Scattering properties of acoustic beams off spinning objects: Induced radiation force and torque

The main research efforts on acoustic metamaterials are based on static objects without any motion.^1–6 Very recently, the topic of time-varying^7–11 and moving metamaterials^12–19 started gaining momentum due to the exciting applications resulted from it, ranging from symmetry breaking,^20,21 time-slabs,²² antennas,²³ to beam-shaping.²⁴ The research on space-time periodic media can be rooted in two visionary papers by Cassedy.^25,26 Similarly, some groups analyzed spinning acoustic objects to realize cloaks,²⁷ topological insulators,²⁸ and negative radiation force.²⁹ These phenomena were based upon the theoretical prediction proposed in Refs. 30–32. Related findings were also observed in the context of electromagnetism for a rotating circular cylinder of isotropic material³³ and optical Magnus radiation force.³⁴ 

On the other hand, exploiting acoustic radiation pressure opens avenues for wave manipulation.^35–39 It characterizes the average force exerted on an object interacting with the acoustic wave and leading to contactless manipulation in acoustics, such as the acoustic analogue of “optical tweezers.” In these devices, the use of radiation pressure of a single laser beam^40–45 makes the optical tweezers efficient and easily maneuverable in space, which constitutes a very powerful tool for many scientific studies.^46,47 However, no device has incorporated the concept of the single beam tweezers for acoustics, which was already demonstrated to be responsible for the precision of optical tweezers.⁴⁸ 

In this Letter, we show that a single acoustic beam^48–51 can lead to positive/negative radiation force and, hence, it may be used for future precision acoustic tweezers. Moreover, the generated torque may be excited by a plane wave if some loss is considered in the spinning, which is otherwise impossible in media at rest. These results are shown with both Mie theory modeling and finite element method (FEM).

We study the scattering of a cylindrical object with radius a spinning at an angular velocity Ω in a surrounding medium. Without loss of generality, we assume time-harmonic waves, with time-dependence of the form $e−iωt$⁠, where ω is the angular wave frequency, and angular dependence of the form $eilφ$⁠, where $φ$ is the polar angle in the plane and l is the azimuthal number. The conservation relations (mass and momentum) are modified accordingly and may be found, for example, in Refs. 27, 28, 30, and 52. By eliminating the velocity variables v_r and $vφ$ (v_z is decoupled, anyways), the following partial differential equation of the pressure field p with the presence of spinning is derived, in cylindrical coordinates,^27,28,30,52

with the modified wavenumber $kl2=−(4Ω2+ζl2)/c2$⁠, where $ζl=−i(ω−lΩ)$ is the complex Doppler frequency in the spinning fluid for an azimuthal order l. We verify that for Ω = 0, we recover the classical dispersion relation $k=ω/c$⁠, with $c=1/ρβ$ being the sound speed when propagating within the scatterer (the properties of the scatterer/surrounding medium are presented without/with subscript). This equation is complemented with appropriate boundary conditions (see the supplementary material). We must further ensure that $Ωa<c0$ to avoid having any portion of the fluid moving faster than the speed of sound in the surrounding fluid.⁵² 

We consider in this study an acoustic beam that propagates inside a nonviscous medium of density ρ₀ and compressibility β₀ (⁠$β0=1/κ0$⁠, with κ₀ being the bulk modulus). The acoustic field in this non-spinning medium satisfies the Helmholtz equation $(∇2+k02)p=0$⁠, with k₀ being the wavenumber in the fluid and p being the pressure field. For instance, the expansion of the incident field in cylindrical coordinates is $pinc(r,φ)=p̃0∑l=−∞+∞blJl(k0r)eilφ$⁠, with J_l being the Bessel function of the first kind and of order l. Here, b_l denotes the beam-shape coefficient (BSC) characterizing the beam's nature.^{37,48,49,53–58} $p̃0$ is the pressure amplitude and is set as 1 for simplicity. As we consider spinning cylinders around their axis of symmetry and normally incident waves, all variations with z are irrelevant. The scattered field is given by $pscatt(r,φ)=∑l=−∞+∞blslHl(1)(k0r)eilφ$⁠, with s_l being the scattering coefficients and $Hl(1)$ being the Hankel functions of the first kind (see the supplementary material for the expressions of s_l). The expression of the time-averaged radiation force is $⟨F⟩=−∬S⟨p−p0⟩dS$⁠, with $dS=dSn$⁠, the normal to the cylinder's outer surface (S) and 〈·〉 denoting time average.^57,59–61 This force can be normalized by the cylinder's diameter 2a and the energy density $p̃02/(2ρ0c02)$⁠,⁶⁰ 

where $*$ denotes the complex-conjugate, for the axial and transverse force components, while $ℑ$/$ℜ$ denote the imaginary and real parts of the radiation force, respectively.

It is also interesting to characterize the torque experienced by the interaction between the spinning object and various beams described above. For instance, the normalized time-averaged torque is shown to be directed along the z-direction, i.e., the axis of rotation of the cylinder,⁶⁰ 

The expression in Eq. (3) is in many ways reminiscent of the expression of the absorption cross section, $σabs=−4k0∑l=−∞+∞|bl|2{ℜ(sl)+|sl|2}$⁠.⁶² 

To get a better understanding of the phenomenon induced by spinning, it is meaningful to isolate the scattering effects attributed to the material contrast. We assume first, except otherwise stated, that $ρ=ρ0$ and $β=β0$⁠. This means that we consider liquid–liquid (or air–air) interfaces, as seen in the supplementary material, which in both cases may be separated by a thin impedance-matched membrane (i.e., a membrane that is transparent to sound and impermeable to fluid particles, e.g., a thin polyethylene film⁶³ or expanded polytetrafluoroethylene⁶⁴). These configurations were also chosen to avoid mixing between the spinning/non-spinning fluids and circumvent issues, such as surface tension breaking and Rayleigh-Plateau instability.⁶⁵ Figure 1 gives the total pressure field [amplitude $|p|$ and phase arg $(p)/π$], i.e., $pscatt+pinc$ in the domain $r∈[a,r0[$ [where Eq. (S4) of the supplementary material is valid]. Here, the object is a cylinder of radius a = 0.35 m that is spinning around its z-axis with a spinning frequency of 300 Hz, at a frequency of 170 Hz. It is spinning in water with a density of 1000 kg/m³ and a bulk modulus $1/β0=2.22$ GPa. In Fig. 1(a), we depict the classical case of a plane wave. For comparison, Figs. 1(b) and 1(c) show the case of a cylindrical diverging and converging acoustic beams, respectively. The beams are centered at $r0=6$ m (distance from the center of the scattering object). The two beams scatter sound differently from a plane wave. Due to the spinning object, the phase of the pressure field exhibits a distortion reminiscent of vortex beams (see Fig. S6 of the supplementary material for the scattered fields). The quasi-Gaussian beam of order l = 1 exhibits even more complex behavior, depicted in Fig. 1(d), which is distinct from the plane wave or the cylindrical beams,^66–68 and demonstrates that the interaction between the quasi-Gaussian beam^69,70 and the spinning cylinder may be of interest to the generation of acoustic vortex beams. For the far field behavior of the scattered beams shown in Fig. 2, we first compute the scattering coefficients by imposing the corresponding BSC for each beam, from which we can compute the scattering cross section (SCS) $σscatt=4k0∑l=−∞+∞|bl|2|sl|2$ and the radiation force from Eq. (2). Two methods are used for computing the scattering properties. In the first method, we make use of the Mie theory succinctly discussed in Sec. 1 of the supplementary material or in Refs. 27, 28, 30, and 52. The second method is based on the linearized Navier–Stokes (LNS) equation model of COMSOL Multiphysics that represents a full-wave numerical validation.⁷¹ Figure 2 shows clear resonance features at frequencies of 96.85, 215.3, and 418.9 Hz, due to the spinning of the cylinder (see Ref. 27 for a detailed analysis). The SCS of the quasi-Gaussian beam mimics that of a plane wave incidence for l = 0 [see Fig. 2(b)]. However, for $l=−1,2$ [see Figs. 2(a) and 2(c)], the resonant peak smears off and the SCS of the beam deviates from that of the plane wave (first resonance for l = – 1 and second and third resonances for l = 2). The radiation force (both its transverse and axial components) is correlated with the scattering coefficients, exhibiting resonant behaviors for the beams l = 0 and $l=−1,2$ that are due to scattering from spinning. The transverse radiation forces also exhibit positive and negative values, which is important for the conception of acoustic tweezers, as can be seen in Figs. 2(g)–2(i). For higher order beams, e.g., l = 5, as the resonant peak is no longer present (see Fig. S7 of the supplementary material), both $Faxial$ and $Ftrans$ go to zero, further manifesting that the origin of the acoustic force takes roots in the intriguing spinning-induced scattering. Figure 2(c) shows that the SCS and force with its axial and transverse components deviate more markedly from the plane wave excitation when the order of the quasi-Gaussian beam is different from zero. While the transverse force takes positive and negative values, the axial force is almost exclusively positive, with resonant peaks for all cases with different orders. $Faxial$ (⁠$Ftrans$⁠) is the imaginary (real) part of a certain quantity shown in Eq. (2) and they satisfy the Kramers–Kronig relations,⁷² which explains their respective shapes. Figure 2 demonstrates the role played solely by the spinning cylinder on the force introduced by the scattering of the incident quasi-Gaussian beam and reveals that spinning offers a degree of freedom to tune the forces. Similar effects are observed in systems with different sizes of the scatterer and frequencies (see Fig. S8 in the supplementary material).

Next, we study the case where the spinning scatterer has different density and/or modulus from those of the surrounding fluid. The case of a cylinder with physical parameters of $ρ/ρ0=2$ and $β/β0=10$ is plotted in Fig. 3(a) for $Ω/(2π)=25$ Hz and in Fig. 3(b) for $Ω/(2π)=50$ Hz, for the SCS for the quasi-Gaussian beam of order l = – 1 and l = 1, respectively. Figures 3(c)–3(f) not only demonstrate the resonances induced by the combination of spinning and physical inhomogeneity but also reveal that more resonant peaks are induced with lower spinning frequency. In the supplementary material (Fig. S4), it is shown that without spinning, the $±l$-th orders possess the same scattering amplitude, while with spinning, they split, which is the acoustic analogues of the Zeeman effect.¹² These results are also verified with FEM (black dotted-dashed curves). These two models agree remarkably up to a small error due to the approximations of the Mie model. For instance, even though the FEM modeling solves the full governing equation with minimal approximations, it gives results fairly close to those of the Mie theory for spinning objects as shown in the supplementary material. The slight deviations between the two methods can be attributed to neglecting some terms in the LNS equations in the Mie model and/or to some viscosity effect (that is present in the LNS modeling). Yet, the Mie model has the advantage of being simple and offering a clearer physical picture of the involved effects. Figures 3(g)–3(i) give a more detailed information by allowing the spinning frequency to continuously vary from 0 to 300 Hz and we plot the same parameters as in Figs. 3(a)–3(f) in a two-dimensional plot. The resonances correspond to yellow bright color in these graphs. The complex behavior from the combination of spinning and inhomogeneity is manifested in these plots. Also reflected by these plots is the tunability of the radiation force resonance.

Equation (3) shows that the non-zero component of the spin-induced torque $Γzrad$ depends on terms of the form $ℜ(sl)+|sl|2$⁠, with s_l being the decoupled scattering coefficient. Due to energy conservation in nonviscous fluids (which we assume everywhere in this Letter, unless otherwise stated) it was already shown in Refs. 48, 73, and 62 that $|sl|2=−ℜ(sl)$⁠, which demonstrates the absence of torque in Hermitian media (without loss or gain). As our structure is spinning, one may expect intuitively that it may experience torque as well. Yet, when the relative density equals 2 and the relative compressibility equals 10 (i.e., both real valued), it is evident from Figs. 4(a) and 4(b) that $Γzrad$ is vanishingly small for both the plane wave and the quasi-Gaussian beam, in the considered frequency range, for the beams of order $l=0,−1$⁠, respectively. This demonstrates that our object behaves as a Hermitian system, by respecting the conservation of energy (⁠$ℜ(sl)+|sl|2=0$⁠, for all $l∈ℤ$⁠). In order to induce torque for our system, unlike the radiation force (transverse and axial), we need to separate the orders l = 0 and $l≠0$⁠. For l = 0, hermiticity (or energy conservation) is broken with the presence of loss in the physical parameters, namely, the density of the spinning object, in addition to spinning [Fig. 4(c) shows the zero torque for l = 0, when the density is $ρ=2×(1+1×i)ρ0$ and Ω = 0]. Figure 4(e) uses $ρ=2×(1+1×i)ρ0$ with a spinning frequency of 250 Hz (⁠$β/β0=10$⁠). Hence, $Γzrad$ undergoes a resonant behavior around 190 Hz, reaching higher values and also some negative values and for higher frequencies beyond 200 Hz, $Γzrad≈0$⁠. If we further increase the loss in the density and the spinning frequency, then $Γzrad$ may reach even higher values, with a Fano resonant line shape (reminiscent of quasi-bound states in the continuum⁷⁴). For the beam l = –1 depicted in Figs. 4(b), 4(d), and 4(f), the situation is quite different. For instance, when the density and compressibility are real-valued, even with spinning [Fig. 4(b)] $Γzrad≈0$⁠. However, when loss is induced and spinning is stopped [Fig. 4(d)], the plane wave cannot result in torque, while the beam gives finite torque. When both spinning and loss are present [Fig. 4(f)], the plane wave results now in torque that is one order of magnitude higher than the case without spinning. Additionally, if spinning is reversed $Ω<0$⁠, then $Γzrad$ flips the sign [yellow and purple colors in Fig. 4(f)] (see Fig. S9 in the supplementary material for more results on the effect of loss on the torque).

We propose a theoretical analysis and a FEM-based full-wave numerical characterization of acoustic beam scattering off spinning cylinders. We apply the Mie scattering formalism, in which a more complex modified boundary condition is considered. The scattering solely induced by the spinning, as shown by our analytic results and numerical simulations, results in acoustic radiation force (resonant and positive/negative) as well as intriguing radiation torque which can open avenues in sound control and precision equipment based on sound and/or ultrasounds. This study considers cylindrical scatterers, yet, it would be useful to investigate three-dimensional scatterers, such as spheres, which is more complicated as the local velocity on the surface of the sphere is a function of the polar angle rather than a constant as in the cylinder case.³⁰ Therefore, our FEM-based simulation may prove useful to characterize such configurations.

See the supplementary material for the details for both Mie theory and the finite-element modeling and further discussions on the scattering and beam-shape coefficients.

This work was supported by King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Grant No. OSR-2020-CRG9-4374 as well as KAUST Baseline Research Fund BAS/1/1626-01-01.