Theory of impedance metasurface

The concept of SAW impedance is first put forward to play a key role as a bridge linking propagating waves and evanescent surface waves. According to the theory of holography⁴⁵, the phase information of the holography could be captured by interfering the scattered (radiation) wave from an object with a reference wave of the same frequency. For a reference wave ψ_ref and an object wave ψ_obj, the holographic interferogram⁴⁶ is given by $$ \mathrm{\Psi }=∣{\psi }_{\mathrm{ref}}+{\psi }_{\mathrm{obj}}∣$$. In our approach, acoustic impedance is introduced to characterize the intensity gradient of the interferogram; thus, it can be rewritten as

where X and M are the average SAW impedance and modulating depth, respectively. When the interference pattern is excited by the reference wave, the desired object wave can be reproduced as $$ {\psi }_{\mathrm{ref}}\left({\psi }_{\mathrm{ref}}^{*}{\psi }_{\mathrm{obj}}\right)={∣{\psi }_{\mathrm{ref}}∣}^2{\psi }_{\mathrm{obj}}$$.

To facilitate the implementation, the ideal continuous holographic impedance metasurface should be discretized into the discrete metasurface consisting of 2D modulated basic unit cells. Each cell as shown in Fig. 2a can be recognized as a rigid body with a subwavelength center-hollow-hole filled with background medium air^47–50, which is designed to couple the SAW modes with surrounding medium. By gradually changing the hole sizes, we can obtain the different surface refractive indexes and SAW impedances without breaking the quasi-periodicity of the whole structure. In this study, the eigen-mode analysis method in COMSOL Multiphysics is applied to extract the dispersive relation between the phase delay of a unit cell (with a fixed period of d) and the frequency (Fig. 2b), which is defined as

where ϕ corresponds to the phase difference across the unit cell, k_t and k₀ are the wave numbers on the metasurface and in free space, respectively, and n is the average surface refractive index of the metasurface. Based on Eq. (2), the effective surface refractive index can be directly acquired. Combining with the equation of SAW impedance (Supplementary Note 1), the relation between the phase offset and SAW impedance can be described as:

Therefore, once the phase offset is acquired through the dispersion curve, the effective surface refractive index and SAW impedance can be calculated correspondingly (Fig. 2c). On the contrary, the hole size can be set using interpolation method, as specific hole size corresponds to a particular phase delay at a certain frequency, provided the phase delay is set by the determined impedance. For illustration, at the intended working frequency of 6000 Hz, the height of the hole is 9 mm, which can be regarded as subwavelength, and the hole size varies from 2 to 4 mm with the values of X and M determined as 170.9 and 99.8, respectively.

To complete the design of a 2D modulated artificial acoustic metasurface, the SAW impedance textures of the surface should be first shaped. Assuming that a cylindrical sound source is located at the origin on the metasurface, which occupies the X–Y plane, the reference wave (or the surface plane wave) can be written as

where r is the radial distance from the center origin. The object wave is defined as⁵¹where θ₀ is the desired radiation elevation angle. Hence, the hologram composed of impedance can be expressed as

To obtain the periodicity of the holographic distribution of surface impedance along one direction, we make the phase item satisfy $$ \left(k_{0}n-k_0\sin {\theta }_0\right)p=2\pi m_0$$, so the corresponding period is represented by

which can be rewritten as

Here the left side of Eq. (8) represents the radiation term, whereas the first and second term of the right side are the wave vector of SAW and the modulation term, respectively. In essence, the shift of wave vector transfers the surface wave to the radiation wave (Supplementary Note 2). It is noteworthy that what we need is the −1 order diffraction term only, that is to say, when m₀ = 1, the left side of the equation should be modulated into the radiation area (i.e., point I to II); but when m₀ = 2, this term need to be removed out of the air cone (i.e., point I to III).

Acoustic spiral field

In a spiral field, the wave will twist along its axis as it travels, forming a beam shape similar to a bottle screw. The propagation phase satisfies:

in which the expression about the exit direction is set to be 0 ($$ \sin 0^{\circ }$$) as the desired direction of radiation is along the normal of the metasurface, whereas mθ acts as an additional phase depending on the azimuth angle, to generate a twisted wavefront. On the basis of the previous derivation, the SAW impedance interferogram here is given bywhere m is called the topological charge or the order of the spiral field, which is defined as the number of twist of the wavefront within one wavelength distance. For generating a first-order vortex beam with m = 1, the holographic pattern is shown in Fig. 3a, which is combined with the dispersion and impedance data shown in Fig. 2b, c, to determine the required hole as a function of spatial position. The pattern is a spiral-shaped strip and the spiral direction of the outgoing beam is consistent with the rotation of the surface spiral impedance strip.

We construct a sample antenna of finite dimension, whose profile of hole size a in the circumference and radial directions along pink dashed circle and blue dashed hatching line labeled in Fig. 3a are shown in Fig. 3b, c, respectively. The whole sample is a round sheet with a radius of 30 cm, composed of 7660 unit cells. Smaller transverse dimensions and the limitations are discussed in Supplementary Note 3. Measurements are designed and carried out to validate the generation of far-field spiral waves by the proposed method in the anechoic chamber and the photograph of experimental setup is illustrated in Fig. 3d. In fabrication, the sample material is chosen to be photosensitive resin (via the 3D printing technology), which can be treated as acoustically rigid for airborne sound. In measurements, we employed a point-like source placed at a quarter wavelength away from the metasurface to excite the SAW, whose directivity is similar to that of an electromagnetic dipole in accordance with mirror image principle. The far-field radiation information is detected by microphone array (G.R.A.S. Type 40PH), moving step by step in the plane parallel to the metasurface, with one additional microphone fixed in the outgoing field as a reference signal. As a result, the amplitude and phase of the sound in each scan point can be recorded in real time.

The corresponding simulation results are shown in Fig. 4 and the right-handed radiation wave can be received in the z direction. The simulation is carried out in a cylindrical calculation domain with the designed metasurface positioned at the bottom of the domain and the spatial lateral extent of the simulation window is selected to be 0.34 m, which is about 0.7λ beyond the radius of the metasurface. The cylindrical wave radiation condition is applied on the exteriors of the domain to prevent boundary diffraction at the edges, whereas a perfect matching layer is employed to absorb the radiated sound waves in the far field. Figure 4a illustrates a cutaway view of the outgoing sound field with the same radius of the metasurface. Good collimation characteristics can be observed along the entire propagation axis, making it possible to form a stable acoustic vortex beam over a considerably long distance. For illustration, we only depicted the spiral sound field within 2.00 m, which has reached 35 spatial wavelengths already. Given that the near field is inevitably subject to the interference of surface wave, the critical distance between the near- and far-field is defined as z = R²/λ = 1.57 m referring to the radiation of a circular piston⁵², and the far field exhibits the perfect acoustic vortex. The sound energy of the vortex is obtained by integrating the intensity I_z of the horizontal section of the cylindrical calculation domain in the far-field and its ratio to the total energy radiated from sound source obtained by integrating the sound intensity over the entire boundary of the model determines the efficiency of energy conversion as 76%. In Fig. 4b, a section of the far-field region is enlarged to manifest the distortion of both amplitude and phase more clearly (see Supplementary Movie 1 for the dynamic view of acoustic vortex beam). From the cross-sectional view, the phases have shifted from −π to π in each turn, revealing the expected topological charge m = 1. Moreover, the superposition of the phases at the center results in an intensity minimum at z axis (point B), which can be also seen in Fig. 4c. By contrast, sound energy is well concentrated in other areas of circular wavefront (i.e., point A) so that the beam can reach the far field, even though for each wavelength the wavefront of the vortex propagates forward, the sound pressure amplitude will decrease by around 4% as a result of the inevitable viscous loss in experiments. In addition, the measured phase distributions and sound amplitudes at the far-field cross-sections z = 1.57 and 2.00 m are shown in Fig. 4d (see Supplementary Note 4 for measured evolution process of phase diagrams within one period). The expected OAM in a first-order spiral beam with a smooth helical phase and null amplitude at the core can be clearly observed. Measurement results have good agreements with the numerical simulations, which demonstrates the performance of metasurface antenna in converting acoustic surface wave to a radiation beam with OAM. The weak asymmetry in the amplitude originates from the discretization differences among the impedances in each direction, which leads to the spiral stripes in impedance, despite the superficial resemblances of cylindrical symmetry. Consequently, the impedances traversed by the surface waves on each path are not identical to bring the corresponding outgoing wave field with an azimuthal angular dependence of e^−jmθ, which introduces some unevenness in the output amplitude.

From the above discussions, we can see that the proposed method achieves the desired first-order vortex beam both theoretically and experimentally. This method can be further extended to higher-order vortex beams by directly assigning a larger value to m. Figure 5 presents the amplitude and phase distributions of wavefront of the far-field vortex beam with topological charges m = 2–4 (see Supplementary Movie 2 for corresponding dynamic view). As in the previous section, the cross-sections at 2 m confirm that the long-distance propagation of the spiral beam could be maintained via adding the number of turns. It is noteworthy that we concentrate on the modulation of the output phase in this work. Although the radial distribution of the amplitude in theoretical design is set to be uniform, the actual amplitude values on each path are not exactly identical due to the impedance modulation and the quite modest difference can be evaluated from the stable vortex beams in far-fields in simulations and experiments. In addition, in the case of m > 1, the phase rotation with the azimuth angle is increased, where the helical wavefront reaches a total phase shift of 2πm over a complete turn. Hence, the phase singularity still emerges at the core without offset, while the hollow part of the central area is extended arising from the variation of the phase, indicating that the vortex of charge −m has been split into m vortices of charge −1 rather than 1. This phenomenon can be ascribed to effect that the high-order vortex has a more unstable structure than the first-order vortex and some slight disturbances could degenerate the unstable high-order vortex into multiple stable first-order vortices^53,54. In our simulations, the disturbances mainly arise from the weak non-cylindrically symmetric structure design and the interference of the point-like sound source with the radiated acoustic vortex^54,55.