Acoustic Radiation Force and Torque on Small Particles as Measures of the Canonical Momentum and Spin Densities
AAcoustic Radiation Force and Torque on Small Particlesas Measures of the Canonical Momentum and Spin Densities
I. D. Toftul,
1, 2
K. Y. Bliokh,
1, 3
M. I. Petrov, and F. Nori
1, 4 Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan ITMO University, Birzhevaya liniya 14, St.-Petersburg 199034, Russia Nonlinear Physics Centre, RSPE, The Australian National University, Canberra, ACT 0200, Australia Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
We examine acoustic radiation force and torque on a small (subwavelength) absorbing isotropicparticle immersed in a monochromatic (but generally inhomogeneous) sound-wave field. We showthat by introducing the monopole and dipole polarizabilities of the particle, the problem can betreated in a way similar to the well-studied optical forces and torques on dipole Rayleigh parti-cles. We derive simple analytical expressions for the acoustic force (including both the gradientand scattering forces) and torque. Importantly, these expressions reveal intimate relations to thefundamental field properties introduced recently for acoustic fields: the canonical momentum andspin angular momentum densities. We compare our analytical results with previous calculations andexact numerical simulations. We also consider an important example of a particle in an evanescentacoustic wave, which exhibits the mutually-orthogonal scattering (radiation-pressure) force, gradientforce, and torque from the transverse spin of the field.
Introduction.—
Optical and acoustic radiation forcesand torques are of great importance from both practi-cal and fundamental points of view. On the one hand,these mechanical manifestations of the radiation powerunderpin optical and acoustic manipulations of smallparticles [1–6], atomic cooling [7–9], optomechanics [10],acoustofluidics [11, 12], etc. On the other hand, radia-tion forces and torques reveal the fundamental momen-tum and angular-momentum properties of the optical andsound wave fields [13–23].Since Kepler’s observation of the comet tail and earlytheoretical works by Euler and Poynting [13, 14], thestudies of optical and acoustic momentum and forceswere developed in parallel ways. Remarkably, despitenumerous works calculating radiation forces and torquesacting on various small particles in optics [24–30] andacoustics [31–41], the explicit proportionality of the forceand torque to the local wave momentum and spin angu-lar momentum densities was properly established in op-tics only recently [42–51]. The reason for this is that, ingeneric inhomogeneous wave fields, the force and torqueon an isotropic small absorbing particle are proprtional tothe canonical momentum and spin densities rather thanthe Poynting (kinetic) momentum and angular momen-tum commonly used for many decades [45–48, 50, 52–54].In acoustics , such explicit connection between the ra-diation force/torque and momentum/spin in generic in-homogeneous fields has not been described so far. More-over, the concepts of the canonical momentum and spinangular momentum densities in sound wave fields havebeen introduced only in very recent works [55–58].In this work, we provide a simple yet accurate theoryof acoustic forces and torques on small (subwavelength)absorbing isotropic particles in generic monochromaticacoustic fields. By employing methods well-established inoptics and involving the monopole and dipole polarizabil- ities of the particle (determined by the leading terms inthe Mie scattering problem), we derive simple analyticalexpressions for the acoustic forces and torque. Most im-portantly, these expressions indeed expose the intimaterelation to the canonical momentum and spin densitiesin the acoustic field. We show that our results agree withspecific previous calculations and exact numerical simu-lations. We illustrate our general theory with an explicitexample of the forces and torque on a small particle inan evanescent acoustic wave.
Properties of acoustic fields.—
We will deal withmonochromatic but arbitrarily inhomogeneous acousticfields of frequency 𝜔 in a homogeneous dense medium(fluid or gas). The complex pressure and velocity fields, 𝑝 ( r ) and v ( r ), obey the wave equations [59] 𝑖𝜔𝛽 𝑝 = ∇ · v , 𝑖𝜔𝜌 v = ∇ 𝑝 , (1)where the medium is characterized by the compressibility 𝛽 , the mass density 𝜌 , and the speed of sound 𝑐 = 1 / √ 𝜌𝛽 .We will characterize the dynamical properties of theacoustic wave field by its energy, canonical momentum,and spin angular momentum densities. The energy den-sity reads [59]: 𝑊 = 14 (︀ 𝛽 | 𝑝 | + 𝜌 | v | )︀ ≡ 𝑊 ( 𝑝 ) + 𝑊 ( v ) . (2)The canonical momentum and spin densities of acousticfields were introduced very recently [56–58]: P = 14 𝜔 Im[ 𝛽 𝑝 * ∇ 𝑝 + 𝜌 v * · ( ∇ ) v ] ≡ P ( 𝑝 ) + P ( v ) , (3) S = 𝜌 𝜔 Im( v * × v ) , (4)where [ v * · ( ∇ ) v ] 𝑖 ≡ Σ 𝑗 𝑣 * 𝑗 ∇ 𝑖 𝑣 𝑗 [42].The energy (2) and momentum (3) densities are repre-sented as symmetric sums of the pressure- and velocity-related contributions, indicated by the corresponding su-perscripts. This is similar to the symmetric electric- and a r X i v : . [ phy s i c s . c l a ss - ph ] S e p magnetic-field contributions in electromagnetism [42, 45–48, 52, 54, 60]. In contrast, the spin density (4) has onlythe velocity contribution because the scalar pressure fieldcannot generate any local vector rotation.Note that the canonical momentum determines the or-bital angular momentum density L = r × P [45, 48, 52,53, 58], and that the more familiar kinetic momentumdensity (the acoustic analogue of the Poynting momen-tum) is given by Π = P + 14 ∇× S = 12 𝑐 Re( 𝑝 * v ) [58, 59].The equivalence of the canonical and kinetic momentumand angular momentum quantities appears for their in-tegral values for localized acoustic fields: ⟨ P ⟩ = ⟨ Π ⟩ and ⟨ S ⟩ + ⟨ L ⟩ = ⟨ r × Π ⟩ [45, 48, 52, 53, 58], where the angularbrackets stand for spatial integration. However, here weare interested in local rather than integral field proper-ties, which are very different in the canonical and kineticpictures; below we show that it is the canonical quanti-ties (3) and (4) which correspond to the force and torqueon small particles. Interaction with a small particle.—
The most straight-forward way to detect the momentum and angular mo-mentum of a wave field is to measure the force andtorque it produces on a probe particle [13–22, 43–46, 48–51, 56, 61]. Therefore, we consider the interaction ofa monochromatic acoustic wave with a small (subwave-length) spherical isotropic particle of the radius 𝑎 , density 𝜌 and compressibility 𝛽 , with its center at r = r . Weallow the particle to be absorbing , i.e., the parameters { 𝜌 , 𝛽 } are generally complex .The wave-particle interaction is directly related to thewave scattering problem. For small isotropic particles,the scattered field is conveniently represented by a mul-tipole expansion [62–64], where the small parameter is 𝑘𝑎 ≪ 𝑘 = 𝜔/𝑐 is the wave number). For electro-magnetic waves, the leading term is the dipole one [1–4, 24–26], because the monopole cannot radiate transver-sal waves. In contrast, for longitudinal acoustic waves,the leading terms are the monopole and dipole ones, andthese generally have the same order in 𝑘𝑎 [11, 31]. There-fore, a small particle in an acoustic wave field can be ap-proximated by a monopole and dipole, which are inducedby the incident field and are interacting with this field (sothe interaction is quadratic with respect to the field).The oscillating monopole and dipole modes of the par-ticle are schematically shown in Fig. 1. The monopolemode is associated with the isotropic compression/ex-pansion of the sphere, while the dipole mode representsoscillations of the particle position along certain direc-tion. It is easy to see that these modes can be excitedby the oscillating pressure 𝑝 and velocity v fields, re-spectively. Therefore, the induced monopole and dipolemoments of the particle can be written as: 𝑄 = − 𝑖 𝜔𝛽 𝛼 𝑚 𝑝 ( r ) , D = 𝛼 𝑑 v ( r ) , (5)where, following optical terminology, 𝛼 𝑚 and 𝛼 𝑑 are the FIG. 1. The monopole and dipole oscillatory modes ofa spherical particle. These modes are associated with anisotropic compression/expansion and a linear oscillatory mo-tion of the particle, which are induced by the oscillating scalarpressure 𝑝 and vector velocity v fields, respectively. monopole and dipole polarizabilities of the particle, andthe prefactor − 𝑖 𝜔𝛽 in the monopole term is introducedfor the convenience in what follows and equal dimension-ality of the polarizabilities. Comparing the leading termsof the multipole expansion of the acoustic Mie scatteringproblem with the standard expressions for the acousticmonopole and dipole radiation [63, 64], we find the ex-pressions for the polarizabilities (see Supplemental Ma-terial [65]): 𝛼 𝑚 = − 𝜋𝑖𝑘 𝑎 ≃ 𝜋 𝑎 (︀ ¯ 𝛽 − )︀ ,𝛼 𝑑 = − 𝜋𝑖𝑘 𝑎 ≃ 𝜋 𝑎 𝜌 − 𝜌 + 1 , (6)Here, ¯ 𝜌 = 𝜌 /𝜌 and ¯ 𝛽 = 𝛽 /𝛽 are the relative density andcompressibility of the particle, 𝑎 and 𝑎 are the first twoMie scattering coefficients, and we approximated thesecoefficients by the leading ( 𝑘𝑎 ) term in 𝑘𝑎 ≪ Absorption rate, force, and torque.—
The interactionof the induced monopole and dipole moments of the par-ticle with the acoustic field can be described via theminimal-coupling model between the moments (5) ( 𝑄, D )and the fields ( 𝑝, v ). Introducing the proper dimen-sional coefficients, the complex interaction energy takesthe form 𝑊 int = 12 (︂ 𝑖𝜔 𝑄 * 𝑝 − 𝜌 D * · v )︂ . Notably, thisenergy is precisely equivalent to the energy of the elec-tric dipole D and charge 𝑄 in the electric field E = 𝜌 v and the corresponding electric potential Φ = 𝑖 𝜔 − 𝑝 ( E = −∇ Φ).The interaction can be characterized by the rates ofthe energy , momentum , and angular momentum transferbetween the field and the particle, which are quantified bythe absorption rate , radiation force , and radiation torque ,respectively [46]. First, the absorption rate is determinedby the imaginary part of the interaction energy: A = 𝜔 Im( 𝑊 int ) = 2 𝜔 [︁ Im( 𝛼 𝑚 ) 𝑊 ( 𝑝 ) + Im( 𝛼 𝑑 ) 𝑊 ( v ) ]︁ . (7)It is naturally proportional to the imaginary parts of theparticle polarizabilities (6) (and, hence, of the parameters¯ 𝜌 and ¯ 𝛽 ) and to the corresponding pressure- and velocity-related energy densities (2) of the field.Second, the radiation force is associated with the gra-dient of the real part of the interaction energy and canbe written as [3, 4, 26, 28, 29, 46]: F = −
12 Re [︂ 𝑖𝜔 𝑄 * ∇ 𝑝 − 𝜌 D * · ( ∇ ) v ]︂ = F grad + F scat . (8)Here the gradient and scattering parts are related to thereal and imaginary parts of the particle polarizabilities: F grad = Re( 𝛼 𝑚 ) ∇ 𝑊 ( 𝑝 ) + Re( 𝛼 𝑑 ) ∇ 𝑊 ( v ) , (9) F scat = 2 𝜔 [︁ Im( 𝛼 𝑚 ) P ( 𝑝 ) + Im( 𝛼 𝑑 ) P ( v ) ]︁ . (10)These laconic expressions reveal the direct relation be-tween the scattering force (which is associated with theabsorption of phonons by the particle) and canonical mo-mentum density (3) of the acoustic field. Importantly,substituting the polarizabilities (6) into Eqs. (9) and (10),one can check that the gradient force exactly coincideswith the force found in earlier calculations for losslessparticles [11, 31, 36, 41] ( F scat = 0 in this approxima-tion), while the scattering-force part is equivalent to thatfound in recent works [35, 37] considering viscous fluids.Remarkably, Eqs. (8)–(9) are entirely similar to the ex-pressions for optical radiation forces on small Rayleighparticles or atoms [3, 4, 24–26, 28–30, 42–48]. In thismanner, the electric- and magnetic-dipole terms in op-tical equations [28, 45–48] (related to the electric andmagnetic fields E and H ) correspond to the monopoleand dipole terms in the acoustic equations (related tothe pressure and velocity fields 𝑝 and v ).Using the above correspondence between the opticaland acoustic interactions, we readily find the acoustictorque on a small particle. The isotropic monopole mo-ment cannot induce any torque, and the torque origi-nates solely from the dipole moment D of the particle.In analogy with an electric dipole in an electric field [44–46, 48, 49], we obtain: T = 12 Re [ 𝜌 D * × v ] = 𝜔 Im( 𝛼 𝑑 ) S . (11)The very simple Eq. (11) reveals the direct connectionbetween the spin angular momentum density (4) of the FIG. 2. A small spherical particle in the acoustic evanes-cent field (12), which can be treated as a plane wave withthe complex wave vector k = 𝑘 𝑧 ¯ z + 𝑖𝜅 ¯ x . The gradient andscattering (radiation pressure) forces (9) and (10) are pro-duced by the energy density gradient and canonical momen-tum (the real part of the wave vector), respectively. Thetorque (11) is produced by the transverse spin of the evanes-cent field [45, 48, 54, 56, 57]. acoustic field and the radiation torque on a small absorp-tive particle. To the best of our knowkedge, this equationhas not been derived before. This general connection (en-tirely similar to the optical case) is very important, be-cause it was implied without rigorous grounds in the veryrecent experiment measuring acoustic spin [56]. Further-more, this connection can be seen by comparing very re-cent numerical simulations of the acoustic torque and an-alytical calculations of the spin density in the particularcase of acoustic Bessel beams [40, 58]. Having the sim-ple expression (11), acoustic torques on subwavelengthisotropic particles can be readily found analytically in an arbitrary acoustic field.Equations (7)–(11) are the main results of our work.Even though some of these are equivalent to thepreviously-known expressions (such as gradient force onlossless particles), here the acoustic absorption, forces,and torque are for the first time presented in a unifiedand physically clear form. All these quantities are de-termined by the fundamental energy, momentum, andangular-momentum properties (2)–(4) of the field, as wellas by the monopole and dipole particle polarizabilities (5)and (6). Note that all the quantities (6)–(11) behave as ∝ ( 𝑘𝑎 ) , i.e., proportionally to the volume of the par-ticle. This makes perfect physical sense and allows oneto discriminate between various calculations of radiationforces and torques (see, e.g., torques in [33, 39] with de-pendences ∝ ( 𝑘𝑎 ) and ∝ ( 𝑘𝑎 ) , respectively). For largeror lossless (Im( 𝛼 𝑚,𝑑 ) = 0) particles, one has to involvehigher-order terms in 𝑘𝑎 (see below). Example: Forces and torques in an evanescent acousticfield.—
To illustrate the above general theory, we con-sider a single evanescent acoustic wave with the pressureand velocity fields given by [56, 57]: 𝑝 = 𝐴 𝑒 𝑖𝑘 𝑧 𝑧 − 𝜅𝑥 , v = 𝐴𝜔𝜌 ⎛⎝ 𝑖𝜅 𝑘 𝑧 ⎞⎠ 𝑒 𝑖𝑘 𝑧 𝑧 − 𝜅𝑥 . (12)Here, 𝐴 is a constant amplitude, 𝑘 𝑧 is the longitudinalpropagation constant, and 𝜅 is the vertical decay con-stant. This example is very simple yet generic. On theone hand, the evanescent wave can be treated as a planewave with the complex wave vector k = 𝑘 𝑧 ¯ z + 𝑖𝜅 ¯ x (theoverbar denotes the unit vectors of the correspondingaxes) [45, 48], Fig. 2, which allows one to use the exactlysolvable Mie scattering problem for numerical calcula-tions of forces and torques [82]. On the other hand, theevanescent wave is inhomogeneous , and it carries the in-tensity gradient ∇ 𝑊 , canonical momentum P , and spin S , which exert the gradient force (9), scattering forces(10), and the radiation torque (11) in the three mutually-orthogonal directions [45, 48, 54, 56, 57], see Fig. 2.Figure 3 shows the dependences of these two forces andtorque in the field (12) on the dimensionless particle ra-dius 𝑘𝑎 for the cases of absorptive and lossles particles.We plot analytical results from Eqs. (9)–(11), valid onlyto leading order, ∝ ( 𝑘𝑎 ) , and the exact numerical calcu-lations using the Mie scattering solutions together withthe momentum and angular momentum fluxes, similar tothe Maxwell stress tensor approach in optics (see Supple-mental Material [65]). Note that the forces and torqueare normalized by 𝐹 = 𝜋𝛽 | 𝐴 | 𝑎 / 𝑇 = 𝐹 /𝑘 , sothe analytical dependences (9)–(11) are linear in Fig. 3.For an absorptive particle, the analytical approximationagrees with the exact calculations for 𝑘𝑎 (cid:46) . 𝑎 and 𝑎 in Eqs. (6) (see Sup-plemental Material [65]). In this case, the monopoleand dipole terms include all orders in 𝑘𝑎 , although thehigher-order multipole terms are still neglected. The cor-responding refined analytical dependences are shown inFig. 3 by dashed curves, and these agree with the exactnumerical calculation for 𝑘𝑎 (cid:46) . lossless particles.First, the scattering (radiation-pressure) force vanishesonly in the ( 𝑘𝑎 ) order but is generally non-zero (seeFig. 3). The higher-order radiation-pressure force origi-nates from the so-called “radiation friction” effect, whichis described by small higher-order imaginary parts in themonopole and dipole polarizabilities [29, 83, 84], and alsofrom the interference between the monopole and dipolefields [29]. Using the Mie coefficients 𝑎 and 𝑎 , wefind that the higher-order imaginary parts of the po-larizabilties can be written as ˜ 𝛼 𝑚 ≃ 𝛼 𝑚 + 𝑖𝑘 𝜋 𝛼 𝑚 and˜ 𝛼 𝑑 ≃ 𝛼 𝑑 + 𝑖𝑘 𝜋 𝛼 𝑑 , where 𝛼 𝑚,𝑑 are the leading-order polar- FIG. 3. Exact numerical and approximate analytical calcu-lations of the gradient force, scattering (radiation-pressure)force, and torque on a spherical particle in the acoustic evanes-cent field, Fig. 2. The field and particle parameters are: 𝑘 𝑧 /𝑘 = 1 . 𝜅/𝑘 = 0 .
89, ¯ 𝜌 = 2 + 0 . 𝑖 , ¯ 𝛽 = 3 + 0 . 𝑖 (theimaginary parts are omited in the lossless-particle case). Seediscussion in the text. izabilities (6) (see Supplemental Material [65]). Second,the radiation torque vanishes exactly for lossless spheri-cal particles of any radius (Fig. 3). This is also similarto optics, where the radiation-friction effect produces theforce but not the torque on the particle [27, 49]. Thus,the simplest analytical approximation (6) and (11) coin-cides with the exact calculations in this case. Conclusion.—
We have presented a general theory ofthe interaction of a monochromatic acoustic wave fieldwith a small absorbing spherical particle. Our theoryis based on the complex monopole and dipole polariz-abilities of the particle, and it provides simple analyticalexpressions for the absorption rate, radiation forces (in-cluding the gradient and scattering forces), and radiationtorque. Most importantly, these expressions reveal closeconnections with the fundamental local properties of theacoustic field: its energy, canonical momentum, and spinangular momentum densities [56–58]. Thus, one can nowuse acoustic forces and torques to measure the canoni-cal momentum and spin densities of sound waves, andvice versa: use canonical momentum and spin to pre-dict radiation forces and torques. Our work also unifiestheoretical approaches to the acoustic and optical field-particle interactions, and reveals close parallels betweenthese. This provides a more fundamental understandingand new physical insights into these important problems.We are grateful to Y. P. Bliokh, A. Y. Bekshaev,and Y. S. Kivshar for fruitful discussions. This workwas partially supported by MURI Center for DynamicMagneto-Optics via the Air Force Office of ScientificResearch (AFOSR) (FA9550-14-1-0040), Army ResearchOffice (ARO) (Grant No. Grant No. W911NF-18-1-0358), Asian Office of Aerospace Research and Develop-ment (AOARD) (Grant No. FA2386-18-1-4045), JapanScience and Technology Agency (JST) (Q-LEAP pro-gram, and CREST Grant No. JPMJCR1676), Japan So-ciety for the Promotion of Science (JSPS) (JSPS-RFBRGrant No. 17-52-50023, and JSPS-FWO Grant No.VS.059.18N), RIKEN-AIST Challenge Research Fund,the John Templeton Foundation, the Foundation for theAdvancement of Theoretical Physics and Mathematics“Basis”, and the Australian Research Council. [1] A. 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Fourier acoustics: sound radiation andnearfield acoustical holography (Elsevier, 1999).[64] D. T. Blackstock,
Fundamentals of physical acoustics (ASA, 2001).[65] See Supplemental Material at ..., which includesRefs. [66–81].,.[66] K. Yosioka and Y. Kawasima, “Acoustic radiation pres-sure on a compressible sphere,” J. Fluid Mech. , 1(1955).[67] W. T. Doyle, “Optical properties of a suspension of metalspheres,” Phys. Rev. B , 9852 (1989).[68] L. Jylh¨a, I. Kolmakov, S. Maslovski, and S. Tretyakov,“Modeling of isotropic backward-wave materials com-posed of resonant spheres,” J. Appl. Phys. , 043102(2006).[69] A. Moroz, “Depolarization field of spheroidal particles,”J. Opt. Soc. Am. B , 517 (2009).[70] A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S.Luk’yanchuk, and B. N. Chichkov, “Optical response fea- tures of Si-nanoparticle arrays,” Phys. Rev. B , 045404(2010).[71] A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. N.Chichkov, “Collective resonances in metal nanoparticlearrays with dipole-quadrupole interactions,” Phys. Rev.B , 245411 (2012).[72] E. C. Le Ru, W. R. C. Somerville, and B. Augui´e, “Ra-diative correction in approximate treatments of electro-magnetic scattering by point and body scatterers,” Phys.Rev. A , 012504 (2013).[73] P. J. Westervelt, “The theory of steady forces caused bysound waves,” J. Acoust. Soc. Am. , 312 (1951).[74] A. J. Livett, E. W. Emery, and S. Leeman, “Acousticradiation pressure,” J. Sound Vib. , 1 (1981).[75] G. Maidanik, “Torques due to acoustical radiation pres-sure,” J. Acoust. Soc. Am. , 620 (1958).[76] L. Zhang and P. L. Marston, “Acoustic radiation torqueand the conservation of angular momentum (L),” J.Acoust. Soc. Am. , 1679 (2011).[77] G. Maidanik, “Acoustical radiation pressure due to in-cident plane progressive waves on spherical objects,” J.Acoust. Soc. Am. , 738 (1957).[78] F. G. Mitri and Z. E.A. Fellah, “New expressions for the radiation force function of spherical targets in stationaryand quasi-stationary waves,” Arch. Appl. Mech. , 1(2007).[79] G. Gouesbet and G. Gr´ehan, Generalized Lorenz-Mie the-ories (Springer, 2011).[80] G. T. Silva, “An expression for the radiation force exertedby an acoustic beam with arbitrary wavefront (L),” J.Acoust. Soc. Am. , 3541 (2011).[81] G. T. Silva, T. P. Lobo, and F. G. Mitri, “Radiationtorque produced by an arbitrary acoustic wave,” EPL , 54003 (2012).[82] A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Mie scatter-ing and optical forces from evanescent fields: A complex-angle approach,” Opt. Express , 7082 (2013).[83] S. H. Simpson and S. Hanna, “Orbital motion of opticallytrapped particles in Laguerre-Gaussian beams,” J. Opt.Soc. Am. A , 2061 (2010).[84] S. Albaladejo, R. Gomez-Medina, L. S. Froufe-Perez,H. Marinchio, R. Carminati, J. F. Torrado, G. Armelles,A. Garcia-Martin, and J. J. Saenz, “Radiative correc-tions to the polarizability tensor of an electrically smallanisotropic dielectric particle,” Opt. Express , 3556(2010). SUPPLEMENTAL MATERIAL:Acoustic radiation force and torque on small particlesas measures of the canonical momentum and spin densities
I. D. Toftul,
1, 2
K. Y. Bliokh,
1, 3
M. I. Petrov, and F. Nori
1, 4 Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan ITMO University, Birzhevaya liniya 14, St.-Petersburg 199034, Russia Nonlinear Physics Centre, RSPE, The Australian National University, Canberra, ACT 0200, Australia Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
1. MIE SCATTERING OF AN ACOUSTIC PLANE WAVE BY A SPHERE
Here we describe the exact Mie-type solution for the acoustic plane wave scattering by a spherical particle [1, 2].Note that throughout this work we consider only monochromatic fields, which are described by the complex coordinate-dependent fields p ( r ) and v ( r ). The real time-dependent fields are obtained by applying the Re[ ... exp( i!t )] operator.Accordingly, all quadratic forms (such as energy density, momentum density, etc.) are considerfed as cycle-averagedquantities. This means that the form f f of real time-dependent fields becomes 12 Re( f ⇤ f ) in terms of complextime-independent fields. Note also that it is often su cient to write the explicit form of the p ( r ) field, and the velocityfield can be obtained as v = ( i!⇢ ) r p .As in the main text, we consider a spherical particle with the parameters ⇢ , , and the radius a , located at r = in a homogeneous medium with parameters ⇢ and . Using spherical coordinates ( r, ✓, ), the incident z -propagatingplane-wave field can be written as [1] p (in) = A e ikr cos ✓ = X n =0 A n j n ( kr ) P n (cos ✓ ) , (1)where A n = A i n (2 n + 1), j n are the spherical Bessel functions of the first kind, and P n are the Legendre polynomials.Taking into account the azimuthal symmetry of the problem, the field inside the spherical particles and scattered fieldoutside the particles can be written as: p (part) = X n =0 A n c n j n ( k r ) P n (cos ✓ ) , p (sc) = X n =0 A n a n h (1) n ( kr ) P n (cos ✓ ) . (2)where k = k p ¯ ⇢ ¯ is the wave number inside the particle, and h (1) n are the spherical Hankel functions of the first kind.The coe cients a n and c n in Eqs. (2) should be determined from the boundary conditions, i.e., the continuity ofthe pressure and normal velocity component at the interface r = a . Using v = ( i!⇢ ) r p , we have: p (in) + p (sc) = p (part) , ⇢ ✓ @p (in) @r + @p (sc) @r ◆ = 1 ⇢ @p (part) @r . (3)Substituting the fields (1) and (2) into the boundary conditions (3), we derive [2, 3]: c n = i/ ( ka ) j n ( k a ) h (1) n ( ka ) j n ( k a ) h (1) n ( ka ) , a n = j n ( k a ) j n ( ka ) j n ( k a ) j n ( ka ) j n ( k a ) h (1) n ( ka ) j n ( k a ) h (1) n ( ka ) , (4)where = ( k ⇢ ) / ( k⇢ ) = p ¯ / ¯ ⇢ , ¯ ⇢ = ⇢ /⇢ , ¯ = / , and the prime stands for the derivative with respect to theargument of the functions.The terms with the coe cients a , a , a , ... in the decomposition (2) of the scattered field can be associated with thecorresponding multipole radiations: the monopole, dipole, quadrupole, ... ones. In the case of a small subwavelengthparticle, ka ⌧
1, the higher- n terms have higher leading orders in ka (but each term a n has all orders higher thanthe leading ones), see Table I. In this work, we restrict our consideration by the leading monopole and dipole terms,which generally have the same order of smallness. Their coe cients (4) can be expanded in the Taylor series as: a = i
3( ¯ ka ) + i ⇥ ¯ (¯ ⇢ + 5)
15 ¯ + 9 ⇤ ( ka )
19( ¯ ( ka ) + . . . ,a = i ⇢ ⇢ + 1( ka ) + i ⇢ ( ¯ ¯ ⇢ + 1(2¯ ⇢ + 1) ( ka ) ✓ ¯ ⇢ ⇢ + 1 ◆ ( ka ) + . . . . (5) TABLE I. The leading orders of di↵erent multipole terms in the decomposition (2) and (4) of the scattered field for the case ofa lossless particle [Im(¯ ⇢ ) = Im( ¯ ) = 0] [4].Coe cient a n O (Re a n ) O (Im a n )Monopole a ⇠ ( ka ) ⇠ ( ka ) Dipole a ⇠ ( ka ) ⇠ ( ka ) Quadrupole a ⇠ ( ka ) ⇠ ( ka ) Octupole a ⇠ ( ka ) ⇠ ( ka ) ... n -th multipole a n ⇠ ( ka ) n +1) ⇠ ( ka ) n +1
2. MONOPOLE AND DIPOLE POLARIZABILITIES OF THE PARTICLE
Using the monopole and dipole coe cients in the Mie plane-wave scattering series, one can determine the genericmonopole and dipole responses of the particle to an arbitrary incident monochromatic field. A similar approach iswell known in optics [5–11].To do this, note that the monochromatic radiation of the oscillating acoustic point monopole and z -oriented dipolecan be written as [1, 2]: p m = i Q ⇢ ! ⇡r e ikr , p d = k D ⇢ ! ⇡r cos ✓ ✓ ikr ◆ e ikr , (6)where Q and D are the monopole and dipole strengths , respectively, which we define with the signs opposite to thosein [1, 2]. These expressions have the same form as the first two terms in the Mie series (2) for the scattered field: A a h (1)0 ( kr ) P (cos ✓ ) and A a h (1)1 ( kr ) P (cos ✓ ). Writing these terms in the form of Eqs. (6) with the monopole anddipole strength presented as Q = i! ↵ m p (in) ( ) , D = ↵ d p (in) ( ) , (7)where p (in) ( ) = A is the incident field at the particle’s position, we obtain the monopole and dipole polarizabilities ofthe particle: ↵ m = ⇡ik a ,↵ d = ⇡ik a . (8)Using the leading terms in the Taylor series (5) yields Eqs. (6) of the main text. For lossless particles, the leading termsin the Taylor series (5) yield purely real polarizabilities, while the third terms in Eqs. (5) provide small imaginarycorrections, responsible for the “radiation friction” e↵ect [5, 12, 13]. Due to this e↵ect, even a lossless particleexperiences a non-zero scattering (radiation pressure) force, while the radiation torque vanishes identically (see themain text).
3. EXACT CALCULATIONS OF THE ACOUSTIC FORCE AND TORQUE
The radiation force and torque acting on a scattering particle can be calculated using the momentum and angularmomentum fluxes through a closed surface ⌃ enclosing the particle [14–18]: F = I ⌃ ˆ P n d ⌃ , T = I ⌃ ˆ M n d ⌃ , (9)where n is the outer normal unit vector to the surface, P ij = 14 | p | ⇢ | v | ij + 12 ⇢ Re( v ⇤ i v j ) is the cycle-averagedkinetic momentum flux density tensor (the acostic analogue of the Maxwell stress tensor), M ij = " jk` r k P `i is thecorresponding angular momentum flux density, whereas ji and " ik` are the Kronecker and Levi-Civita symbols.Here, the acoustic wave field is the sum of the incident and scattered fields outside the particle: p = p (in) + p (sc) , v = v (in) + v (sc) . { r = R > a } . In this case, the expression for the torque (9) can be simplified to [19, 20]: T = ⇢ R Re I ( n · v ⇤ ) [ n ⇥ v ] d ⌦ , (10)where d ⌦= sin ✓d d✓ is the element of the spherical solid angle.Note that the integral (9) for the radiation force on a spherical particle from the incident plane wave (1) can beevaluated analytically [21–23]: F z = ⇡⇢ ! | A | X n =0 [(2 n + 1) Re ( a n ) + 2( n + 1) Re ( a ⇤ n a n +1 )] . (11)For an absorbing small particle, the leading-order approximation ⇠ ( ka ) yields: F z ' ⇡⇢ ! | A | [Re (¯ a ) + 3 Re (¯ a )] , (12)where we denoted the first terms in the Taylor series (5) as ¯ a and ¯ a . Using Eqs. (3) and (6) of the main text, wefind that the canonical momentum of the plane-wave field (1) is P ( p ) z = P ( v ) z = k/ ! , and Eq. (12) coincides withthe scattering (radiation-pressure) force expression (10) of the main text.For a lossless particle, Re ¯ a = Re ¯ a = 0, the approximate expression (12) vanishes, and one has to involve higher-order terms from the exact Eq. (11). Using the Taylor series (5), where the third terms equal ¯ a and ¯ a , the firstnon-vanishing approximation for the radiation-pressure force can be written as F z ' ⇡⇢ ! | A | h (Im ¯ a ) a ) + 2 Im(¯ a ) Im(¯ a ) i . (13)Here the first two terms can be associated with the “radiation-friction” corrections to the monopole and dipolepolarizabilities (6) in the main text [12, 13, 24], while the third term originates from the interference of the monopoleand dipole responses (the Re ( a ⇤ n a n +1 ) term in Eq. (11)). An analogous higher-order force from the interference ofelectric and magnetic dipoles plays an important role in optics [12, 25–28]. All the three terms in Eq. (13) are generally ⇠ ( ka ) .
4. COMPLEX-ANGLE APPROACH FOR THE EVANESCENT INCIDENT WAVE
To apply the Mie scattering solutions (1)–(4) to the case of the evanescent incident wave, we use the approachdescribed in [29]. Namely, we note that the incident plane wave (1), p (in) ( r ) = Ae ikz , can be transformed to theevanescent wave, Eq. (12) in the main text, by the rotation of its argument on the imaginary angle: p (in evan) ( r ) = p (in) ⇣ ˆ R ( i ) r ⌘ = Ae ikz cosh kx sinh . (14)Here k cosh = k z , k sinh = , i.e., = tanh ( /k z ), andˆ R ( i ) = cosh i sinh i sinh (15)is the rotational operator of the imaginary argument.Since the Mie scattering problem is linear, the field scattered from the evanescent wave can be obtained by thesame transformation (14) of the plane-wave scattered field (2) and (4) [29]: p (sc evan) ( r ) = p (sc) ⇣ ˆ R ( i ) r ⌘ . (16)Notice that the transformation (15) is written for the Cartesian coordinates: ( x, y, z ) ! ( x , y , z ) = ( x cosh iz sinh , y, ix sinh + z cosh ). The corresponding transformation of the spherical coordinates is: ( r, ✓, ) ! r, cos ( z /r ) , tan ( y/x ) . [1] E. G. Williams, Fourier acoustics: sound radiation and nearfield acoustical holography (Elsevier, 1999).[2] D. T. Blackstock,
Fundamentals of physical acoustics (ASA, 2001).[3] K. Yosioka and Y. Kawasima, “Acoustic radiation pressure on a compressible sphere,” J. Fluid Mech. , 1 (1955).[4] G. T. Silva, “Acoustic radiation force and torque on an absorbing compressible particle in an inviscid fluid,” J. Acoust.Soc. Am. , 2405 (2014).[5] C. F. Bohren and D. R. Hu↵man,
Absorption and scattering of light by small particles (John Wiley & Sons, 2008).[6] W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B , 9852 (1989).[7] L. Jylh¨a, I. Kolmakov, S. Maslovski, and S. Tretyakov, “Modeling of isotropic backward-wave materials composed ofresonant spheres,” J. Appl. Phys. , 043102 (2006).[8] A. Moroz, “Depolarization field of spheroidal particles,” J. Opt. Soc. Am. B , 517 (2009).[9] A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B , 045404 (2010).[10] A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. N. Chichkov, “Collective resonances in metal nanoparticle arrayswith dipole-quadrupole interactions,” Phys. Rev. B , 245411 (2012).[11] E. C. Le Ru, W. R. C. Somerville, and B. Augui´e, “Radiative correction in approximate treatments of electromagneticscattering by point and body scatterers,” Phys. Rev. A , 012504 (2013).[12] M. Nieto-Vesperinas, J. J. S´aenz, R. G´omez-Medina, and L. Chantada, “Optical forces on small magnetodielectric parti-cles,” Opt. Express , 11428 (2010).[13] S. H. Simpson and S. Hanna, “Orbital motion of optically trapped particles in Laguerre-Gaussian beams,” J. Opt. Soc.Am. A , 2061 (2010).[14] P. J. Westervelt, “The theory of steady forces caused by sound waves,” J. Acoust. Soc. Am. , 312 (1951).[15] L. P. Gor’kov, “On the forces acting on a small particle in an acoustical field in an ideal fluid,” Sov. Phys. Dokl. , 773(1962).[16] A. J. Livett, E. W. Emery, and S. Leeman, “Acoustic radiation pressure,” J. Sound Vib. , 1 (1981).[17] H. Bruus, “Acoustofluidics 7: The acoustic radiation force on small particles,” Lab Chip. , 1014 (2012).[18] L. Zhang and P. L. Marston, “Angular momentum flux of nonparaxial acoustic vortex beams and torques on axisymmetricobjects,” Phys. Rev. E , 065601(R) (2011).[19] G. Maidanik, “Torques due to acoustical radiation pressure,” J. Acoust. Soc. Am. , 620 (1958).[20] L. Zhang and P. L. Marston, “Acoustic radiation torque and the conservation of angular momentum (L),” J. Acoust. Soc.Am. , 1679 (2011).[21] G. Maidanik, “Acoustical radiation pressure due to incident plane progressive waves on spherical objects,” J. Acoust. Soc.Am. , 738 (1957).[22] T. Hasegawa, “Comparison of two solutions for acoustic radiation pressure on a sphere,” J. Acoust. Soc. Am. , 1445(1977).[23] F. G. Mitri and Z. E.A. Fellah, “New expressions for the radiation force function of spherical targets in stationary andquasi-stationary waves,” Arch. Appl. Mech. , 1 (2007).[24] S. Albaladejo, R. Gomez-Medina, L. S. Froufe-Perez, H. Marinchio, R. Carminati, J. F. Torrado, G. Armelles, A. Garcia-Martin, and J. J. Saenz, “Radiative corrections to the polarizability tensor of an electrically small anisotropic dielectricparticle,” Opt. Express , 3556 (2010).[25] K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. ,3300 (2014).[26] A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X ,011039 (2015).[27] M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis,A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transversespin-dependent force using a nano-cantilever,” Nat. Phys. , 731 (2016).[28] L. Liu, A. Di Donato, V. Ginis, S. Kheifets, A. Amirzhan, and F. Capasso, “Three-Dimensional Measurement of theHelicity-Dependent Forces on a Mie Particle,” Phys. Rev. Lett. , 223901 (2018).[29] A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Mie scattering and optical forces from evanescent fields: A complex-angleapproach,” Opt. Express , 7082 (2013).[30] G. Gouesbet and G. Gr´ehan, Generalized Lorenz-Mie theories (Springer, 2011).[31] G. T. Silva, “An expression for the radiation force exerted by an acoustic beam with arbitrary wavefront (L),” J. Acoust.Soc. Am. , 3541 (2011).[32] G. T. Silva, T. P. Lobo, and F. G. Mitri, “Radiation torque produced by an arbitrary acoustic wave,” EPL97