Introduction

Vortices are ubiquitous phenomena across various domains of physics and have been extensively studied and generated in diverse wave fields, including electromagnetic waves1,2,3,4,5, acoustic waves6,7,8,9,10,11,12, electron waves13, and neutron waves14. Optical and acoustic vortices are characterized by a doughnut-shaped intensity distribution and a spiral phase distribution, and they can carry longitudinal orbital angular momentum (OAM) aligned with their propagation direction. The longitudinal OAM carried by vortex beams provides a new degree of freedom and has been widely utilized in applications such as information transmission15,16,17, super-resolution imaging18, quantum key distribution19, and particle manipulation20,21,22,23.

Recently, there has been a growing interest in spatiotemporal (ST) vortices where the wave packets are structured in both space and time24,25,26,27,28,29,30,31,32,33,34. Differing from conventional monochromatic vortices with a spiral phase in the spatial domain, these vortices are polychromatic wave packets and exhibit a spiral phase in the ST domain. The energy circulation in the ST domain results in a transverse OAM perpendicular to the propagation direction. This transverse OAM provides a new degree of freedom which is expected to extend the functionalities of vortices, and leads to the discovery of novel physical phenomena, such as spin-orbit interaction between transverse spin and orbital angular momentum28, transverse shifts and time delays of vortices at interfaces35 and nonlinear optical harmonic generation with transverse OAM36,37,38. Generating such transverse OAM requires simultaneously spatial and temporal modulations of wave packets. In optics, ST vortices were generated experimentally using sophisticated optical systems containing components like spatial light modulators and lenses39. ST vortices can also be generated by designed structures such as photonic crystal slabs and resonant diffractive gratings. The transmission function of these structures exhibits nodal lines in the wavevector-frequency space40,41,42,43,44. Moreover, arbitrarily oriented optical vortices could be constructed40,45,46,47, promising more controllable degree of freedoms and advanced applications.

Wave fields with OAM have also been widely investigated in the area of acoustics. Acoustic vortices can be generated by using a planar array of transducers7,48 or passive metasurface structures10. Several acoustic OAM-based applications were proposed, such as acoustic trapping49, rotation50, levitation51, and sound asymmetric transmission52. Acoustic communication by multiplexing OAM was also experimentally demonstrated, enabling high-speed data transmission for underwater applications16. Very recently, ST vortices have been explored in acoustics, bringing novel functionalities to acoustic OAM-based applications. The ST acoustic vortices are generated via a one-dimensional (1D) acoustic phased array53 or a periodic meta-grating54. These ST vortices propagate within the acoustic waveguide, which can be treated as a two-dimensional (2D) spatial geometry.

In this work, we advance our understanding of acoustic vortices by synthesizing three-dimensional (3D) ST acoustic vortices with OAM oriented in arbitrary directions, aiming for more comprehensive control over acoustic OAM. Compared to the 2D case, adding an additional dimension allows for the realization of more complex ST acoustic vortices. Unlike the methods used in optics, we decompose the ST vortices into a series of plane wave modes and generate them using an acoustic phased array. The 3D profile of ST acoustic vortices can be measured directly and precisely. We initially utilize a 2D acoustic phased array to generate 3D ST vortices carrying longitudinal and transverse OAM, respectively. To generate acoustic OAM with arbitrary orientations, we employ two distinct approaches. The first approach involves the direct rotation of wave packets carrying longitudinal or transverse OAM in 3D space, as depicted in Fig. 1a. The orientation of OAM will change accordingly. The second approach is achieved through the intersection of vortices carrying longitudinal and transverse OAM, as shown in Fig. 1b. The longitudinal and transverse OAM components can be modified, thereby altering the orientation of the OAM. The tilted OAM may be employed for more sophisticated particle manipulation through the transfer of acoustic OAM to particles. Our work establishes a platform for investigating 3D ST acoustic vortices and may open new avenues for exploring novel structured acoustic fields in the ST domain.

Fig. 1: Synthesizing three-dimensional (3D) spatiotemporal (ST) acoustic vortices with orbital angular momentum (OAM) oriented in arbitrary directions.
figure 1

a Direct rotation of ST acoustic vortices in 3D space. A ST wave packet carrying longitudinal OAM is rotated around the \(x\)-axis, and the direction of OAM will change accordingly, which is indicated by the red arrow in the coordinate system. b Intersection of ST vortices carrying transverse and longitudinal OAM yields a ST wave packet carrying tilted OAM. c Schematic of experimental setup. Spatiotemporal vortices are generated by an acoustic phased array controlled by the sound cards. On the scanning plane at a distance of \({z}_{0}=20\,{cm}\) from the acoustic phased array, we measure the time-domain acoustic pressure signals using a microphone array. A trigger signal is generated simultaneously with the wave packet signal by the sound card and transmitted to the data acquisition module to initiate data collection, ensuring that each captured time-domain acoustic pressure signal is located within the same time interval. d The acoustic phased array consists of 121 loudspeakers with a diameter of D = 4.3 cm, arranged in a two-dimensional rectangular grid with a spacing of L = 5.5 cm. e Photograph of the experimental setup.

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Results

Generation of three-dimensional (3D) spatiotemporal (ST) vortices with longitudinal and transverse orbital angular momentum (OAM)

Assuming that all wave packets propagate along the \(z\)-axis, we initially construct 3D ST acoustic wave packets carrying longitudinal and transverse OAM, respectively. The procedure is as follows: First, the analytical expression of the ST acoustic wave packet is derived, and a Fourier transform is applied to obtain its spatial frequency spectrum. From this spectrum, the intrinsic OAM of the ST acoustic wave packet is theoretically calculated. Next, the spatial frequency spectrum is sampled to extract a series of plane wave modes with specific frequencies and propagation directions. Experimentally, these plane wave modes are generated using an acoustic phased array. Finally, the superposition of these plane waves constructs the target ST acoustic wave packet.

To generate a 3D ST wave packet possessing longitudinal OAM along the \(z\)-axis, a spiral phase factor, \({[x+i\cdot {{\mathrm{sgn}}}\left(l\right)y]}^{\left|l\right|}\), is introduced into the \(x-y\) plane of a 3D Gaussian wave packet. Such a spiral phase will give rise to a phase singularity line along the \(z\)-axis, and \(l\) describes the number of times the phase wraps around the singularity line. Consequently, the amplitude at the center of the \(x-y\) plane must be zero, which manifests as a tunnel along the \(z\)-axis on the 3D iso-intensity profile of the wave packet. The presence of such a tunnel results in the iso-intensity surface taking on a toroidal geometry. The acoustic pressure field of this ST wave packet can be expressed as follows:

$$p\left(x,y,z,t\right)\propto {[x+i \cdot {{\mathrm{sgn}}}\left(l\right)y]}^{\left|l\right|}{{e}^{-\frac{{x}^{2}+{y}^{2}+{\zeta }^{2}}{{w}^{2}}}}{e}^{i{k}_{0}\zeta }$$
(1)

where \({k}_{0}\) is the central wavevector, \(\zeta=z-{ct}\) is the pulse-accompanying coordinate, and \(c\) is the speed of acoustic wave. The widths of the wave packet along \(x\), \(y\), and \(z\) axes are identical and are defined by the parameter \(w\). Similarly, we can add the phase factor \({[\zeta+i \, \cdot \, {{\mathrm{sgn}}}\left(l\right)x]}^{\left|l\right|}\) into the \(z-x\) plane (or the \(t-x\) plane) of a 3D Gaussian wave packet, and the acoustic pressure field of the ST wave packet can be expressed as follows:

$$p\left(x,y,z,t\right)\propto {[\zeta+i\cdot {{\mathrm{sgn}}}\left(l\right)x]}^{\left|l\right|}{{e}^{-\frac{{x}^{2}+{y}^{2}+{\zeta }^{2}}{{w}^{2}}}}{e}^{i{k}_{0}\zeta }$$
(2)

The ST wave packet is a polychromatic beam that can be constructed through the superposition of a series of plane wave modes \(P\left({{\bf{k}}}\right){e}^{i{{\bf{k}}\cdot{\bf{r}}}-i\omega \left({{\bf{k}}}\right)t}\), where \(P({{\bf{k}}})\) is the complex plane-wave amplitudes. \(P({{\bf{k}}})\) can be obtained through a Fourier transform, and represents the spatial frequency spectrum of the acoustic pressure field. The normalized integral values of OAM can be calculated using \(P({{\bf{k}}})\) as55:

$${{\bf{L}}}=\frac{\int {\omega }^{-1}{P}^{*}\left(-i{{\bf{k}}}\times {\nabla }_{{{\bf{k}}}}\right)P{d}^{3}{{\bf{k}}}}{\int {\omega }^{-1}{\left|P\right|}^{2}{d}^{3}{{\bf{k}}}}.$$
(3)

It is worth noting that the OAM consists of both intrinsic and extrinsic components. The extrinsic component originates from the transverse shift of the pulse probability centroid \({{\bf{R}}}\), which is defined as55:

$${{\bf{R}}}=\frac{\int {\omega }^{-1}{P}^{*}{e}^{i\omega t}\left(i{\nabla }_{{{\bf{k}}}}\right)P{{e}^{-i\omega t}d}^{3}{{\bf{k}}}}{\int {\omega }^{-1}{\left|P\right|}^{2}{d}^{3}{{\bf{k}}}}.$$
(4)

The extrinsic component satisfies \({{{\bf{L}}}}^{{{\boldsymbol{ext}}}}={{\bf{R}}}{{\boldsymbol{\times }}}{{{\boldsymbol{k}}}}_{{{\boldsymbol{0}}}}\), and we can obtain the intrinsic component as:

$${{{{\bf{L}}}}^{{{\boldsymbol{int}}}}={{\bf{L}}}-{{\bf{L}}}}^{{{\boldsymbol{ext}}}}.$$
(5)

The calculated \({{{\bf{L}}}}^{{{\boldsymbol{int}}}}\) of the two wave packets defined by Eqs. (1) and (2) are \((0,0,l)\) and \((0,l,0)\), which confirms that these two wave packets carry longitudinal and transverse OAM, respectively.

We then sample the spatial frequency spectrum to extract a series of plane wave modes, each defined by specific frequencies and propagation directions. In the spatial frequency domain, centered at \((0,0,{k}_{0})\), with \({k}_{0}=75.1\,m^{-1}\), we sample 14 values in each of the \({k}_{x}\), \({k}_{y}\), and \({k}_{z}\) directions with an interval of \(0.046{k}_{0}\), resulting in a total of 2744 modes. The corresponding frequency range is \(2873-5607\) Hz. These 2744 plane wave modes are then generated using a 2D acoustic phased array to construct the ST wave packet through their superposition. The resulting wave packet will exhibit a spatial width of approximately \(70\) cm and a temporal duration of around \(2\) ms.

Our experimental setup is schematically shown in Fig. 1c–e. The experiment is conducted in an anechoic chamber to mitigate the impact of reflections on the measurements. The 121 speakers are arranged in an \(11\times 11\) 2D square grid with a spacing of 5.5 cm between each row and column, forming a 2D acoustic phased array. Each speaker is controlled by an individual channel of a sound card, allowing independent adjustment of its amplitude and phase [see Methods for details]. By adjusting the phase of each speaker element in the array, the propagation direction of the acoustic wave can be controlled, enabling the generation of a plane wave mode with the desired direction. By simultaneously generating the selected 2744 plane wave modes, the ST wave packet can be constructed.

The 3D ST acoustic wave packet can be characterized either in the \(x-y-z\) space by a time snapshot or the \(x-y-t\) spatial-temporal domain at a fixed \(z\) position. Here, we adopt the latter characterization method, measuring the time-domain signals using microphones at each point within the \(z=0\) plane (the orange region in Fig. 1c), as depicted in Fig. 1c, e. The measurement step size in the \(x\)- and \(y\)- directions is 1.5 cm. In the experiment, the central frequency of 4100 Hz corresponds to a wavelength of 8.4 cm. The scanning step size is approximately 0.18 times the corresponding wavelength, ensuring accurate characterization of the acoustic field. For each measurement, the phased array generates the ST wave packet, and a segment of the time-domain acoustic pressure signal is recorded using the microphone. We employ a separate pulse signal from the sound card to synchronously trigger each recording session. The measured signal corresponds to the real part of the acoustic pressure field. By applying the Hilbert transform to the time-domain signal, the imaginary part can be obtained, providing the full complex representation of the acoustic pressure field [see Supplementary Section 3 for details]. This approach enables direct determination of the amplitude and phase distribution of acoustic field.

The 3D iso-intensity profiles of the ST wave packets carrying longitudinal and transverse OAM, obtained through both theoretical calculations and experimental measurements, are shown in Fig. 2a and b, respectively. The \(l\) term for both vortices are set to 1. The time \(t=0\) corresponds to the moment when the center position of the wave packet reaches \(z=0\). The orange arrows in Fig. 2a, b indicate the direction of the OAM carried by the corresponding wave packets. It can be observed that the experimental measurements are consistent with the theoretical predictions. Due to the spiral phase within the \(x-y\) plane, there is a phase singularity line along the \(z\)-axis, resulting in a tunnel in the direction of the \(z\)-axis for the wave packet carrying longitudinal OAM, as shown in Fig. 2a. Similarly, for the wave packet carrying transverse OAM, it exhibits a helical phase in the \(t-x\) plane and a singularity line along the \(y\)-axis, leading to a tunnel parallel to the \(y\)-axis, as depicted in Fig. 2b. We present theoretical calculations and experimental measurements of the amplitude and phase distributions of cross-sections passing through the centers of the two types of wave packets (the cyan planes in Fig. 2a, b). As shown in Fig. 2c, d, both cross-sections exhibit annular amplitude distributions, which arise from the presence of phase singularities at the centers of the cross-sections for both types of vortices, resulting in zero intensity. The phase distributions of the cross-sections of the two types of wave packets are provided in Fig. 2e and f, respectively. As illustrated in Fig. 2e, the wave packet carrying longitudinal OAM exhibits a phase screw dislocation in the \(x-y\) plane. In contrast, the wave packet carrying transverse OAM exhibits a characteristic edge phase dislocation in the \(t-x\) plane, as depicted in Fig. 2f. We demonstrate that 3D ST wave packets carrying longitudinal and transverse OAM can be generated in acoustics, laying the foundation for manipulating the direction of OAM. Next, we follow the same procedure to generate rotated and intersected ST acoustic vortices.

Fig. 2: Three-dimensional (3D) spatiotemporal (ST) vortices with longitudinal and transverse orbital angular momentum (OAM).
figure 2

a, b The 3D iso-intensity profiles of the ST wave packets carrying longitudinal OAM and transverse OAM. The maximum intensity is set to 1, with isovalue at 0.0625, meaning that the iso-intensity profile includes all points with intensity greater than 0.0625. The orange arrows indicate the direction of OAM carried by the wave packet. c, e The amplitude and phase distributions of the cross-section passing through the center of the wave packets carrying longitudinal OAM (the cyan plane in (a)), which show the screw phase dislocation. d, f The amplitude and phase distributions of the cross-section passing through the center of the wave packets carrying transverse OAM (the cyan plane in (b)), which show the edge phase dislocation.

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Direct rotation of spatiotemporal (ST) vortices in three-dimensional (3D) space

While previous studies on acoustic OAM have primarily focused on longitudinal and transverse OAM, it is essential to recognize that acoustic OAM can exhibit arbitrary orientations. To alter the direction of the OAM, we can directly rotate the wave packet carrying transverse or longitudinal OAM in 3D space. Here, we take the example of rotating a wave packet carrying longitudinal OAM. Considering a 45-degree rotation around the \(x\)-axis of the acoustic pressure field described in Eq. (1) and setting \(l=1\), the rotated wave packet is expressed as:

$$p\left(x,y,z,t\right)\propto \left[x+i\left(\frac{\sqrt{2}}{2}y-\frac{\sqrt{2}}{2}\zeta \right)\right]{{e}^{-\frac{{x}^{2}+{y}^{2}+{\zeta }^{2}}{{w}^{2}}}}{e}^{i{k}_{0}\zeta }.$$
(6)

The wave packet carrying longitudinal OAM is rotated without altering its propagation direction along the \(z\)-axis. The intrinsic OAM carried by the tilted wave packet can be calculated using Eqs. (3)–(5) as \((0,\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})\), indicating that the OAM direction also rotates 45 degrees around the \(x\)-axis. The 3D iso-intensity profiles, derived from both theoretical calculations and experimental measurements for the rotated wave packet, are illustrated in Fig. 3a, b, including perspective, front, side, and top views. The direction of the OAM after rotation is indicated by orange arrows in Fig. 3a. From the three views, it is evident that there is a tunnel tilted at a 45-degree angle upwards relative to the \(z\)-axis. This is because the singularity line is rotated from being aligned with the z-axis to a direction at a 45-degree angle to the z-axis. Figure 3c, d presents the amplitude distributions in the \(t-x\), \(t-y\), and \(x-y\) cross-sections passing through the center of the wave packet, and Fig. 3e, f illustrate the corresponding phase distributions, based on both theoretical calculations and experimental measurements. Due to the rotation of the wave packet, the previously circular intensity distribution on the \(x-y\) plane is now rotated onto a plane with a normal vector of \((0,\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})\). The tilted annular intensity distribution obtained after rotation exhibits an elliptical intensity distribution on both the \(t-x\) and \(x-y\) cross-sections, while the diagonal line with zero intensity in the \(t-y\) cross-section corresponds to the tilted tunnel passing through the center of the wave packet, as illustrated in Fig. 3c, d. The phase distribution in the \(t-x\) cross-section exhibits an edge phase dislocation, corresponding to the transverse OAM component in the \(y\)-direction, while the phase distribution in the \(x-y\) cross-section manifests a helical phase dislocation, corresponding to the longitudinal OAM component in the \(z\)-direction, as shown in Fig. 3e, f. Here, only an example of a wave packet rotating 45 degrees around the \(x\)-axis is shown. More generally, by controlling the direction and angle of rotation of the wave packet, the orientation of OAM can be precisely manipulated in the 3D space.

Fig. 3: Direct rotation of spatiotemporal (ST) vortices in three-dimensional (3D) space.
figure 3

a, b The 3D iso-intensity profiles of the rotated ST wave packet, along with its front, side, and top views, obtained from theoretical calculations and experimental measurements. The maximum intensity is set to 1, with isovalue at 0.0625. The orange arrows indicate the direction of the orbital angular momentum (OAM) carried by the wave packet, pointing towards \((0,\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})\). The amplitude and phase distributions of three cross-sections passing through the center of the wave packet, derived from both theoretical calculations and experimental measurements, are shown in (c), (d) and (e), (f), respectively.

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Intersection of vortices carrying longitudinal and transverse orbital angular momentum (OAM)

To vary the direction of the OAM, apart from rotating a wave packet carrying OAM in a particular direction, another method involves integrating vortices carrying different types of OAM within a single wave packet. For instance, the intersection of a vortex carrying longitudinal OAM and another carrying transverse OAM within a 3D Gaussian wave packet yields a wave packet carrying tilted OAM. The acoustic pressure distribution of the wave packet generated by the intersection of two types of vortices is illustrated as follows:

$$p\left(x,y,z,t \right)\propto {[x+i \cdot {{\mathrm{sgn}}}({l}_{1})y]}^{\left|{l}_{1}\right|}{[\zeta+i \cdot {{\mathrm{sgn}}}\left({l}_{2}\right){{x}}]^{\left|{l}_{2}\right|}{e}^{-\frac{{x}^{2}+{y}^{2}+{\zeta }^{2}}{{w}^{2}}}}{e}^{i{k}_{0}\zeta }$$
(7)

where \({l}_{1}\) and \({l}_{2}\) denote the spiral phase in the \(x-y\) domain and \(z-x\) domain, respectively. The intrinsic OAM carried by the wave packet is calculated as \((0,{l}_{2},{l}_{1})\). By adjusting the values of \({l}_{1}\) and \({l}_{2}\), we can change the orientation of the OAM in the \(y-z\) plane. When we set \(l\) terms of both vortices to 1, the tilted OAM can be calculated using Eqs. (3)–(5), resulting in \((0,1,1)\). Figure 4a, b illustrates the 3D iso-intensity profiles of the wave packet represented by Eq. (7), obtained through both theoretical calculations and experimental measurements. The wave packet features two tunnels, with the \(t\)-axis direction tunnel arising from the helical phase in the \(x-y\) plane and the \(y\)-axis direction channel originating from the helical phase in the \(t-x\) plane. Figures 4c, d illustrates the acoustic pressure distribution in the \(t-x\), \(t-y\), and \(x-y\) cross-sections passing through the center of the wave packet.

Fig. 4: Intersection of vortices carrying longitudinal and transverse orbital angular momentum (OAM).
figure 4

a, b The three-dimensional iso-intensity profiles of the spatiotemporal wave packets combining two vortices carrying different types of OAM, obtained through theoretical calculations and experimental measurements. The maximum intensity is set to 1, with isovalue at 0.04. The orange arrows indicate the direction of the OAM carried by the wave packet, pointing towards \((0,1,1)\). c, d Theoretical calculation and experimental measurement results of the acoustic pressure field on the cross-sections passing through the center of the wave packet.

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Furthermore, we demonstrate the intersection of three vortices carrying OAM in mutually perpendicular directions. By changing the \(l\) terms of the three vortices, the orientation of the OAM can be arbitrarily changed within the \(x-y-z\) space. Figure 5a, b depicts the theoretical calculation results and experimental measurements of the iso-intensity profiles of the wave packet combining three vortices, each with a \(l\) term of 1. The calculated intrinsic OAM carried by the wave packet is \(({\mathrm{1,1,1}})\), as depicted by the orange arrow in Fig. 5a. The tunnel along the \(t\)-axis is attributed to the helical phase in the \(x-y\) plane, whereas the tunnels along the \(x\)-axis and \(y\)-axis is caused by the helical phase in the \(t-y\) plane and \(t-x\) plane. This demonstrates that we can create complex zero-amplitude tunnel structures through the superposition of 3D ST vortices. The acoustic pressure fields of three cross-sections passing through the center of the wave packet, are depicted in Fig. 5c, d. In comparison to the wave packet illustrated in Fig. 4, the wave packet depicted in Fig. 5 introduces an extra vortex in the \(t-y\) domain, which induces an additional phase difference of π between the regions of \(t \, < \, 0\) and \(t \,>\, 0\) in the \(t-x\) cross-section, as well as between the regions of \(y\,>\,0\) and \(y \, < \, 0\) in the \(x-y\) cross-section. It is worth noting that, to achieve more precise control over the orientation of OAM in 3D space, the \(l\) terms of the intersected vortices can take fractional values rather than being restricted to integer values.

Fig. 5: Intersection of three vortices carrying orbital angular momentum (OAM) in mutually perpendicular directions.
figure 5

a, b The three-dimensional iso-intensity profiles of the spatiotemporal wave packets combining three vortices, obtained through theoretical calculations and experimental measurements. The maximum intensity is set to 1, with isovalue at 0.0841. The orange arrows indicate the direction of the OAM carried by the wave packet, pointing towards \((1,1,1)\). c, d Theoretical calculation and experimental measurement results of the acoustic pressure field on the cross-sections passing through the center of the wave packet.

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Discussion

In summary, we report the generation of ST acoustic vortices with arbitrarily oriented OAM. We first extend ST acoustic vortices from 2D to 3D space, constructing ST wave packets carrying longitudinal and transverse OAM, respectively. The ST wave packet carrying tilted OAM can be obtained through the rotation of vortices in 3D space or the intersection of vortices carrying distinct types of OAM. The direction of OAM can be altered in the first method by adjusting the rotation angle or, in the second method, by modifying the OAM components in orthogonal directions. We experimentally generated ST wave packets using an acoustic phased array. By combining 2D spatial field scanning with 1D time-domain signal measurement, we achieved precise characterization of 3D wave packets in the space-time domain. The direction of OAM serves as a new degree of freedom, expanding the capabilities for manipulating acoustic waves, with potential applications across various fields. Through the interaction between ST vortices and particles, it is possible to transfer OAM in arbitrary directions to particles, thereby achieving transient and arbitrary-directional particle manipulation. Shaping waves in the spatiotemporal domain with increasingly complex structures offers a promising avenue for research in both fundamental physics and applied sciences. Our work provides a platform for studying 3D acoustic wave fields with novel space-time structures, which will aid in the discovery and study of more exotic acoustic wave physics56,57,58,59,60,61,62.

Methods

Generation of spatiotemporal (ST) vortices using the acoustic phased array

Using Fourier integration, a ST wave packet can be represented as a superposition of a series of plane wave modes:

$$p\left(x,y,z,t\right)=\int\int\int P\left({k}_{x},{k}_{y},{k}_{z}\right){e}^{{ik}\cdot r-i\omega \left(k\right)t}d{k}_{x}d{k}_{y}d{k}_{z}.$$
(8)

The spatial frequency spectrum can be obtained by performing a Fourier transform on the acoustic pressure field at \(t=0\), expressed as follows:

$$P\left({k}_{x},{k}_{y},{k}_{z}\right)=\int\int\int p\left(x,y,z,t \,=\, 0\right){e}^{-{ik}\cdot r}{dxdydz}.$$
(9)

By setting the width of the wave packet to \(\frac{14}{{k}_{0}}\), the non-zero plane wave modes in the spatial frequency spectrum of ST wave packets mainly distribute within the intervals \([-0.3{k}_{0},0.3{k}_{0}]\) for \({k}_{x}\), \([-0.3{k}_{0},0.3{k}_{0}]\) for \({k}_{y}\), and \([{k}_{0}-0.3{k}_{0},{k}_{0}+0.3{k}_{0}]\) for \({k}_{z}\). To generate ST wave packets using an acoustic phased array, we sample points in the spatial frequency spectrum. The wave vectors \({k}_{x}\), \({k}_{y}\), and \({k}_{z}\) are sampled at equal intervals within the ranges of [\(-0.3{k}_{0}\), \({0.3k}_{0}\)], [\(-0.3{k}_{0}\), \({0.3k}_{0}\)], and [\({k}_{0}-{0.3k}_{0}\), \({k}_{0}+0.3{k}_{0}\)], with each interval containing 14 values. In this manner, we acquire 2744 plane wave modes represented by \(({k}_{x},{k}_{y},{k}_{z})\). We drive the speaker array with the appropriately phased and amplitude-adjusted signals to generate acoustic plane wave modes steered in the desired directions. By superimposing these 2744 plane wave modes, the ST wave packet can be constructed.

Experimental setup

The experimental setup and measurement process are illustrated in Fig. 1c, e. We conduct experiments in an anechoic chamber covered with sound-absorbing materials on all sides to avoid unwanted reflections that could impact the experimental measurements. The 121 1.5-inch speakers are arranged in an \(11\times 11\) 2D square grid with a spacing of 5.5 cm between each row and column [see Supplementary Fig. S1 for details]. The phased array is driven by four 32-channel sound cards (PreSonus Quantum 4848), all synchronized and controlled by a computer. Each speaker corresponds to a single channel on the sound card, allowing for individual control of each speaker’s amplitude and phase. To improve measurement efficiency, we use the microphone array to capture the acoustic pressure field. The microphone array consists of 8 microphones (Knowles SPM0687LR5H−1), with a spacing of 9 cm between each microphone [see Supplementary Fig. S2 for details]. The microphone array is fixed on a 2D translation stage to measure the acoustic pressure field on the scan plane. The scan plane is parallel to the phased array and is positioned 20 cm away from it, with dimensions of approximately 70 cm by 70 cm. The translation stage moves in both the \(x\) and \(y\) directions with a step size of 1.5 cm. At each step, the phased array emits the same spatiotemporal wave packet, and the microphone array measures the time-domain acoustic pressure signals. Ultimately, the measurement points form a two-dimensional square grid with a spacing of 1.5 cm, covering the entire scan plane. The time-domain acoustic pressure signals are collected by the NI data acquisition module (NI 9234) at a sampling rate of 51.2 kHz. A trigger signal is generated simultaneously with the wave packet signal by the sound card and transmitted to the data acquisition module to initiate data collection, ensuring that each captured time-domain acoustic pressure signal is located within the same time interval. After the measurement is completed, the acoustic pressure field \(p(x,y,t)\) can be obtained.