PHYSICAL REVIEW B 75, 195447 ͑2007͒ Surface resonant states and superlensing in acoustic metamaterials Muralidhar Ambati, Nicholas Fang, Cheng Sun, and Xiang Zhang* Center for Scalable and Integrated Nanomanufacturing (SINAM), University of California, Berkeley, California 94720, USA ͑Received 19 April 2007; published 31 May 2007͒ We report that the negative material responses of acoustic metamaterials can lead to a plethora of surface resonant states. We determine that negative effective-mass density is the necessary condition for the existence of surface states on acoustic metamaterials. We offer the microscopic picture of these unique surface states; in addition, we find that these surface excitations enhance the transmission of evanescent pressure fields across the metamaterial. The evanescent pressure fields scattered from an object can be resonantly coupled and enhanced at the surface of the acoustic metamaterial, resulting in an image with resolution below the diffrac- tion limit. This concept of acoustic superlens opens exciting opportunities to design acoustic metamaterials for ultrasonic imaging. DOI: 10.1103/PhysRevB.75.195447 PACS number͑s͒: 43.20.ϩg, 43.35.ϩd, 46.40.Ff I. INTRODUCTION metamaterial half-space ͓Fig. 1͑a͔͒. Medium I is considered as a fluid, with positive mass density and bulk modulus, Advances in electromagnetic metamaterials1,2 offer excel- whereas medium II is taken as an acoustic metamaterial. In a lent opportunities to realize new and exotic phenomena, such homogeneous medium with mass density ␳ and bulk modu- as negative refraction3 and artificial magnetism.4,5 Its acous- lus B, the equilibrium equation and mass continuity equation tic counterpart with negative material properties has been are written in the index notation to relate the pressure P and recently explored.6–9 Acoustic metamaterials with locally velocity field v at any given point, resonant elements can be designed to display anomalous re- i sponse at selected frequencies so that effective dynamic ma- P k P − ␳␻v =0, k v − ␻ =0, ͑1͒ terial properties can be negative. It should be noted that the i i i i B effective negative properties of the metamaterials are dy- ␻ namic and highly dispersive. Such acoustic metamaterials where ki and are the wave vector and angular frequency of offer a possible route to the realization of intriguing phenom- the plane wave. ena; negative refraction was reported earlier in double nega- Let us consider the necessary and sufficient conditions tive acoustic metamaterials.8 that allow the existence of surface states at the interface be- Recent work on phononic crystals—periodic modulation tween a fluid and an acoustic metamaterial, the latter being of density and modulus—also showed that negative refrac- modeled as a structured isotropic fluid. Structured fluid con- tion and focusing of acoustic waves10–12 can be obtained sists of the same fluid as medium I as the matrix with acous- from band-folding effects due to Bragg scattering. Phononic tic resonators in it. The surface states represent elastic per- crystals require that the spatial modulation of impedance is turbations with pressure—maximum at the interface and of the same order as the acoustic wavelength.13–19 Metama- exponentially decaying in the direction perpendicular to the terials provide a major step toward an effective-medium de- interface—schematically shown in Fig. 1͑a͒. By matching scription, as the spatial periodic modulation of impedance is pressure and normal velocity fields at the interface ͑z=0͒,we much smaller than the wavelength. Acoustic metamaterials, obtain the relation between the normal components of wave also referred as sonic crystals, have been considered as spec- vectors: tral gap materials6,20–22 and refractory acoustic devices.23,24 kI kII Metamaterials broaden the range of material responses found z + z =0. ͑2͒ in nature, which can be especially useful in realizing new ␳I ␳II phenomena and in designing novel devices. Yet to the best of The necessary condition for a bound state to exist at the our knowledge, there has been very limited investigation in interface requires that both k ’s have positive imaginary this direction. In this paper, we report a surface resonance on z parts, which implies that the mass density of the acoustic an acoustic metamaterial and examine its unique character- metamaterial should be negative ͑␳IIϽ0͒. Negative density istics. As a result of this surface resonance, we propose and implies that the response of fluid—acceleration—is out of numerically demonstrate an acoustic superlens—analogous phase with the dynamic pressure gradient, and it is obtained to Pendry’s perfect lens25—for sub-diffraction-limited acous- when such negative responses of the oscillators constituting tic imaging. the acoustic metamaterial dominate the background. The dy- namic effective-mass density of the metamaterials can be II. SURFACE STATES ON AN significantly different from the volume averaged mass 26,27 ACOUSTIC METAMATERIAL density, as the multiple scattering is treated in an average sense.26 Negative effective-mass density is reported for fluid- We consider a plane harmonic longitudinal wave analysis solid composites, e.g., soft rubber spheres in water.8 In gen- at a boundary between a fluid half-space and an acoustic eral, acoustic metamaterial can be modeled as a set of har- 1098-0121/2007/75͑19͒/195447͑5͒ 195447-1 ©2007 The American Physical Society AMBATI et al. PHYSICAL REVIEW B 75, 195447 ͑2007͒ III. DISPERSION RELATION OF THE SURFACE STATES We derive the surface-state dispersion between the angu- ␻ lar frequency of the surface states and the wave vector ky by combining Eq. ͑2͒ with the dispersion relations of media ͑ 2͒I,II ͑ 2͒I,II ␻2͑␳ ͒I,II I and II, ky + kz = /B , The tangential wave vector ky is required to be continuous across the interface, resulting in the dispersion relation of surface states, ␳I␳II ␳II ␳I k2 = ␻2 ͩ − ͪ. ͑3͒ y ͑␳II͒2 − ͑␳I͒2 BI BII In addition to the necessary condition ͑␳IIϽ0͒, surface 2 Ͼ 2 ␻2͑␳I I͒ bound states require that ky k0 = /B . This inequality and Eq. ͑3͒ are used to determine the conditions on the prop- erties of acoustic metamaterial that support the surface states. In the case ͉␳II͉Ͼ␳I, the bulk modulus of the acoustic metamaterial should be either positive, BIIϾ0, or negative satisfying the inequality BIIϽ␳IIBI /␳I. On the other hand, for the case ͉␳II͉Ͻ␳I, the bulk modulus of the acoustic metama- terial should be negative and BIIϾ␳IIBI /␳I. We find that when the bulk moduli of two media are the same ͑B͒, the dispersion relation of surface states reduces to ␻2 ␳I␳II 2 ͩ ͪ ͑ ͒ ky = . 4 FIG. 1. ͑Color online͒. Schematic of the surface state at an B ␳I + ␳II interface separating two semi-infinite media. Medium I is consid- Similar to the surface-plasmon resonance in electromagnet- ered to have positive material properties ͑mass density ␳ and bulk modulus B͒, whereas medium II is taken as an acoustic metamate- ics, negative mass density in acoustic metamaterials signifi- rial with generic material properties. All the physical quantities per- cantly broadens the surface resonant band of ky in the limit ␳II͑␻͒→͑ ␳I͒ taining to the medium are designated by the respective medium of − . This broadening is represented by the flat superscript. ͑a͒ Schematic of the pressure profile of surface bound dispersion curve that is wave vector independent ͑Fig. 2͒. states; it is maximum at the interface and decaying exponentially The bulk modulus of both the media is taken to be the same, perpendicular to the interface. The motion trajectory of the material BII=BI. The effective-mass density of medium II—a struc- points is an ellipse; the elliptical rotation of the particles near the tured fluid—is taken in Lorentz form.7 Linear-response func- ͑ ͒ ͑␻2 interface is the same in both media shown clockwise . The eccen- tions such as mass density take the Lorentz form 1/ o tricity of the ellipses depends on the wave vector of the surface state −␻2͒, in the case where a wave of angular frequency ␻ in- with major axes parallel to the interface. ͑b͒ The pattern of the teracts with a medium consisting of localized resonator with displacements of the surface wave is illustrated ͑top view͒. The ␻ 6 resonant frequency o. Here, we consider only the fre- schematic shows that the displacements in y and z directions are 90° quency interval in which the surface state exists: effective- out of phase, with slipping at the interface. mass density of medium II is negative and greater in magni- tude than medium I, ␳II͑␻͒Ͻ0 and ͉␳II͑␻͉͒Ͼ␳I. Dispersion curve for the surface resonance is shown in red, and the monic oscillators. Here, surface resonance corresponds to the linear dispersion curve of medium I is shown in blue. It is oscillation of harmonic oscillators such that there is constant evident that the wave vector of the surface wave is always slipping at the interface. Given effective negative mass den- greater than that of the bulk wave in medium I. sity, the slip arises due to the difference in tangential veloci- ties that changes the direction across the interface. By calcu- IV. CHARACTERISTICS OF THE SURFACE STATES lating the components of the velocities along the y and z axes by Eq. ͑1͒, we obtain that the velocity components are We consider a thin slab of acoustic metamaterial whose shifted in phase by ␲/2. Therefore, the trajectories of the effective-mass density is negative in the near field of an ob- particle motions are ellipses with major axes parallel to the ject. We find that the evanescent pressure fields, scattered interface, and the elliptical rotation of the particles near the from the object, can be resonantly coupled onto the surface interface remains the same in both the media ͓Fig.