Acoustic orbital angular momentum transfer to matter by chiral scattering

We report on orbital angular momentum exchange between sound and matter mediated by a non-dissipative chiral scattering process. An experimental demonstration is made possible by irradiating a three-dimensional printed, spiral-shaped chiral object with an incident ultrasonic beam carrying zero orbital angular momentum. Chiral refraction is shown to impart a nonzero orbital angular momentum to the scattered field and to rotate the object. This result constitutes a proof of concept of a novel kind of acoustic angular manipulation of matter.

The mechanical effects of waves on material media can achieve contactless manipulation in a controlled manner, and the technique has already found numerous applications in optics and acoustics. Wave-matter exchange of linear and/or angular momentum that results from radiation force and/or torque can be classified in two categories, dissipative and non-dissipative. To date, the demonstration of all kinds of exchanges mediated by dissipative processes have been carried out experimentally. In optics, this includes the transfer of both spin and/ or orbital angular momentum (OAM) [1][2][3]. Notably in acoustics, the angular manipulation of objects resulting from the absorption of (pseudo [4]) OAM was demonstrated only recently. This included both angular deviation [5,6] and spinning [7][8][9] of sound-absorbing systems.
Non-dissipative exchanges of linear and angular momentum between waves and matter are also possible. In optics, this was experimentally realized in 1936 for spin angular momentum [10] and the OAM case was explicitly identified much later [11]. Today, the field of optical angular manipulation is mature; see [12] for a recent review. Intriguingly however, the demonstration of exchange of angular momentum between sound and matter that is not mediated by absorption has not yet been reported. Here, we aim to fill this gap by observing the direct signature of the phenomenon experimentally, both for the material medium and the acoustic field.
Our demonstration relies on the use of a chiral object having the form of a spiral phase plate (SPP). This is the acoustic analog of an optical SPP [13] but for sound waves. It formally consists of a plate having an azimuthally varying thickness with f the frequency of the sound wave, c 1 the sound speed in the medium surrounding the SPP and c 2 the sound speed in the material that constitutes the SPP. A normally incident axisymmetric acoustic wave thus emerges from the SPP with a pressure field having an azimuthal dependence of the form f ℓ ( ) exp i , i.e. an acoustic vortex with topological charge ℓ. Such a field carries nonzero acoustic OAM [4,14] leading to the appearance of a nonzero acoustic radiation torque exerted on the SPP due to angular momentum conservation [15]. This is the acoustic analog of optical OAM carried by light beams endowed with phase singularities [16], which is a general feature of wave physics. The mechanical action of the acoustic torque is here demonstrated from the direct observation of SPP rotation around the beam propagation axis, whereas OAM generation is identified from pressure field measurements.
We designed a SPP operating at frequency ) as depicted in figure 1(b). The SPP is therefore sustained by buoyancy and capillarity, and is either free to rotate (for rotation experiments) or fixed using a holder (for pressure scans). With present parameters,  = 1.7 mm, a picture of the = ℓ 4 SPP is shown in figure 1(b). In practice, the pedestal ensures a reproducible pinning of the triple contact line along its perimeter and enhances the mechanical stability of the SPP. In addition, an axial hole of diameter = d 0.8 mm h is drilled through the SPP in order to maintain its axis along the beam axis owing to a needle.
The SPP is irradiated from below by an acoustic beam produced by a spherical, piezoelectric, mono-element, high-power transducer with radius of curvature F = 38 mm, diameter D = F, central frequency 2.55 MHz and bandwidth 600 kHz. The ultrasonic transducer is immersed in the lower aqueous layer, as sketched in figure 1(c), and is fed with sinusoidal wave trains with voltage amplitude U at carrier frequency f, with N cycles duration at 1 kHz repetition rate (N = 10 for pressure scans). Optimization of matter-field interaction generation requires both the incident acoustic power propagating through the hole and that outside the SPP cross-section to be minimized. In practice, the SPP is placed at a distance F − H from the transducer, with H = 10 mm as a trade-off value for which 90% of the incident acoustic power is intercepted by the SPP.
The first experiment is dedicated to the demonstration of SPP rotation using the setup sketched in figure 1(c). We choose a freely rotating SPP with = ℓ 4 and N = 510 for the number of cycles of the wave trains emitted by the transducer at a 1 kHz repetition rate. This corresponds to a 20% duty cycle and we may thus refer to a quasi-continuous acoustic irradiation. At steady state, the SPP rotates at constant angular velocity Ω in the clockwise direction (when looking at the SPP from the top), namely W = W z with W < 0. In addition, Ω is found to be proportional to U 2 , hence proportional the incident acoustic power P, as shown in figure 2. Unfortunately, because the manufacturer calibrated the transducer in pure water whereas the transducer is immersed in saturated brine in the present experiment-and the emission efficiency of our high-power transducer notably depends on the acoustic impedance of the liquid in contact with its emitting surface-we were unable to determine the acoustic power of the incident beam quantitatively. In addition, the viscous torque exerted on the rotating SPP could not be quantitatively determined theoretically due to the complex geometry of the SPP.
We have also established that the effect is due to the shape of the SPP, by checking that a disk (i.e. an SPP with = ℓ 0) does not rotate. Moreover, the observed direction of rotation and the linearity of the SPP angular velocity with incident power are both consistent with the balance of angular momentum between the SPP and the emerging pressure field proportional to . Indeed, the output field carries an OAM whose projection along the z-axis has the sign of ℓ and its magnitude is proportional to ℓ | | P , where P is the beam power [4,17]. Consequently, from angular momentum conservation, the SPP experiences a recoil torque with a sign equal to that of -ℓ and a magnitude proportional to ℓ | | P . Moreover, since the viscous torque from the creeping flow assumption varies linearly with Ω, balance of the viscous and radiation torques results in W µ P, in agreement with the data presented in figure 2. The latter assumption is realistic since the value of the in is omitted. Neglecting the reflected field at the brine-SPP and SPP-oil interfaces for the time being (the effect of reflection due to acoustic impedance mismatch will be detailed in the last part of the paper), within the paraxial approximation the transmitted pressure at an altitude > z z in from the SPP is In the ideal case of a pure phase mask, i.e.
the spatial distribution of the output pressure magnitude exhibits an axisymmetric, doughnut shape as a consequence of the on-axis phase singularity with topological charge ℓ. This is illustrated, for = ℓ 4, in figures 3(a) and (b) that show the calculated magnitude and phase of p out at altitude =z h 2. The second experiment consists of the direct observation of acoustic vortex generation at =z h 2, which is performed using the same setup but holding the SPP. The transverse distribution of the maximum of the pressure magnitude of the transmitted wavetrain is acquired using a needle hydrophone with a 85 μm-diameter In fact, this observation can be explained by accounting for the actual features of the experiment: PLA attenuates sound, the SPP has a hole at its center and the SPP is placed in a holder that completely absorbs sound for > r d 2.
s This leads to an azimuthally dependent pressure transmittance expressed as is the pressure transmission factor of the pedestal. The computed spatial distribution of the pressure magnitude, which is displayed in figure 3(d), qualitatively reproduces the experimentally observed square-shaped doughnut-like pressure magnitude pattern. Such non-axisymmetric spatial distribution of the pressure magnitude is associated with splitting of the on-axis phase singularity of topological charge ℓ into ℓ off-axis single charge phase singularities, see figure 3(e), which is a generic feature of the structural instability of highorder vortices [18,19].
Note that in our experiments the noise of the transmitted wavetrain phase due to unavoidable bulk and surface imperfections of the 3D printing process prevents direct measurement of the transverse phase distribution. This also prevents a quantitative study of the sound driven rotation of the SPP as a function of the topological charge. Still, the above results allow experimental ascertainment of the generation of acoustic vortex generation by the SPP as a result of chiral scattering of sound.
We next review possible mechanical consequences of steady flows induced by sound absorption on SPP rotation. First, let us consider the flow due to the absorption of the incident beam below the SPP ( <z h), which may contribute to the total torque exerted on the SPP if its bottom facet is not perfectly flat or not coaxial with the incident beam. Since an irradiated disk (i.e. = ℓ 0) does not rotate, acoustic linear streaming is not at play in our experiment. Second, let us consider acoustic rotational streaming [9], which consists of steady fluid rotation due to absorption of OAM. In our experiment, this occurs above the SPP ( >z h) as a result of absorption of the generated acoustic vortex. Angular momentum conservation implies that the angular momentum acquired by the SPP and the angular momentum carried by the generated acoustic vortex are opposite. Hence, viscous drag exerted by rotational acoustic streaming on the SPP results in a torque opposite to the acoustic radiation torque induced by OAM transfer. Acoustic rotational streaming is therefore also not at play in our experiment.