High-efficiency Ultrathin Nonlocal Waterborne Acoustic Metasurface

High-performance and lightweight waterborne acoustic devices have provided important support for the development of ocean exploration, health monitoring, etc. In this work, we propose a high-efficiency ultrathin nonlocal waterborne acoustic metasurface. Due to strong nonlocal interaction between the waterborne unit cells induced by the fluid-solid interaction, a nonlocal design concept based on diffraction theory and the optimization approach is proposed to design waterborne acoustic metasurfaces, in contrast to the local design based on the generalized Snell’s law. The theoretical efficiency limitation of the generalized Snell’s law can be broken, and unitary efficiency could be obtained even for large-angle anomalous reflection. The high-efficiency modulation is achieved owing to the energy flow along the metasurface resulting from the nonlocal interaction. The thickness of the waterborne acoustic metasurface can be minimized to one tenth or even one sixtieth of the working wavelength. The nonlocal design with fluid-solid interaction paves the way for high-efficiency and deep-subwavelength waterborne acoustic wave manipulation.

Figure 1

Schematic of the local design for a WAM based on the GSL. (a) WAM based on the local design is composed of some discrete unit cells that provide local phase shifts. Based on the GSL, the incident waves can be reflected at an angle ${\theta }_r$ through the WAM with phase gradient on the basement. Panel (b) illustrates the cross section of the waterborne unit cell. The depth and width of the ith (i = 1, …, 5) rectangular cavity are $L_i$ and $W_i$, respectively. Panel (c) shows a full $2\pi$ range variation of the phase shift (green dashed lines) with the parameter $L_1(=L_5)$ for the unit cell with the FSI boundaries in (b), which is calculated by the finite-element model of a single unit cell with the periodic boundary condition. The analytical and simulated phase shifts for the unit cell with rigid walls are represented by blue dotted lines and red solid lines, respectively. Panel (d) shows the variation of the linear phase profile with coordinate x for anomalous reflection with $D=\lambda /|\sin {\theta }_i-\sin {\theta }_r|$, where the green solid line and purple squares represent the continuous and gradient phase profiles approximated by the local discrete unit cells, respectively.

Figure 2

Local design of the WAM for anomalous reflection. Panels (a)–(c) illustrate the cross sections of the WAMs based on the local design for ${\theta }_i=0^{\circ }$ and ${\theta }_r={36}^{\circ }$, ${72}^{\circ }$, and ${81}^{\circ }$, respectively, the thickness of which is $\lambda /10$. The corresponding real parts of the reflected acoustic pressure fields are shown in (d) ${\theta }_r={36}^{\circ }$, (e) ${\theta }_r={72}^{\circ }$, and (f) ${\theta }_r={81}^{\circ }$, where the white arrows indicate the local power intensity vector of the reflected acoustic pressure fields with $P_0=1\mathrm{Pa}$ and $max=\sqrt{\cos {\theta }_i/\cos {\theta }_r}$ in the color bar; and perfectly matched layers with thickness $2D$ and typical wavelength $3\lambda$ are added at the top regions (not shown in figure) to eliminate the reflection at the boundaries. The corresponding FFT amplitudes of the reflected pressure on the transverse line located at $z=2.5D$ as a function of $-k_x/k\sin {\theta }_r$ are presented for ${\theta }_r={36}^{\circ }$ (g), ${\theta }_r={72}^{\circ }$ (h), and ${\theta }_r={81}^{\circ }$ (i), respectively.

Figure 3

Nonlocal interaction between the waterborne unit cells. Panels (a) to (c) show the real parts of the reflected acoustic pressure fields and total displacements for single unit cells no. 1–5, a homogeneous UCA of unit cell no. 3 and an inhomogeneous UCA of unit cells no. 1–5 at $f=10\mathrm{kHz}$, respectively, where unit cells no. 1–5 in panels (a)–(c) are selected from five unit cells in Fig. 2; point A is an observation point $0.75\lambda$ away from the upper surface of the unit cell; and the phase shift and reflection coefficient are retrieved and normalized as $[\arg (P_r)+\pi ]/2\pi$ and $|P_r|/P_0$, respectively, with $P_0=1\mathrm{ Pa}$; $w_0=10^{-11}\mathrm{m}$; and the scale factor of the structural deformation is $2\times 10^8$. Panels (d) and (e) show the phase shifts at point A and reflection coefficients varying with the parameter $L_1(=L_5)$ for the single unit cell in (a) (red solid lines), unit cell no. 3 in the homogeneous UCAs in (b) (blue dotted lines) and the inhomogeneous UCAs in (c) (green dashed lines).

Figure 4

Schematic of the nonlocal design for a WAM based on diffraction theory and optimization. (a) WAM based on the nonlocal design is composed of periodic supercells, where the period length of a supercell along the x direction is D. Based on diffraction theory, when an acoustic wave is incident from the negative z axis direction, there will be reflected waves of 0th, ±1st,..., ±Nth order diffractive components above the WAM on the basement. Panel (b) illustrates the sketch of the nonlocal design combining diffraction theory and optimization. A background acoustic field at the specific incident angle $+{\theta }_i$, 0, or $-{\theta }_i$ serves as incident acoustic waves and is exerted on a supercell with Floquet periodicity. Reflection acoustic pressure on the transverse line is extracted to analyze the reflected waves in different directions ($+{\theta }_r$, 0, or $-{\theta }_r$, etc.) with FFT. Driven by the objective function of reflection efficiency, a parameter optimization procedure with genetic algorithm is applied to reverse design the WAM. Panel (c) shows the variation of the real and imaginary parts of the acoustic pressure with $x/D$ for $p_1(x)=exp(ik\sin {\theta }_{r}x)$ (1st column), $p_2(x)=1$ (2nd column), and $p_3(x)=exp(-ik\sin {\theta }_{r}x)$ (3rd column). The corresponding FFT amplitudes of the above acoustic pressure are presented in (d).

Figure 5

Nonlocal design of the WAM for anomalous reflection. Panels (a)–(c) illustrate the cross sections of the WAMs based on the nonlocal design for ${\theta }_i=0^{\circ }$ and ${\theta }_r={36}^{\circ }$, ${72}^{\circ }$, and ${81}^{\circ }$, respectively, the thickness of which is $\lambda /10$. The corresponding real parts of the reflected acoustic pressure fields are shown in (d) ${\theta }_r={36}^{\circ }$, (e) ${\theta }_r={72}^{\circ }$, and (f) ${\theta }_r={81}^{\circ }$, where the white arrows indicate the local power intensity vector of the reflected acoustic pressure fields with $P_0=1\mathrm{Pa}$ and $max=\sqrt{\cos {\theta }_i/\cos {\theta }_r}$ in the color bar; and perfectly matched layers with thickness $2D$ and typical wavelength $3\lambda$ are added at the top regions (not shown in figure) to eliminate the reflection at the boundaries. The corresponding FFT amplitudes of the reflected pressure on the transverse line located at $z=2.5D$ as a function of $-k_x/k\sin {\theta }_r$ are presented for ${\theta }_r={36}^{\circ }$(g), ${\theta }_r={72}^{\circ }$ (h), and ${\theta }_r={81}^{\circ }$ (i), respectively.

Figure 6

High-efficiency ultrathin WAM for anomalous reflection under oblique incidence. Real parts of the reflected acoustic pressure fields (1st column) and the corresponding FFT amplitudes (2nd column) of the reflected pressure on a transverse line located at $z=2.5D$ are presented for two ultrathin WAMs with thicknesses of $\lambda /30$ (a) and $\lambda /60$ (c), respectively, where $P_0=1\mathrm{Pa}$ and $max=\sqrt{\cos {\theta }_r/\cos {\theta }_i}$ with ${\theta }_i=-{72}^{\circ }$ and ${\theta }_r={72}^{\circ }$ in the color bar; and perfectly matched layers with thickness $2D$ and typical wavelength $3\lambda$ are added at the top regions (not shown in figure) to eliminate the reflection at the boundaries. Panels (b) and (d) illustrate the cross sections of the WAMs with thicknesses of $\lambda /30$ and $\lambda /60$, respectively.

Figure 7

Impedance and intensity distribution on the WAM. Panel (a) shows the impedance profiles of the WAM with the unitary reflection efficiency for ${\theta }_r={72}^{\circ }$ with ${\theta }_i=0^{\circ }$. Panel (b) shows the corresponding normal components of the incident, reflected, and total intensity vectors as functions of coordinate x, where $D=\lambda /|\sin {\theta }_i-\sin {\theta }_r|$ and $I_0=P_0^2/(2Z_w)$. The theoretical results are obtained from Eq. (6a), and simulated ones are retrieved from the transverse line located at $z=0.5D$ within $0.1375D\le x\le 1.1375D$ in Fig. 5.

Figure 8

Scaling law between the unit cell size and operating wavelength. Panel (a) shows the variation of the reflection efficiency $\eta$ of the scaled WAM in Fig. 5 with operating frequency ${\log }_{10}f$ and ${\log }_{10}a$. Panels (b) and (c) present the real parts of the reflected acoustic pressure fields within one supercell ($0\le x\le D$ and $z\le 5D$) when $f=1\mathrm{kHz}$ and $f=100\mathrm{kHz}$, respectively, where D is the corresponding width of a supercell; $P_0=1\mathrm{Pa}$; and $max=\sqrt{\cos {\theta }_r/\cos {\theta }_i}$ with ${\theta }_i=0^{\circ }$ and ${\theta }_r={72}^{\circ }$ in the color bar.

Figure 9

Theoretical effective model of the unit cell. Panel (a) shows the schematic diagram of the unit cell with rigid walls. Panel (b) illustrates the effective circuit model of the unit cell with rigid walls to derive the phase shift.

Figure 10

Influence of FSI on the local WAM. Panel (a) shows the variation of the phase shifts of the channel-type unit cell with rigid and FSI boundaries. The reflection efficiency $\eta$ of the local WAM as a function of the reflected angle ${\theta }_r$ is presented in (b), where the purple solid line represents the GSL theoretical efficiency limitation, and squares and stars denote the efficiencies of the local WAMs with rigid and FSI boundaries for 12 phase constants C ($\pi /6-2\pi$ with a step of $\pi /6$), respectively. Panels (c) and (d) show the FFT amplitudes of the reflected pressure on a transverse line located at $z=1.5D$ as a function of $-k_x/k\sin {\theta }_r$, which are presented to calculate the reflection efficiency of the local WAMs with rigid (c) and FSI (d) boundaries for ${\theta }_r={36}^{\circ }$ (1st column), ${\theta }_r={72}^{\circ }$ (2nd column) and ${\theta }_r={81}^{\circ }$ (3rd column) at $C=2\pi$, respectively. The solid and dashed lines represent the cases with rigid and FSI boundaries, respectively. The insets show the real parts of the reflected acoustic pressure fields within one supercell ($0\le x\le D$ and $z\le 3D$), where D is the width of a supercell.

Figure 11

Theoretical analysis on the FSI. Coupled plate-fluid layer system consists of the plate-fluid-plate sandwich model. There exists a propagation wave on one side, such as areas II and IV.

Figure 12

Locality between the airborne unit cells. Panels (a)–(c) show the real parts of the reflected acoustic pressure fields of single unit cells no. 1–5, a homogeneous UCA of unit cell no. 3 and an inhomogeneous UCA of unit cells no. 1–5 at $f=2.2867\mathrm{kHz}$, respectively, where unit cells no. 1–5 in panels (a)–(c) are selected from five unit cells in Fig. 2 and $P_0=1\mathrm{Pa}$.

Figure 13

Panels (a) and (b) present the phase shifts (thick dashed line) and reflection coefficients (thick solid line) of unit cell no. 3 in an inhomogeneous UCA in Fig. 3 as functions of scaling factor $\zeta$ and Poisson’s ratio $\upsilon$ for $L_1(=L_5)=12\mathrm{mm}$, respectively. For comparison, the phase shifts and reflection coefficients for the rigid walls are also presented by thin dashed lines and thin solid lines, respectively.