Acoustic metamaterials: From local resonances to broad horizons

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Science Advances  26 Feb 2016:
Vol. 2, no. 2, e1501595
DOI: 10.1126/sciadv.1501595


  • Fig. 1 The origin of anomalous constitutive parameters in acoustics.

    (A) A spring-coupled mass-in-mass oscillator. M2 is assumed to slide without friction inside a cavity formed by M1, and K is the spring constant. (B) The oscillator’s apparent mass Embedded Image plotted as a function of angular frequency ω, where Embedded Image is the resonant frequency. Embedded Image is divergent at ω0 and can take negative values in a narrow frequency region that is shaded gray. (C) If there are two resonances, the average displacement 〈X〉 can cross zero at an antiresonance frequency Embedded Image, at which the effective mass/mass density displays a frequency dispersion similar to that shown in (B). Here, the red and green dashed curves show the displacement associated with the first and second resonances (denoted ω1 and ω2), respectively. The black solid curve represents the sum of the two displacements, and it crosses zero at Embedded Image.

  • Fig. 2 Initial realizations of locally resonant acoustic metamaterials.

    (A) Images of the sample that first realized a local resonance-induced anomalous mass effect (16). Left: The cut-away view of a sample unit cell consisting of a small metallic sphere coated by a thin uniform layer of silicone rubber. Right: The sample made by using epoxy to glue together the units shown on the left. The effective frequencies for total reflection by the sample were shown to correspond to a wavelength that is between one and two orders of magnitude larger than the size of the lattice constant, which is 1.55 cm. (B) An illustration of the sample, comprising a series of Helmholtz resonators connected to one side of a conduit, that realized the frequency dispersion for the bulk modulus (18).

  • Fig. 3 Single membrane with negative effective mass density.

    (A) A schematic drawing of a typical DMR (36). (B) Out-of-plane displacement amplitude |W(x)| of the low-frequency eigenmodes (measured with a laser Doppler vibrometer) of two DMRs (194). Light yellow indicates a large normal displacement amplitude, whereas darker colors indicate small or no normal displacement. Cyan circles delineate the edge of the membrane and the position of the platelet. The first mode is characterized by the large up-and-down oscillation of the platelet, pulling along the entire structure (left). In the second mode, the platelet is almost motionless (right), and the oscillation amplitude is largest in the surrounding membrane. (C) A schematic illustration indicating that, as a result of the deep-subwavelength size of the DMR, the large in-plane wave vectors k only contribute to evanescent waves (blue dashed lines), owing to the fact that the lateral fast variations of the up-and-down displacements tend to cancel each other in air, and the net amplitude decays exponentially as a result. The far-field propagating wave is determined by the k of the surface-averaged component of the normal displacement (red dashed lines). In the system shown here, the DMR (black) is blocking a one-dimensional waveguide, and a planar sound wave impinges from the bottom. The reflected field is not shown. (D) Measured amplitude transmission coefficient of a DMR (black solid curve, left axis) and the real part of the calculated effective mass density Embedded Image (red dashed curve, right axis). Various features are explained in the text.

  • Fig. 4 Coupled membranes giving rise to both mass and modulus dispersions.

    (A) A schematic drawing of a DMR with two coupled membranes. Two identical membranes decorated with platelets are placed closely together, sealing a layer of gas in between. A rigid ring is added for extra tunability. (B) Normal displacement profiles of three low-frequency eigenmodes. Red symbols delineate the measured profiles using the laser vibrometer, whereas the black solid curves are the results of finite-element simulations. (C) Left: The transmission coefficient of the structure shown in (A), plotted as a function of frequency. Here, the circles denote the measured result, and the solid curve indicates the calculated result. Middle: Effective mass density Embedded Image and bulk modulus Embedded Image, plotted as a function of frequency. Right: The real part of the effective wave vector Embedded Image, plotted as a function of frequency. The total thickness of the DMR is 2d. Double negativity is seen in the region shaded in gray (22). The effective parameters here are extracted using the method detailed in Yang et al. (31).

  • Fig. 5 Superresolution with local resonances.

    (A) The type of dispersion relation that is commonly used for superresolution and deep-subwavelength focusing with acoustic metamaterials. Here, the blue dotted-dashed line is the “sound line” (that is, the dispersion of acoustic wave in the homogeneous background medium), and the red dashed line is that of the local resonance with eigenfrequency f0. Coupling between the two induces anticrossing in the vicinity of f0 and gives rise to the dispersion delineated by the two black curves. The region shaded in gray is a bandgap. In free space, only the k components within the blue-shaded region are accessible. However, for the lower branch, the k components much larger than those available in free space (the blue-shaded region) become accessible. (B) Superresolution focusing (left; showing the pressure distribution) and subwavelength wave guiding (right; showing the normalized pressure amplitude) achieved by using a two-dimensional array of air-filled cavity resonators (soda cans). The resonances of the soda cans yield a dispersion, as shown in (A) (47, 48). (C) A two-dimensional array of subwavelength waveguides is shown on the left. The thickness of the lens h is the same as the operating wavelength in air (not drawn to scale in this schematic drawing). These waveguides can support Fabry-Pérot resonances that have flat dispersions, which are useful for achieving superresolution imaging (49, 50). In the imaging result shown on the right, light color represents stronger pressure intensity. The object is in the shape of the character “E” (inset). The stroke width of “E” is ~λ/50. The lens is placed close to the object, and an image is formed on the other side, where the shape “E” can be recognized (50).

  • Fig. 6 Acoustic realizations of superlens and hyperlens.

    (A) Top: A slab of doubly negative medium can bring diverging waves into two foci: one inside the slab, and the other one outside. Bottom: The same slab can amplify evanescent waves, thereby theoretically enabling the formation of a perfect image. (B) Experimental demonstration of the imaging capability of an acoustic superlens (30). Here, cavity resonators (soda cans, represented by black circles) were arranged into a honeycomb lattice. It is clearly seen that the near field was sustained and even amplified by the metamaterial slab. Negative refraction and the consequent foci are delineated by white dashed arrows. The normalized pressure intensity field is displayed as a color map. The source (red dots) had an amplitude full width at half maximum of λ/5, whereas the measured image size was λ/15. (C) Three distinctive equifrequency contours. The black circle represents that of a homogeneous material. Anisotropy along Embedded Image and Embedded Image can distort the contour into an ellipse (red), in which a large kr can be accessed. However, if the material’s parameter is negative along Embedded Image but positive along Embedded Image, then the contour becomes hyperbolic, wherein kr and kθ are no longer bounded (blue curves). (D) The strong anisotropy in kr and kθ, such as that indicated by the red ellipse in (C), used for superresolution imaging, as shown experimentally in Li et al. (63). Here, the fan-like structure has stripes alternating between air and brass. From the effective medium theory, the effective mass is highly anisotropic along the Embedded Image and Embedded Image directions. The center circle of the device has a diameter of about one wavelength, in which three sound sources were closely placed with subwavelength separations, represented by the red dots. An image of the sound sources is also shown in the lower-right inset. Red/blue represents positive/negative pressure, with the three clearly resolved regions being representative of the magnified image of the three sound sources.

  • Fig. 7 Space-coiling and acoustic metasurfaces.

    (A) An example of the space-coiling structure and the relevant sound pressure field inside (84). The color indicates the phase of the propagating sound wave. Here, a large phase delay Δφ ≫ Δφ0 = k0a can be caused by the coiled channels, where k0 is the wave vector in the background fluid and a is the exterior dimension of the cell. With proper design, such structures can be used for many novel effects, such as the negative-refracting prism shown in (B) (84). Here, space-coiling unit cells are assembled into a prism. Sound incidents from the left, and encounters of negative refraction as it emerges from the prism, are indicated by white arrows. Red/blue represents positive/negative pressure. (C) A design of reflective acoustic metasurface that is capable of generating phase changes up to 2π. This can be seen from the color (red and blue represent positive pressure and negative pressure, respectively) and the undulations of the stripes (108). The metasurface is shown on the lower left, and the wave is incident normally on the surface. The ridge of red stripe sections indicates the emergence of a reflected wave at a nonspecular angle. Inset: Implementation of a slightly modified space-coiling design of the metasurface, with a lateral gradient of phase delays (109). (D) A design of the metasurface that generates negative refraction for the transmitted wave. Left: The simulation result (white arrows delineate incident and refracted beam directions). Right: The experimentally measured pressure map (113) (red and blue represent positive pressure and negative pressure, respectively) (top) and a photographic image of a section of the actual sample (bottom).

  • Fig. 8 Acoustic absorption by DMRs.

    (A) Left: Numerically simulated elastic bending energy density of a soft membrane decorated with rigid platelets, delineated by blue curves. Light color represents high energy density. The energy is highly concentrated in the small areas along the perimeters of the platelets and along the sample boundary (119). Right: A photographic image of such a membrane absorber. (B) Total absorption of low-frequency sound at multiple frequencies using hybrid resonances. The left panel shows the absorption coefficient of a three-unit membrane metasurface (structure shown in the right inset). The markers are measured data, and the solid curve represents simulation. The membrane metasurface unit cell comprises a stretched membrane decorated with a rigid platelet, backed by a thin layer of sealed gas (thickness expressed in centimeters). The hybrid resonance structure is fabricated on top of a hard reflecting surface (left inset). The right panel of (B) shows the vibration profile of the membrane at the hybrid resonant frequency. Red markers represent experimentally measured data. The surface-averaged normal displacement (blue dashed line; not drawn to scale) matches the amplitude of the incident sound—an indication of impedance matching. However, the normal displacement can be much larger locally, suggesting a very large energy density in the form of the deaf component that couples only to evanescent waves (120).

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