Reconfigurable acoustic coding metasurface for bidirectional wavefront manipulation across the water-air interface

Highlights

• •A reconfigurable coding metasurface enables efficient bidirectional wavefront control across the water-air interface.
• •Sidewall-induced flexural resonance in discrete units simultaneously ensures high transmission and tunable phase modulation.
• •The compact single-layer design supports programmable functions like focusing, splitting, and Airy-beam generation.
• •Experiments confirm enhanced transmission and reconfigurable field control in both propagation directions.

Graphical abstract

Abstract

Bidirectional high-efficiency sound control across the water-air interface is essential for cross-medium communication and remote sensing. However, conventional quarter-wavelength impedance matching imposes a strong trade-off between transmission efficiency and phase tunability, limiting flexible control in both directions. Here, we propose a reconfigurable acoustic coding metasurface that enables efficient bidirectional wavefront manipulation within a single-layer architecture. Discrete solid units with vibro-acoustic coupling harness sidewall flexural vibrations, facilitating effective velocity transfer between water and air. This underlying mechanism overcomes the conventional constraint, enabling simultaneous high transmission and phase modulation. Through mechanical reconfiguration of the coding units, the metasurface supports flexible switching among various acoustic functions, such as focusing, beam splitting, and Airy-beam generation, for both propagation directions. Full-wave simulations and experimental results validate its versatility and reconfigurability. This work opens new possibilities for compact, programmable platforms in intelligent cross-domain acoustic communication and adaptive sensing.

1. Introduction

Bidirectional wavefront manipulation [1], [2] of sound across the water-air interface enables energy propagation along controlled pathways between the ocean and the atmosphere. This capability accelerates the development of emerging engineering applications, such as cross-medium communication [3], [4], [5], remote environmental sensing [6], [7], [8], and intelligent coordination of underwater robots [9]. However, the significant acoustic impedance mismatch between air and water, approximately 3600 times, results in near-total reflection of sound at the interface. Such an extremely poor transmission efficiency imposes a significant challenge to the precise control of transmitted wavefronts.

To address this, the quarter-wavelength impedance matching theory [10] suggests that complete sound transmission can be achieved by incorporating a layer with the geometric mean of the impedances of the two media [11], [12]. Although it is difficult to find such a matching layer in nature due to extreme structural properties, acoustic metasurfaces [13], [14] have emerged as a promising alternative in recent years. By delicately designing unit cells, various metasurfaces with tailored acoustic responses have significantly enhanced water-to-air sound transmission. Representative designs include mass-loaded membranes [15], coupled resonant bubbles [16], [17], [18], [19], hydrophobic lotus-like surfaces [20], and elastic plates separated by periodically spaced ribs [21]. Despite these valuable efforts, existing studies have predominantly focused on improving transmission efficiency [22], whereas effective wavefront manipulation across the water-air interface remains an open challenge.

Actually, according to the quarter-wavelength impedance matching theory [10], there is a strong trade-off between transmission efficiency and phase tunability. Since the matching layer thickness $d_{\text{m}}$ is required to be an odd multiple of one quarter of its effective wavelength ${\lambda }_{\text{m}}$ [11], [12], [23], the resulting transmission phase shift $\varphi =2\pi d_{\text{m}}/{\lambda }_{\text{m}}$ is fixed at $\pm \pi /2$. Consequently, full transmission at the interface allows only two discrete phase states, which limits the flexibility of bidirectional wavefront control. To address this challenge, recent studies on water-air acoustic metasurfaces have introduced composite structural designs. For example, Liu et al. [24] proposed a three-layer unit cell, where the upper and lower resonant components enhance transmission by coupling with incident waves, while the middle labyrinth channel modulates phase through tailored path length. This structure enabled remote water-to-air eavesdropping with a high signal-to-noise ratio. Similarly, Zhou et al. [11] developed a dual-layer hybrid metasurface [25], in which transmission enhancement and wavefront shaping are decoupled into separate functional layers. Various customized wavefront patterns were experimentally demonstrated, facilitating practical applications in ocean-atmosphere communication. However, most reported designs address transmission enhancement and phase modulation independently. Whether a unified mechanism within a single-layer configuration can achieve both functions remains an open question.

In addition, existing studies primarily focus on water-to-air propagation and typically adopt monolithic, spatially continuous structures fabricated via integrated manufacturing. These designs impose inherent limitations on reconfigurability and provide limited flexibility for controlling bidirectional wavefronts. Recently, digital acoustic coding metasurfaces [26] have emerged as a promising alternative. By discretizing phase profiles into programmable coding units, these metasurfaces simplify structural design and enable dynamic modulation [27], [28]. Such strategies open new avenues for developing reconfigurable and multifunctional acoustic platforms. Nevertheless, current coding approaches are largely restricted to homogeneous media, such as air, and their application in water-air coupled scenarios remains underexplored.

In this study, we propose a mechanically reconfigurable acoustic coding metasurface (RACM) for bidirectional wavefront manipulation across the water-air interface. The RACM consists of spatially discrete unit cells inversely designed [29] by topology optimization. Benefiting from sidewall-induced flexural resonances, additional acoustic energy transmission pathways are created through the coupling of flexural vibration modes with the propagating sound under wavevector-matching conditions [30]. This hybrid transmission mechanism enables unit cells with vibro-acoustic coupling to exhibit nearly full transmission and varied phase shifts, thus breaking the inherent trade-off between high transmission and phase tunability. By manually adjusting the coding sequence of these unit cells, the RACM enables switching among various acoustic functionalities, such as focusing [31], [32], [33], [34], beam splitting [35], [36] and Airy-beam generation [37], [38], [39], in both directions of wave propagation. Full-wave simulations and experimental validations confirm the feasibility of the concept. This work opens new possibilities for programmable acoustic devices with adaptable wavefront control across the water-air interfaces, thus expanding the potential for applications in cross-medium communication and remote sensing.

2. Theoretical model and design of RACM

Fig. 1 illustrates the concept of the RACM designed for tunable bidirectional wavefront manipulation across the water-air interface. The RACM consists of multiple spatially discrete unit cells precisely arranged along the interface. Each unit cell is engineered to enhance acoustic transmission between water and air while imparting a tailored phase shift for dynamic wavefront control. When sound waves incident from the air, they are efficiently transmitted into the water and received by an underwater robotic platform. Conversely, sound signals emitted by the robot propagate through the RACM and are directed to a designated aerial location. This bidirectional process is enabled by reconfigurable wavefront manipulation under enhanced acoustic transmission across the water-air interface. When further integrated with a modulated control signal, the system can establish a robust acoustic communication channel between underwater robots and aerial drones. Therefore, integrated air-sea communication and sensing can be achieved through the reconfigurable single-layer system.

Fig. 1.

Schematic illustration of the RACM enabling bidirectional wavefront manipulation across the water-air interface. By precisely controlling both the amplitude and phase of acoustic waves, the RACM facilitates a direct and efficient acoustic transmission pathway between the atmosphere and the ocean.

To achieve efficient bidirectional wavefront modulation across the water-air interface, continuous-integrated and spatially discrete architectures are commonly employed, as illustrated in Fig. 2a. However, compared with continuous designs, the spatially discrete architecture introduces water-air gaps along the sidewalls to effectively decouple adjacent units, enabling unit-wise structural reconfiguration. More importantly, the sidewall gaps open additional pathways for acoustic energy exchange. As shown in Fig. 2b, the energy flux is no longer limited to vertical transmission but also propagates laterally through flexural vibrations of the sidewalls induced by vibro-acoustic coupling. This enhanced energy redistribution significantly improves the tunability of both transmission amplitude and phase. Therefore, the trade-off dictated by conventional quarter-wavelength impedance matching can be broken, since the uniform equivalent impedance assumption no longer holds.

Fig. 2.

Operating principle and optimization results of the discrete unit cells. (a) Schematic of the integrated and reconfigurable metasurfaces designed to operate across the water-air interface. (b) Illustration of acoustic transmission pathways and the underlying mechanisms for both continuous and discrete unit configurations. (c) Vibration modes of the 2-bit coding unit cells, where the normalized velocity amplitude is $u_0=3\times {10}^{-10}$. (d) Normalized acoustic energy fluxes across various boundaries: $E_{\text{b}}$, $E_{\text{s}}$, and $E_{\text{t}}$ denote the acoustic energy fluxes through the bottom surface, sidewalls, and top surface of the solid structure, respectively, while $E_{\text{out}}$ represents the total transmitted energy through the top surface of the unit cell. The normalization factor is $7.6\times {10}^{-9}$. (e) Spatial distribution of the horizontal displacement along the outer surface of the left sidewall of the unit cell, normalized by the maximum real part of the displacement amplitude; $d$ indicates the unit cell thickness. (f) Normalized acoustic pressure fields for both water-to-air and air-to-water transmission scenarios. Here, the sound pressures ($P_{\text{t}}$) are normalized by the square root of the acoustic impedance ($\sqrt{Z}$) of the host medium (water or air), respectively.

To further elucidate the underlying mechanism, the flexural vibration of thin plates is employed as a representative model for analysis. Each unit cell comprises a central cavity enclosed by thin plates on the top, bottom, and side surfaces. Under external excitation, pressure differentials across these surfaces induce significant flexural vibrations. The flexural response of each plate can be described by

$$D{\nabla }^4{w}_j-m_{\rho }{\omega }^2{w}_j=\Delta p=p_{\text{in},j}-p_{\text{out},j},j\in \{\text{b},\text{t},\text{s}\},$$ (1)

where $D=E{h}_{\mathrm{p}}^3/\left[12(1-v^2)\right]$ is the bending stiffness of the plates with $E$ being Young’s modulus; $h_{\mathrm{p}}$ being plate thickness; $\nu$ being Poisson’s ratio; $m_{\rho }$ is the mass per unit area; $w_j$ denotes the flexural displacement; and the pressure difference term $p_{\text{in},j}-p_{\text{out},j}$ accounts for the acoustic load on the bottom (b), top (t), and side (s) plates.

In particular, for continuous units with lateral symmetry, the internal acoustic pressure field also exhibits symmetric distribution under normal plane wave incidence and periodic boundary conditions. Therefore, the net pressure difference across the sidewalls satisfies

$$p_{\text{in},\text{s}}-p_{\text{out},\text{s}}=p_{\text{in},\text{s}}-p_{\text{in},\text{s}}=0,$$ (2)

As a result, the sidewall flexural vibrations are negligible. Moreover, given that the unit width is typically subwavelength, higher-order flexural modes fail to support propagating pressure fields.

This results in only the fundamental translational modes of the top and bottom plates contributing to the coupling with propagating sound waves. However, when unit cells are spatially discretized, the presence of water-air gaps relaxes the constraint of lateral pressure symmetry. This relaxation permits nonzero net pressure differences across the sidewalls, even in geometrically symmetric configurations:

$$p_{\text{in},\text{s}}-p_{\text{out},\text{s}}\ne 0,$$ (3)

Consequently, higher-order flexural modes along the $x$-direction are effectively excited on the sidewalls. These lateral vibrations induce pressure perturbations along the $z$-axis, which are coupled with propagating acoustic modes and introduce additional transmission channels for impedance matching. Moreover, the continuity of normal velocity at the fluid-solid interface requires the lateral wavenumber of the sidewall vibration to match that of the sound field in the water-air gap. Such wavenumber matching enables precise phase modulation via engineered sidewall resonances. This greatly expands the design freedom for phase control with high transmission efficiency.

Based on this modulation mechanism, topology optimization based on the genetic algorithm is applied to inversely design the unit cells with uniform thickness. The width of the solid component in each discrete unit can be adjusted to modulate its flexural vibration modes. Transmission efficiency is initially chosen as the optimization objective, and the fitness function is defined as

$$F_1=1-\frac{P_{\text{trans}}^2}{P_{\text{in}}^2}\frac{Z_{\text{w}}}{Z_{\text{a}}},$$ (4)

where $P_{\text{trans}}$ and $P_{\text{in}}$ denote the transmitted and incident acoustic pressures in air and water, respectively; $Z_{\text{w}}$ and $Z_{\text{a}}$ represent their corresponding characteristic acoustic impedances. This optimization process yields several unit cells with high transmission efficiency. Two unit cells, denoted as U01 and U11 (as shown in Fig. 2(c)), exhibit sound energy transmission coefficients of 0.993 and 0.999, respectively. They are found to have a phase difference of approximately $\pi$. These pairs can serve as a reference for designing other unit cells with different phase shifts.

To further broaden the accessible phase range, the transmission phase is incorporated as an additional optimization variable. Accordingly, the fitness function is reformulated in a weighted form that considers both transmission efficiency and phase shift:

$$F_2=w_1(1-\frac{P_{\text{trans}}^2}{P_{\text{in}}^2}\frac{Z_{\text{w}}}{Z_{\text{a}}})+w_2\left(\frac{|\varphi -{\varphi }_{\text{tar}}|}{2\pi }\right),$$ (5)

where $w_1$ and $w_2$ represents the weighting factors of transmission and phases, respectively; $\varphi$ and ${\varphi }_{\text{tar}}$ represent the actual and target transmission phases, respectively; and ${\varphi }_{\text{tar}}$ are set to differ by $\pi /2$ from those of the initially optimized units. In our optimization, $w_1=0.15$ and $w_2=0.85$ are empirically chosen to prioritize phase accuracy while maintaining high transmission. This inverse design framework captures discrete unit cells with distinct phase responses. Two additional unit cells with other phase shifts, denoted as U00 and U10 (Fig. 2c), achieve sound energy transmission coefficients of 0.973 and 0.955, respectively. According to their phase sequence, the four optimized unit cells are categorized as 2-bit elements, and their corresponding flexural modes are illustrated in Fig. 2c. The transmission amplitudes and phases of these units are provided in the Supplementary Note S1 (online). It can be observed that significant flexural deformations occur along the sidewalls, generating pressure perturbations in the surrounding air. These results are in good agreement with the theoretical analysis.

To explore the mechanism of the energy transfer, Fig. 2d illustrates the distributions of normalized energy flux across different surfaces for U01 and U11. The acoustic energy flux is calculated by [11]:

$$E_j={\int }_{S_j}(I_x·{\mathrm{n}}_x+I_y·{\mathrm{n}}_y)\mathrm{d}s,j\in \{\text{b},\text{t},\text{s},\text{out}\},$$ (6)

where $I_x$ and $I_y$ denote the acoustic intensity associated with the $x$ and $y$ components, respectively; ${\mathrm{n}}_x$ and ${\mathrm{n}}_y$ correspond to the unit vectors aligned in those directions; the term $S_j$ represents the integral boundary for the bottom (b), top (t), and side (s) surfaces of the solid structure; $S_{\text{out}}$ refers to the whole integral boundary encompassing the top surface of the unit cell. It can be observed that acoustic energy enters the resonant cavity from the bottom surface and is subsequently radiated into the air through interactions between the sidewalls and the top surface. Similar patterns of energy flow are also observed for U00 and U10, as detailed in the Supplementary Note S2 (online). This behavior stands in sharp contrast to that of conventional continuous-structure unit cells, where energy transmission typically follows the top and bottom surfaces of the solid body. Furthermore, Fig. 2e presents the flexural displacement profiles along the $x$-axis on the left sidewalls of all four unit cells. Under identical incident wave conditions, the distinct internal resonator geometries result in varying degrees of sidewall deformation. These variations, in turn, directly induce the differences in transmission phase among the unit cells, as illustrated in Fig. 2f. A more intuitive presentation is also provided in Movie S1 (online). The coexistence of high transmission and multiple phase states, together with diverse energy transmission pathways, confirms that the spatially discrete design effectively breaks through the conventional quarter-wavelength impedance matching limitation.

Furthermore, the out-of-plane velocity component along the $z$-direction is extracted to examine the velocity transfer performance across the water-air interface. For all four unit cells, the translational velocity profiles on both the bottom and top surfaces are processed using Fast Fourier Transform (FFT) to extract the zeroth-order component, which mainly contributes to far-field transmission. At the same time, the theoretical velocity values at the corresponding positions are calculated based on the target complex transmission coefficients of each unit. As shown in Table 1, the FFT results show good agreement with the theoretical values on both surfaces, with only small differences. This confirms that acoustic energy is effectively transferred from the water to the air, while keeping the designed amplitude and phase. These results demonstrate that sidewall flexural vibrations induced by vibro-acoustic coupling play an important role in enabling reliable and controllable velocity transfer through the discrete solid units.

Table 1.

Velocity of the$\mathit{z}$component of theoretical and simulated magnitude and phase.

Unit	Position	Theory		FFT (0th Order)
		$|v_z|$ ($\mathrm{m}/\mathrm{s}$)	Arg $\left(v_{z,0}\right)$ (deg)	$|v_{z,0}|$ ($\mathrm{m}/\mathrm{s}$)	Arg $\left(v_{z,0}\right)$ (deg)
U00	Bottom	$6.67\mathrm{e}-7$	0	$6.02\mathrm{e}-7$	8.2
	Top	$4.01\mathrm{e}-5$	91.8	$3.94\mathrm{e}-5$	92.5
U01	Bottom	$6.67\mathrm{e}-7$	0	$7.12\mathrm{e}-7$	$-1.7$
	Top	$4.01\mathrm{e}-5$	167.4	$4.01\mathrm{e}-5$	167.3
U10	Bottom	$6.67\mathrm{e}-7$	0	$7.36\mathrm{e}-7$	$-2.1$
	Top	$4.01\mathrm{e}-5$	$-100.8$	$3.93\mathrm{e}-5$	$-101.3$
U11	Bottom	$6.67\mathrm{e}-7$	0	$6.75\mathrm{e}-7$	$-0.6$
	Top	$4.01\mathrm{e}-5$	$-14.4$	$3.99\mathrm{e}-5$	$-14.1$

3. Simulation results for bidirectional wavefront manipulation

3.1. Acoustic focusing

To demonstrate the capability of the RACM for bidirectional wavefront manipulation across the water-air interface, U01 and U11, which exhibit an approximate $\pi$ phase difference, are selected as binary encoding elements to construct a 1-bit coding metasurface. For simplification, these two unit cells are denoted as ‘Unit 0’ and ‘Unit 1’, respectively. We initially investigate the performance of bidirectional axial acoustic focusing across the water-air interface using simulations based on COMSOL Multiphysics. Fig. 3a and e illustrate schematic diagrams of axial focusing for plane waves incident from water to air and from air to water, respectively. In both cases, when the incident plane waves propagate through the alternately arranged Unit 0 (gray regions) and Unit 1 (green regions), phase modulation occurs along with enhanced transmission. As a result, the acoustic energy is effectively focused in the opposite medium. According to the generalized Snell’s law [40], the ideal phase distribution at each coding position under normal incidence is given by

$$\varphi =k_{\text{t}}[\sqrt{{(x-x_0)}^2+{(z-z_0)}^2}-\sqrt{x_0^2+z_0^2}]+C,$$ (7)

where $k_{\text{t}}$ is the transmitted wave number of main domain (air or water); $(x_0,z_0)$ denotes the coordinate of the focal point; $(x,z)$ denotes the coordinates of each coding units; and $C$ represents an arbitrary constant that determines the initial phase shift of the central coding unit. Here, we set $(x_0,z_0)=(0,15{\lambda }_{\text{a}})$ as an example, the corresponding phase is calculated as $\varphi =201.5\sqrt{x^2+{(z-0.47)}^2}-94.25+C$ [rad]. Different values of $C$ generate distinct coding sequences, resulting in slight differences in focusing performance. In this case, $C=\pi /2$ is selected.

Fig. 3.

Acoustic focusing demonstration of the RACM across the water-air interface. (a,e) Schematic diagrams illustrating axial acoustic focusing from water to air and from air to water, respectively. (b) Numerical simulation of acoustic focusing from water to air, with a focal length of 15${\lambda }_a$. (c,d) Normalized sound intensity profiles extracted along the vertical and horizontal white dashed lines indicated in panel (b), respectively. (f) Numerical simulation of acoustic focusing from air to water, with a focal length of 8${\lambda }_w$. (g,h) Normalized sound intensity profiles along the vertical and horizontal white dashed lines shown in panel (f), respectively. Here, the arrows indicate the propagation direction of the incident plane wave at a frequency of 11 kHz. The normalized sound pressure amplitudes used in panels (b) and (f) are $p_0=54.171$ Pa and $p_0=3066.3$ Pa, respectively.

Fig. 3b presents the simulated water-to-air acoustic focusing, with normalized sound intensity and pressure fields shown in the air and water domains, respectively. A background pressure field is used to emulate a vertically incident plane wave. Plane-wave radiation boundaries are applied on all sides to suppress reflections. The RACM, 0.5 m wide and composed of 22 coding units, is designed with a focal length of $15{\lambda }_{\text{a}}$. The coding sequence is shown in Figure S12a (online). An elliptical focal spot forms along the $z$-axis in the air domain. This indicates that strong axial focusing is achieved by the RACM. To evaluate the focusing performance of the RACM, Fig. 3c plots the sound intensity along the focal axis (white dashed line in Fig. 3b). The peak intensity occurs at 0.41 m, showing a relative deviation of 13% from the ideal focal length of $15{\lambda }_{\text{a}}=0.47$ m. This deviation is influenced by factors such as the number of coding units, target focal position, and initial phase value $C$ in Eq. 7. Fig. 3d further shows the intensity profile along the lateral focal plane (horizontal dashed line in Fig. 3b), with a full width at half-maximum (FWHM) of 0.03 m $\left(0.96{\lambda }_{\text{a}}\right)$, confirming subwavelength resolution.

To validate bidirectional wavefront manipulation, the background pressure field is switched to the air domain to enable acoustic focusing from air to water. Fig. 3f displays the normalized sound pressure field in the air domain and the normalized sound intensity field in the water domain. In this configuration, the RACM has a focal length of $8{\lambda }_{\text{w}}$ and a width of 1.5 m, consisting of 66 coding units. The corresponding coding sequence is shown in Figure S12(b) (online). A distinct focal spot appears in the water domain, confirming the effectiveness of the RACM in achieving bidirectional focusing. Fig. 3g presents the acoustic intensity values along the focal axis, extracted from the interface of water and air, as indicated by the axial white dashed line in Fig. 3f. The maximum intensity is located at a distance of 1.17 m, resulting in a relative deviation of 7% compared to the ideal focal length of $8{\lambda }_{\text{w}}=1.09$ m. Fig. 3(h) shows the intensity distribution along the horizontal line at the location of the peak intensity, as indicated by the horizontal white dashed line in Fig. 3f. The measured FWHM is 0.11 m ($0.80{\lambda }_{\text{w}}$). It demonstrates the superior subwavelength focusing capability of the RACM and its efficient bidirectional manipulation across the water-air interface. More examples of bidirectional sound focusing are provided in Figure S8 (online). By changing the coding sequence, RACM designed at other focal lengths also exhibits good manipulation performance.

3.2. Acoustic beam splitting

Next, we demonstrate the capability of RACM for acoustic beam splitting across the water-air interface. In a single medium, beam splitting can be achieved using an acoustic grating [35], which consists of two types of logical units alternately arranged with a phase difference of $\pi$. The beam splitting angle can be adjusted by changing the grating period. Based on this principle, coding units 0 and 1 are used to construct an acoustic grating across the water-air interface. Fig. 4a shows a schematic of the plane-wave beam splitting through the interface. A supercell is formed by grouping two or more identical coding units (either Unit 0 or Unit 1). According to grating theory, the relationship between the incident and transmitted angles is given by

$$k_{\text{t}}\sin {\theta }_{\text{t}}-k_{\text{i}}\sin {\theta }_{\text{i}}=\frac{2n\pi }{D},$$ (8)

where ${\theta }_{\text{i}}$ and ${\theta }_{\text{t}}$ denote the angles of the incident and transmitted acoustic beams, respectively; $n=0,\pm 1,\pm 2,\cdots ,\pm \infty$ represents the order of the diffracted waves; $D=md$ is the grating period with $d$ being the width of the coding units; and $m$ is the number of units forming one supercell. For normally incident plane waves, when $n=\pm 1$, the RACM symmetrically redirects the wave into two distinct directions, thereby enabling beam splitting. The transmitted angle is given by

$${\theta }_{\text{t}}^{\pm }={\sin }^{-1}\left[(\frac{\sin {\theta }_{\text{i}}}{{\lambda }_{\text{i}}}\pm \frac{1}{D}){\lambda }_{\text{t}}\right],$$ (9)

where ${\lambda }_{\text{i}}$ and ${\lambda }_{\text{t}}$ represent the wavelengths of the incident and transmitted waves, respectively.

Fig. 4.

Simulation demonstration of acoustic beam splitting across the water-air interface. (a) Schematic illustration of the acoustic beam-splitting mechanism across the water-air interface. Plane acoustic waves incident from either the water or air domain are guided by the RACM to split into two distinct directions in the opposite medium. (b,c) Simulated normalized acoustic pressure fields demonstrating beam splitting from water to air and from air to water, respectively. The corresponding metasurface coding sequences are ”$\cdots 0011001100\cdots$” for (b) and ”$\cdots 111110000011111\cdots$” for (c). (d,e) Polar plots of the far-field transmitted acoustic pressure amplitudes corresponding to panels (b) and (c), respectively. The radial coordinate represents the amplitude of the transmitted pressure, while the angular coordinate indicates the beam deflection angle. The directions of the primary transmitted beams are marked. Gaussian beams are employed to approximate plane wave incidence at a frequency of 11 kHz. Arrows indicate the direction of acoustic wave propagation. All acoustic pressure values are normalized by the square root of the acoustic impedance of the respective host medium.

Fig. 4b presents the simulated normalized sound pressure field under a Gaussian beam incident from the water side. The RACM in this configuration consists of 42 coding units, with a grating period set to $D=2d$. It is clearly shown that the transmitted wave is symmetrically deflected to both sides in the air domain. The beam splitting behavior for a Gaussian beam incident from the air side is illustrated in Fig. 4(c). In this case, the RACM is composed of 45 coding units, and the grating period is $D=5d$. It can be seen that the transmitted wave generates two distinct beams in the water domain, further verifying the beam splitting functionality of the RACM. To quantitatively evaluate the performance, Fig. 4d and e plot the far-field radiation patterns for the two cases. According to Eq. 9, the corresponding theoretical deflection angles are $\pm {20}^{\circ }$ and $\pm {37}^{\circ }$, respectively. It can be observed that normalized acoustic pressure amplitudes reach their maximum only near the deflection angles predicted by theoretical analysis. The good agreement with theoretical predictions further validates the excellent beam-splitting performance of the RACM.

By decreasing the grating period $D$, the diffraction angle of sound splitting increases. Two examples of air-to-water sound splitting are shown in Figure S9 (online).

3.3. Airy-beam generation

In addition, we explore the capability of the RACM to generate Airy-beams across the water-air interface. An Airy-beam is a type of non-diffraction wave characterized by an amplitude distribution governed by the Airy function. Such beams exhibit a distinctive self-accelerating trajectory during propagation. To generate an Airy-beam, a specific phase profile must be applied to shape the desired wavefront after diffraction. This typically requires continuous phase modulation to smoothly reproduce the ideal Airy profile. Although 1-bit phase modulation binarizes the ideal phase distribution, the essential features of the wavefront are largely retained. This is mainly attributed to the inherent smoothing of quantization errors in the Fourier domain. As a result, a comparable Airy-beam can still be formed in the far field. Fig. 5a and b illustrate the relationship between the Airy function and the phase values of the coding units for the water-to-air and air-to-water scenarios, respectively, along with the resulting binary coding sequences. The black dashed line represents the inverse Fourier transform of the ideal Airy-beam, which is used to compute the initial phase distribution $\psi (x,z)$[39]:

$$\psi (x,z=0)=Ai\left(bx\right)e^{ax},$$ (10)

where $Ai\left(bx\right)=\frac{1}{\pi }{\int }_0^{\infty }\cos (t^3/3+bxt)\mathrm{d}t$ is the Airy function; the variable $x$ denotes the transverse coordinate of the coding units; $b$ is the transverse scale; and the decay factor $a>0$ is introduced to truncate the infinite tail of the Airy function, thereby enabling the physical realization of Airy-beams.

Fig. 5.

Simulation demonstration of Airy-beam generation across the water-air interface. (a,b) Phase profiles of the coding units (blue dots) used for Airy-beam generation from water to air and from air to water, respectively. The target phase distributions follow the Airy source function (black dashed line). The decay factor is set to $a=1$, with transverse scaling parameters $b=20$ and $b=10$ for (a) and (b), respectively. (c,d) Simulated normalized sound fields demonstrating Airy-beam generation from a plane wave propagating from water to air and from air to water, respectively. The RACM coding sequences correspond to the phase profiles shown in panels (a) and (b). Gaussian beams are employed to approximate plane waves at a frequency of 11 kHz. Arrows indicate the propagation direction of the acoustic waves. The sound pressure is normalized by the square root of the impedance of the host medium.

In Fig. 5a and b, the decay factor is set to $a=1$ with transverse scaling factors $b=20$ and $b=10$, respectively. For the green step curve, the continuous phase profile is quantized into 1-bit (0 or $\pi$) values according to the following rule:

$$\varphi (x,z=0)=\{\begin{array}{c}0,\psi (x,z=0)<0\hfill \\ \pi ,\psi (x,z=0)>0.\hfill \end{array}$$ (11)

The resulting coding sequence is formed by alternating Unit 0 and Unit 1 at zero crossings of $\psi$, forming the coding sequence to modulate the Airy-beam. Each blue dot in the figure represents a coding unit. Units located along the lower edge of the step function correspond to Unit 0, while those along the upper edge correspond to Unit 1.

Fig. 5c shows the simulated normalized acoustic pressure field for Airy-beam generation from water to air. The RACM consists of 47 coding units. A Gaussian plane wave is incident from below, and an Airy-beam forms in the air domain after passing through the RACM. Similarly, Fig. 5d illustrates the air-to-water case, where the RACM contains 70 coding units. A Gaussian plane wave is incident from above, and an Airy-beam emerges in the water domain below. In both cases, the generated Airy-beams exhibit clear parabolic trajectories. The acoustic energy is concentrated in the main lobe, although slight sidelobes are visible. This is mainly due to the special nature of the Airy-beam, which ideally requires both phase and amplitude modulation to achieve the characteristic exponential decay at the interface [41]. In addition, when only phase modulation is applied, the selected Airy function parameters also affect the sidelobe behavior. The amplitude variation of the Airy function influences the arrangement of the encoded unit cells. Less abrupt phase changes across the units are beneficial for reducing sidelobes. As illustrated in Fig. 5, in the air-to-water case, the longer wavelength in water and the smaller parameter $b$ result in gentler phase alternations of 0 and $\pi$. Therefore, fewer sidelobes are observed than the water-to-air case. More Airy-beam generation results are shown in Figure S10 (online). It shows distinct Airy beam trajectories with energy mainly concentrated along the main lobe. Nevertheless, the RACM demonstrates excellent bidirectional wavefront manipulation performance.

3.4. Non-diffraction Bessel beam and anomalous refraction

Furthermore, we investigate the wavefront control capability of the RACM using 2-bit coding units (U00, U01, U10, and U11, as shown in Fig. 2c and f) across the water-air interface. This finer phase resolution enables richer acoustic functionalities, such as generating non-diffraction Bessel beams [42] and anomalous refraction [43]. The required phase distributions ${\varphi }_{\text{B}}$ and ${\varphi }_{\text{AR}}$ for the two functions are given by

$${\varphi }_{\text{B}}\left(x\right)=k_{\text{t}}\sin {\theta }_{\text{t}}\left|x\right|,{\varphi }_{\text{AR}}\left(x\right)=k_{\text{t}}\sin {\theta }_{\text{t}}x,$$ (12)

where $k_{\text{t}}\sin {\theta }_{\text{t}}=2\pi /D$ is set by the grating period $D$. Fig. 6a shows the 2-bit coding sequences for non-diffraction Bessel beam generation and anomalous refraction from water to air. The corresponding simulated normalized sound fields are shown in Fig. 6b. The generated Bessel beam exhibits a well-defined central main lobe with a consistent width along the propagation direction. At the same time, the incident sound wave is effectively deflected along the expected direction for the anomalous refraction case. Similarly, Fig. 6c and d show similar wavefront manipulation from air to water. Both the Bessel beam and anomalous refraction patterns are clearly visible with moderate side lobes.

Fig. 6.

Simulation results of non-diffraction Bessel beams and anomalous refraction across the water-air interface using 2-bit coding unit cells. (a) 2-bit coding sequences for non-diffraction Bessel beam generation and anomalous refraction from water to air, corresponding to U00, U01, U10 and U11, respectively. (b) Simulated normalized sound intensity/pressure fields corresponding to the coding sequences in panel (a). (c) 2-bit coding sequences for non-diffraction Bessel beam generation and anomalous refraction from air to water. (d) Simulated normalized sound intensity/pressure fields corresponding to the coding sequences in panel (c). The sound pressure is normalized by the square root of the impedance of the host medium.

It is noted that, although the solid parts of the unit cells are spatially discrete, the air domains on both sides of the solids remain interconnected. As a result, a certain degree of acoustic coupling is unavoidable [44]. To evaluate the potential influence of coupling between adjacent unit cells, we arrange the assumed decoupled unit cells to form the same metasurface for Bessel beam generation (see Supplementary Note S7 online). The results show that the original design exhibits nearly identical wavefront manipulation performance compared with the case without internal coupling. This indicates that these discrete unit cells have a relatively weak internal coupling. Overall, these findings suggest that the proposed metasurface framework is not limited to binary coding and can be scaled to higher-bit implementations, enabling a broader range of practical acoustic applications.

4. Experimental verification

4.1. Experimental setup

Fig. 7a shows a photograph of the experimental setup used for evaluating the performance of the proposed RACM. To suppress unwanted reflections, the rubber wedge absorbers were placed in the water tank. The sample is fabricated via 3D printing and spans 500 mm along the cross-section, floating at the water-air interface. The 3D-printed structures were fabricated using a commercial photosensitive resin with a Young’s modulus of 2.964 GPa, a density of 1.12 g/cm, and a Poisson’s ratio of 0.41. Fig. 7b presents cross-sectional views of Unit 0 and Unit 1. The underwater transducer and the airborne microphone are vertically aligned at the coordinate origin, which corresponds to the geometric center of the RACM upper surface. Fig. 7c and d display the acoustic sources and receivers of the water-to-air and air-to-water transmission experiments, respectively. In the water-to-air test, a piezoelectric transducer of Model T313 (Neptune Sonar Limited) is submerged at a depth of 0.59 m to emit acoustic waves. A free-moving microphone of Type 2670 (Brüel $\&$ KjȪr), positioned above the RACM, records the transmitted signals in air. In the air-to-water test, a loudspeaker array of Model B1S (HiVi) serves as the airborne source, while a free-moving hydrophone of Model TC4013 (Teledyne Marine), placed below the RACM, measures the transmitted waves in water.

Fig. 7.

Experimental setup and acoustic response of the fabricated sample. (a) Photograph of the experimental setup, showing the underwater acoustic transducer, RACM, and air microphone arranged from bottom to top. The RACM is suspended above the water surface using four adjustable-length soft ropes, with its lower surface fully contacting the water to maintain a shallow immersion depth. (b) Cross-sectional views of the fabricated Unit 0 and Unit 1 samples. (c,d) Photographs of the acoustic sources and receivers used in the water-to-air and air-to-water transmission experiments, respectively. (e,h) Measured transmission amplitudes of Unit 0 and Unit 1 in air and water, respectively, across the frequency range of 8-14 kHz, compared with those at a bare water surface. (f,i) Measured transmitted phase distributions of Unit 0 and Unit 1 at 11.2 kHz in air and water, respectively. (g) Averaged phase values along the same vertical ($z$) coordinate for Unit 0 and Unit 1, measured in air over different frequencies. (j) Comparison of phase profiles along the longitudinal centerline for the two types of coding unit, corresponding to the phase fields in panel (i).

4.2. Acoustic property characterization of coding units

As a reference, we initially measure the transmitted sound fields at the bare water-air interface, as shown in Supplementary Note S4 (online). It suggests that the underwater acoustic source generates a spherical wavefront. The enhanced transmission performance of the RACM is subsequently evaluated by measuring transmitted sound pressure amplitudes across the water-air interface. Fig. 7e presents the experimental results of water-to-air transmission amplitudes for Unit 0, Unit 1, and the bare interface over the frequency range of 8 kHz–14 kHz. The receiving microphone is positioned at a fixed height of $3{\lambda }_{\text{a}}$ above the RACM. During each measurement, the RACM is composed of an identical array of coding units. Data are collected over a $9\times 9$ grid covering a $160\times 160$ mm area in the $x-y$ plane, centered at (0, 0, $3{\lambda }_{\text{a}}$). Both Unit 0 and Unit 1 exhibit significant transmission enhancement in the 11 kHz-12 kHz range compared to the bare interface. At 11.2 kHz, transmission enhancement reaches 14.1 dB for Unit 0 and 10.7 dB for Unit 1. A slight shift in the peak frequency from the designed 11 kHz is observed, which may be attributed to several factors, including fabrication tolerances, material inconsistencies, measurement uncertainties [12], and the non-planar nature of the emitted wavefront. The simulation results in Figure S3 (online) show that increasing damping reduces transmission efficiency, while having a limited effect on the phase response. Meanwhile, Figure S4 (online) demonstrates that, within a certain range, an increase in Young’s modulus or Poisson’s ratio shifts the resonant peak frequency upward, whereas an increase in density shifts it downward.

The phase modulation characteristics of the RACM are further examined by measuring the transmitted phase fields. Fig. 7f presents the experimentally measured phase distributions of Unit 0 and Unit 1 at 11.2 kHz in the vertical $x-z$ plane. A clear phase difference between the two units is observed, indicating effective binary modulation. To assess the stability of this modulation, additional measurements and simulation results of transmission phases at nearby frequencies are provided in Supplementary Note S5 (online). Fig. 7g shows the averaged transmitted phases of Unit 0 and Unit 1 over the frequency range. A nearly constant phase contrast close to $\pi$ is maintained across the working band. This suggests the robustness of the 1-bit coding design for practical wavefront control applications.

To further verify the bidirectional control capability of the RACM, additional experiments are conducted under air-to-water transmission. Fig. 7h presents the measured transmitted sound pressure amplitudes of Unit 0, Unit 1, and the bare interface in water. Compared with the water-to-air case, the amplitude curves exhibit similar transmission enhancement near the peak frequency range. In addition, noticeable periodic fluctuations are observed, which may be attributed to the limited absorption performance of the rubber wedges used in the experiments. Fig. 7i shows the transmitted phase fields of Unit 0 and Unit 1 at 11.2 kHz in the vertical $x-z$ plane. Due to the increased wavelength in water, the wavefronts appear relatively smooth while still maintaining a clear phase difference between the two units. Fig. 7j displays the extracted phase values along the vertical centerlines of both units. The phase contrast close to $\pi$ further confirms the effectiveness and bidirectional robustness of the binary phase modulation.

4.3. Manipulation performance measurement of the RACM

To further evaluate the wavefront shaping capability of the RACM, we conduct acoustic focusing experiments across the water-air interface. Owing to the spherical wavefront radiated from the underwater source, a tailored phase compensation is required to ensure constructive interference at the target focal point. The binary coding sequences applied to the RACM are presented in Figures S12c, d (online). Due to spatial constraints in the experimental setup, the RACM comprises 22 discrete coding units. The measurement plane spans $400\times 500$ mm${}^2$, with its lower boundary located 200 mm above the RACM.

Fig. 8a presents the normalized sound intensity field in the focal region for a designed focal length of $14{\lambda }_{\text{a}}$, with both experimental and simulated results shown for comparison. A distinct focal spot appears at the target location, accompanied by symmetrically distributed sidelobes. Although the sidelobe levels observed in the experiment are notably higher than those in the simulation, the focal position and overall beam shape demonstrate good agreement. This thereby confirms the focusing functionality of the RACM. The discrepancy in sidelobe suppression is primarily attributed to two factors: non-uniform energy radiation from the sound source, as evidenced in Figures S5(a-c) (online), and phase distortions arising from slight inconsistencies in the acoustic responses of individual coding units.

Fig. 8.

Measurements on acoustic focusing across the water-air interface using the RACM. (a) Experimentally measured and numerically simulated normalized sound intensity fields for a designed focal length of $14{\lambda }_{\text{a}}$. A side view of the RACM is shown above the intensity maps. (b,c) Normalized intensity profiles along the vertical ($z$-axis) and horizontal ($y$-axis) directions through the focal point, as indicated by the white dashed lines in panel (a). (d-f) Corresponding experimental and simulation results for a designed focal length of $19{\lambda }_{\text{a}}$. All data are obtained at a frequency of 11.2 kHz. Sound pressure values are normalized with respect to the peak pressure $P_{\text{max}}$ at the focal point.

To further quantify the focusing performance, Fig. 8b shows the normalized intensity distributions along the $z$- and $y$-axes through the focal center. The experimental and simulated axial profiles reveal consistent trends, with the measured focal spot located at approximately $13.7{\lambda }_{\text{a}}$, representing a 2% deviation from the designed value. It indicates a good consistency in axial resolution. Fig. 8c presents the lateral intensity profiles, where the full widths at half maximum (FWHM) are measured as $0.8{\lambda }_{\text{a}}$ and $1.1{\lambda }_{\text{a}}$ in the experiment and simulation, respectively, confirming satisfactory lateral confinement.

To demonstrate the reconfigurable wavefront modulation capability of the RACM, another acoustic focusing experiment is conducted using a different coding sequence corresponding to a target focal length of $19{\lambda }_{\text{a}}$. Fig. 8d–f show the experimentally measured intensity fields and the corresponding axial/lateral profiles. The new focal spot is clearly formed in the expected region, and the experimental results align well with the simulation. The measured focal length is approximately $16.9{\lambda }_{\text{a}}$, corresponding to a 11% deviation from the target. This is likely due to accumulated phase errors and increased environmental disturbances along the extended propagation path. As shown in Fig. 8f, the measured and simulated FWHMs are both $1.2{\lambda }_{\text{a}}$. The good consistency further confirms the ability of the RACM to flexibly manipulate acoustic wavefronts and achieve dynamic focusing across the water-air interface.

5. Conclusion

In summary, we propose a mechanically reconfigurable acoustic coding metasurface that enables high-efficiency and bidirectional wavefront manipulation across the water-air interface. In contrast to conventional impedance-matching strategies, the presented approach overcomes the inherent trade-off between transmission and phase control by exploiting sidewall-induced flexural resonance in discrete solid units with strong vibro-acoustic coupling. This mechanism allows for simultaneous high transmission and tunable phase modulation within a compact single-layer design. By implementing binary coding based on a $\pi$-phase contrast between two unit types, the RACM supports reconfigurable wavefront shaping functionalities, including focusing, beam splitting, and Airy-beam generation, for both air-to-water and water-to-air propagation. Systematic experiments across the water-air interface validate the versatility of the RACM under different coding configurations. Transmission measurements confirm that the designed units significantly enhance sound transmission compared to a bare interface, while phase measurements demonstrate a robust $\pi$-level phase contrast across the working frequency band. Additionally, acoustic focusing experiments at two distinct focal lengths further suggest the tunable field modulation capabilities of the RACM, with measured focal patterns showing good agreement with full-wave simulations. Although the current implementation relies on manual operation, the reconfigurable design framework provides a foundation for the future development of a fully automated system capable of real-time, dynamic wavefront control. In addition, the discrete unit cells are expected to be extended to three-dimensional geometries through stretching or rotation. With phase gradients along two orthogonal directions, the RACM enables three-dimensional arbitrary wavefield manipulation, such as the generation of acoustic vortex beams. This work provides a compact and adaptable platform for cross-medium acoustic control, and opens new avenues for intelligent underwater-air communication and reconfigurable acoustic sensing systems.

CRediT authorship contribution statement

Yan-Long Du: Writing – review & editing, Supervision, Project administration, Conceptualization. Hong-Tao Zhou: Writing – review & editing, Visualization, Supervision, Software, Methodology, Conceptualization. Chen-Yang Li: Visualization, Software, Methodology. Lei Shi: Validation, Software, Methodology. Yan-Feng Wang: Writing – review & editing, Supervision, Project administration, Conceptualization. Yue-Sheng Wang: Writing – review & editing, Supervision, Project administration, Conceptualization.

Declaration of competing interest

The authors declare that they have no conflicts of interest in this work.

Biographies

Yan-Long Du is currently a master student in Mechanical Engineering at Tianjin University, China. He is with the Department of Mechanics, School of Mechanical Engineering, under the supervision of Prof. Yan-Feng Wang. His current research focuses on wave propagation in acoustic metasurfaces, with particular interest in wavefront manipulation across different media interfaces.

Hong-Tao Zhou is currently a postdoctoral researcher at Tianjin University. He received his BE from Huazhong Agricultural University in 2019 and his PhD in Solid Mechanics from Tianjin University in 2024, including joint PhD training at the National University of Singapore during 2022-2023. His research focuses on wave modulation mechanisms and the design of metasurfaces with vibro-acoustic coupling.

Yan-Feng Wang is a full professor at Department of Mechanics, Tianjin University. He began independent research career since 2016, after obtaining BS degree in Southwest Jiaotong University in 2009 and PhD degree in Beijing Jiaotong University in 2015, and then working as a postdoctoral associate in Institut FEMTO-ST CNRS in 2015–2016. He has been working in the area of wave mechanics, including wave propagation in acoustic/elastic metamaterials/metasurfaces.

Appendix A. Supplementary materials

Supplementary Data S1

Supplementary Raw Research Data. This is open data under the CC BY license http://creativecommons.org/licenses/by/4.0/

Supplementary Data S2

Supplementary Raw Research Data. This is open data under the CC BY license http://creativecommons.org/licenses/by/4.0/