Equivalence between angular spectrum-based and multipole expansion-based formulas of the acoustic radiation force and torque
aa r X i v : . [ phy s i c s . a pp - ph ] F e b Gong & Baudoin, JASA
Equivalence between angular spectrum-based and multipole expansion-basedformulas of the acoustic radiation force and torque
Zhixiong Gong a and Michael Baudoin
1, 2 Univ. Lille, CNRS, Centrale Lille, Yncr´ea ISEN, Univ.Polytechnique Hauts-de-France, UMR 8520 - IEMN, F- 59000 Lille,France Institut Universitaire de France, 1 rue Descartes, 75005 Paris (Dated: 5 February 2021)
Two main methods have been proposed to derive the acoustical radiation force andtorque applied by an arbitrary acoustic field on a particle: The first one relies on theplane wave angular spectrum decomposition of the incident field (see [Sapozhnikov &Bailey, J. Acoust. Soc. Am. 133, 661–676 (2013)] for the force and [Gong & Baudoin,J. Acoust. Soc. Am. 148, 3131–3140 (2020)] for the torque), while the second onerelies on the decomposition of the incident field into a sum of spherical waves, the so-called multipole expansion (see [Silva, J. Acoust. Soc. Am. 130, 3541–3544 (2011)]and [Baresh et al. , J. Acoust. Soc. Am. 133, 25–36 (2013)] for the force, and [Silva etal. , EPL 97, 54003 (2012)] and [Gong et al. , Phys. Rev. Applied 11, 064022 (2019)]for the torque). In this paper, we formally establish the equivalence between theexpressions obtained with these two methods for both the force and torque.PACS numbers: 43.25.Qp, 43.20.Fn, 43.20.Ks a Corresponding author: [email protected] . INTRODUCTION Since the seminal works of Rayleigh , Langevin and Brillouin , many expressionsof the acoustic radiation force and torque applied by various acoustic fields on differenttypes of particle have been derived. King was the first to propose an expression of theacoustic radiation force applied on a rigid sphere by a plane (standing or progressive) wave.This expression was extended later on by Yosika & Kawasima for compressible particle andHasegawa & Yiosika for an elastic sphere. The case of spherical and focused incident waveswas addressed by Embleton and Chen & Apfel for rigid and elastic spheres respectively.Nevertheless all these cases assume axisymmetric incident fields centered on the particle,which considerably simplifies the problem and does not enable to compute the 3D trappingforce applied by a selective tweezer on an object . The case of arbitrary acoustic fieldwas at this point only treated in the framework of the Long Wavelength Regime (LWR),i.e. for particle much smaller than the wavelength . Concerning the torque, the veryexistence of a torque applied on a spherical particle requires the existence of a momentumcarried out by the wave, which cannot be obtained with an axisymmetric acoustic field.Busse & Wang demonstrated the role played by the viscous boundary layer on the Torqueapplied by orthogonal acoustic waves on a spherical particle in the LWR. Later on, Zhang& Marston proposed an expression of the axial acoustic radiation torque acting on anaxisymmetric particle centered on the axis of a cylindrical acoustical vortex. But again theproposed expressions assume certain symmetry of the incident beam and specific locationof the scatterer. 2he treatment of the general problem of the acoustic radiation force and torque applied ona spherical particle of arbitrary size requires to solve three major issues: First, the incidentfield must be decomposed into a sum of elementary waves suitable for the treatment of thescattering problem and then the caculation of the force and torque. In the angular spectrummethod (ASM) , the incident field is decomposed into a sum of plane wave assuming theprior knowledge of the incident field in a source plane. In the multipole expansion method(MEM) the incident field is decomposed into a sum of spherical waves , whose respectivecontribution (the beam shape coefficients) can be calculated by different methods . Second,the scattering problem must be solved. For an arbitrary wave this task is complexified bythe non axisymmetry of the incident acoustic field. The angular spectrum method alleviatesthe problem by using the fact that the solution of the scattering problem for a plane waveis known. Nevertheless, each plane wave of the angular spectrum decomposition have adifferent incident angle. This problem was solved by Sapozhnikov and Bailey by usingthe Legendre addition theorem. In the multipole expansion method, the scattering problemwas solved for an arbitrary spherical wave. It was shown by Baresch et al. that theproblem degenerates to the one of an incident plane wave so that the classical scatteringcoefficients can be used (see appendix A in ref ). Third, the force and torque can becalculated by integrating the time-averaged linear and angular radiation stress tensor overthe particle surface, respectively. Such integration over the particle surface can be tediousto perform directly since (i) the particle surface is vibrating and hence is varying over time,(ii) the particle geometry may be complex in the case of non-spherical particles and (iii) theexistence of viscous and thermal boundary layers must be considered in the near field. It3as first shown by Brillouin , that the integral over the vibrating surface of the particlecan be transferred to a still surface by replacing the stress tensor by the so-called Brillouintensor, which includes the flux of momentum through this steady surface. Later on, it wasshown that the integral can be transferred to a closed surface in the far field by using themomentum and angular momentum balances in the surrounding fluid and the Gaussdivergence theorem or the Reynolds transport theorem . Hence, by choosing a sphericalsurface in the far field, (i) the integration is conducted over a simpler surface (concentric withthe particle center) and (ii) the far-field approximation enables to use asymptotic expressionsfor Bessel and Hankel functions which simplifies the integration procedure. This also enablesthe treatment of non spherical particles, e.g. using T-matrix method .Of course, the values of the acoustic radiation force and torque must be independent ofthe method used to calculate them. While some links between some of the expressions of theARF available in the literature have been previously evoked , there is no explicit demon-stration of the link between these complex formula. The present paper aims at clarifyingthis point and establishing formally the equivalence between the different expressions of theARF and ART derived with different approaches. II. DECOMPOSITION OF THE INCIDENT FIELD
In the multipole expansion method (MEM) , the incident acoustic potential is directlydecomposed in the spherical waves basis as follows:Φ i = Φ ∞ X n =0 n X m = − n a mn j n ( kr ) Y mn ( θ, ϕ ) e − iωt , (1)4ith a mn the incident beam-shape coefficients (BSC), which sets the weight of each sphericalwaves, Y mn ( θ, ϕ ) the normalized spherical harmonics defined by: Y mn ( θ, ϕ ) = s n + 14 π ( n − m )!( n + m )! P mn (cos θ ) e imϕ , (2)with ( r, θ, ϕ ) the spherical coordinates, Φ the potential amplitude, j n the Bessel functionof the first kind, k the wavenumber, and P mn the associated Legendre functions. Note thatonly the Bessel functions of the first kind appear in this expression since the incident fieldexists in absence of the scatterer and hence must be finite in ( r = 0), hence eliminating theBessel functions of the second kind which are singular at this point. A general method todetermine the beam shape coefficients (inspired by previous work in optics) for an arbitraryfield was introduced by Baresch et al. and various methods were tested and compared byZhao et al. .In the angular spectrum method (ASM) , the calculation starts from the prior knowl-edge of the incident pressure field in a source plane ( z = 0) p i | z =0 = p i ( x, y, p i ( x, y, z ) = 14 π × Z Z k x + k y ≤ k S ( k x , k y ) e ik x x + ik y y + i √ k − k x − k y z dk x dk y . (3)using the angular spectrum decomposition (2D spatial Fourier transform) of the source planefield: S ( k x , k y ) = Z + ∞−∞ Z + ∞−∞ p i ( x, y, e − ik x x − ik y y dxdy, (4)with k x and k y are the lateral components of the wavenumber k in Cartesian coordinates, k z = k − k x − k y and k = ω/c . 5f the angle γ between the position vector r = x x + y y + z z and the wavevector k isintroduced, we see clearly that the incident field is nothing but the sum of plane waves p k i with different incident angles γ : p k i ( x, y, z ) = S ( k x , k y ) e ikr cos( γ ) Using (i) the known decomposition of a plane wave with an incident angle γ into sphericalwaves and (ii) the Legendre addition theorem to express the final result as a function of theabsolute spherical coordinate ( θ, ϕ ) instead of the auxiliary angle γ , Sapozhnikov & Bailey were able to express the incident field into a sum of spherical waves: p i = 1 π ∞ X n =0 n X m = − n i n H nm j n ( kr ) Y mn ( θ, ϕ ) , (5)with the ASM-based BSC H nm describing the respective weight of each spherical wave: H nm = Z Z k x + k y ≤ k S ( k x , k y ) [ Y mn ( θ k , ϕ k )] ∗ dk x dk y , (6)The asterisk ∗ designates the complex conjugate, and the angle parameters ( θ k , ϕ k ) in theFourier space have the relation: cos θ k = [1 − (cid:0) k x + k y (cid:1) /k ] / and ϕ k = arctan ( k y /k x ).This decomposition into a sum of spherical waves is necessary to compute the force andtorque since the total field (incident + scattered) needs to be integrated over an arbitraryclosed surface surrounding the particle, which for commodity will be chosen as a sphericalsurface in the far field as discussed in section IV. The H nm coefficients can be easily obtainedwhen the field is known in a source plane by using the Spatial Fast Fourier Transform of theincident field, which makes the ASM method very convenient to compute the force appliedon a particle by a field generated by a planar transducer .6he comparison of Eqs. (1) and (5) and use of the relationship between the velocitypotential and pressure p i = iωρ Φ i (with ω the angular frequency and ρ the fluid density),enables to establish the relationship between the incident BSC a mn and the angular-spectrumbased BSC H nm a mn = 1 πωρ Φ i n − H nm . (7)which is essential to prove the equivalence of MEM and ASM based ARF and ART formulas.Note that an equivalent form of Eq. (7) has been given in Eq. (15) of Ref. by comparingtwo expressions of acoustic pressure. III. RESOLUTION OF THE SCATTERING PROBLEM
In the MEM, the scattered field, as the incident field is decomposed directly into a sumof spherical waves: Φ s = Φ ∞ X n =0 n X m = − n s mn h (1) n ( kr ) Y mn ( θ, ϕ ) e − iωt , (8)with s mn the beam shape coefficient of the scattered field. Note that this time the scatteredfield is expressed in terms of the Hankel function of the first kind since the scattered field isan outgoing wave, hence eliminating the Hankel function of the second kind (correspondingto converging wave in the convention used here for the temporal part of the wave e − iωt ).The expression of the scattered beam shape coefficients as a function of the incident beamshape coefficients requires to solve the scattering problem, i.e. to determine the partial wavecoefficients A mn defined by s mn = A mn a mn . These coefficients depend on the particle shape,material composition and surface boundary condition. In the MEM, the solution of thescattering problem is a priori not known since the axisymmetry and resulting simplifications7sed in the case of plane waves can no longer be invoked. The complete problem was solvedby Baresh et al. for an elastic sphere through the introduction of three scalar potentials,(one for the longitudinal wave and the two Debye potential for the shear wave, solutionsof the wave equation and then applying the boundary conditions). It was shown that infact the problem degenerates to the one of plane incident wave, so that the partial wavescoefficient A n computed for a plane wave, which do not depend on the index m , can beused. Note that in this simplified case, people sometime introduce the so-called scatteringcoefficients S n linked to the partial wave coefficients by the formula A n = ( S n − /
2. Notealso that in the general case of nonspherical particles, the partial wave coefficients can bedetermined using the transition matrix method which makes the theory operable fornonspherical shapes, such as spheroids and finite cylinders .In the angular spectrum method, the treatment relies on known results for the scatteringof a plane wave by a sphere. Indeed, (i) the incident field has been decomposed into asum of plane waves and (ii) the solution of the scattering problem is known for each planewave. Hence using these solution for each plane wave and then using (i) the decompositionof a plane wave into a sum of spherical waves and (ii) the Legendre addition theorem, thescattered field can also be decomposed into a sum of spherical waves: p s = 1 π ∞ X n =0 n X m = − n i n H nm A mn h (1) n ( kr ) Y mn ( θ, ϕ ) , (9) IV. CALCULATION OF THE FORCE AND TORQUE
The last step, which is common to ASM and MEM is to compute the integral of thestress tensor or angular stress tensor over the surface of the particle to compute the force8nd torque, respectively. One major difficulty comes from the fact that the surface of theparticle is vibrating. This problem can be overcome in two ways: firstly, using Lagrangiancoordinates instead of Eulerian coordinates, and secondly, transferring the integral to a stillsurface by substracting the flux of momentum (flux of angular momentum) to the stresstensor (angular stress tensor) for the force and torque respectively as first demonstrated byBrillouin (for the Force) . To simplify the calculation, these integrals can be transported toany surface surrounding the particle, e.g. for simplicity a spherical surface in the far field asdemonstrated by Westervelt for the force and Maidanik and others for the torque .Using these results, the integrals to compute the force F and torque T can be written underthe following form in terms of the acoustic potential (Φ i,s ) of the incident and scattered fieldas: F = ρ k Z Z S Re (cid:26)(cid:18) ik ∂ Φ i ∂r − Φ i (cid:19) Φ ∗ s − Φ s Φ ∗ s (cid:27) n dS, (10) T = ρ (cid:26)Z Z S (cid:18) ∂ Φ ∗ i ∂r L Φ s + ∂ Φ ∗ s ∂r L Φ i + ∂ Φ ∗ s ∂r L Φ s (cid:19) dS (cid:27) , (11)where S is a closed spherical surface in the far field centered at the mass center of theparticle, ρ is the density at rest, ‘Re” means the real part of a complex number, “Im”designates the imaginary part, n is the outward unit normal vector, and the differentialsurface area is dS = r sin θdθdϕ with θ and ϕ the polar and azimuthal angles, L = − i ( r ×∇ )is the angular momentum operator, with its components in the three directions L x,y,z andthe recursion relations of the normalized spherical harmonics with ladder operators L ± givenin detail in Appendix D.In the next section we establish the link between the different formulas obtain in theliterature. 9 . EQUIVALENCE OF THE THREE-DIMENSION ARF FORMULAS Expressions of the ARF exerted by an arbitrary field on an arbitray located sphericalscatterer has been established independently by 3 different groups: Silva and Baresch et al. with a MEM, and Sapozhnikov & Bailey based on the ASM . The equivalencebetween the formulas obtained by Baresch et al. and Sapozhnikov & Bailey has beenbriefly discussed by Thomas et al. , while the equivalence with Silva’s work has not beeninvestigated yet. In this section, the reason for the different forms of ARF formulas by Silva and Baresch et al. is provided (since both use the MEM), while pointing out some minorexisting issues in the formula and at the same time, for the first time, providing detailedproof of the equivalence of the ARF formulas for the three work. A. Equivalence between MEM formula and compact formulation
1. MEM formula by Silva and Gong et al. and reindexing
Following the work of Silva and of Gong et al. , the dimensionless ARF formulas interms of the incident a mn and scattered s mn BSC are obtained by substituting Eq. (1) and (8)into Eq. (10) and conducting several algebraic calculations given in Eqs. (11-13) of Ref. bySilva or Eqs. (12-14) of Ref. by Gong et al. . The ARF formulas can be therefore obtainedbased on the relation with the dimensionless ARF according to Eq. (10) in Ref. . Notethat for the two separate derivations, different asymptotic expressions of velocity potentialsin the far-field are used: Silva uses trigonometric functions [see Eq. (4) in his paper],while Gong et al. use the exponential functions to approximate the Bessel and Hankel10unctions. In addition, Gong et al. ’s work is an extension of numerical implementation fornon-spherical shapes by using the T-matrix method .However, the ARF formulas by Silva and Gong et al. missed a re-indexing step in thescattered BSC ( s m +1 n − , s m − n − and s − m − n ), as pointed out recently . Here, we explicit thereason for the index issue and provide the good expressions: Silva and Gong et al. use thesimplified double summation symbol P nm to represent P ∞ n =0 P nm = − n to conduct the integralprocess involving the product of two spherical harmonics [see Eq. (11) in Ref. ]. A mistakeappears since the regime of m should be correctly chosen for the spherical harmonics Y m ± n − and Y mn − [as given in Eqs. (A2) and (A4)] based on the definition of spherical harmonics Y mn with | m | ≤ n , which means P nm is not always P ∞ n =0 P nm = − n .In this work, we re-derive the formulas following the right indexes ( n, m ) and thereforeget the correct forms as (see details in Appendix B) F x = ρ Φ ( ∞ X n =0 n X m = − n (cid:20) b − mn +1 [( a mn + s mn ) s m − ∗ n +1 − (cid:0) a m − n +1 + s m − n +1 (cid:1) s m ∗ n ] (12a)+ b mn +1 [ (cid:0) a m +1 n +1 + s m +1 n +1 (cid:1) s ∗ nm − ( a mn + s mn ) s m +1 ∗ n +1 ] (cid:21)) ,F y = ρ Φ ( ∞ X n =0 n X m = − n (cid:20) b − mn +1 (cid:2) ( a mn + s mn ) s m − ∗ n +1 + (cid:0) a m − n +1 + s m − n +1 (cid:1) s m ∗ n (cid:3) (12b)+ b mn +1 [ (cid:0) a m +1 n +1 + s m +1 n +1 (cid:1) s m ∗ n + ( a mn + s mn ) s m +1 ∗ n +1 ] (cid:21)) ,F z = ρ Φ ( ∞ X n =0 n X m = − n c mn +1 (cid:2)(cid:0) a m ∗ n +1 + s m ∗ n +1 (cid:1) s mn + ( a mn + s mn ) s m ∗ n +1 (cid:3) ) . (12c)where n ∈ [0 , ∞ ] and m ∈ [ − n, n ], and the coefficients b mn and c mn defined in terms of n and m are given in the Appendix A. 11ote that despite the index issue, the numerical computations in Ref. are correct sincethe erroneous additional terms were cancelled in the numerical procedure. This can befurther verified by the comparison of results by Gong et al. with the partial wave basedresults by Marston . Note also that this set of formulas can be written in a much morecompact form using the relation s mn = A mn a mn which will be given in Sec. V A 2.
2. The ARF formulas by Baresh et al.
Another set of ARF formulas based on the MEM, was derived by Baresh, Thomas &Marchiano for 3D ARF on an arbitrarily located elastic sphere, as given by Eqs. (14-16)in Ref. [reorganized as Eqs. (1-3) by Zhao, Thomas & Marchiano in Ref. ]. Note thatthere is a typo for the regime of index m : it should be | m | ≤ n instead of | m | < n (otherwise the formulas are not equivalent to those by Sapozhnikov & Bailey ). The ARFformulas with the right index regimes by Thomas and colleagues are equivalent to the abovecorrected version (see Eq. 12) of Silva and Gong et al. ’s formulas.The difference between formulas by Silva (or Gong et al. ) and Baresh, Thomas &Marchiano are the following: (i) Silva uses the incident a mn and scattered s mn BSC. Baresch et al. solved the scattering problem for an elastic sphere insonified by an arbitrary incidentbeam and showed that the problem degenerates to the one of the scattering of an incidentplane wave, so that the corresponding partial wave coefficients A n can be used leading tothe relation: s mn = A n a mn . (ii) Silva uses the orthogonality and recursion relationship ofnormalized spherical harmonics directly based on Arfken’s textbook (see Appendix A) ,while Baresch et al. use the orthogonality relationship of associated Legendre functions12 P mn ) and exponential functions, and also the recursion relationship of associated Legendrefunctions. This leads to the fact that there are four terms for the lateral forces and twoterms for the axial force in Silva’s work (without reindexing) , while only two terms forthe lateral and one term for the axial forces by Zhao et al. (with reindexing during thederivation procedure) . (iii) Silva uses the normalized spherical harmonics, while Baresch et al. use the unnormalized spherical harmonics to derive the ARF formulas , which havebeen re-organized with normalized spherical harmonics to be compact by Zhao, Thomas &Marchiano [Eqs. (1-3)] .
3. Compact expression of the ARF for arbitrary shaped particles.
If we substitute the relation s mn = A mn a mn for a particle with an arbitrary shape, the correctversion of ARF formulas in terms of a mn and s mn in Eq. (12) is further written in a compactmanner as: F x = ρ Φ ( ∞ X n =0 n X m = − n (cid:0) C m − n b − mn +1 a mn a m − ∗ n +1 − C m +1 n b mn +1 a mn a m +1 ∗ n +1 (cid:1)) , (13a) F y = ρ Φ ( ∞ X n =0 n X m = − n (cid:0) C m − n b − mn +1 a mn a m − ∗ n +1 + C m +1 n b mn +1 a mn a m +1 ∗ n +1 (cid:1)) , (13b) F z = ρ Φ ( ∞ X n =0 n X m = − n C mn c mn +1 a mn a m ∗ n +1 ) . (13c)with C m ∓ n = A mn + 2 A mn A m ∓ ∗ n +1 + A m ∓ ∗ n +1 and C mn = A mn + 2 A mn A m ∗ n +1 + A m ∗ n +1 . Note that thesecompact equations (13) are equivalent to the re-organized ones (using normalized sphericalharmonics instead of spherical harmonics in Ref. ) by Zhao et al. in a direct way whenthe particle shape is considered as a sphere (so that A mn = A n and C m ∓ n = C mn = C n = A n + 2 A n A ∗ n +1 + A ∗ n +1 ) and the index m is with the right regime | m | ≤ n .13 . Equivalence analysis of the three sets of ARF formulas As claimed above, the different forms of ARF formulas derived by Thomas and colleagues (compact form of correct version of ARF formulas by Silva and Gong et al. ) and Sapozh-nikov & Bailey come from the different elementary wave expansion of velocity potentialor pressure. The explicit relation between the beam coefficient a mn based on MEM and H nm based on ASM is given by Eq. (7) in Sec. II, which can be used to substitute into Eq. (13)to derive the 3D ARF formulas in terms of the notation H nm introduced by Sapozhnikov& Bailey based on the ASM. The equivalence between the two sets of formulas will beverified immediately if A mn = A n is set for a spherical shape (see details in Appendix C).The question raised by Sapozhnikov & Bailey in their paper between their formula and theone by Silva is now solved. All in all, considering the correction of the index issues pointedout above, all the three sets of original 3D ARF formulas are proved to be equivalent. VI. EQUIVALENCE OF THREE-DIMENSIONAL ART FORMULAS
The ART on a particle in an ideal fluid can be calculated by the integral of the time-averaged angular stress tensor minus the angular momentum flux over a far-field standardspherical shape centered at the mass center of the particle (see Eq. 11). Explicitexpressions of 3D ART formulas have been derived by Silva et al. and Gong et al. basedon the MEM and Gong & Baudoin based on the ASM .As for the ARF, there are also index issues in the expression obtained by Silva et al. and Gong et al. . Here we provide the correct expressions of ART formulas by Silva et al. et al. based on the multipole expansion method (see details in Appendix E): T x = − ρ Φ k Re ( ∞ X n =0 n X m = − n +1 b mn (cid:2) ( a m ∗ n + s m ∗ n ) s m − n + (cid:0) a m − ∗ n + s m − ∗ n (cid:1) s mn (cid:3)) , (14a) T y = − ρ Φ k Im ( ∞ X n =0 n X m = − n +1 b mn (cid:2) ( a m ∗ n + s m ∗ n ) s m − n − (cid:0) a m − ∗ n + s m − ∗ n (cid:1) s mn (cid:3)) , (14b) T z = − ρ Φ k Re ( ∞ X n =0 n X m = − n m ( a m ∗ n + s m ∗ n ) s mn ) , (14c)with the coefficients b mn given in Appendix D. Again, it is noteworthy that the numericalcomputations in Ref. are correct since they use the definition for the scattered BSC that s mn = 0 when n < | m | > n .The relationship s mn = A mn a mn can be introduced into Eq. (14) to obtain a set of compactformulas in terms of the incident BSC only: T x = − ρ Φ k Re ( ∞ X n =0 n X m = − n +1 b mn C mn a m ∗ n a m − n ) , (15a) T y = − ρ Φ k Im ( ∞ X n =0 n X m = − n +1 b mn C mn a m ∗ n a m − n ) , (15b) T z = − ρ Φ k Re ( ∞ X n =0 n X m = − n mD mn a m ∗ n a mn ) . (15c)where C mn = A m − n + A m ∗ n + 2 A m − n A m ∗ n , D mn = A mn + A mn A m ∗ n . The above compact ARTformulas are identical to Eqs. (10-12) of Ref. by using the relation between a mn and H nm given by Eq. (7) in Sec II (see details in Appendix F). Hence, the equivalence of the ARTformulas between the correct form [see Eq. (14)] of Silva et al. and Gong et al. ’s workbased on the MEM and those derived by Gong and Baudoin based on the ASM isdemonstrated in this section. 15 II. CONCLUSIONS AND DISCUSSIONS
In summary, we provide in this paper a clear proof of the equivalence of the three sets ofthe 3D acoustic radiation force (ARF) formulas derived independently by Silva (extendedlater on by Gong et al. to arbitrary shape particles), Thomas and associates , andSapozhnikov & Bailey , and the 3D acoustic radiation torque (ART) formulas derived bySilva et al. (extended by Gong et al. to arbitrary shape particles) and Gong & Baudoin .The reasons for the different forms of ARF and ART expressions are discussed completelyin Secs. V and VI, respectively.The advantage of the MEM-based ARF and ART formulas is that the calculationsof 3D ARF and ART are direct by using the incident BSC a mn of a known acoustic field on aparticle with available A mn which has a long research history in the literature for scatteringproblems. For a non planar beam such as Bessel beam, the incident BSC a mn is affectedby the relative position of the beam axis with respect to the particle center. In Silva’swork, the off-axis incidence of a Bessel beam was not studied . This was accomplished byBaresch et al. and Gong et al. with two different ideas: Under the off-axis incidence,there are two way to calculate the incident BSC a mn : (i) the first one is to involve the off-setinformation inside the elementary wave [i.e., j n ( kr ) Y mn ( θ, φ )] as demonstrated by Baresch etal. using the translocation and rotation matrices for spherical harmonics and illustrated onthe example of a helicoidal Bessel beam; (ii) the second one is to put the off-set informationin the BSC a mn directly using the addition theorem for the Bessel functions for a cylindricalBessel beam by Gong et al. , which is limited to certain kinds of ideal beams. The latter way16an be taken as a special case of the former. The a mn can be also calculated through numericalintegration , however, with the drawback of the computational cost and parameter selectionof the beams .The advantage of the ASM-based ARF and ART is that they are easy to set upwhen the field is known (e.g. measured) in a transverse plane (e.g. for planar holographictransducers ). Note that this set of formulas can also be used for ideal beams whoseintroduced coefficients H nm are available either by using the angular spectrum of the beam S ( k x , k y ) or the relation given in Eq. (7).To finalize the calculation of the ARF and ART with all these formulas, the key pointis to obtain the partial wave coefficients A mn of the particle exactly. Silva , Thomas andcolleagues and Sapozhnikov & Bailey discuss particles with spherical shapes so that A mn only depends on the index n , having A mn = A n . Gong et al. derive the formulas with A mn depending on the indexes of ( n, m ) with several numerical computation for arbitrary-sizednonspherical shapes by a semi-analytical T-matrix method . For a rigid spheroidalparticle in the so-called long-wavelength limit, Silva and colleagues gives the A mn with theTaylor expansion up to the dipole ( n = 1) in spheroidal coordinates and obtain conciseanalytical ARF and ART expressions using the partial wave expansion . Note also thatthe overall formulas discussed in the present work are generally applied for a particle in anideal fluid but are still applicable for a particle in a viscous fluid if the viscous effect in thefluid can be accounted in the expression of scattering (partial wave) coefficients .From a perspective viewpoint, the present work on the ARF and ART formulas may beextended for multiple particles if the partial wave coefficients are available, which can17e used for the manipulation and assembly of large particles beyond Rayleigh regime .Based on the Eqs. (13) and (15), the ARF and ART are closely related to the scatteringfrom the particle in a fluid. Hence, the scattering characteristics are essential to the actingforce and torque of acoustic field on the particle. For example, the resonance scattering froman elastic sphere may be suppressed under a on-axis Bessel beam of selected parameters andbe not with an off-axis incidence , which could be used to tune the ARF and ART,such as a stable tractor (pulling) beam , or a 3D stable trapping with suppressedspinning rotation. More importantly, the present work will help to build acoustical tweezersnumerical toolbox as an analogy to its optical counterpart . APPENDIX A: ORTHOGONALITY AND RECURRENCE RELATIONS OF SPHER-ICAL HARMONICS
The orthogonality relationship of normalized spherical harmonics is given in Eq. (15.138)by Arfken et al. Z π dϕ Z π sin θdθY m ∗ n Y m ′ n ′ = δ nn ′ δ mm ′ , (A1)The recurrence relations of normalized spherical harmonics involved with trigonometricand exponential functions are given in Eqs. (15.150) and (15.151) by Arfken et al. ,respectively cos θY mn = c mn Y mn − + c mn +1 Y mn +1 , (A2)18ith c mn = s ( n + m )( n − m )(2 n − n + 1) , (A3)which is based on a recurrence relation of associated Lengendre functions [Eq. (15.88) inArfken et al. ’s textbook], as also used by Baresch et al. in Eq. (C5) in Appendix C . And e ± iϕ sin θY mn = ± b ∓ m − n Y m ± n − ∓ b ± mn +1 Y m ± n +1 (A4)with b mn = s ( n + m )( n + m + 1)(2 n − n + 1) , (A5)which is based on two recurrence relations of associated Lengendre functions [Eqs. (15.89-90) in Arfken et al. ’s textbook], with Eq. (15.89) also used by Baresch et al. in AppendixD .By using the Euler’s formula e ± iϕ = cos ϕ ± i sin ϕ , the terms of normalized sphericalharmonics involved with trigonometric functions (cos ϕ sin θY nm and sin ϕ sin θY nm ) can beobtained, which can be further applied into Eq. (11) in Gong et al. for the final 3D ARFexpressions. The relation used for the derivation of F x ϕ sin θY mn = b − m − n Y m +1 n − − b mn +1 Y m +1 n +1 + b − mn +1 Y m − n +1 − b m − n Y m − n − . (A6)and the expression for the derivation of F y i × sin ϕ sin θY mn = b − m − n Y m +1 n − − b mn +1 Y m +1 n +1 − b − mn +1 Y m − n +1 + b m − n Y m − n − . (A7)19 PPENDIX B: DETAILED DERIVATION OF ARF WITH CORRECT INDEX1. Detailed derivation of F x Based on the ARF formulas of Eq. (9) from Ref. , the expression of x -component ofARF is F x = 12 ρ k Φ Z Z S Re ( − ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( kr ) ( a mn + s mn ) s m ′ ∗ n ′ Y mn Y m ′ ∗ n ′ ) × r sin θ cos ϕ sin θdθdϕ = − ρ Φ Re ( ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ Z Z S Y mn Y m ′ ∗ n ′ sin θ cos ϕ sin θdθdϕ ) , (B1)Substituting Eqs. (A6) into (B1), F x can be divided into 4 terms: F x = − ρ Φ Re ( ∞ X n =1 n − X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ Z Z S b − m − n Y m +1 n − Y m ′ ∗ n ′ sin θdθdϕ + ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ Z Z S − b mn +1 Y m +1 n +1 Y m ′ ∗ n ′ sin θdθdϕ + ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ Z Z S b − mn +1 Y m − n +1 Y m ′ ∗ n ′ sin θdθdϕ + ∞ X n =1 n X m = − n +2 ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ Z Z S − b m − n Y m − n − Y m ′ ∗ n ′ sin θdθdϕ ) , (B2)It is important to note that the index regimes of ( n, m ) for different terms are differentbecause the correct index regime of Y mn should be n ∈ [0 , ∞ ] and m ∈ [ − n, n ] for theindexes. In addition, based on the definition in Eq. (1), the intersection of regime of ( n, m )is listed in Table I. 20 ABLE I. Regime of ( n, m ) in normalized spherical harmonics for derivaiton of F x and F y . Notethat based on the definition in Eq. ( ?? ), we have n ∈ [0 , ∞ ] and m ∈ [ − n, n ]. n m Intersection Y m +1 n − n ∈ [1 , ∞ ] m ∈ [ − n, n − n ∈ [1 , ∞ ], m ∈ [ − n, n − Y m +1 n +1 n ∈ [ − , ∞ ] m ∈ [ − n − , n ] n ∈ [0 , ∞ ], m ∈ [ − n, n ] Y m − n +1 n ∈ [ − , ∞ ] m ∈ [ − n, n + 2] n ∈ [0 , ∞ ], m ∈ [ − n, n ] Y m − n − n ∈ [1 , ∞ ] m ∈ [ − n + 2 , n ] n ∈ [1 , ∞ ], m ∈ [ − n + 2 , n ] Using the orthogonality relation of the normalized spherical harmonics in Eq. (A1),Eq.(B2) can be further written as F x = − ρ Φ Re ( ∞ X n =1 n − X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ b − m − n δ n − ,n ′ δ m +1 ,m ′ + ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ − i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ b mn +1 δ n +1 ,n ′ δ m +1 ,m ′ + ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ b − mn +1 δ n +1 ,n ′ δ m − ,m ′ + ∞ X n =1 n X m = − n +2 ∞ X n ′ =0 n ′ X m ′ = − n ′ − i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ b m − n δ n − ,n ′ δ m − ,m ′ ) = − ρ Φ Re ( ∞ X n =1 n − X m = − n − i ( a mn + s mn ) s m +1 ∗ n − b − m − n + ∞ X n =0 n X m = − n − i ( a mn + s mn ) s m +1 ∗ n +1 b mn +1 + ∞ X n =0 n X m = − n i ( a mn + s mn ) s m − ∗ n +1 b − mn +1 + ∞ X n =1 n X m = − n +2 i ( a mn + s mn ) s m − ∗ n − b m − n ) = − ρ Φ Im ( ∞ X n =1 n − X m = − n ( a mn + s mn ) s m +1 ∗ n − b − m − n + ∞ X n =0 n X m = − n ( a mn + s mn ) s m +1 ∗ n +1 b mn +1 + ∞ X n =0 n X m = − n − ( a mn + s mn ) s m − ∗ n +1 b − mn +1 + ∞ X n =1 n X m = − n +2 − ( a mn + s mn ) s m − ∗ n − b m − n ) (B3)21ote that Re { X } = Im { iX } with X an arbitrary complex number. Here, a re-index isapplied with p = n − ∈ [0 , ∞ ] for the first and fourth term of Eq. (B3) F x = − ρ Φ Im ( ∞ X p =0 p − X m = − p − (cid:0) a mp +1 + s mp +1 (cid:1) s m +1 ∗ p b − m − p +1 + ∞ X n =0 n X m = − n ( a mn + s mn ) s m +1 ∗ n +1 b mn +1 + ∞ X n =0 n X m = − n − ( a mn + s mn ) s m − ∗ n +1 b − mn +1 + ∞ X p =0 p +1 X m = − p +1 − (cid:0) a mp +1 + s mp +1 (cid:1) s m − ∗ p b m − p +1 ) (B4)Now, using a re-index for m : for the first term q = m + 1 ∈ [ − p, p ], and for the fourthterm q = m − ∈ [ − p, p ], we have F x = − ρ Φ Im ( ∞ X p =0 p X q = − p (cid:0) a q − p +1 + s q − p +1 (cid:1) s q ∗ p b − qp +1 + ∞ X n =0 n X m = − n ( a mn + s mn ) s m +1 ∗ n +1 b mn +1 + ∞ X n =0 n X m = − n − ( a mn + s mn ) s m − ∗ n +1 b − mn +1 + ∞ X p =0 p X q = − p − (cid:0) a q +1 p +1 + s q +1 p +1 (cid:1) s q ∗ p b qp +1 ) = 14 ρ Φ Im ( ∞ X n =0 n X m = − n h b − mn +1 (cid:2) ( a mn + s mn ) s m − ∗ n +1 − (cid:0) a m − n +1 + s m − n +1 (cid:1) s m ∗ n (cid:3) + b mn +1 (cid:2) (cid:0) a m +1 n +1 + s m +1 n +1 (cid:1) s m ∗ n − ( a mn + s mn ) s m +1 ∗ n +1 (cid:3)i ) (B5)which is Eq. (12a) in Sec. V A 1.
2. Derivation of F y The expression of y -component of ARF is F y = 12 ρ k Φ Z Z S Re ( − ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( kr ) ( a mn + s mn ) s m ′ ∗ n ′ Y mn ( θ, ϕ ) Y m ′ ∗ n ′ ( θ, ϕ ) ) × r sin θ sin ϕ sin θdθdϕ = − ρ Φ Re ( ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ Z Z S Y mn Y m ′ ∗ n ′ sin θ sin ϕ sin θdθdϕ ) , (B6)The detailed derivation of F y is similar as that for the x -component F x by substituting Eq.(A7) replacing of (A6) into (B6), and using of the orthogonality relationship of normalized22pherical harmonics of Eq. (A1). The final expression of F y in terms of a mn and s mn is givenin Eq. (12b) in Sec. V A 1, which is not given here for brevity.
3. Detailed derivation of F z The expression of z -component of ARF is F z = 12 ρ k Φ Z Z S Re ( − ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( kr ) ( a mn + s mn ) s m ′ ∗ n ′ Y mn ( θ, ϕ ) Y m ′ ∗ n ′ ( θ, ϕ ) ) × r cos θ sin θdθdϕ = − ρ Φ Re ( ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ Z Z S Y mn Y m ′ ∗ n ′ cos θ sin θdθdϕ ) , (B7)Substituting Eq. (A2), we have F z = − ρ Φ Re ( ∞ X n =1 n − X m = − n +1 ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ Z Z S c mn Y mn − Y m ′ ∗ n ′ sin θdθdϕ + ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ Z Z S c mn +1 Y mn +1 Y m ′ ∗ n ′ sin θdθdϕ ) , (B8)For the definition of ( n, m ) in Eq. (1), it has n ∈ [0 , ∞ ] and m ∈ [ − n, n ]. Since Y mn − ( n ∈ [1 , ∞ ] and m ∈ [ − n + 1 , n − Y mn +1 ( n ∈ [ − , ∞ ] and m ∈ [ − n − , n + 1]) areintroduced here, the final regimes of indexes ( n, m ) are the intersection and given differentlyfor the first and second part.By using Eq. (A1), the expression of F z is F z = − ρ Φ Re ( ∞ X n =1 n − X m = − n +1 ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ c mn δ n − ,n ′ δ m,m ′ + ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n ′ − n ( a mn + s mn ) s m ′ ∗ n ′ c mn +1 δ n +1 ,n ′ δ m,m ′ ) = − ρ Φ Re ( ∞ X n =1 n − X m = − n +1 i − ( a mn + s mn ) s m ∗ n − c mn + ∞ X n =0 n X m = − n i ( a mn + s mn ) s m ∗ n +1 c mn +1 ) , (B9)23 ABLE II. Regime of ( n, m ) in normalized spherical harmonics for derivation of F z . Note thatbased on the definition in Eq. (1), we have n ∈ [0 , ∞ ] and m ∈ [ − n, n ]. n m Intersection Y m +1 n − n ∈ [1 , ∞ ] m ∈ [ − n, n − n ∈ [1 , ∞ ], m ∈ [ − n, n − Y m +1 n +1 n ∈ [ − , ∞ ] m ∈ [ − n − , n ] n ∈ [0 , ∞ ], m ∈ [ − n, n ] Re-index for the first part of Eq. (B9) using p = n − ∈ [0 , ∞ ], so that m ∈ [ − p, p ]. Thefinal form of F z in terms of a mn and s mn is F z = − ρ Φ Re ( ∞ X p =0 p X m = − p i − (cid:0) a mp +1 + s mp +1 (cid:1) s m ∗ p c mp +1 + ∞ X n =0 n X m = − n i ( a mn + s mn ) s m ∗ n +1 c mn +1 ) = − ρ Φ Re ( ∞ X n =0 n X m = − n i (cid:0) a m ∗ n +1 + s m ∗ n +1 (cid:1) s mn c mn +1 + ∞ X n =0 n X m = − n i ( a mn + s mn ) s m ∗ n +1 c mn +1 ) = 12 ρ Φ Im ( ∞ X n =0 n X m = − n c mn +1 h (cid:0) a m ∗ n +1 + s m ∗ n +1 (cid:1) s mn + ( a mn + s mn ) s m ∗ n +1 i) . (B10)which is Eq. (12c) in Sec. V A 1. Note that Re { X } =Re { X ∗ } and Re { iX } = − Im { X } . APPENDIX C: EQUIVALENCE OF EQ. (11) AND FORMULAS BY SAPOZH-NIKOV & BAILEY
By substituting Eq. (7) into (13), we can prove that the three components of ARFformulas for a sphere (with A mn = A n ) are equivalent to those in terms of H nm by Sapozhnikov& Bailey [see Eqs. (46-48) in Ref. ], respectively. The detailed derivations are given below.24ecall that for a sphere, one has C m ∓ n = C n . The x -component of ARF: F x = ρ Φ ( ∞ X n =0 n X m = − n C n (cid:0) b − mn +1 a mn a m − ∗ n +1 − b mn +1 a mn a m +1 ∗ n +1 (cid:1)) = 14 π ρ k c Im ( ∞ X n =0 n X m = − n C n (cid:2) b − mn +1 (cid:0) i n − H nm (cid:1)(cid:0) i n H n +1 ,m − (cid:1) ∗ − b mn +1 (cid:0) i n − H nm (cid:1)(cid:0) i n H n +1 ,m +1 (cid:1) ∗ (cid:3)) = 14 π ρ k c Im ( ∞ X n =0 n X m = − n iC n (cid:0) − b − mn +1 H nm H ∗ n +1 ,m − + b mn +1 H nm H ∗ n +1 ,m +1 (cid:1)) = 14 π ρ k c Re ( ∞ X n =0 n X m = − n C n (cid:0) − b − mn +1 H nm H ∗ n +1 ,m − + b mn +1 H nm H ∗ n +1 ,m +1 (cid:1)) (C1)Note that ω = kc with the sound speed in fluid c , and Im { iX } = Re { X } . By replacing − m with m for the first part, Eq.(C1) is further written as F x = 14 π ρ k c Re ( ∞ X n =0 n X m = − n C n b mn +1 (cid:0) − H n, − m H ∗ n +1 , − m − + H nm H ∗ n +1 ,m +1 (cid:1)) . (C2)which is Eq.(46) in Ref. .The y -component of ARF: F y = ρ Φ ( ∞ X n =0 n X m = − n C n (cid:0) b − mn +1 a mn a m − ∗ n +1 + b mn +1 a mn a m +1 ∗ n +1 (cid:1)) = 14 π ρ k c Re ( ∞ X n =0 n X m = − n C n (cid:2) b − mn +1 (cid:0) i n − H nm (cid:1)(cid:0) i n H n +1 ,m − (cid:1) ∗ + b mn +1 (cid:0) i n − H nm (cid:1)(cid:0) i n H n +1 ,m +1 (cid:1) ∗ (cid:3)) = 14 π ρ k c Re ( ∞ X n =0 n X m = − n iC n (cid:0) − b − mn +1 H nm H ∗ n +1 ,m − − b mn +1 H nm H ∗ n +1 ,m +1 (cid:1)) = 14 π ρ k c Im ( ∞ X n =0 n X m = − n C n (cid:0) b − mn +1 H nm H ∗ n +1 ,m − + b mn +1 H nm H ∗ n +1 ,m +1 (cid:1)) (C3)As similar as the derivation for F x , taking C m ∓ n = C n for a sphere and replacing − m with m , F y can be also written as F y = 14 π ρ k c Im ( ∞ X n =0 n X m = − n C n b mn +1 (cid:0) H n, − m H ∗ n +1 , − m − + H nm H ∗ n +1 ,m +1 (cid:1)) . (C4)which is Eq. (47) in Ref. . 25he z -component of ARF: F z = ρ Φ ( ∞ X n =0 n X m = − n C mn c mn +1 a mn a m ∗ n +1 ) = 12 π ρ k c Im ( ∞ X n =0 n X m = − n C mn c mn +1 (cid:0) i n − H nm (cid:1)(cid:0) i n H n +1 ,m (cid:1) ∗ ) = 12 π ρ k c Im ( ∞ X n =0 n X m = − n ( − i ) C mn c mn +1 H nm H ∗ n +1 ,m ) = − π ρ k c Re ( ∞ X n =0 n X m = − n C mn c mn +1 H nm H ∗ n +1 ,m ) (C5)which is Eq. (48) in Ref. by replacing C mn with C n for a sphere. APPENDIX D: ANGULAR MOMENTUM AND LADDER OPERATORS
The ladder operators L ± has the relationship with the lateral components of the angularmomentum operator L x,y : L ± = L x ± iL y . The recursion relations of ladder operators L ± (or axial component of angular momentum operator L z ) and normalized spherical harmonicsare L + Y mn = b − mn Y m +1 n , (D1a) L − Y mn = b mn Y m − n , (D1b) L z Y mn = mY mn . (D1c)with b mn = p ( n + m )( n − m + 1). 26 PPENDIX E: DETAILED DERIVATION OF ART WITH CORRECT INDEX1. Detailed derivation of T x Based on the ART formulas of Eq. (7) from Ref. , the expression of x -component ofART is T x = − ρ Φ k Z Z S Re ( ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n − n ′ ( a m ∗ n + s m ∗ n ) s m ′ n ′ Y m ∗ n L x Y m ′ n ′ sin θdθdϕ ) (E1)With insertion of Eqs. (D1a) and (D1b) into (E1) and since L x = ( L + + L − ) / T x = − ρ Φ k Re ( ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n − n ′ ( a m ∗ n + s m ∗ n ) s m ′ n ′ Z Z S Y m ∗ n ( L + + L − ) Y m ′ n ′ sin θdθdϕ ) = − ρ Φ k Re ( ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ − X m ′ = − n ′ i n − n ′ ( a m ∗ n + s m ∗ n ) s m ′ n ′ Z Z S Y m ∗ n ¯ b − m ′ n ′ Y m ′ +1 n ′ sin θdθdϕ + ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ +1 i n − n ′ ( a m ∗ n + s m ∗ n ) s m ′ n ′ Z Z S Y m ∗ n ¯ b m ′ n ′ Y m ′ − n ′ sin θdθdϕ ) (E2)The regime of ( n ′ , m ′ , )in the summation symbol is listed in Table III. TABLE III. Regime of ( n ′ , m ′ ) in normalized spherical harmonics for derivation of F x and F y . Notethat based on the definition in Eq. (1), we have n ′ ∈ [0 , ∞ ] and m ′ ∈ [ − n ′ , n ′ ]. n ′ m ′ Intersection Y m ′ +1 n ′ n ′ ∈ [0 , ∞ ] m ′ ∈ [ − n ′ − , n ′ − n ′ ∈ [0 , ∞ ], m ′ ∈ [ − n ′ , n ′ − Y m ′ − n ′ n ′ ∈ [0 , ∞ ] m ′ ∈ [ − n ′ + 1 , n ′ + 1] n ′ ∈ [0 , ∞ ], m ′ ∈ [ − n ′ + 1 , n ′ ] T x is T x = − ρ Φ k Re ( ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ − X m ′ = − n ′ i n − n ′ ( a m ∗ n + s m ∗ n ) s m ′ n ′ ¯ b − m ′ n ′ δ nn ′ δ m,m ′ +1 + ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ +1 i n − n ′ ( a m ∗ n + s m ∗ n ) s m ′ n ′ ¯ b m ′ n ′ δ nn ′ δ m,m ′ − ) = − ρ Φ k Re ( ∞ X n =0 n X m = − n +1 ( a m ∗ n + s m ∗ n ) s m − n ¯ b − m +1 n + ∞ X n =0 n − X m = − n ( a m ∗ n + s m ∗ n ) s m +1 n ¯ b m +1 n ) (E3)A re-index is necessary for the second part of Eq. (E3) by using q = m + 1 ∈ [ − n + 1 , n ],and note that ¯ b − m +1 n = ¯ b mn , we have T x = − ρ Φ k Re ( ∞ X n =0 n X m = − n +1 ( a m ∗ n + s m ∗ n ) s m − n ¯ b − m +1 n + ∞ X n =0 n X q = − n +1 (cid:0) a q − ∗ n + s q − ∗ n (cid:1) s qn ¯ b qn ) = − ρ Φ k Re ( ∞ X n =0 n X m = − n +1 ¯ b mn h ( a m ∗ n + s m ∗ n ) s m − n + (cid:0) a m − ∗ n + s m − ∗ n (cid:1) s mn i) (E4)which is Eq. (14a) in Sec. VI.
2. Derivation of T y The expression of y -component of ART is T y = − ρ Φ k Z Z S Re ( ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n − n ′ ( a m ∗ n + s m ∗ n ) s m ′ n ′ Y m ∗ n L y Y m ′ n ′ sin θdθdϕ ) (E5)As similar as the derivation for T x , the final expression of T y in terms of a mn and s mn can beobtained by using Eqs. (D1a) and (D1b) into (E5) and L y = ( L + − L − ) / i instaed of L x ,as given in Eq. (14b) and omitted here for brevity.28 . Detailed derivation of T z The expression of z -component of ART is T z = − ρ Φ k Z Z S Re ( ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n − n ′ ( a m ∗ n + s m ∗ n ) s m ′ n ′ Y m ∗ n L z Y m ′ n ′ sin θdθdϕ ) (E6)Insertion of Eq. (D1c) into (E6), we have T z = − ρ Φ k Re ( ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n − n ′ ( a m ∗ n + s m ∗ n ) s m ′ n ′ Z Z S Y m ∗ n mY m ′ n ′ sin θdθdϕ ) (E7)By substituting Eq. (A1) into (E7), the final expression of T z in terms of a mn and s mn can bederived as T z = − ρ Φ k Re ( ∞ X n =0 n X m = − n ∞ X n ′ =0 n ′ X m ′ = − n ′ i n − n ′ ( a m ∗ n + s m ∗ n ) s m ′ n ′ mδ nn ′ δ mm ′ ) = − ρ Φ k Re ( ∞ X n =0 n X m = − n m ( a m ∗ n + s m ∗ n ) s mn ) (E8)which is Eq. (14c) in Sec. VI. APPENDIX F: EQUIVALENCE OF EQ. (14) AND FORMULAS BY GONG &BAUDOIN
By substituting Eq. (7) into (15), we can prove that the three components of ARTformulas are equivalent to those in terms of H nm by Gong & Baudoin [see Eqs. (10-12) inRef. ], respectively. The detailed derivations are given below.29he x -component of ART: T x = − ρ Φ k Re ( ∞ X n =0 n X m = − n +1 b mn C mn a m ∗ n a m − n ) = − π ρ k c Re ( ∞ X n =0 n X m = − n +1 b mn C mn (cid:0) i n − H nm (cid:1) ∗ (cid:0) i n − H n,m − (cid:1)) = − π ρ k c Re ( ∞ X n =0 n X m = − n +1 b mn C mn H ∗ nm H n,m − ) (F1)which is Eq. (10) in Ref. .The y -component of ART: T y = − ρ Φ k Im ( ∞ X n =0 n X m = − n +1 b mn C mn a m ∗ n a m − n ) = − π ρ k c Im ( ∞ X n =0 n X m = − n +1 b mn C mn (cid:0) i n − H nm (cid:1) ∗ (cid:0) i n − H n,m − (cid:1)) = − π ρ k c Im ( ∞ X n =0 n X m = − n +1 b mn C mn H ∗ nm H n,m − ) (F2)which is Eq. 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