Acoustic Focusing and Energy Confinement Based on Multilateral Metasurfaces

Metamaterial-based acoustic wave manipulation shows great potential in effective acoustic energy confinement and low-frequency acoustic isolation. We numerically and theoretically propose here a concept based on multilateral metasurfaces for reflected acoustic focusing and energy confinement. The theoretical phase-shift profile required for reflected wave focusing and governed by the generalized Snell’s law can be discretely realized by appropriately arraying the labyrinthine units in the right sequences. Based on this design, multilateral metasurfaces for acoustic wave focusing and energy confinement under point-source incidence are considered and sufficiently investigated. The coupling effects and multiple reflections between or among metasurfaces, which play a significant role in the energy confinement, are initially analyzed and discussed. We show that the acoustic focusing and confinement increase with the sides of the multilateral metasurfaces as anticipated. In addition to the contribution of the first reflection, multiple reflections also contribute to the acoustic focusing and energy confinement, especially when the metasurfaces are configured in parallel. The proposed multilateral metasurfaces should have excellent performance in acoustic energy confinement in various situations due to the variable designs and strong acoustic focusing capabilities.

Figure 1

(a) Schematic diagram of a typical labyrinthine unit (width $l_x$ and length $l_y$) for the case $(m,n)=(3,2)$, with $m$ and $n$ referring to the number of identical bars (width $w$ and length $l$) in the upper and lower boundaries, respectively. $d$ is the gap width between any two bars, and $t$ is the thickness of the upper and lower boundary plates. (b) Four phase-shift curves towards thickness $h$ evolution curves corresponding to labyrinthine units with four different ($m$, $n$) configurations. Eight units with individual phase shift (solid circles) are carefully chosen to continuously cover the phase range from 0 to $2\pi$ with a step of $\pi /4$ for the minimum viscosity loss.

Figure 2

(a) Schematic illustration of a partially enclosed metasurface structure composed of two perpendicular metasurfaces designed for the acoustic confinement at arbitrary focusing position ($x_f$, $y_f$) in an acoustic field with a point source ($x_s$, $y_s$) centering at (0.5 m, 0.5 m), and the corresponding phase profiles (theory and metasurfaces) for the left (b) and upper (c) metasurfaces at a specific focusing position $(x_f,y_f)=(0.7\mathrm{m},0.3\mathrm{m})$.

Figure 3

(a) Normalized sound pressure field (numerical simulation) of the two perpendicularly connected metasurfaces composed of labyrinthine units for the focusing position (0.7 m, 0.3 m). The magnified labyrinthine units in the dashed red rectangles provide sound distribution details inside the metasurface structure. (b) Normalized theoretical sound pressure field. (c) and (d) are the normalized numerical and theoretical reflected sound-intensity fields $|p{|}^2$, respectively. (e) Comparison of the numerical and theoretical sound intensities of the cross-section lines illustrated in (c) and (d). The blue and red circles and curves, correspondingly, stand for the normalized numerical and theoretical sound intensity along the horizontal ($x=0.7\mathrm{m}$) and vertical ($y=0.3\mathrm{m}$) cross sections. The dashed black line is the normalized incidence intensity of the point source. The two red frame lines in (b) and (d) are theoretical designs for the phase-shifting manipulations.

Figure 4

Normalized reflected sound-intensity mapping for the two-sided perpendicular configuration (left and top) within the area $(x,y)\in [0.1\mathrm{m},0.9\mathrm{m}]$.

Figure 5

(a) Normalized reflected sound-intensity field of the two-sided parallel configuration (up and down) for a typical focusing position (0.37 m, 0.18 m). (b) Normalized sound-intensity curves at the horizontal ($x=0.37\mathrm{m}$) and vertical ($y=0.18\mathrm{m}$) cross sections. (c) Normalized sound-intensity mapping of the area $(x,y)\in [0.1\mathrm{m},0.9\mathrm{m}]$ for the parallel configuration.

Figure 6

(a) Normalized reflected sound-intensity field of the three-sided configuration (left, top, and right) for a typical focusing position (0.84 m, 0.34 m). (b) Normalized sound-intensity curves at the horizontal ($x=0.84\mathrm{m}$) and vertical ($y=0.34\mathrm{m}$) cross sections. (c) Normalized sound-intensity mapping of the area $(x,y)\in [0.1\mathrm{m},0.9\mathrm{m}]$ for the three-sided configuration.

Figure 7

(a) Normalized sound pressure field of the enclosed configuration for a typical focusing position (0.21 m, 0.24 m). (b) Normalized reflected sound-intensity field of the enclosed configuration for a typical focusing position (0.21 m, 0.24 m). (c) Normalized sound-intensity curves at the horizontal ($x=0.21\mathrm{m}$) and vertical ($y=0.24\mathrm{m}$) cross sections. (d) Normalized sound-intensity mapping of the area $(x,y)\in [0.1\mathrm{m},0.9\mathrm{m}]$ for the enclosed configuration.

Figure 8

Normalized maximum reflected intensities and average focusing spot sizes achieved in multilateral metaurfaces with various numbers of sides.

Figure 9

Phase-shift profiles with and without considering the thermoviscosity towards plate thickness $h$ in four $(m,n)=(1,1)$, (2, 1), (2, 2), and (3, 2) configurations, which are demonstrated in (a), (b), (c), and (d), respectively.

Figure 10

Reflection coefficients with and without considering the thermoviscosity towards plate thickness $h$ in four $(m,n)=(1,1)$, (2, 1), (2, 2), and (3, 2) configurations, which are demonstrated in (a), (b), (c), and (d), respectively.

Figure 11

Reflected loss coefficients and phase variations of the eight labyrinthine units caused by thermoviscous effects.

Figure 12

(a) Representative focusing positions $A$ (0.36 m, 0.73 m) and $B$ (0.7 m, 0.3 m) in the upper-left and lower-right areas. (b) Normalized acoustic pressure field of the focusing position $A$ at time $t=14.6T_0$ with a generation signal width of $6T_0$. (c) Normalized acoustic pressure field of the focusing position $B$ at $t=17T_0$ with a generation signal width of $6T_0$.

Figure 13

(a)–(f) are the focusing pressure curves towards time at position $A$ with generation signals of $1T_0–3T_0$ and $4T_0$, $10T_0$, $15T_0$, respectively. (g)–(l) are the focusing pressure curves towards time at position $B$ with generation signals of $1T_0–3T_0$ and $4T_0$, $10T_0$, $15T_0$, respectively.