Abstract
Spin angular momentum enables fundamental insights for topological matters, and practical implications for information devices. Exploiting the spin of carriers and waves is critical to achieving more controllable degrees of freedom and robust transport processes. Yet, due to the curlfree nature of longitudinal waves distinct from transverse electromagnetic waves, spin angular momenta of acoustic waves in solids and fluids have never been unveiled only until recently. Here, we demonstrate a metasurface waveguide for sound carrying nonzero acoustic spin with tight spinmomentum coupling, which can assist the suppression of backscattering when scatters fail to flip the acoustic spin. This is achieved by imposing a soft boundary of the π reflection phase, realized by comblike metasurfaces. With the specialboundarydefined spin texture, the acoustic spin transports are experimentally manifested, such as the suppression of acoustic cornerscattering, the spinselected acoustic router with spinHalllike effect, and the phase modulator with rotated acoustic spin.
Introduction
The spin angular momenta (SAM) of classical waves, ranging from elastic to optical waves, are essentially associated with their local chiral polarized profiles. Many wellknown waveforms, such as circularly polarized plane wave, twowave interference, spatial confined Gaussian beam, and surface evanescent wave, have been shown to possess nonzero SAM^{1,2,3,4,5,6,7,8}. The SAM exhibits a family of spinrelated effects with remarkable properties for waves, i.e., spinHall effect (SHE)^{2,9}, quantum spin Hall effect (QSHE)^{10,11} and spinlocked scattering in bianisotropic media^{12}. Meanwhile, these findings give rise to some emergent classes of research fields, like the chiral quantum optics^{13}. Many applications have then been proposed to control classical waves with SAM, such as spinselective plasmonics excitations^{14}, chiral dependencies of quantum optical devices^{15}, hyperbolic metamaterial wave routers^{5,16}, and so on. The advantages of robustness and flexible control of the spinbased wave devices would have impact on many areas of wave theories, experiments, and devices^{17,18,19,20,21,22,23,24,25}.
In the past, vast previous research focused on the orbital angular momentum of acoustics^{26} without thinking about the existence possibility of SAM due to the curlfree nature of longitudinal waves. Because conventionally it is enough to represent the acoustic wave by scalar pressure fields only, and classical field theory concludes that scalar fields possess zero SAM^{27}. Therefore, people widely believed that the curlfree longitudinal wave cannot shape into circularly polarized from as transverse optical waves so that the acoustic waves cannot possess SAM. Just recently, after fully treating the acoustics as vectorial velocity fields, the fact that the acoustics can process the SAM intrinsically has been unveiled theoretically and observed experimentally^{5,6,7,8}. The acoustic SAM can thus be naturally associated with the circularly (elliptically, in general) polarized profile of acoustic velocity fields (nonzero v* × v), which does not conflict with the zero vorticity ∇ × v = 0 of the curlfree acoustic waves. After these improvements in understanding the longitudinal wave SAM, exploiting the acoustic SAM in practice becomes of significance and attracts considerable attention, including spin–orbit coupling in acoustic Bessel beams^{28}, acoustic spininduced torque^{29}. In this work, we propose metasurface waveguides to demonstrate several acoustic SAM applications on spinrelated robust transport. The key of our work is based on the nonzero acoustic SAM resulting from the specific waveguide modes after introducing metasurface boundary conditions. As shown in Fig. 1a) we find that the SAM of modes is strongly coupled to the propagation direction (momentum), resembling the relation of spinmomentum locking in QSHE states^{2,25}. Owing to these properties, the acoustic waveguide mode exhibits backscatteringsuppressed transport if scatters do not flip the SAM, and becomes spinselective when facing the multiple channels due to different SAMmomentum dependences. We perform several experiments to exemplify these spinrelated phenomena. Finally, we obtain the rotated acoustic spin as a phase modulator induced by gradually rotating the metawaveguide boundary. Our work would give more fundamental understandings about wave spin physics and provide new insights about controlling waves via SAM degree of freedom.
Results
Nonzero acoustic spin in waveguide with symmetry breaking
According to the physical meanings possessed in angular momenta of acoustic waves, the SAM density can be described as (refs. ^{5,6,7,8}):
where ρ is the mass density, ω is the frequency, c is acoustic velocity, and v is acoustic velocity field. According to the definition in Eq. (1), the acoustic SAM density will be nonzero for the local circularly (or in general, elliptically) polarized profile of acoustic velocity fields. To satisfy the circularly polarized velocity field conditions, we introduce nonsymmetric boundary settings to break the xmirror symmetry of waveguide, shown in Fig. 1b, which will led to significant phase delay for v_{x} compared with v_{z}. By general symmetry argument, setting \({{\mathcal{M}}}_{y}\)(y → −y) as the ymirror operation, we have \({{\mathcal{M}}}_{y}{s}_{x}(y){{\mathcal{M}}}_{y}^{1}={s}_{x}(y)\), \({{\mathcal{M}}}_{y}{s}_{y}(y){{\mathcal{M}}}_{y}^{1}={s}_{y}(y)\) and \({{\mathcal{M}}}_{y}{s}_{z}(y){{\mathcal{M}}}_{y}^{1}={s}_{z}(y)\). So for the nondegenerate waveguide mode, when the crosssection has the ymirror symmetry, the crosssection SAM S_{x} = ∫s_{x}dxdy and S_{z} = ∫s_{z}dxdy will definitely vanish due to the cancellation of oddsymmetric SAM densities on the waveguide section. Similarly, the xmirror symmetry of the system will lead to the vanishing S_{y} = 0 and S_{z} = 0 due to the oddfunction s_{y,z}(x) = −s_{y,z}(−x). Therefore, to induce the nonzero SAM, e.g., S_{y} ≠ 0 for nondegenerate modes, one could break the \({{\mathcal{M}}}_{x}\) symmetry.
Considering the combination of the sound hard boundary (ϕ = 0) and the metasurfaceinduced soft boundary (ϕ = π) in Fig. 1b, we can analytically obtain that the waveguide mode indeed processes the nonzero total SAM as (see “Method”):
where J_{0}(κr) is the zeroorder Bessel function of the first kind, e_{y} is the unit vector in ydirection, k is the longitudinal momentum component with k = ke_{z} the wave vector along zdirection, \(\kappa =\sqrt{{\omega }^{2}/{c}^{2}{k}^{2}}\) is the transverse wave vector, c is the sound speed in the air, R is the radius of the waveguide, and p_{0} is the pressure field amplitude. Based on these nonsymmetric boundary conditions, the metawaveguide eigenmodes will naturally carry nonzero SAM S_{y} ≠ 0 in the ydirection, depicted in Fig. 1c. Specially, this SAM is tightly locked to the momentum direction that reversing the momentum k will flip the spin, which resembles the spinmomentum locking in QSHE^{2,11}. By using the spin sources (circularly polarized acoustic dipoles)^{5,6} corresponding to different SAM, we can excite the waveguide modes selectively, as shown in Fig. 1d.
It should be mentioned that this spinmomentum locking feature of metawaveguide modes is different from the cases in the surface evanescent modes^{1,30} due to nontrivial bulk topology and bulkedge correspondence^{25,31}, where the field strength is strongly localized at (meta)surfaces. For our waveguide modes, the SAMdistinguished dispersion is attributed to the structuredependent wave interference and the resulting chiral velocity field wrapped by the phasedelayed metasurface. Specifically, the acoustic wave is spatially confined well in the area near the normal surface but severely attenuated in the area near the metasurface (see Fig. 1c, d), the latter of which is caused by the destructive interference due to the phasedelayed reflective metasurfaces. This transverse attenuation induces effective chiral fields perpendicular to the waveguide propagation direction, similar to the case in Gaussian longitudinal wave beams^{5}.
Acoustic metasurface structures
We exploit the side bar structures to realize arbitrary phasedelayed reflective acoustic wave metasurfaces, as shown in Fig. 2a, b. The side bar arrays on the waveguide boundary can be regarded as acoustic metaatoms^{32}. These metaatoms can reflect the acoustic wave with arbitrary phase delay when they become resonant^{33}. As such, we can find that the waveguide mode can carry nontrivial acoustic SAM S_{y} ≠ 0, and the SAM is strongly associated with the linear momentum responsible to the propagation shown in Fig. 2c. This momentumdependent SAM will led to the opposite SAM texture of modes for different momentum excitations in Fig. 2d. Specially, the SAM densities is associated with the nearcircularly (elliptically) polarized velocity fields. As demonstrated in Fig. 2e, metasurface waveguide modes with k > 0 at different frequencies will result in the topviewed anticlockwise elliptically polarized velocity fields corresponding to s_{y} > 0, which make opposite propagating modes of opposite polarizations become approximately orthogonal to each other. This momentumlocked SAM profile of bulk modes within the waveguide will apparently reduce couplings between forward propagating and reflected scattering modes. To experimentally verify the SAM densities inside the metasurface waveguide, we perform experimental measurements (see “Method”) and compare them with analytic theory predictions Eq. (2) and numerical simulations, as shown in Fig. 3. One can see that the experimental measurements are in good agreements with theoretical and numerical results. Indeed, the ymirror symmetry makes the oddsymmetric s_{x} with \({s}_{x}(y)={{\mathcal{M}}}_{y}{s}_{x}(y){{\mathcal{M}}}_{y}^{1}={s}_{x}(y)\), resulting in the vanishing S_{x} = ∫s_{x}dxdy = 0 due to the cancellation in the integral. However, the xmirror symmetry breaking leads to the nonperfect cancellation s_{y}(x) ≠ −s_{y}(−x), facilitating the nonzero S_{y} ≠ 0.
Experimental observations of acoustic spin transport
With this specialboundarydefined spin texture and the inherent spinmomentum locking shown in Eq. (2) and Fig. 3, the transport in the metasurface waveguide would be robust compared with conventional spinless waveguide modes. As demonstrated in Fig. 4a, we bend the metasurface waveguide with the bending angle θ. In the Fig. 4b, the simulated transmission T of the bended metasurface waveguide will become better than the conventional circular duct waveguide, especially for θ ∈ [0, π/2], which will be nearly 100% transmission. This is assisted by the tight acoustic spinmomentum locking of the acoustic spinful mode in the metawaveguide (see Eq. (2) and Fig. 2), which for opposite transport directions (momentum) has opposite SAMs. In particular, the nonsymmetric transmission T(θ) ≠ T(−θ) can be found for the metasurface waveguides with opposite bending angles, which indicates that the spinful waveguide mode will pass more easily through corners decorated by metasurfaces. This might be attributed to the asymmetric distribution of the waveguide field, i.e., the pressure fields mainly localize on one half of the waveguide crosssection without metasurface decorations. In the following, we are going to experimentally explore the acoustic spin transport in several typical examples, including robust transport, spinselective routing and rotating spin under effective magnetic field.
Robust transport with cornerscattering suppression due to tight spinmomentum coupling: We first simulate the acoustic field of a Ushape normal acoustic waveguide without metasurface when incident wave imposed at one port. Results show that the strong backscattering will happen at bending corners, as depicted in Fig. 5a. As a contrast, we simulate the acoustic field when incident acoustic wave into the similar Ushape waveguide but with metasurface. From the simulation results shown in Fig. 5a, we can see the acoustic waves successfully pass through the Ushape metawaveguide with low scattering loss at corners.
To verify these spinrelated backscatteringsuppressed effects, we perform experiments to measure the transport of Ushape acoustic waveguide with/without metasurfaces as shown in Fig. 5b. From the experimental results in Fig. 5c, we can see that the Ushape has little effects on the transmission of the metasurface waveguide at frequency region 2.85–2.95 kHz, which is exactly around the resonant frequency of the side bar structures. But in contrast, the waveguide without metasurface does not hold a clear acoustic SAM texture and it will suffer strong scatterings leading to severely attenuated transmission. Compared with conventional waveguides, this robustness transport of acoustic spinful mode in metasurface waveguides is assisted by the nonzero SAM and its tight spinmomentum locking relation, resulting in that the backpropagation requires spin flips to opposite one. Acoustic SAM textures represent polarization profiles of velocity fields. To ensure the spin (polarization) matching between opposite propagating acoustic modes of opposite SAMs, backscattering requires strong scatters that reverse spin. Indeed, scattering behaviors highly depend on the type of scatters and the interaction with wave modes. For example, certain types of impurities and defects will break topological transport protections in topological insulators without preserving timereversal symmetry so that backscattering occurs with spin flip^{34,35}. In fact, many scatters in metasurfaces will possess spinrelated scattered/reflected properties^{14,36,37}. From experimental results, one can see that here the metasurface waveguide modes with nonzero SAM is insensitive to the present corner defects.
Spinselective wave router based on opposite acousticspindependences of different directions: As discussed before, the momenta of metasurface waveguide modes are tightly coupled to the SAM of modes, which means that the wave will propagate along the direction selectively based on the SAM match, when facing multiple transport channels with opposite SAMs. To observe the selective phenomena, we design a Tshape structure and confirm the spinselective router by simulation results as shown in the Fig. 5d. From Fig. 1, we know that the acoustic mode to the left export is the same upspin state (S_{y} > 0) as the incident acoustic wave, while the right export has opposite downspin state, so that the acoustic wave will choose the left direction with the same upspin. If we rotate the incident waveguide around incident axis for 180°, the incident acoustic wave will possess a flipped SAM. The right export will be the same downspin state (S_{y} < 0) as the incident acoustic wave, while the left export has opposite upspin state. As such, the acoustic wave now will choose the right direction with the same downspin as expected. This is reminiscent of the spinHalllike effect for electrons and optics^{36,37}.
Experimentally, we input the acoustic wave from the port and measure the transmission at sides A/B shown in Fig. 5e. The experiment results in Fig. 5f show that the acoustic wave chooses one side (A) to propagate and leave no output for the other one (B). At the crossing point, the SAM of modes will make the acoustic wave propagate along the direction with the matched SAM. Moreover, it shows that this spinselective transport will also be insensitive to the backscattering at T corner, due to the asymmetric spatial distribution of acoustic spinful waveguide modes and the assistance of spinmomentum locking.
Gradually rotated metasurface waveguide for rotating acoustic spin and phase modulator: As is known from Fig. 1, the acoustic wave polarization confined within the metawaveguide forms complicated SAM texture, which highly depends on the boundary geometry settings of metasurfaces. This indicates that following a gradually rotating metawaveguide boundary, the evolving polarization of acoustic wave will result in rotating SAM textures, as shown in Fig. 6a. By rotating metasurface boundary with an angular velocity Ω along zaxis, we find that the variation of acoustic SAM texture in xy plane will follow the rotating evolution of velocity polarized profile (\(\frac{\partial }{\partial z}{{\bf{e}}}_{i}={\mathbf{\Omega}} \,\times {{\bf{e}}}_{i}\)) as: \(\frac{\partial }{\partial z}{\bf{S}}={\mathbf{\Omega}} \, \times {\bf{S}}\). This is a reminiscent of “acoustic spin Bloch equation” that resembles the equation of motion of electron spin procession under an effective “magnetic field” Ω^{38}. The zaxis plays the role of a pseudotime so that the wave propagation along this pseudotime will experience effective Larmor precession for acoustic spin with the Larmor frequency Ω.
As the consequence of this effective Larmor precession of acoustic spin, the total acoustic wave phase accumulated during the propagation in rotating metasurface waveguide will have contributions from two individual phases Θ = Θ_{0} + γ: the unperturbed phase Θ_{0} = kL related with waveguide length L and wave vector k; and the additional phase \(\gamma =\frac{k{R}_{0}^{2}}{2L}{\theta }^{2}\) induced by adiabatic variation of metaboundary conditions, where θ = ΩL is the final rotation angle (See Supplementary Note 3). To experimentally observe the phase γ induced by rotating SAM texture, we rotate the boundary settings of metasurface waveguide slowly along z direction and measure the corresponding transmitted phase, as shown in Fig. 6b at frequency f = 2.85 and 2.9 kHz, respectively. The measured phases γ with different rotating rate Ω after the pseudotime L are in good agreement with theoretical results. Results indicate that the gradually rotating metasurface waveguide is also a phase modulator that has implications in acoustic interferometers.
Discussion
To summarize, by introducing the special metasurface boundary to induce polarizationdependent phase delay, we have realized the nonzero acoustic SAM and associated tight spinmomentum locking. Owing to these properties, the acoustic waveguide mode exhibits backscatteringsuppressed transport if scatters do not flip the SAM. Besides, the wave transport becomes spinselective by choosing the right channel with matched SAMs when facing multiple channels with different SAMs, i.e., a spinHalllike feature. Finally, we have demonstrated a rotating metasurface waveguide and measured the additional phase modulation, resembling the spin precession under an effective magnetic field. Our work demonstrates spinrelated acoustic transport mechanisms within the bulk, rather than at the interface^{5,6}. This pure spinful bulk mode is determined by the special boundary with nontrivial reflection phase. Our results would supply new insights about the spin mechanism of vibrational waves and pave the way about acoustic SAMrelated wave control, especially in the onchip phononic devices^{39} and even coupling to electron spin and quantum magnetism.
Methods
Spin angular momentum in the metasurface waveguide
In the cylindrical coordination system (r, φ, z) as shown in Fig. 7a, the solution of the acoustic field in the metasurface waveguide mode will be the linear composition of the Bessel function J_{l}(r) as: p = p_{0}∑_{l}c_{l}J_{l}(κr)e^{ilφ}e^{i(kz−ωt)}, where c_{l} is the modal amplitude, \(\kappa =\sqrt{\frac{{\omega }^{2}}{{c}^{2}}{k}^{2}}\), k = ke_{z} is the longitudinal wave vector along the z direction, ω is the frequency and c is the sound speed in the air. Due to the nonsymmetric boundary conditions in Fig. 7a, we will have: p_{r=R} = 0 for φ ∈ [−π/2, π/2] and ∂_{r}p∣_{r=R} = 0 for φ ∈ [−π, −π/2) ∪ (π/2, π], where R is the radius of the waveguide. For the lowest order state for the metasurface waveguide, we consider only l = 0, ±1 here. After some maths, the acoustic pressure field for the ground state in the metasurface waveguide can be approximately represented as:
with the dispersion \({k}^{2}+{\kappa }^{2}=\frac{{\omega }^{2}}{{c}^{2}}\), α is the constant. With the boundary conditions, we will have that: κR = ξ, \(\alpha =\frac{{J}_{0}(\xi )}{{J}_{1}(\xi )}\), and ξ is the first root of the equation \({J}_{1}^{2}(\xi )\frac{1}{2}{J}_{0}(\xi )({J}_{0}(\xi ){J}_{2}(\xi ))=0\) and ξ > 0. Thus, the velocity field \({\bf{v}}=\frac{i}{\rho \omega }\nabla p\) will be :
and the corresponding SAM density \({\bf{s}}=\frac{\rho }{2\omega }{\rm{Im}}[{{\bf{v}}}^{* }\times {\bf{v}}]\) will be:
The s_{x} and s_{y} will be: \({s}_{x}={s}_{r}\cos (\varphi ){s}_{\varphi }\sin (\varphi )\), \({s}_{y}={s}_{r}\sin (\varphi )+{s}_{\varphi }\cos (\varphi )\). As such, the total SAMs as the integral of acoustic SAM densities on crosssection S = ∫sd^{2}r are obtained as:
and S_{x} = S_{z} = 0.
We note that from Fig. 3 one can find some unimportant mismatchings near boundaries between exact simulation results and analytical theory results. These are due to our lowest order truncation of the Bessel basis as approximations. When incorporating more higherorders of Bessel functions, the theoretical field will coincide with the exact simulations very well (see Supplementary Note 1). Nevertheless, the present simplest approximation with lowest order up to 1 is sufficient to express the SAM values in a neat analytical form, which clearly uncovers the physics and reflects the key information: nonzero SAM with tight spinmomentum locking.
Sample fabrication and experimental measurements
All the elements are fabricated using 3D printing with the photosensitive resin material (modulus 2765 MPa, density 1.3 g/cm^{3}), and can be treated as acoustically rigid. The elements were 3Dprinted, one unit cell at a time, and were connected to form different configurations. The 1/4in. microphone (GRAS Type 46BE) is placed at the measuring terminal of the sample to detect amplitude and phase of the sound. The experimental environment was an open space to avoid unnecessary reflection. The detected signals are acquired by the NI 9234 data acquisition module. A sine signal is driven by the waveform generator (Keysight 33500B) with frequency range from 2.7 to 3.2 kHz. The sine signal was fed into a loudspeaker as the sound source. The sound waves emitted by the loudspeaker are guided into the tube. The position of the microphone is controlled by the automatic stage.
The acoustic field inside the waveguide is measured by placing the acoustic detector into the waveguide (the scale of the microphone is much smaller than the radius of the crosssection of the waveguide), moving the acoustic detector with the automatic stage and recording the acoustic pressure field \({p}_{\exp }\), shown in Fig. 7b. The experimental SAM densities will be obtained by \({{\bf{s}}}_{\exp }=\frac{\rho }{2\omega }{\rm{Im}}[{{\bf{v}}}_{\exp }^{* }\times {{\bf{v}}}_{\exp }]\) and \({{\bf{v}}}_{\exp }=\frac{i}{\rho \omega }\nabla {p}_{\exp }\).
The wave vector k inside the waveguide is measured by moving the acoustic detector along the zaxis gradually and recording the phase change during the relocation. The wave vector k will be the ratio between the phase change and the position change along zaxis. The additional phase detections are performed by recording phase data 3 times in each 3 points (with distances 6.5 mm, 16.5 mm, 26.5 mm to the center of the waveguide respectively) firstly and then averaging these detected phase data. Five rotation settings are experimentally prepared: ΩL = 0° (straight duct), 90°, 180°, 270°, and 360°.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work is supported by the National Key R&D Program of China (Grant No. 2016YFA0301101), the National Natural Science Foundation of China (No. 61621001, 11935010, and 11775159), the Natural Science Foundation of Shanghai (Nos. 18ZR1442800 and 18JC1410900), and the Opening Project of Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology.
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These two authors Y.L. and D.Z. contributed equally to this work. Y.L., D.Z., and C.Y. carried out the numerical simulations. D.Z., C.Y., and J.G. performed the experimental measurements. Y.L. and J.R. derived the theory and developed the analysis. J.R. and H.C. conceived the project. All the authors contributed to discussion, interpreting the data and the writing.
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Long, Y., Zhang, D., Yang, C. et al. Realization of acoustic spin transport in metasurface waveguides. Nat Commun 11, 4716 (2020). https://doi.org/10.1038/s4146702018599y
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