Figure 1

(a) Schematic representation of an acoustic metamaterial consisting of stacked hard plates perforated by a periodic hole array. For simplicity, we consider a square array of period $d_x$ and holes of diameter $D$. The plates of thickness $t$ are separated by a distance $d_z-t$ (i.e., the period of the stack is $d_z$). The holes and the interstitial regions between the plates are filled with a sound-supporting medium in which the sound wavelength is $\lambda$. Perfect reflection of sound at the plate hard material is assumed. (b) Dependence of the refraction angle ${\theta }_{\mathrm{ref}}$ and group refraction index $n_{\mathrm{g}}$ on incidence angle ${\theta }_{\mathrm{inc}}$ for $d_z/d_x=2.55$, $t/d_x=1.9$, $D/d_x=0.6$, and $\lambda /d_x=1.95$ (this corresponds to 175 kHz in air for $d_x=1\mathrm{mm}$). The inset shows equifrequency curves both in the homogeneous interstitial medium (circle of radius $k_0=2\pi /\lambda$) and in the metamaterial (hyperbolic dispersion curves) as a function of wave-vector components parallel and perpendicular to the plates, $k_{\parallel }$ and $k_z$, respectively.Reuse & Permissions

Figure 2

Pressure-field simulations for a finite metamaterial containing 250 layers with the same geometrical parameters as in Fig. 1. (a) Negative refraction for a Gaussian beam incident with angle ${\theta }_{\mathrm{inc}}=42°$ from the left with wavelength $\lambda =1.95d_x$ (175 kHz in air for $d_x=1\mathrm{mm}$.). Broken lines indicate the slab interfaces. (b),(c) Phase (b) and group (c) front propagation are both along the incidence direction in this case. (d) The same as (a) for $\lambda =3.56d_x$ (95 kHz in air for $d_x=1\mathrm{mm}$.), giving rise to positive refraction. (e),(f) Backward-wave propagation is observed at this new wavelength.Reuse & Permissions

Figure 3

Evidence of sound focusing in the calculated pressure-field distribution for an anisotropic metamaterial slab. (a) Frequency and parallel wave-vector dependence of the transmittance through a metamaterial slab consisting of 50 steel plates immersed in air with parameters $t/d_x=2.5$, $d_z/d_x=3.5$, $D/d_x=0.6$, and $d_x=1\mathrm{mm}$. The transmittance is defined as the squared amplitude of the far-side zero-order wave upon irradiation with a wave of unit amplitude at the near-side interface. (b) Focal length from the far side of the slab to the image (right) and FWHM of the image spot (left) as a function of sound frequency. The source is placed at a distance from the slab equal to the slab thickness $W$ in all cases. (c) Near-field intensity distribution showing the image at a distance $\sim 0.6\mathrm{W}$ from the far side of the slab for a frequency of 62 kHz, which corresponds to $\lambda /d_x=5.52$. The inset illustrates the transversal intensity profiles around the source (a pointlike monopole) and the image ($\mathrm{FWHM}\approx 0.2\lambda$).Reuse & Permissions