force \(F(\omega ),\) we have

*M*

_{1}–

*M*

_{2}can be considered as a homogenous one-object system with the resonant frequency \(\omega_{0}\) and an effective mass is

One can deduce from this equation that the effective mass *M*
_{
eff
} can be negative if the external force oscillates near the resonant frequency of the system, particularly, in the range \(\omega_{0} < \omega < \sqrt {K/M_{1} + \omega_{0}^{2} }\) as can be seen in Fig. 2b. Finally, we have an effective mass density \(\rho_{eff}\) by dividing *M*
_{
eff
} by the system volume. The term “effective” will be often omitted when describing effective mass density and effective bulk modulus in this review.

*G*and unit cells are connected to each other by a spring

*K*. The dynamic system is finally excited with a harmonic external force with non-friction condition by the MTS Tytron 250 and air track. Actual picture of the experimental setup is shown in Fig. 3b and corresponding measurement results for a single unit cell (Fig. 3c) indicates a strong resonance near 6 Hz. Harmonic movement of the whole system with seven unit cells is also measured as shown in Fig. 3d. As a result, negative mass density was found with a ban-gap near the resonant frequency from about 6 to 7.6 Hz and transmittance defined as the amplitude ratio of

*X*

_{ N }/

*X*

_{0}was obtained as around −30 dB.

Generally, negative mass density in acoustic metamaterials can be realized by replacing the mass-spring system to any kind of system having constitutive compositions corresponding to a mass and a spring. For example, a membrane system having a unit cell made up of a rigid grid is reported in [23] where the rigid grid and membrane play the role of the mass and spring, respectively. Such a membrane system with negative mass density has been applied to realize sound absorbers [24–29].

### 2.2 Negative bulk modulus

*P*, Δ

*V*/

*V*and

*B*denote the pressure change, volume strain and bulk modulus, respectively. Like negative mass density, negative bulk modulus can also be realized by introducing the definition of negative effective bulk modulus in acoustic metamaterials. A simple example of the negative bulk modulus system is a Helmholtz resonator that is basically made up of a large cavity and a narrow neck as shown in Fig. 4a. The effective bulk modulus is expressed by Fang et al. [30]

*F*is the geometrical factor, \(\omega_{0}\) is the resonant angular frequency and \(\varGamma\) is the dissipation loss in the resonating Helmholtz elements. Once again, we can see from the above equation that effective bulk modulus can reach a negative value when the external force oscillates near the resonant frequency. This phenomenon relates to the fact that the cavity is expanded due to an outward restoring force in near the resonant frequency, which indicates the negative bulk modulus. Whereas, being shrunk of the cavity due to an external compressive force indicates the positive bulk modulus in a conventional case. The incoming sound through the neck and the cavity inside are analogous to a mass and a spring, respectively.

The negative bulk modulus system was experimentally demonstrated by Fang et al. and Lee et al. [30, 31]. Fang’s group conducted an underwater ultrasonic transmission experiment composed of Helmholtz resonators in series. The experimental setup shown in Fig. 4b consists of a transducer for underwater sound source and two hydrophones for detection of the signals. Extremely low transmission was observed as visualized in Fig. 4c, indicating that the propagation wave was transformed to evanescent form due to the negative bulk modulus of the metamaterials. Moreover, a formation of negative phase velocity was also confirmed in this experiment due to the loss of friction in the system. Other related works for different types of Helmholtz resonators can be found in [32–35].

### 2.3 Double negative parameters

We have explained in previous sections that either effective mass density or effective bulk modulus of acoustic parameters can be negative near resonant frequency of a periodic artificial structure and then a fully opaque acoustic material is possible. However, an inverse effect in which sound wave energy propagates instead of attenuation will occur when both these two parameters are negative simultaneously.

The above methods are limited to a extremely narrow frequency range and more recent researches have continued to overcome this limitation, leading to novel class of acoustic metamaterials so called “space-coiling metamaterials” having negative refractive index over broad range of frequency [43–48]. This kind of metamaterial is realized by coiling up space with curled channels and no requirements for creating local resonances, and can be constructed easily not only for two dimensions but also for three dimensions. We will go back to this type of metamaterial later in Sect. 2.5. Another method for obtaining metamaterials with negative refractive index is to stack several holey plates forming hyperbolic dispersion with highly anisotropic structure [10, 11]. The hyperbolic acoustic metamaterials will be discussed in more detail in Sect. 2.7.2.

### 2.4 Near-zero and approaching-infinity mass density

Another interesting characteristic can be achieved when the effective mass density approaches to infinity. In this case, the impedance in the slab would be very large, leading to large impedance mismatch between the slab and background, and therefore resulting in the nearly total reflection on the interface. This characteristic is demonstrated based on membrane-type acoustic metamaterials and could be exploited in noise control [25, 55].

### 2.5 Space-coiling metamaterials

### 2.6 Acoustic cloaking

*x*,

*y*) plane of the Cartesian coordinates generated by conformal mapping.

By emerging this powerful tool into the acoustic wave science [67–69], acoustic applications for cloaking and super-resolution that require metamaterials containing complicated and hard to implement properties are now possible. The term “acoustic cloaking” refers to a phenomenon that a shell makes the surrounded object invisible from any directions of the incoming sound waves. In fact, the idea of acoustic cloaking was inspired from electromagnetics and optics where experimental cloaking phenomena have been realized at radio [59, 70] and optical [71] frequency range.

*A*is the Jacobian matrix of coordinate transformation and,

*A*

^{ T }and det(

*A*) denote transpose and determinant of

*A*, respectively. Consequently, we can map the normal Cartesian space to a distorted space for the purpose of bending of wave propagation trajectories, resulting in the cloaking phenomenon. More theoretical details are presented in the work of Milton et al. [72] which described how to apply the cloaking phenomenon in electromagnetic waves to other types of waves, especially in the acoustic waves. Numerical studies for acoustic cloaking in two dimensions [13, 14] and three dimensions [15] have also been conducted. The first experiment of acoustic cloaking was realized by Zhang et al. [16] with a design of 2D array of sub-wavelength cavities filling with water and connected channels with spatially tailored geometry (Fig. 10a). The design of cavities is referred to the concept of lumped acoustic elements which are analogous to electronic circuit elements (Fig. 10b). As a result, 2D acoustic cloaking with a proper array of the unit cells composed of cavities and connected channels was achieved with almost no scattering in front and rare of the steel cylinder as shown in Fig. 10c.

Another approach for acoustic cloaking inspired from carpet cloaking suggested by Li and Pendry in electromagnetic field [73]. With this concept, the first experimental 2D acoustic carpet cloaking was demonstrated by Popa et al. [74]. Subsequently, a 3D carpet cloaking which is an extension of the 2D one was demonstrated by Zigoneanu et al. [75]. The setups of 2D and 3D carpet cloaking are made of arrays of the perforated plastic plates with sub-wavelength holes that allow the penetration of airborne sounds. Metamaterials with highly anisotropic mass density are required for this approach so that it can uncover high-loss scattering on the perforated plastic plates. For example, a 3D omnidirectional acoustic carpet cloaking was designed with a pyramid-shaped structure (Fig. 10d). The scheme of experimental setup is illustrated in Fig. 10e and the results of instantaneous scattered pressure field are shown in Fig. 10f. Besides the cloaking devices based on transformation acoustics, acoustic cloaking can also be realized by using scattering cancellation method to eliminate the scattered acoustic field between background and system [76–82].

### 2.7 Acoustic lenses

Concepts of optical or electromagnetic lenses can also be applied to acoustics. In this sub-section, we will review multiple designs of acoustic metamaterials for realization of acoustic lenses, including superlens and hyperlens for sub-diffraction imaging, Luneburg lens for focusing acoustic waves without aberration and Eaton lens for control and manipulation of acoustic waves with arbitrary refraction angles in spherical geometry.

#### 2.7.1 Superlens and hyperlens

*k*

_{ r }, \(k_{\theta }\) are wavevectors in the radial and azimuthal direction, respectively. In conventional medium, since both radial and tangential mass density are positive, the dispersion profile representing

*k*

_{ r }as a function of \(k_{\theta }\) will be circular according to Eq. (11) leading to the existence of a cutoff wavevector that limits the tangential spatial frequency, resulting in the diffraction limit. In the case of hyperlens, since \(\rho_{r}\) is negative, the dispersion described in Eq. (11) will have a hyperbolic form in which the radial wavevector

*k*

_{ r }can still be positive for a very large value of the tangential wavevector \(k_{\theta } .\) In other words, the high frequency information of objects which cannot be resolved in the conventional system is transformed to propagating waves and brought to the far-field. Consequently, a magnified fine feature information can be acquired by using the hyperlens.

*θ*direction (Fig. 13a). Because of the huge difference of mass densities between brass and air, highly anisotropic dispersion relation is obtained, leading to imaging enhancement as shown in Fig. 13b. The negative refractive index and enhanced imaging were also achieved by arranging proper layers of perforated plates with hyperbolic dispersion [10, 11]. More recently, Shen et al. [12] realized a hyperlens utilizing multiple arrays of clamped thin plates similar to membranes with the negative mass density, yielding a hyperbolic dispersion.

#### 2.7.2 Luneburg and Eaton lens

*R*is the radius of the lens and \(0 \le r \le R.\) The wave equation of acoustic Luneburg lens is governed by mass density and bulk modulus. But, the bulk modulus inside and outside of the lens is assumed to be constant. Therefore, variable mass density inside of the lens is the main factor for acoustic Luneburg lens which can control the refractive index gradually. Recently, three-dimensional Luneburg lens was demonstrated at optical frequency range [106]. This kind of lenses in acoustics could be considered as a candidate for harvesting energy or sonar system in practical use.

Eaton lens as an extension of GRIN lens for arbitrary refraction angles in spherical geometry can also be realized in acoustics by controlling the mass density inside of the lens with constant bulk modulus. 180° acoustic Eaton lens has been recently reported but, the complete demonstration still seems to be remained [107]. More efforts of metamaterial engineering are necessary for realization of Eaton lens which is able to work with various refraction angles.

## 3 Conclusion

Together with the advent of electromagnetic and optical metamaterials, the field of acoustic metamaterials has expanded marvelously over the past 15 years. Although theoretical studies including analytical models and numerical tools have been well explored, many of significant challenges remain in the practical implementation of acoustic metamaterials. With the purpose to have a unified overview of the study progress, we have described research highlights with particular attention given to the sound waves in this review. Acoustic parameters, the mass density and bulk modulus, which are analogous to the permittivity and permeability in electromagnetic waves are identified as key parameters for acoustic wave science. We now know that various values of effective mass density and bulk modulus including negative values can be achieved by engineering mass-spring systems (or membranes) and Helmholtz resonators, respectively. Implementation of these structures for metamaterials with a single negative parameter, double negative parameters, near-zero and approaching-infinity mass density were then reviewed. In addition, space-coiling metamaterials were presented to realize negative, higher and zero refrative index not utilizing local resonance systems. We also reviewed some applications of acoustic cloaking with different approaches such as transformation acoustics, highly anisotropic parameters and scattering cancellation method. And then, superlens and hyperlens for diffraction limit breaking were well explained. Lastly, Luneburg and Eaton lens based on gradient index profile for manipulation of sound waves were introduced in terms of focusing and arbitrary refraction angles, respectively. Nowadays, acoustic metamaterials inspired by electromagnetic and optical metamaterials recently started influencing to not only elasticity but also seismology and even thermodynamics. Although our review didn’t include other fields of metamaterials, we also hope all research area of metamaterials will lead to advanced science and technology.

## Declarations

### Authors’ contributions

DL and DMN wrote the manuscript. JR guided manuscript preparation. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Funding

The research was supported by Young Investigator Research program (NRF-2015R1C1A1A02036464) and Global Frontier program (CAMM-2014M3A6B3063708) through the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science, ICT and Future Planning (MSIP) of Korean government.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- V.G. Veselago, The electrodynamics of substances with simultaneously negative values of ε and μ. Sov. Phys. Uspekhi
**10**, 509–514 (1968)View ArticleGoogle Scholar - J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, Magnetism from conductors and enhanced nonlinear phenomena. IEEE Trans. Microw. Theory Tech.
**47**, 2075–2084 (1999)View ArticleGoogle Scholar - J.B. Pendry, A.J. Holden, W.J. Stewart, I. Youngs, Extremely low frequency plasmons in metallic mesostructures. Phys. Rev. Lett.
**76**, 4773 (1996)View ArticleGoogle Scholar - D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Composite medium with simultaneously negative permeability and permittivity. Phys. Rev. Lett.
**84**, 4184 (2000)View ArticleGoogle Scholar - R.A. Shelby, D.R. Smith, S. Schultz, Experimental verification of a negative index of refraction. Science
**292**, 77–79 (2001)View ArticleGoogle Scholar - J Bassan, V Srinivasan, in
*3D Printing and the Future of Manufacturing*. CSC Web, 2012. http://www.csc.com/innovation/insights/92142-3d_printing_and_the_future_of_manufacturing. Accessed 29 Nov 2012 - H. Lipson, M. Kurman,
*Fabricated: The New World of 3D Printing*, 1st edn. (Wiley, Hoboken, 2013), p. 320Google Scholar - E Matias, B Rao, in 3D
*Printing: On Its Historical Evolution and the Implications for Business*. Paper Presented at the 15th Portland international conference on management of engineering and technology (PICMET), Hilton Portland and Executive Tower, Portland, 2–6 August 2015 - J. Li, L. Fok, X. Yin, G. Bartal, X. Zhang, Experimental demonstration of an acoustic magnifying hyperlens. Nat. Mater.
**8**, 931–934 (2009)View ArticleGoogle Scholar - J. Christensen, F.J.G. de Abajo, Anisotropic metamaterials for full control of acoustic waves. Phys. Rev. Lett.
**108**, 124301 (2012)View ArticleGoogle Scholar - V.M. Garcia-Chocano, J. Christensen, J. Sanchez-Dehesa, Negative refraction and energy funneling by hyperbolic materials: an experimental demonstration in acoustics. Phys. Rev. Lett.
**112**, 144301 (2014)View ArticleGoogle Scholar - C. Shen, Y. Xie, N. Sui, W. Wang, S.A. Cummer, Y. Jing, Broadband acoustic hyperbolic metamaterial. Phys. Rev. Lett.
**115**, 254301 (2015)View ArticleGoogle Scholar - S.A. Cummer, D. Schurig, One path to acoustic cloaking. New J. Phys.
**9**, 45 (2007)View ArticleGoogle Scholar - D. Torrent, J. Sanchez-Dehesa, Acoustic cloaking in two dimensions: a feasible approach. New J. Phys.
**10**, 063015 (2008)View ArticleGoogle Scholar - H. Chen, C.T. Chan, Acoustic cloaking in three dimensions using acoustic metamaterials. Appl. Phys. Lett.
**91**, 183518 (2007)View ArticleGoogle Scholar - S. Zhang, C. Xia, N. Fang, Broadband acoustic cloak for ultrasound waves. Phys. Rev. Lett.
**106**, 024301 (2011)View ArticleGoogle Scholar - Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Locally resonant sonic materials. Science
**289**, 1734–1736 (2000)View ArticleGoogle Scholar - G.W. Milton, J.R. Willis, On modifications of Newton’s second law and linear continuum elastodynamics. Proc. R. Soc. A
**463**, 855–880 (2007)View ArticleGoogle Scholar - G Ma, Dissertation, Hong Kong University of Science and Technology, 2012
- S. Yao, X. Zhou, G. Hu, Experimental study on negative effective mass in a 1D mass–spring system. New J. Phys.
**10**, 043020 (2008)View ArticleGoogle Scholar - J. Mei, G. Ma, M. Yang, J. Yang, P. Sheng, in
*Acoustic Metamaterials and Phononic Crystals*, vol. 173, ed. by P.A. Deymier (Springer, Berlin, 2013), pp. 159–199 - S.H. Lee, O.B. Wright, Origin of negative density and modulus in acoustic metamaterials. Phys. Rev. B
**93**, 024302 (2016)View ArticleGoogle Scholar - Z. Yang, J. Mei, M. Yang, N.H. Chan, P. Sheng, Membrane-type acoustic metamaterial with negative dynamic mass. Phys. Rev. Lett.
**101**, 204301 (2008)View ArticleGoogle Scholar - S.H. Lee, C.M. Park, Y.M. Seo, Z.G. Wang, C.K. Kim, Acoustic metamaterial with negative density. Phys. Lett. A
**373**, 4464–4469 (2009)View ArticleGoogle Scholar - C.J. Naify, C.M. Chang, G. McKnight, S. Nutt, Transmission loss and dynamic response of membrane-type locally resonant acoustic metamaterials. J. Appl. Phys.
**108**, 114905 (2010)View ArticleGoogle Scholar - C.J. Naify, C.M. Chang, G. McKnight, S. Nutt, Transmission loss of membrane-type acoustic metamaterials with coaxial ring masses. J. Appl. Phys.
**110**, 124903 (2011)View ArticleGoogle Scholar - C.J. Naify, C.M. Chang, G. McKnight, F. Scheulen, S. Nutt, Membrane-type metamaterials: transmission loss of multi-celled arrays. J. Appl. Phys.
**109**, 104902 (2011)View ArticleGoogle Scholar - G. Ma, M. Yang, S. Xiao, Z. Yang, P. Sheng, Acoustic metasurface with hybrid resonances. Nat. Mater.
**13**, 873–878 (2014)View ArticleGoogle Scholar - M. Yang, C. Meng, C. Fu, Y. Li, Z. Yang, P. Sheng, Subwavelength total acoustic absorption with degenerate resonators. Appl. Phys. Lett.
**107**, 104104 (2015)View ArticleGoogle Scholar - N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, X. Zhang, Ultrasonic metamaterials with negative modulus. Nat. Mater.
**5**, 452–456 (2006)View ArticleGoogle Scholar - S.H. Lee, C.M. Park, Y.M. Seo, Z.G. Wang, C.K. Kim, Acoustic metamaterial with negative modulus. J. Phys. Condens. Matter
**21**, 175704 (2009)View ArticleGoogle Scholar - Y. Cheng, J.Y. Xu, X.J. Liu, One-dimensional structured ultrasonic metamaterials with simultaneously negative dynamic density and modulus. Phys. Rev. B
**77**, 045134 (2008)View ArticleGoogle Scholar - C. Ding, L. Hao, X. Zhao, Two-dimensional acoustic metamaterial with negative modulus. J. Appl. Phys.
**108**, 074911 (2010)View ArticleGoogle Scholar - N.R. Mahesh, P. Nair, in
*Experimental and Theoretical Investigation of Acoustic Metamaterial with Negative Bulk*-*Modulus*. Paper presented at the Proceedings of the COMSOL Users Conference, Bangalore, 4–5 November, 2011 - J. Li, W. Wang, Y. Xie, B.I. Popa, S.A. Cummer, A sound absorbing metasurface with coupled resonators. Appl. Phys. Lett.
**109**, 091908 (2016)View ArticleGoogle Scholar - S.H. Kim, S.H. Lee, Air transparent soundproof window. AIP Adv.
**4**, 117123 (2014)View ArticleGoogle Scholar - J. Li, K. Fung, Z. Liu, P. Sheng, C.T. Chan, in
*Physics of Negative Refraction and Negative Index Materials*, vol. 98, eds. by C.M. Krowne, Y. Zhang (Springer, Heidelberg, 2007), pp. 183–215 - Z. Liu, C.T. Chan, P. Sheng, Analytic model of phononic crystals with local resonances. Phys. Rev. B
**71**, 014103 (2005)View ArticleGoogle Scholar - Y. Wu, Y. Lai, Z.Q. Zhang, Effective medium theory for elastic metamaterials in two dimensions. Phys. Rev. B
**76**, 205313 (2007)View ArticleGoogle Scholar - X. Zhou, G. Hu, Analytic model of elastic metamaterials with local resonances. Phys. Rev. B
**79**, 195109 (2009)View ArticleGoogle Scholar - S.H. Lee, C.M. Park, Y.M. Seo, Z.G. Wang, C.K. Kim, Composite acoustic medium with simultaneously negative density and modulus. Phys. Rev. Lett.
**104**, 054301 (2010)View ArticleGoogle Scholar - L. Fok, X. Zhang, Negative acoustic index metamaterial. Phys. Rev. B
**83**, 214304 (2011)View ArticleGoogle Scholar - Z. Liang, J. Li, Extreme acoustic metamaterial by coiling up space. Phys. Rev. Lett.
**108**, 114301 (2012)View ArticleGoogle Scholar - Y. Li, B. Liang, X. Tao, X.F. Zhu, X.Y. Zou, J.C. Cheng, Acoustic focusing by coiling up space. Appl. Phys. Lett.
**101**, 233508 (2012)View ArticleGoogle Scholar - Y. Li, B. Liang, X.Y. Zou, J.C. Cheng, Extraordinary acoustic transmission through ultrathin acoustic metamaterials by coiling up space. Appl. Phys. Lett.
**103**, 063509 (2013)View ArticleGoogle Scholar - Z. Liang, T. Feng, S. Lok, F. Liu, K.B. Ng, C.H. Chan, J. Wang, S.H. Han, S.Y. Lee, J. Li, Space-coiling metamaterials with double negativity and conical dispersion. Sci. Rep.
**3**, 1614 (2013)Google Scholar - Y. Xie, B.I. Popa, L. Zigoneanu, S.A. Cummer, Measurement of a broadband negative index with space-coiling acoustic metamaterials. Phys. Rev. Lett.
**110**, 175501 (2013)View ArticleGoogle Scholar - S.K. Maurya, A. Pandey, S. Shukla, S. Saxena, Double negativity in 3D space coiling metamaterials. Sci. Rep.
**6**, 33683 (2016)View ArticleGoogle Scholar - Y. Jing, J. Xu, N. Fang, Numerical study of a near-zero-index acoustic metamaterial. Phys. Lett. A
**376**, 2834–2837 (2012)View ArticleGoogle Scholar - G. Ma, M. Yang, Z. Yang, P. Sheng, Low-frequency narrow-band acoustic filter with large orifice. Appl. Phys. Lett.
**103**, 011903 (2013)View ArticleGoogle Scholar - S. Xiao, G. Ma, Y. Li, Z. Yang, P. Sheng, Active control of membrane-type acoustic metamaterial by electric field. Appl. Phys. Lett.
**106**, 091904 (2015)View ArticleGoogle Scholar - R. Fleury, A. Alu, Extraordinary sound transmission through density-near-zero ultranarrow channels. Phys. Rev. Lett.
**111**, 055501 (2013)View ArticleGoogle Scholar - Q. Wei, Y. Cheng, X.J. Liu, Acoustic total transmission and total reflection in zero-index metamaterials with defects. Appl. Phys. Lett.
**102**, 174104 (2013)View ArticleGoogle Scholar - J.J. Park, K.J. Lee, O.B. Wright, M.K. Jung, S.H. Lee, Giant acoustic concentration by extraordinary transmission in zero-mass metamaterials. Phys. Rev. Lett.
**110**, 244302 (2013)View ArticleGoogle Scholar - J. Mei, G. Ma, M. Yang, Z. Yang, W. Wen, P. Sheng, Dark acoustic metamaterials as super absorbers for low-frequency sound. Nat. Commun.
**3**, 756 (2012)View ArticleGoogle Scholar - A.J. Ward, J.B. Pendry, Refraction and geometry in Maxwell’s equations. J. Mod. Opt.
**43**, 773–793 (1996)View ArticleGoogle Scholar - U. Leonhardt, Optical conformal mapping. Science
**312**, 1777–1780 (2006)View ArticleGoogle Scholar - J.B. Pendry, D. Schurig, D.R. Smith, Controlling electromagnetic fields. Science
**312**, 1780–1782 (2006)View ArticleGoogle Scholar - D. Schurig, J.J. Mock, B.J. Justice, S.A. Cummer, J.B. Pendry, A.F. Starr, D.R. Smith, Metamaterial electromagnetic cloak at microwave frequencies. Science
**314**, 977–980 (2006)View ArticleGoogle Scholar - D. Schurig, J.B. Pendry, D.R. Smith, Calculation of material properties and ray tracing in transformation media. Opt. Express
**14**, 9794–9804 (2006)View ArticleGoogle Scholar - J.B. Pendry, in
*Metamaterials and the Control of Electromagnetic Fields*. Paper Presented at the 9th Rochester Conference on Coherence and Quantum Optics, Rochester, 10–13 June, 2007 - B. Wood, J.B. Pendry, Metamaterials at zero frequency. J. Phys.: Condens. Matter
**19**, 076208 (2007)Google Scholar - H. Chen, C.T. Chan, P. Sheng, Transformation optics and metamaterials. Nat. Mater.
**9**, 387–396 (2010)View ArticleGoogle Scholar - L. Xu, H. Chen, Conformal transformation optics. Nat. Photon.
**9**, 15–23 (2014)View ArticleGoogle Scholar - M. Kraft, Y. Luo, S.A. Maier, J.B. Pendry, Designing plasmonic gratings with transformation optics. Phys. Rev. X
**5**, 031029 (2015)Google Scholar - Y.S. Kim, S.Y. Lee, J.W. Ryu, I.B. Kim, J.H. Han, H.S. Tae, M.H. Choi, B.K. Min, Designing whispering gallery modes via transformation optics. Nat. Photon.
**10**, 647–652 (2016)View ArticleGoogle Scholar - H. Chen, C.T. Chan, Acoustic cloaking and transformation acoustics. J. Phys. D Appl. Phys.
**43**, 113001 (2010)View ArticleGoogle Scholar - A.N. Norris, Acoustic cloaking theory. Proc. R. Soc. A
**464**, 2411–2434 (2008)View ArticleGoogle Scholar - A.N. Norris, Acoustic cloaking. Acoust. Today
**11**, 38–46 (2015)Google Scholar - R. Liu, C. Ji, J.J. Mock, J.Y. Chin, T.J. Cui, D.R. Smith, Broadband ground-plane cloak. Science
**323**, 366–369 (2009)View ArticleGoogle Scholar - J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, An optical cloak made of dielectrics. Nat. Mater.
**8**, 568–571 (2009)View ArticleGoogle Scholar - G.W. Milton, M. Briane, J.R. Willis, On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys.
**8**, 248 (2006)View ArticleGoogle Scholar - J. Li, J.B. Pendry, Hiding under the carpet: a new strategy for cloaking. Phys. Rev. Lett.
**101**, 203901 (2008)View ArticleGoogle Scholar - B.I. Popa, L. Zigoneanu, S.A. Cummer, Experimental acoustic ground cloak in air. Phys. Rev. Lett.
**106**, 253901 (2011)View ArticleGoogle Scholar - L. Zigoneanu, B.I. Popa, S.A. Cummer, Three-dimensional broadband omnidirectional acoustic ground cloak. Nat. Mater.
**13**, 352–355 (2014)View ArticleGoogle Scholar - M.D. Guild, A. Alu, M.R. Haberman, Cancellation of acoustic scattering from an elastic sphere. J. Acoust. Soc. Am.
**129**, 1355–1365 (2011)View ArticleGoogle Scholar - V.M. Garcia-Chocano, L. Sanchis, A. Diaz-Rubio, J. Martinez-Pastor, F. Cervera, R. Llopis-Pontiveros, J. Sanchez-Dehesa, Acoustic cloak for airborne sound by inverse design. Appl. Phys. Lett.
**99**, 074102 (2011)View ArticleGoogle Scholar - W. Kan, B. Liang, X. Zhu, R. Li, X. Zou, H. Wu, J. Yang, J. Cheng, Acoustic illusion near boundaries of arbitrary curved geometry. Sci. Rep.
**3**, 1427 (2013)Google Scholar - L. Sanchis, V.M. Garcia-Chocano, R. Llopis-Pontiveros, A. Climente, J. Martinez-Pastor, F. Cervera, J. Sanchez-Dehesa, Three-dimensional axisymmetric cloak based on the cancellation of acoustic scattering from a sphere. Phys. Rev. Lett.
**110**, 124301 (2013)View ArticleGoogle Scholar - M.D. Guild, A. Alu, M.R. Haberman, Cloaking of an acoustic sensor using scattering cancellation. Appl. Phys. Lett.
**105**, 023510 (2014)View ArticleGoogle Scholar - W. Kan, V.M. Garcia-Chocano, F. Cervera, B. Liang, X.Y. Zou, L.L. Yin, J. Cheng, J. Sanchez-Dehesa, Broadband acoustic cloaking within an arbitrary hard cavity. Phys. Rev. Appl.
**3**, 064019 (2015)View ArticleGoogle Scholar - C.A. Rohde, T.P. Martin, M.D. Guild, C.N. Layman, C.J. Naify, M. Nicholas, A.L. Thangawng, D.C. Calvo, G.J. Orris, Experimental demonstration of underwater acoustic scattering cancellation. Sci. Rep.
**5**, 13175 (2015)View ArticleGoogle Scholar - J.B. Pendry, Negative refraction makes a perfect lens. Phys. Rev. Lett.
**85**, 3966 (2000)View ArticleGoogle Scholar - S. Zhang, L. Yin, N. Fang, Focusing ultrasound with an acoustic metamaterial network. Phys. Rev. Lett.
**102**, 194301 (2009)View ArticleGoogle Scholar - G.V. Eleftheriades, A.K. Iyer, P.C. Kremer, Planar negative refractive index media using periodically LC loaded transmission lines. IEEE Trans. Microw. Theory Tech.
**50**, 2702–2712 (2002)View ArticleGoogle Scholar - A.K. Iyer, P.C. Kremer, G.V. Eleftheriades, Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial. Opt. Express
**11**, 696–708 (2003)View ArticleGoogle Scholar - A. Grbic, G.V. Eleftheriades, Overcoming the diffraction limit with a planar left-handed transmission-line lens. Phys. Rev. Lett.
**92**, 117403 (2004)View ArticleGoogle Scholar - Z. Jacob, L.V. Alekseyev, E. Narimanov, Optical hyperlens: far-field imaging beyond the diffraction limit. Opt. Express
**14**, 8247–8256 (2006)View ArticleGoogle Scholar - A. Salandrino, N. Engheta, Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations. Phys. Rev. B
**74**, 075103 (2006)View ArticleGoogle Scholar - B. Wood, J.B. Pendry, D.P. Tsai, Directed subwavelength imaging using a layered metal-dielectric system. Phys. Rev. B
**74**, 115116 (2006)View ArticleGoogle Scholar - R.K. Luneburg, M. Herzberger,
*Mathematical Theory of Optics*, 1st edn. (Brown University, Providence, 1944), p. 448Google Scholar - N. Kundtz, D.R. Smith, Extreme-angle broadband metamaterial lens. Nat. Mater.
**9**, 129–132 (2010)View ArticleGoogle Scholar - A.D. Falco, S.C. Kehr, U. Leonhardt, Luneburg lens in silicon photonics. Opt. Express
**19**, 5156–5162 (2011)View ArticleGoogle Scholar - T.M. Chang, G. Dupont, S. Enoch, S. Guenneau, Enhanced control of light and sound trajectories with three-dimensional gradient index lenses. New J. Phys.
**14**, 035011 (2012)View ArticleGoogle Scholar - J. Hunt, T. Tyler, S. Dhar, Y.J. Tsai, P. Bowen, S. Larouche, N.M. Jokerst, D.R. Smith, Planar, flattened Luneburg lens at infrared wavelengths. Opt. Express
**20**, 1706–1713 (2012)View ArticleGoogle Scholar - Y.L. Loo, Y. Yang, N. Wang, Y.G. Ma, C.K. Ong, Broadband microwave Luneburg lens made of gradient index metamaterials. J. Opt. Soc. Am. A
**29**, 426–430 (2012)View ArticleGoogle Scholar - S.H. Kim, in
*Cylindrical Acoustic Luneburg Lens*. Paper presented at the 8th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics (Metamaterials), Copenhagen, 25–30 August 2014 - A.S. Gutman, Modified Luneburg lens. J. Appl. Phys.
**25**, 855–859 (1954)View ArticleGoogle Scholar - S.P. Morgan, General solution of the Luneburg lens problem. J. Appl. Phys.
**29**, 1358–1368 (1958)View ArticleGoogle Scholar - C.A. Boyles, Wave theory of an acoustic Luneburg lens. II. The theory of variable density lenses. Nat. Nanotechnol.
**45**, 351–364 (1969)Google Scholar - T. Zentgraf, Y. Liu, M.H. Mikkelsen, J. Valentine, X. Zhang, Plasmonic Luneburg and Eaton lenses. Nat. Nanotechnol.
**6**, 151–155 (2011)View ArticleGoogle Scholar - A. Climente, D. Torrent, J. Sanchez-Dehesa, Sound focusing by gradient index sonic lenses. Appl. Phys. Lett.
**97**, 104103 (2010)View ArticleGoogle Scholar - J.T. Welter, S. Sathish, D.E. Christensen, P.G. Brodrick, M.R. Cherry, Focusing of longitudinal ultrasonic waves in air with an aperiodic flat lens. J. Acoust. Soc. Am.
**130**, 2789–2796 (2011)View ArticleGoogle Scholar - A. Climente, D. Torrent, J. Sanchez-Dehesa, Gradient index lens for flexural waves based on thickness variations. Appl. Phys. Lett.
**105**, 064101 (2014)View ArticleGoogle Scholar - S.H. Kim,
*Sound focusing by acoustic luneburg lens*(2014). arXiv:1409.5489 - Y.Y. Zhao, Y.L. Zhang, M.L. Zheng, X.Z. Dong, X.M. Duan, Z.S. Zhao, Three-dimensional Luneburg lens at optical frequencies. Laser Photon. Rev.
**10**, 665–672 (2016)View ArticleGoogle Scholar - S.H. Kim, V.S. Pham, M.P. Das,
*Acoustic eaton lens array and its fluid application*(2016). arXiv:1607.02913