Reconfigurable acoustic vortex via a 3D reflective metagrating

Abstract

Acoustic vortices carrying orbital angular momentum, often generated using metagratings, provide an effective approach for wavefront engineering, particle manipulation, and underwater information transfer. However, most existing acoustic metagratings rely on fixed structural configurations, which inherently restrict their tunability and multifunctionality. Here, we propose a three-dimensional reconfigurable reflective acoustic metagrating for dynamic manipulation of vortex fields. The device consists of two gradient-depth air-groove supercells, in which the reflection response can be actively reconfigured by tuning the depth of paired grooves. Based on a generalized conservation principle of topological charge in metagrating diffraction, we demonstrate controllable conversion of incident acoustic vortex beams into reflected vortex states with different diffraction channels. In particular, near-perfect switching between anomalous reflection and specular reflection is achieved with high conversion efficiency. In addition to structural reconfiguration, frequency-dependent modulation provides an additional degree of freedom for dynamically tailoring the reflected vortex field. Our results reveal a simple yet robust mechanism for tunable vortex-beam reflection and offer a compact platform for multifunctional acoustic devices, with potential extensions to other wave systems such as electromagnetic and elastic waves.

Graphical abstract

Keywords

acoustic vortex / reconfigurable metagrating / orbital angular momentum / topological charge

1 Introduction

Vortices are ubiquitous in nature, ranging from atmospheric tornadoes to fluidic whirlpools, and are intrinsically associated with rotational motion and angular momentum. Beyond classical fluid dynamics, vortex states can also emerge in wave systems. In acoustics, such states are referred to as acoustic vortices and are characterized by helical phase fronts, a central phase singularity, and the ability to carry orbital angular momentum (OAM) [1]. These unique properties enable acoustic vortices to transfer angular momentum to matter, thereby enabling applications such as particle trapping [2, 3], acoustic levitation [4, 5], and precise manipulation of cells or micro-objects [6, 7]. Acoustic waves also remain the dominant carrier for underwater communication because electromagnetic and optical waves experience strong attenuation and limited propagation distances in aqueous environments [8]. Therefore, the efficient generation and flexible manipulation of acoustic vortex beams are of considerable fundamental and practical importance.

Artificial metasurfaces have emerged as a versatile platform for wave manipulation by planar arrays of engineered subwavelength meta-atoms. Through tailoring the local phase, amplitude, or impedance responses, metasurfaces enable unprecedented control over wavefronts in both optical and acoustic systems [9–16]. In particular, phase-gradient metasurfaces can impart designed momentum shifts to incident waves, enabling functionalities such as anomalous reflection and refraction [17, 18], vortex-beam generation [19–23], asymmetric transmission [24–28], flat focusing [29–32], and advanced information-processing functionalities [33–36]. Nevertheless, conventional metasurfaces often suffer from impedance mismatch with the surrounding medium, thereby limiting the conversion efficiency toward the desired scattering channel. Metagratings offer an alternative and highly efficient route for wavefront control. Unlike deeply subwavelength metasurfaces that rely on local phase accumulation, metagratings employ carefully designed scatterers within a supercell to redistribute energy among discrete diffraction orders. By suppressing undesired diffraction channels while enhancing a selected diffraction order, metagratings can achieve high-efficiency anomalous reflection or transmission [37, 38]. Recently, phase-gradient metagratings (PGMs) have been shown to exhibit reflection and transmission characteristics governed by the parity configuration of unit cells within a supercell [21, 39]. This phenomenon, known as the parity-reversal diffraction law, provides a compact mechanism for controlling diffraction channels and has been explored in both acoustic and electromagnetic systems. These advances highlight the potential of metagratings for efficient and flexible wavefront engineering.

Despite these developments, most reported metasurfaces and metagratings [40–43] are based on fixed and often geometrically complex architectures. Once fabricated, their scattering responses are usually predetermined, resulting in single-function or limited-function operation. To overcome this limitation, dynamically tunable acoustic metasurfaces and metagratings have been proposed for beam steering and adaptive wave control. However, many existing tunable designs rely on multilayer configurations, complicated unit-cell geometries, or intricate modulation mechanisms, which may compromise compactness, fabrication simplicity, and energy efficiency [44, 45]. Recently, studies have shown that broadband acoustic vortices can be manipulated by tuning the reconfigurable multilayered metasurface with rotatable blades [46]. These constraints pose challenges for miniaturized acoustic information-processing systems and large-scale practical deployment. Therefore, it remains highly desirable to develop a simple, compact, and efficient acoustic metagrating capable of reconfigurable vortex manipulation.

In this work, we propose a three-dimensional reconfigurable reflective acoustic metagrating for dynamic manipulation of acoustic vortex beams. The metagrating is composed of two gradient-depth air-groove supercells arranged in a volute-like reflective configuration. By exploiting a generalized conservation principle of topological charge (TC) in metagrating diffraction [21], we reveal how the OAM carried by an incident vortex beam can be redistributed among selected reflected diffraction channels. More importantly, the reflected vortex state can be dynamically reconfigured by simply adjusting the depth of an air groove within each supercell. This structural modulation enables efficient switching between distinct reflection modes, including anomalous reflection and specular reflection, with near-perfect conversion efficiency [42]. Furthermore, we demonstrate that the reflected vortex field can also be tuned through the operating frequency, providing an additional degree of freedom for dynamic wavefront control. The proposed mechanism combines the high-efficiency diffraction control of metagratings with the reconfigurability required for multifunctional vortex manipulation. Compared with conventional fixed acoustic metagratings, the present design offers a simpler and more compact platform with high degree of freedom and excellent conversion efficiency for tunable three-dimensional vortex reflection. The underlying principle is general and can be extended to other wave systems, including electromagnetic and elastic waves. Our findings deepen the understanding of vortex-beam reflection in PGMs and provide a promising strategy for developing reconfigurable acoustic devices for particle manipulation, underwater communication, and acoustic information processing.

2 Methods

2.1 Numerical simulations

All numerical simulations were performed using the Pressure Acoustics, Frequency Domain interface in COMSOL Multiphysics®. A port boundary condition was applied at the entrance of the cylindrical waveguide to excite the incident acoustic vortex, with the pressure distribution prescribed as p\propto J_l(k_{r}r){\mathrm{e}}^{\mathrm{i}l\varphi }. The walls of the cylindrical waveguide and the rigid skeleton of the metagrating were modeled as sound-hard boundaries. The reflection coefficients for different diffraction orders were extracted using the coupled-mode theory described in Eqs. (1)–(5). The mesh was carefully refined with a maximum element size smaller than one-sixth of the operating wavelength. The reflection coefficients were evaluated at the interface between the metagrating and the air region, i.e., the plane z=0.

2.2 Mechanical implementation

In the practical device, ∆d_1 is continuously tunable via a sliding insert. The sliding insert consists of two fan-shaped end plates whose geometries precisely match the cross sections of the corresponding air grooves. The two end plates are rigidly connected to two push rods, which are in turn jointly fixed to a mounting plate. By manually driving the mounting plate or using a stepper motor to move it axially forward or backward, the two end plates slide synchronously within their respective grooves, thereby continuously and equally varying the effective groove depth to d_1+∆d_1. The possible effects of gaps, leakage, mechanical tolerance, and wall compliance are negligible.

3 Results

3.1 Schematics of the proposed 3D PGM and the mechanism of reconfigurable vortex

We first present the schematic design and operating mechanism of the proposed reconfigurable PGM for the manipulation of the vortex, as shown in Fig. 1. The sign of the topological charge is defined with respect to the propagation direction of the wave: a positive (negative) topological charge indicates that the phase increases counterclockwise (clockwise) when viewed against the propagation direction, corresponding to right-handed (left-handed) chirality. An incident acoustic vortex with TC l^{in}=1 impinges on the proposed PGM composed of two air-groove supercells (m=4). When the depth of the tunable grooves is set to d_1, a reflected vortex wave with the same TC (l^r=1) is generated, corresponding to anomalous reflection with the same chirality as the incident vortex, as shown in Fig. 1(a). By further tuning the groove depth to d=d_1+\mathrm{\Delta }{d}_1, the reflected vortex is converted into a state with the opposite TC (l^r=-1) under the same incident excitation, corresponding to specular reflection with the opposite chirality to the incident vortex, as displayed in Fig. 1(b).

We next describe the design principle of the acoustic PGM, as illustrated in Figs. 2(a) and (b). The PGM consists of two azimuthally arranged supercell groups, where each supercell contains m=4 fan-shaped unit cells. Figure 2(a) shows the cross section of the PGM with radius R. Each unit cell has an angular width of \theta =2\pi /(m{l}^{\xi }), where l^{\xi } denotes the intrinsic topological charge (ITC) provided by the PGM. Each unit cell is composed of a fan-shaped sound-hard wall and a fan-shaped air groove with angular width \eta \theta, where \eta is the filling ratio. The inner and outer radii of the grooves are R_1=0.05\lambda and R_2=0.38\lambda, respectively, where \lambda =\text{10 cm} is the designed wavelength in air. For simplicity, the PGM is implemented by assigning gradient depths to the air grooves, as shown in Fig. 2(b). The depth of the j-th groove is defined as d_j=\left(g+\frac{j}{2m}\right)\lambda ,j=1,2,\cdots ,m, where g is a scaling factor used to tune the overall phase distribution of the PGM. In the present case, with g=0.963, the groove depths are d_1=8.18\text{cm}, d_2=9.43\text{cm}, d_3=10.68\text{cm}, and d_4=11.93\text{cm}. The depth difference between two adjacent grooves is therefore \mathrm{\Delta }d=d_{j+1}-d_j=\lambda /(2m). When an acoustic vortex beam impinges on the PGM at the operating wavelength \lambda, with the corresponding wave number k_0=2\pi /\lambda, the wave entering the j-th groove accumulates a round-trip propagation phase before being reflected back to the air-PGM interface at z=0. The resulting phase delay is u_j=2k_0{d}_j-{\varphi }_r, 2k_0{d}_j is the propagation phase and {\varphi }_r denotes the additional phase delay introduced when the acoustic vortex is reflected at the interface between the air groove and the waveguide, i.e., at the plane z=0. The total phase accumulated by the acoustic wave during its round-trip propagation within the groove includes not only the propagation phase 2k_0{d}_j determined by the groove depth, but also phase discontinuities arising from impedance mismatch at the groove opening, edge scattering, and non-ideal rigid boundary conditions. {\varphi }_r is precisely the correction term that accounts for this deviation from ideal reflection behavior. Since {\varphi }_r is identical for all grooves, the phase difference between adjacent grooves is determined solely by the depth gradient \mathrm{\Delta }\varphi =2k_0\mathrm{\Delta }d=2\pi /m. Thus, the phase delay over one supercell spans a complete 2\pi range, enabling the PGM to carry an ITC of l^{\xi } [21, 27]. As a result, an azimuthal phase gradient is established at the interface.

The introduced phase gradient enables effective manipulation of the scattered wavefront and gives rise to anomalous reflection governed by a generalized conservation principle of TC, l^{in}+n{l}^{\xi }=l^t, [21] where l^{in} and l^t denote the TCs of the incident and transmitted vortex fields, respectively, and n is the diffraction order. Because the device operates in reflection, the reflected vortex undergoes an axial reversal of propagation direction relative to the transmitted vortex (+z\to -z). In the fixed laboratory coordinate system, the helical chirality (phase rotation direction) of the vortex is mirror-reversed with respect to the propagation direction. Therefore, to correctly describe the topological charge of the reflected vortex in a unified coordinate system, the reflected topological charge l^r and the hypothetical transmitted topological charge l^t are opposite in sign, i.e., l^t=-l^r. Substituting this into the transmission formula yields the reflected topological charge conservation relation: l^{in}+n{l}^{\xi }=-l^r. In this work, the incident vortex carries l^{in}=1, while the PGM provides an ITC of l^{\xi }=2. Under the conventional phase-gradient configuration, the dominant diffraction order is n=-1, thus yielding l^r=1, which corresponds to anomalous reflection of the vortex. Although the n=0 order corresponds to specular reflection, this channel is generally suppressed because of the phase-matching condition imposed by the well-defined azimuthal phase gradient [40, 47]. Consequently, a single incident vortex mode is usually converted into a single fixed reflected mode.

To overcome this limitation, we propose a simple reconfiguration strategy by tuning the depth variation \mathrm{\Delta }{d}_1 of an air groove within each supercell, namely the groove #1 highlighted in light blue and indicated by the red arrow in Fig. 2(b). Physically, this perturbation modifies the local phase response at the two selected grooves and introduces an additional phase difference, thereby breaking the local phase-gradient symmetry of the PGM [48]. As a consequence, the otherwise suppressed n=0 diffraction channel is opened. The coupling efficiency of this channel is strongly governed by the resonant behavior of the guided mode inside the air grooves. In particular, when the depth of the selected grooves satisfies the Fabry−Pérot (FP) resonance condition (i.e., the phase change produced by the vortex beam traveling back and forth once in the groove is equal to an integer multiple of 2\pi), a standing wave is formed inside the grooves, resulting in significant energy localization. At resonance, the reflected vortex is efficiently coupled into the n=0 diffraction order, leading to nearly perfect specular reflection. According to the generalized conservation principle of TC, the reflected vortex then carries a TC of l^r=-1 for the n=0 order.

The reflection coefficient is a key quantity for characterizing wave transport in the PGM. Here, the reflection coefficients are calculated using coupled-mode theory [25, 49]. By placing the PGM inside a cylindrical waveguide with radius R=\text{0.4λ}, the total acoustic pressure field can be expanded as a superposition of the incident vortex mode and all reflected vortex modes:

(1)\begin{array}{rl}{p}^1=& {\displaystyle \sum _{l,\alpha }[{\delta }_{l,l^{in}}\exp \left(\mathrm{i}{k}_z^{l,\alpha }z\right)}\\ & +r^{l,\alpha }\exp \left(-\mathrm{i}{k}_z^{l,\alpha }z\right)]J_l\left(k_r^{l,\alpha }r\right)\exp \left(\mathrm{i}l\varphi \right),\end{array}

where {\delta }_{l,l^{in}} is the Kronecker delta function, J_l(k_r^{l,\alpha }r) is the l-th Bessel function, r^{l,\alpha } is the reflection coefficient of the reflected vortex mode for the α-th order, and k_r^{l,\alpha } and k_z^{l,\alpha }=\sqrt{{(k_0)}^2-(k_r^{l,\alpha }{)}^2} are the transverse and longitudinal wave numbers of the corresponding vortex mode, respectively. Since only the fundamental mode and first-order mode are allowed to propagate, the pressure field within the j-th air groove ( and ) can be expressed as

(2)\begin{array}{rl}{p}_j^2=& \sum _{k,v}\left[A_j^{k,v}\exp \left(\mathrm{i}{k}_z^{k,v}z\right)+B_j^{k,v}\exp \left(-\mathrm{i}{k}_z^{k,v}z\right)\right]\\ & \times Q_k\left(k_r^{k,v}r\right)\cos \left(k\left(\varphi -{\theta }_j\right)\right),\end{array}

where j=1,2,\cdots ,m{l}^{\xi }, A_j^{k,v} and B_j^{k,v} represent the complex amplitude of forward and backward waves in the unit cell, Q_k(k_r^{k,v}r) is the radial wave mode that is a mixture function of the Bessel function and the Neumann function (Q_k(k_r^{k,v}r)=J_k(k_r^{k,v}r)+{\mu }_{k,v}{N}_k(k_r^{k,v}r)), k=m\pi /\theta (m=0,1,2,\cdots ) is the angular mode number, and the transverse wave number k_r^{k,v} and the coefficient {\mu }_{k,v} are the v-th solutions of the equation {Q^{\prime }}_k(k_r^{k,v}r){|}_{r=R_1}={Q^{\prime }}_k(k_r^{k,v}r){|}_{r=R_2}=0, respectively. The longitudinal wave number k_z^{k,v} is described as k_z^{k,v}=\sqrt{{(k_0)}^2-(k_r^{k,v}{)}^2}. Using the acoustic impedance relation v=-\frac{1}{i\omega \rho }\frac{\mathrm{\partial }p}{\mathrm{\partial }z} at z=d_j, and applying the continuous boundary conditions of the pressure and velocity fields at z=0, we have the following equations:

(3)\begin{array}{rl}& \sum _{l,\alpha }\left({\delta }_{l,l^{in}}+r^{l,\alpha }\right){\int }_{R_1}^{R_2}{J}_l\left(k_r^{l,\alpha }r\right)Q_k\left(k_r^{k,v}r\right)r\text{d}r\cdot \mathrm{i}l\exp \left(\mathrm{i}l{\theta }_j\right)\frac{\exp (\mathrm{i}l\theta )\cos (k\theta )-1}{k^2-l^2}\\ & =C_j^{k,v}\cos (k_z^{k,v}{d}_j)\cdot \frac{\theta }{2}(1+{\delta }_{k,0})\\ & \cdot \left[\frac{1}{2}\left({R_2}^2-{\left(\frac{k}{k_r^{k,v}}\right)}^2\right){\left(Q_k\left(k_r^{k,v}{R}_2\right)\right)}^2-\frac{1}{2}\left({R_1}^2-{\left(\frac{k}{k_r^{k,v}}\right)}^2\right){\left(Q_k\left(k_r^{k,v}{R}_1\right)\right)}^2\right],\end{array}

(4)\begin{array}{rl}& 2\pi \mathrm{i}({\delta }_{l,l^{in}}-r^{l,\alpha })k_z^{l,\alpha }\cdot \frac{1}{2}[R^2-{\left(\frac{l}{k_r^{l,\alpha }}\right)}^2]{\left[J_l\left(k_r^{l,\alpha }R\right)\right]}^2\\ & =\sum _j\sum _{k,v}{k}_z^{k,v}{C}_j^{k,v}\sin \left(k_z^{k,v}{d}_j\right)\cdot {\int }_{R_1}^{R_2}{J}_l\left(k_r^{l,\alpha }r\right)Q_k\left(k_r^{k,v}r\right)r\text{d}r\cdot -\mathrm{i}l\mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{i}l{\theta }_j\right)\frac{\exp (-\mathrm{i}l\theta )\cos (k\theta )-1}{k^2-l^2}.\end{array}

By numerically solving Eqs. (3) and (4), the reflection coefficients of the reflected vortex modes can be obtained. The reflection efficiency of the reflected vortex mode R^{l,\alpha }, defined as the ratio between the reflected and incident power, is given by

(5)\begin{array}{rl}{R}^{l,\alpha }=& \frac{\text{real}\left[{\displaystyle \iint p^{l,\alpha }{\left(v_z^{l,\alpha }\right)}^{\ast }{|}_{(z=0)}\text{d}S}\right]}{{\displaystyle \iint p^{in}{\left(v_z^{in}\right)}^{\ast }{|}_{(z=0)}\text{d}S}}\\ =& \frac{\text{real}\left[k_z^{l,\alpha }\right]{\displaystyle {\int }_0^R{\left[J_l\left(k_r^{l,\alpha }r\right)\right]}^{2}r\text{d}r}}{k_z^{in}{\displaystyle {\int }_0^R{\left[J_l\left(k_r^{in}r\right)\right]}^{2}r\text{d}r}}{|r^{l,\alpha }|}^2.\end{array}

For the lossless system, the total efficiencies of all reflected vortex modes are naturally normalized, i.e., {\sum }_{l,\alpha }{R}^{l,\alpha }=1.

Next, we evaluate the reflection efficiencies of the two reflected vortex modes as functions of the air groove depth variation \mathrm{\Delta }{d}_1\in [-0.5\lambda ,0.5\lambda ] and the operating frequency f in the range 3.3-3.7\text{kHz} for the n=-1 and n=0 orders, as shown in Figs. 2(c) and (d). The results reveal a pronounced transition in reflection efficiency at the operating frequency f=\text{3.43 kHz}, switching between the two vortex states with reflected TCs of 1 and -1. That is, the reflected vortex state switches between state 1 (marked with a yellow circle) and state 2 (marked with a gray circle), as illustrated in Figs. 2(c) and (d). We therefore focus on the case of f=\text{3.43 kHz} for the subsequent analysis.

3.2 Illustration for the reconfigurable 3D acoustic vortex

We calculate the variations in diffraction efficiencies for different orders and the achievable multiple reflection effect as the depth of a pair of air grooves (\mathrm{\Delta }{d}_1) is varied, using analytical equations and finite element simulations (FES), with obtained results shown in Fig. 3. Specifically, Fig. 3(a) shows the dependence of the reflection efficiencies on \mathrm{\Delta }{d}_1/\lambda for various diffraction orders, where n=-1 and n=0 are marked in blue and red, respectively. In the simulations, the entire system is immersed in air with a density of \rho =\text{1.21 kg/}{\text{m}}^3 and a sound speed of c=343\text{m/s}. The solid curves and discrete symbols correspond to analytical solutions and simulation results, respectively, which are in good agreement. Clearly, when \mathrm{\Delta }{d}_1=l\lambda /2(l=0,\pm 1,\pm 2,\cdots ), the gradient phase structure remains intact, and diffraction is dominated by the n=-1 order, achieving nearly perfect anomalous reflection with an efficiency exceeding 96.0%. By tuning \mathrm{\Delta }{d}_1 to adjust the accumulated phase of u_1, the FP resonance condition is satisfied, i.e.,

(6)2k_0(d_1+\mathrm{\Delta }{d}_1)-{\varphi }_r=2\pi q,

where {\varphi }_r=0.752\pi represents the additional phase delay induced by acoustic vortex reflection at the interface, and q=0,\pm 1,\pm 2,\cdots. {\varphi }_r is obtained independently using both analytical estimation and full-wave numerical simulations, and can be further fine-tuned and validated through fitting. In this case, when \mathrm{\Delta }{d}_1=(q-1.26)\frac{\lambda }{2}, the diffraction is dominated by the n=0 order, yielding near-perfect specular reflection with an efficiency exceeding 98.7%. It can be observed from Fig. 3(a) that the resonance peak has a certain width. Even with a fabrication deviation of \pm 0.03\lambda (approximately 3\text{mm}) in \mathrm{\Delta }{d}_1, the reflection efficiency can still remain above 90%, indicating that the design exhibits high robustness.

Next, we present the simulated pressure field and phase distribution associated with multiple high-efficiency reflections for the tunable PGM with varying \mathrm{\Delta }{d}_1, as shown in Figs. 3(b)–(d). Evidently, upon comparing the incident acoustic pressure field and phase depicted in Fig. 3(b) with the reflected fields and phases shown in Figs. 3(c) and (d), pronounced anomalous reflection is observed for the case of \mathrm{\Delta }{d}_1=0 under acoustic vortex beam incidence, as demonstrated in Fig. 3(c). Subsequently, by setting \mathrm{\Delta }{d}_1=-0.13\lambda, specular reflection can be achieved, as displayed in Fig. 3(d). In addition to visual inspection, the topological charge of the reflected field should also be quantitatively verified. First, the phase loop integration method can be applied: on a cross section of the reflected field, a closed path C encircling the central phase singularity is chosen. According to the definition of topological charge, its value is l^r=\frac{1}{2\pi }{\oint }_C^{}\mathrm{\nabla }\varphi \cdot \text{d}l. Second, modal decomposition of the reflected field can be employed. In this work, the total reflected field is expanded into a superposition of reflected vortex modes using coupled-mode theory, p={\sum }_{l,\alpha }{r}^{l,\alpha }\exp \left(-\mathrm{i}{k}_z^{l,\alpha }z\right)J_l\left(k_r^{l,\alpha }r\right)\exp \left(\mathrm{i}l\varphi \right), where each mode is labeled by the topological charge l and the radial order \alpha. By solving the boundary-condition equations, the reflection coefficients r^{l,\alpha } of each mode are obtained, and subsequently the reflection efficiencies R^{l,\alpha }. It is evident that the vortex reflection behavior transitions from anomalous to specular reflection merely by tuning the depth of a pair of air grooves. The aforementioned discussion indicates that precise manipulation of specific diffraction behaviors can be realized by disrupting the local phase of imperfect PGMs. In contrast, under the condition of perfect PGMs with well-defined phase gradient, the diffraction is invariably governed by the n=-1 order. Conversely, under the condition of imperfect PGMs lacking well-defined phase gradient, when the groove depth satisfies the FP resonance condition, the diffraction becomes dominated by specular reflection of the n=0 order, with a reflection efficiency approaching 98.7%. It should be emphasized that in perfect PGMs, FP resonances occasionally emerge within the grooves. However, they do not alter the diffraction characteristics of the PGMs, specifically, the n=0 order remains suppressed. This discovery provides theoretical support for the development of tunable acoustic devices based on phase gradient structures.

3.3 Dynamically tunable frequency on manipulating the vortex

Furthermore, for \mathrm{\Delta }{d}_1=-0.13\lambda, the variations in diffraction efficiencies for different orders and the associated multi-reflection characteristics are investigated by tuning the operating frequency f via the FES, as illustrated in Fig. 4. Specifically, Fig. 4(a) depicts the relationship between the reflection efficiencies of various diffraction orders and f, i.e., n=-1 and n=0, which are denoted by blue and red curves, respectively. As the operating frequency is gradually increased from 3.3\text{kHz} to 3.7\text{kHz}, the reflection efficiency of the n=0 diffraction order initially rises to its peak value. Thereafter, both the n=0 and n=-1 diffraction orders contribute jointly to the overall response, and ultimately, the n=-1 diffraction order gradually becomes the dominant component.

To intuitively demonstrate the frequency-dependent dynamic manipulation of acoustic vortices, we present the simulated pressure field and phase distribution associated with multiple high-efficiency reflections of the frequency-tunable PGM at distinct operating frequencies, as shown in Figs. 4(b)–(d). As clearly observed from the reflected acoustic pressure and phase distribution, high-efficiency anomalous reflection is achieved in the designed PGM at f=3.585\text{kHz} under the incidence of an acoustic vortex beam, as shown in Fig. 4(d). When the frequency is further tuned to 3.500\text{kHz}, high-efficiency specular reflection emerges and coexists with the anomalous reflection, as displayed in Fig. 4(c). Pure specular reflection alone can be attained by adjusting the frequency to 3.441\text{kHz}, as shown in Fig. 4(b). It can be clearly concluded that the vortex reflection behavior undergoes a transition from anomalous reflection to specular reflection via simple adjustment of the operating frequency. For imperfect PGMs operating at f=3.585\text{kHz}, the diffraction behavior is consistently governed by the n=-1 diffraction order, resulting in a reflection efficiency close to 99.9%. In this case, the 90% efficiency bandwidth is approximately 150\text{Hz} (from 3.55\text{kHz} to 3.70\text{kHz}), indicating a broadband operating range. Conversely, for imperfect PGMs at f=3.441\text{kHz}, the diffraction becomes dominated by specular reflection of the n=0 order, achieving a reflection efficiency of approximately 98.4%. Under this condition, the 90% efficiency bandwidth is approximately 40\text{Hz} (from 3.42\text{kHz} to 3.46\text{kHz}), revealing a narrowband resonant response. Apart from structural reconfiguration capabilities, this finding provides an innovative strategy to dynamically manipulate the reflected vortex field, and greatly promotes the development of frequency-tunable acoustic devices built upon phase gradient metasurface structures.

4 Conclusion

In summary, we have proposed a three-dimensional reconfigurable reflective acoustic metagrating for dynamic and multifunctional manipulation of acoustic vortex beams. Based on the generalized conservation principle of topological charge in metagrating diffraction, the proposed gradient-depth air-groove supercells enable controllable conversion of an incident vortex beam into distinct reflected diffraction channels. By simply tuning the depth of a selected pair of grooves, the local phase-gradient symmetry can be modified, allowing efficient switching between anomalous reflection and specular reflection. Theoretical calculations and finite element simulations verify the high-efficiency performance of the proposed design. Nearly perfect anomalous vortex reflection with an efficiency exceeding 96.0% is obtained when the original phase-gradient distribution is maintained, whereas near-perfect specular reflection with an efficiency up to 98.7% is achieved when the tuned grooves satisfy the Fabry-Pérot resonance condition. We also demonstrate that frequency modulation provides an additional mechanism for tailoring the reflected vortex state. These results provide a simple, compact, and robust strategy for reconfigurable acoustic vortex manipulation. The proposed concept advances the physical understanding of topological-charge conversion in reflective metagratings and may find applications in acoustic particle manipulation, underwater communication, and acoustic information processing. The design principle may also be extended to other wave systems, including electromagnetic, elastic, and water waves.

References

[1] K. Y. Bliokh and F. Nori , Spin and orbital angular momenta of acoustic beams, Phys. Rev. B 99(17), 174310 (2019)

[2] A. Marzo , M. Caleap , and B. W. Drinkwater , Acoustic virtual vortices with tunable orbital angular momentum for trapping of Mie particles, Phys. Rev. Lett. 120(4), 044301 (2018)

[3] Y. Jia , X. Zhang , S. Zhang , G. Xu , T. Chen , H. Zhou , Y. Bai , Y. Cheng , D. Wu , X. Liu , and C. W. Qiu , Synthesized acoustic vortex-frequency comb via rotational Doppler effect, Phys. Rev. Lett. 134(13), 137001 (2025)

[4] T. Feng , T. Meng , Y. Li , G. Guo , J. Tu , D. Zhang , and Q. Ma , Stable acoustic levitation based on coaxial confocal dual-frequency focused ultrasound and vortex beams, Ultrasonics 155, 107738 (2025)

[5] J. Zhou , S. Xu , L. Huang , S. Bao , F. Liao , B. Li , W. Qiu , F. Li , and H. Zheng , Large-scale airborne holographic acoustic tweezers for synthesizing acoustic fields and particle manipulation, Ultrasonics 158, 107810 (2026)

[6] Z. Gong and M. Baudoin , Particle assembly with synchronized acoustic tweezers, Phys. Rev. Appl. 12(2), 024045 (2019)

[7] M. Baudoin , J. L. Thomas , R. A. Sahely , J. C. Gerbedoen , Z. Gong , A. Sivery , O. B. Matar , N. Smagin , P. Favreau , and A. Vlandas , Spatially selective manipulation of cells with single-beam acoustical tweezers, Nat. Commun. 11(1), 4244 (2020)

[8] L. Mullen , D. Alley , and B. Cochenour , Investigation of the effect of scattering agent and scattering albedo on modulated light propagation in water, Appl. Opt. 50(10), 1396 (2011)

[9] A. V. Kildishev , A. Boltasseva , and V. M. Shalaev , Planar photonics with metasurfaces, Science 339(6125), 1232009 (2013)

[10] N. Yu and F. Capasso , Flat optics with designer metasurfaces, Nat. Mater. 13(2), 139 (2014)

[11] N. Meinzer , W. L. Barnes , and I. R. Hooper , Plasmonic meta-atoms and metasurfaces, Nat. Photonics 8(12), 889 (2014)

[12] Y. Xu , Y. Fu , and H. Chen , Planar gradient metamaterials, Nat. Rev. Mater. 1(12), 16067 (2016)

[13] B. Assouar , B. Liang , Y. Wu , Y. Li , J. C. Cheng , and Y. Jing , Acoustic metasurfaces, Nat. Rev. Mater. 3(12), 460 (2018)

[14] S. Sun , Q. He , J. Hao , S. Xiao , and L. Zhou , Electromagnetic metasurfaces: Physics and applications, Adv. Opt. Photonics 11(2), 380 (2019)

[15] W. Wang , C. Hu , J. Ni , Y. Ding , J. Weng , B. Liang , C. Qiu , and J. Cheng , Efficient and high-purity sound frequency conversion with a passive linear metasurface, Adv. Sci. (Weinh. ) 9(33), 2203482 (2022)

[16] H. W. Dong , C. Shen , Z. Liu , S. D. Zhao , Z. Ren , C. X. Liu , X. He , S. A. Cummer , Y. S. Wang , D. Fang , and L. Cheng , Inverse design of phononic meta-structured materials, Mater. Today 80, 824 (2024)

[17] T. He , T. Liu , S. Xiao , Z. Wei , Z. Wang , L. Zhou , and X. Cheng , Perfect anomalous reflectors at optical frequencies, Sci. Adv. 8(9), eabk3381 (2022)

[18] H. Zhu and F. Semperlotti , Anomalous refraction of acoustic guided waves in solids with geometrically tapered metasurfaces, Phys. Rev. Lett. 117(3), 034302 (2016)

[19] N. Yu , P. Genevet , M. A. Kats , F. Aieta , J. P. Tetienne , F. Capasso , and Z. Gaburro , Light propagation with phase discontinuities: Generalized laws of reflection and refraction, Science 334(6054), 333 (2011)

[20] H. Ren , G. Briere , X. Fang , P. Ni , R. Sawant , S. Héron , S. Chenot , S. Vézian , B. Damilano , V. Brändli , S. A. Maier , and P. Genevet , Metasurface orbital angular momentum holography, Nat. Commun. 10(1), 2986 (2019)

[21] Y. Fu , C. Shen , X. Zhu , J. Li , Y. Liu , S. A. Cummer , and Y. Xu , Sound vortex diffraction via topological charge in phase gradient metagratings, Sci. Adv. 6(40), eaba9876 (2020)

[22] H. Ahmed , H. Kim , Y. Zhang , Y. Intaravanne , J. Jang , J. Rho , S. Chen , and X. Chen , Optical metasurfaces for generating and manipulating optical vortex beams, Nanophotonics 11(5), 941 (2022)

[23] Q. Chen , G. Qu , J. Yin , Y. Wang , Z. Ji , W. Yang , Y. Wang , Z. Yin , Q. Song , Y. Kivshar , and S. Xiao , Highly efficient vortex generation at the nanoscale, Nat. Nanotechnol. 19(7), 1000 (2024)

[24] Y. Xu , C. Gu , B. Hou , Y. Lai , J. Li , and H. Chen , Broadband asymmetric waveguiding of light without polarization limitations, Nat. Commun. 4(1), 2561 (2013)

[25] Y. Li , C. Shen , Y. Xie , J. Li , W. Wang , S. A. Cummer , and Y. Jing , Tunable asymmetric transmission via lossy acoustic metasurfaces, Phys. Rev. Lett. 119(3), 035501 (2017)

[26] Y. Cao , Y. Fu , Q. Zhou , X. Ou , L. Gao , H. Chen , and Y. Xu , Mechanism behind angularly asymmetric diffraction in phase-gradient metasurfaces, Phys. Rev. Appl. 12(2), 024006 (2019)

[27] Y. Fu , Y. Tian , X. Li , S. Yang , Y. Liu , Y. Xu , and M. Lu , Asymmetric generation of acoustic vortex using dual-layer metasurfaces, Phys. Rev. Lett. 128(10), 104501 (2022)

[28] C. Chen , Y. Liu , L. Zhao , X. Hu , and Y. Fu , Asymmetric nonlinear-mode-conversion in an optical waveguide with PT symmetry, Front. Phys. (Beijing) 17(5), 52504 (2022)

[29] X. Chen , L. Huang , H. Mühlenbernd , G. Li , B. Bai , Q. Tan , G. Jin , C. W. Qiu , S. Zhang , and T. Zentgraf , Dual-polarity plasmonic metalens for visible light, Nat. Commun. 3(1), 1198 (2012)

[30] M. Khorasaninejad , W. T. Chen , R. C. Devlin , J. Oh , A. Y. Zhu , and F. Capasso , Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging, Science 352(6290), 1190 (2016)

[31] W. T. Chen , A. Y. Zhu , V. Sanjeev , M. Khorasaninejad , Z. Shi , E. Lee , and F. Capasso, , A broadband achromatic metalens for focusing and imaging in the visible, Nat. Nanotechnol 13(3), 220 (2018)

[32] Y. Luo , K. Ou , Z. Wang , K. Li , S. Dong , J. Zhu , P. Wang , Y. Wang , X. Dun , and Z. Wei , High-efficiency broadband achromatic orbital-angular-momentum metalens for focusing and edge-enhanced imaging in mid-wavelength infrared, Results Phys. 79, 108536 (2025)

[33] K. Wang , J. G. Titchener , S. S. Kruk , L. Xu , H. P. Chung , M. Parry , I. I. Kravchenko , Y. H. Chen , A. S. Solntsev , Y. S. Kivshar , D. N. Neshev , and A. A. Sukhorukov , Quantum metasurface for multiphoton interference and state reconstruction, Science 361(6407), 1104 (2018)

[34] R. Bekenstein , I. Pikovski , H. Pichler , E. Shahmoon , S. F. Yelin , and M. D. Lukin , Quantum metasurfaces with atom arrays, Nat. Phys. 16(6), 676 (2020)

[35] A. S. Solntsev , G. S. Agarwal , and Y. S. Kivshar , Metasurfaces for quantum photonics, Nat. Photonics 15(5), 327 (2021)

[36] Z. Guo , Z. Tan , X. Zang , T. Zhang , G. Wang , H. Li , Y. Wang , Y. Zhu , F. Ding , and S. Zhuang , Polarization-selective unidirectional and bidirectional diffractive neural networks for information security and sharing, Nat. Commun. 16(1), 4492 (2025)

[37] Y. Ra'di , D. L. Sounas , and A. Alù , Metagratings: Beyond the limits of graded metasurfaces for wave front control, Phys. Rev. Lett. 119(6), 067404 (2017)

[38] Y. Chen , J. Quan , Y. Gao , B. Sun , F. Gao , Y. Fu , and Y. Xu , Reconfigurable multifunctional acoustic metagratings enabled by local phase harnessing, Appl. Phys. Lett. 126(20), 201704 (2025)

[39] Y. Fu , C. Shen , Y. Cao , L. Gao , H. Chen , C. T. Chan , S. A. Cummer , and Y. Xu , Reversal of transmission and reflection based on acoustic metagratings with integer parity design, Nat. Commun. 10(1), 2326 (2019)

[40] Y. Xu , Y. Fu , and H. Chen , Steering light by a sub-wavelength metallic grating from transformation optics, Sci. Rep. 5(1), 12219 (2015)

[41] C. W. Qiu , T. Zhang , G. Hu , and Y. Kivshar , Quo vadis, metasurfaces, Nano Lett. 21(13), 5461 (2021)

[42] X. Li , C. Hu , Y. Tian , Y. Liu , H. Chen , Y. Xu , M. H. Lu , and Y. Fu , Maximum helical dichroism enabled by an exceptional point in non-Hermitian gradient metasurfaces, Sci. Bull. (Beijing) 68(21), 2555 (2023)

[43] Z. Hao , H. Chen , Y. Yin , C. W. Qiu , S. Zhu , and H. Chen , Efficient conversion of acoustic vortex using extremely anisotropic metasurface, Front. Phys. (Beijing) 19(4), 42202 (2024)

[44] W. K. Cao , C. Zhang , L. T. Wu , K. Q. Guo , J. C. Ke , T. J. Cui , and Q. Cheng , Tunable acoustic metasurface for three-dimensional wave manipulations, Phys. Rev. Appl. 15(2), 024026 (2021)

[45] Y. Tang , Y. Zhang , B. Xie , H. Cheng , J. Tian , and S. Chen , Transmission-reflection-integrated multifunctional continuously tunable metasurfaces for decoupled modulation of acoustic waves, Phys. Rev. Appl. 17(4), 044027 (2022)

[46] S. Zuo , Y. Wang , L. Zhang , Y. Guo , C. Cai , Y. Tian , and X. Liu , Generation of tunable broadband acoustic vortex via reconfigurable multilayered metasurface, J. Appl. Phys. 139(5), 053102 (2026)

[47] Y. Xie , W. Wang , H. Chen , A. Konneker , B. I. Popa , and S. A. Cummer , Wavefront modulation and subwavelength diffractive acoustics with an acoustic metasurface, Nat. Commun. 5(1), 5553 (2014)

[48] S. Li , C. Wang , M. Jiang , Z. Sun , L. Gao , Y. Fu , and Y. Xu , Subwavelength aperture light diffraction controlled by resonant metagrating with local symmetry-breaking, Front. Phys. (Beijing) 21(1), 14201 (2026)

[49] Y. Gao , J. Quan , B. Sun , L. Xu , Y. Fu , H. Chen , and Y. Xu , Frequency-doubling perfect negative reflection in phase gradient metasurfaces, Appl. Phys. Lett. 124(19), 191701 (2024)

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