arXiv:1210.2116v3 [.class-ph] 15 Feb 2014 clDvcs cutc esr ehooyTa,M D429, 87545. MS USA Team, NM, Technology Sensors Alamos, & Elect Devices, & ical Sensors MPA-11, Division, Applications and Physics diinTerm igeba cutclTweezer. Acoustical Single-beam Theorem, spherical Addition a trap regime. may scattering beam Rayleigh regimes. focused the scattering the in resonant that particle the shown and is Rayleigh It carrie the is case analysis in The we the out fluid. compressible the which is in absorbing placed for an arbitrarly This on sphere beam, force available. radiation are the focused compute BSCs spherically pressure a its incident of but useful the known, of particularly the expression to is is closed-form relative approach coefficients no new when Such theorem the position. addition obtain translational to sphere’s the used Given BSCs, be hand, surface. known can other sphere’s of acoustic set the the the a across On from conditions center. obtained which are boundary sphere’s frame in coefficients reference the scattering the mode by the of multipole Moreover, defined choice a the beam. is on incident of depend the weight they of complex expansion beam-shape the partial-wave the of is coefficien terms fluid. beam-shape (BSC) Each in inviscid coefficients. arbitrary given scattering an the is in and an force suspended by sphere radiation a The produced on force beam shaped radiation calculat acoustic exactly to employed the is functions spherical for rem aeFdrld lga,Mcio L50290 rzl Ema Brazil. 57072-900, Un F´ısica, AL Macei´o, de Alagoas, Instituto [email protected] de Group, Federal Acoustics dade Physical with are acoustical Single-beam [4]. phased-arrays employ- by circular and [3] ing devices tweezers lab-on-a-chip Two-dimensional in achieved [2]. used are cells of is and trapping wave particles and small standing separation, the manipulation, ultrasound acoustophoresis, contactless a In in of [1]. tweez- Wu force acoustical by radiation of introduced concept was the ers after trigged been has nashr sn h rnltoa diintheorem addition translational the using sphere a on ai .Mtii ihLsAao ainlLbrtr,Mater Laboratory, National Alamos Los with is Mitri G. Farid lue .Sla nreL ago n .Hniu Lopes Henrique J. and Baggio, L. Andr´e Silva, T. Glauber xc opttoso h cutcrdainforce radiation acoustic the of computations Exact ne Terms Index nicesn neeto lrsudrdainforce radiation ultrasound on interest increasing An Abstract I hsppr h rnltoa diintheo- addition translational the paper, this —In lue .Silva, T. Glauber Aosi aito oc,Translational Force, Radiation —Acoustic .I I. NTRODUCTION sehttp://www.if.ufal.br/ (see ebr IEEE Member, ∼ gaf) Mitri, nreL ago .Hniu oe,adFrdG. Farid and Lopes, Henrique J. Baggio, L. Andr´e , rochem- iversi- ebr IEEE Member, Los ials il: d e t ocmueBC ntecneto pia scattering optical of context the methods in quadrature BSCs other compute of transform to description Fourier harmonics discrete A the on spherical transform. based is which discrete [26], [25], the (DSHT) [23], and rule integral midpoint to [24] with the schemes include host BSC Numerical BSC the the the position. compute particle’s calculating in the by to located obtained respect arbitrarly be can sphere medium a on transducer scatter. to the of related incident properties is the mechanical coefficients the of scattering the geometry Similarly, carry the beam. BSCs boundary to The regarding acoustic surface. information sphere’s the the from across scattering conditions obtained the while are partial-wave, am- coefficients incident complex an [23]. the of is coefficients plitude (BSC) scattering coefficient the beam-shape and Each radiation terms beam-shape the in the expressed is method, of sphere suspended this a on In partial-wave exerted force the [22]. using method computed expansion be can force radiation n h eoat( resonant the and ais o te iissc steRyeg ( Rayleigh the as such limits other For radius. ofr hsaayi a enol efre o the for which performed for only [21], been regime point. has scattering focus analysis transducer’s geometric this sphere the the far, of on So vicinity force the radiation in how behaves of analysis an quires focus the in the placed beams, sphere focused the is point. For to sphere beam. restricted the the a was of studies, analysis and axis these [19], the all in [18], In located focused [17], beam. spherically [16], [20] a beams Gaussian [14], by Bessel [13], sphere [15], a beam beam on piston exerted planar force the a scattered radiation on the axial Based and last the [12]. incident waves, [11], the the [10], of over [9], expansion partial-wave [8], investigated nonviscous [7], extensively a [6], in century been sphere has suspended fluid a tightly on wave a spherical using by developed [5]. been beam focused also has tweezer where h aito oc rdcdb peial focused spherically a by produced force radiation The h eino igeba cutcltezr re- tweezers acoustical single-beam of design The a or plane a by exerted force radiation acoustic The k steicdn aeubrand wavenumber incident the is ka ∼ 1 cteigrgms the regimes, scattering ) a stesphere’s the is ka ka ≫ ≪ 1 1 1 ) , 2 is provided in Ref. [27]. Both methods require that the (see Fig. 1) The origin of the coordinate system O is set incident pressure amplitude should be sampled over a at the transducer’s focus. When the center of the sphere virtual a sphere which encloses the beam propagation coincides with O, the scattering of the incident beam is region, which contains the spherical target. For highly referred to as the on-focus scattering configuration [30]. oscillating functions, the midpoint rule requires a large number of sampling points to ensure proper converge In the on-focus scattering formalism, the normalized of the BSC computation. Yet the DSHT renders more amplitude of the incident pressure beam can be described accurate results with less sampling points, it may develop in spherical coordinates r = rer(θ, ϕ), where er is the numerical errors related to aliasing due to undersampling radial unit-vector, θ and ϕ are the polar and the azimuthal and spectral leakage caused by function domain trunca- angles, respectively. The incident pressure partial-wave tion. expansion is given by [31] In order to circumvent numerical approximations, m m we propose a method to exactly calculate the acous- pi = an jn(kr)Yn (θ, ϕ), (1) n,m tic radiation force based on the partial-wave expan- X sion method [22], [26] and the translational addition ∞ m m where n,m = n=0 n=−m, an are the BSCs to theorem [28]. A similar method has been divised to be determined, k = ω/c0, jn is the nth-order spherical calculated the radiation pressure generated by an elec- P P mP Bessel function, and Yn is the spherical harmonic func- tromagnetic wave [29]. The proposed method is used to tion of nth-order and mth-degree. Note that the ampli- calculate the radiation force produced by a spherically tude is normalized to the pressure magnitude p0. Using focused transducer on a silicone-oil droplet. The incident the orthogonality properties of the spherical harmonics, beam is generated considering the parameters of a typical the BSCs are obtained from (1) as biomedical focused transducer with an F-number of 1.6 1 and a driving frequency of 3.1 MHz. Using closed-form am = p (kR,θ,ϕ)Y m∗(θ, ϕ)dΩ, (2) n j (kR) i n expressions of the BSCs with respect to the beam focal n ZΩ point [30], the radiation force is computed along the where R is the radius of a virtual spherical region where beam’s axis and on the transducer’s focal plane. Both the beam propagates, dΩ is the differential solid angle, Rayleigh and resonant scattering regimes are considered and Ω= (θ, ϕ) : 0 θ π, 0 ϕ 2π . { ≤ ≤ ≤ ≤ } in this analysis. The results obtained for the Rayleigh regime are compared to those computed from Gorkov’s Suppose now that the sphere is translated to a new theory [10]. A significant deviation from Gorkov’s theory point denoted by a vector d as shown in Fig. 1. This is noted in the axial radiation force when ultrasound corresponds to the off-focus scattering by the sphere. absorption inside the droplet is taken into account. In In spherical coordinates, the translational vector d is addition, transverse trapping is achieved in both Rayleigh represented with respect to the system O by (d, θd, ϕd). and resonant scattering regimes. Nevertheless, simultane- The sphere’s center defines a new coordinate system ous axial and transverse trapping only occurs for droplets denoted by O′. The incident beam can be described in a in the Rayleigh scattering regime. new spherical coordinate system (r′,θ′, ϕ′) as

µ′ ′ µ ′ ′ II. PHYSICALMODEL pˆi = aν jν (kr )Yν (θ , ϕ ), (3) ν,µ Consider an acoustic beam of arbitrary wavefront X µ′ with angular frequency ω that propagates in an inviscid where aν are called the translational BSCs. A similar infinite fluid. The fluid has ambient density ρ0 and expression to (2) can be obtained for the translational speed of c0. The acoustic beam is described by BSCs by using the orthogonal property of the spherical the excess of pressure p as a function of the position harmonics. vector r, with respect to a defined coordinate system. −iωt The time-dependence e is suppressed for the sake of The relation between the position vectors of the O and simplicity. An spherical scatterer with radius a, density O′ coordinate systems is r = r′ + d. The translational ρ1, and c1 is placed in the beam path. addition theorem between the wave functions in the systems O and O′ states that [28] A. Scattering problem m mµ d ′ µ ′ ′ jn(kr)Yn (θ, ϕ)= Snν (k )jν (kr )Yν (θ , ϕ ), A spherically focused transducer with diameter 2b and ν,µ X curvature radius z0 is used to produce a focused beam (4) 3

the aforementioned boundary conditions, one obtains m m  sn = snan , (8) where

! ! ) # γjn(ka) jn(k1a) sn = det ′ ′ − jn(ka) jn(k1a) "   (1) −1   γhn (ka) jn(k1a) det ′ , (9) (1) ′ × hn (ka) j (k1a) !"#'*$+(,-#*"!'( " n # with γ = ρ0k1/(ρ1k). The prime symbol indicates dif- !" #" ferentiation with respect to the argument. According to (8) the scattering coefficient depends on the BSCs with "  !"#$#%&'(' respect to the system O and on the sphere’s mechanical parameters sn, which is independent of the choice of Fig. 1. Sketch of the acoustic scattering by a sphere placed anywhere the coordinate system. Therefore, in the system O′ the ′ in the host medium. The systems O and O correspond to the on- scattering coefficient becomes and off-focus scattering configurations. m′ m′ sn = snan . (10) Ultrasound absorption inside the sphere is considered where the separation matrix is given by by adding an imaginary part to the wavenumber as follows [33] n+ν ω Sµm(kd) = 4π( 1)m iq+n−ν (ν,µ; n, m; q) k1 = + iα0, (11) νn − G − c1 q=|n−ν| X where α0 is the absorption coefficient. It is remarked that j (kd)Y µ−m(θ , ϕ ), (5) × q q d d shear wave propagation inside the sphere is neglected. with being the Gaunt coefficient [28]. This coefficient G is zero if any of the following conditions happens n + ν + q is odd, q>ν + n, or q < n ν . | − | B. Beam-shape coefficients of a focused beam Now, substituting (4) into (1) and use the result in (2) one obtains the translational BSCs as The amplitude of the pressure generated by a trans- µ′ m µm d aν = an Sνn (k ), (6) ducer is given in terms of the Rayleigh integral by [34] n,m ik|r−r′| X ikρ0c0v0 e ′ pi = dS , (12) where ν = 0, 1,..., and ν µ ν. Note that each − 2π r r′ translational BSC is∞ a combination− ≤ of≤ all possible BSC ZS | − | with respect to the system O. where S is the surface of the transducer’s active ele- ment and v0 is the uniform velocity distribution on S. Now, the scattered pressure is considered. Assume Physically, the Rayleigh integral expresses the Huygens’ that the sphere is located at the transducer’s focus principle in which the pressure at a position r is the (i.e. on-focus scattering configuration). The amplitude sum of wavelets generated on the surface S. Hereafter, of the scattered pressure is described by the following the pressure amplitude is normalized to p0 = ρ0c0v0. expansion [31] To derive the on-focus BSCs, the origin of the coor- m (1) m ps = sn hn (kr)Yn (θ, ϕ), (7) dinate system is centered at the transducer focal point. ′ n,m By assuming that r z , with r = z , the following X 0 0 expansion is used [35]≪ k k m (1) where sn is the scattering coefficient, hn is the nth- ′ ik|r−r | ikz0 order spherical Hankel function of first type. Considering e e n = 4π ( i) jn(kr) a compressional sphere, the scattering coefficient is de- r r′ z − | − | 0 n,m termined from the continuity condition for the pressure X Y m(θ′, ϕ′)Y m∗(θ, ϕ) (13) and the particle velocity across the sphere’s surface. × n n Further details are provided in Ref. [32]. Thus, using into (12). Thus, integrating in the angular variables 4

(θ′, ϕ′) yields the control surface r lies in the farfield, i.e. kr 1. ≫ ∞ Therefore, the radiation force is given by ikz0 n 4π pi = ikz e i − 0 2n + 1 f 2 S e n=0 r = r rdΩ, kr 1. (19) X − Ω · ≫ [P (cos α) P (cos α)]j (kr)Y 0(θ, ϕ), Z n+1 n−1 n n One can show that the Cartesian components of the × − (14) radiation force can be expressed as [22], [?] −1 where α = sin (b/r0) is the half-spread angle of the 2 f = πa E0(Yxex + Yyey + Yzez), (20) transducer and Pn is the Legendre polynomial of order . The on-focus BSCs are found by comparing Eqs. (1) 2 2 n where E0 = p0/(2ρ0c0) is the characteristic energy and (14). Accordingly, density of the incident wave, and ex, ey and ez are the Cartesian unit-vectors. The radiation force functions are m (n+1) ikz0 4π an = i kz0e given by − r2n + 1 [Pn+1(cos α) Pn−1(cos α)]δm,0. (15) i (n + m + 1)(n + m + 2) × − Y + iY = x y 2π(ka)2 (2n + 1)(2n + 3) The partial-wave expansion of the focused beam can n,m s be compared to results based on the paraxial approx- m mX+1∗ ∗ −m∗ −m−1 Sna a + S a a , (21) imation in (12). In adopting this approximation, it × n n+1 n n n+1 2 2 is assumed that b z0 . Let the radial distance in 1 (n m + 1)(n + m + 1) ≪ Yz = Im − cylindrical coordinates be given by ̺ = x2 + y2. The π(ka)2 (2n + 1)(2n + 3) n,m s pressure produced by the transducer in the focal plane X p S amam∗ , (22) is given by [36] × n n n+1 where ‘Im’ denotes the imaginary part, the symbol ∗ b ̺2 pi(̺, 0) = i exp ik + z0 J1(k̺ sin α), means complex conjugation, and ̺ z0      (16) ∗ ∗ Sn = sn + sn+1 + 2snsn+1. (23) where J1 is the first-order Bessel function. Along the transducer’s axis, we have It is worth to note that the radiation force functions 2 Yx,Yy, and Yz are real-valued quantities. iz0 ikb 1 1 pi(0, z)= 1 exp . z z0 − 2 z − z0 IV. RESULTS AND DISCUSSION −    (17) Consider an acoustic beam that is generated in water These equations will be compared to the result obtained for which c = 1500 m/s and ρ = 1000 kg/m3. by the partial-wave expansion given in (14). 0 0 The focused transducer has a radius b = 22mm, a F-number of 1.6, and operates at 3.1 MHz. Note that III. ACOUSTICRADIATIONFORCE 2 (b/z0) = 0.1, which ensures the paraxial approximation After calculating both the incident and scattered for the incident beam. The magnitude of the pressure acoustic fields through the partial-wave expansion 5 generated by the transducer is p0 = ρ0c0v0 = 10 Pa. method, the radiation force can be calculated by integrat- A droplet made out of silicone-oil (c1 = 974 m/s, ing the radiation stress tensor S over the scatter’s sur- 3 ρ1 = 1004 kg/m , and α0 = 21 Np/m at 3.1 MHz) is face. In second-order approximation, the time-averaged used as the target object. radiation stress tensor is given by [9] The truncation error in computing the translational 2 2 BSCs in (6) is analyzed in Appendix A. Accordingly, ρ0v p S = ρ vv I, (18) the truncation order for the index n necessary to achieve 0 − 2 − 2ρ c2 0 0 ! a certain error ǫ is given by where the overbar denotes time-average, ρ vv is the 0 N = ν + kd + 1.8(log ǫ−1)2/3(kd)1/3. (24) Reynolds’ stress tensor, v = v , and I is the 3 3- k k × unit matrix. Considering a nonviscous fluid, the radiation The error considered in the translational BSC computa- stress tensor is a zero divergent quantity, i.e. S = 0. tions is ǫ = 10−6. The truncation order L of the radiation Thus, by using the Gauss divergence theorem∇ one· can force series in (22) is established by the scattering show that the radiation force is given by integrating coefficient ratio sm/s0 < 10−7. | L 0| the radiation stress tensor on a control spherical surface In Fig. 2, the pressure amplitude (normalized to p0) centered at the target object. Furthermore, the radius of produced by the focused transducer is shown along both 5 x and z directions. The pressure is computed using the Figure 3 exhibits the axial radiation force versus partial-wave expansion method and the paraxial approxi- the droplet’s position along the transducer’s axis (z mation based on (16). Good agreement is found between direction). The droplet’s size factors are ka = 0.1 the methods in the transverse direction and in the vicinity (Rayleigh regime) and ka = 0.5 (resonant regime). The of the focal region. Though moving away from the focal solid line represents the radiation force computed with region, the partial-wave expansion method deviates from Gorkov’s method. Good agreement between the methods the paraxial approximation result. This happens because is found when no ultrasound absorption is considered the partial-wave expansion is valid in the vicinity of the in the droplet. However, when ultrasound absorption is focal region, i.e. r z0. taking into account, a significant deviation between these ≪ methods is observed. This result is expected since the acoustic radiation force caused by a plane traveling wave on a sphere depends on the sum of the scattered and 40 (a) paraxial PWE absorbed power [9]. The Gorkov’s theory does not take 20 into account absorption inside the particle. Therefore, if absorption is considered, the radiation force will be 0 norm. pressure larger than in the case where absorption is neglected [37], −2 −1 0 1 2 [38]. The droplet with ka = 0.1 is axially trapped along x−axis [mm] the transducer axis at z = 2.5mm (with absorption). Note that a nonabsorptive droplet would be trapped at (b) 40 paraxial z = 0. Gorkov’s method no longer describes the behavior PWE of the radiation force generated by the focused transducer 20 when ka = 0.5. Furthermore, it is not possible to axially trap the droplet with the analyzed transducer at this size norm. pressure 0 factor. This can be understood as follows. The radiation −20 −10 0 10 20 z−axis [mm] force exerted on a small particle by the spherically focused beam is formed by two contributions [10]: “the Fig. 2. Pressure amplitude generated by the spherically focused scattering force” caused mostly by the traveling wave transducer with aperture 44 mm and F-number of 1.6, operating at part of the beam, while “the gradient force” is due to 3.1 MHz . The pressure is evaluated along (a) the transverse and (b) the spatial variation of the potential and kinetic energy the axial directions using the partial-wave expansion (PWE) method and the paraxial approximation. densities of the beam. As the size factor increases, so does the scattering force. When this force overcomes the gradient force, the axial trapping is no longer possible. This might be prevented using tightly focused beams [5]. (a) ka = 0.1 The transverse radiation force versus the droplet’s 2 Gorkov position along x direction in the focal plane is shown in PWE 1 PWE (absorption) Fig. 4. Excellent agreement is found between the partial- wave expansion and Gorkov’s methods when ka = 0.1. 0 Some deviation between the methods arises when ka =

axial force [nN] 0.5. Note that the results with and without absorption −1 −10 −5 0 5 10 are very alike. The transverse trapping happens for both (b) ka = 0.5 ka = 0.1 and 0.5, because no scattering radiation force

1500 Gorkov is present in the transverse direction. Only the gradient PWE radiation force appears in this direction. Based on Figs. 3 1000 PWE (absorption) and 4 we conclude that the focused transducer forms 500 a 3D acoustical tweezer for silicone-oil droplets in the

axial force [nN] 0 Rayleigh scattering regime.

−10 −5 0 5 10 The axial radiation force versus the droplet’s position z-axis [mm] along z direction for the resonant scattering regime (ka = 1 and 5) is shown in Fig. 5. In this case, the Fig. 3. Axial radiation force versus the droplet’s position along z direction in the focal plane. The force is computed through the partial- scattering force totally overcomes the gradient force. wave expansion (PWE) and Gorkov’s methods. The size factors of Thereby, no trapping is possible in the axial direction the sphere are (a) ka = 0.1 and (b) ka = 0.5. with the analyzed transducer. The axial radiation force 6

(a) ka = 0.1 (a) ka = 1

N] 20 Gorkov µ 10 PWE PWE PWE (absorption) 10 PWE (absorption) 0 0 −10 −10

transverse force [nN] −15 −10 −5 0 5 10 15 transverse force [ −15 −10 −5 0 5 10 15 (b) ka = 0.5 (b) ka = 5

2000 N] Gorkov µ 400 PWE 1000 PWE 200 PWE (absorption) PWE (absorption) 0 0 −200 −1000 −400

transverse force [nN] −15 −10 −5 0 5 10 15 transverse force [ −15 −10 −5 0 5 10 15 x - axis [mm] x - axis [mm]

Fig. 4. Transverse radiation force versus the droplet’s position along Fig. 6. Transverse radiation force on a silicone oil droplet (with and x direction in the focal plane. The force is computed through the without attenuation) in the resonant scattering regime (a) ka = 1 and partial-wave expansion and Gorkov’s methods. The size factors are (b) ka = 5. (a) ka = 0.1 and (b) ka = 0.5.

(a) ka = 1 axial radiation force exerted on the droplet. If the droplet 150 N] PWE lies in the circular region with radius of 0.5mm around µ 100 PWE (absorption) the focus point, it will be attracted and trapped along 50 the transducer’s axis. The droplet is transversely trapped by a force of about 10 µN (see Fig. 4), but it will be 0 further pushed axially by a force of 60 µN. Therefore, the axial force−50 [ −10 −5 0 5 10 focused transducer operates as a 2D acoustical tweezer (b) ka = 5 for droplets in the resonant scattering regime (ka = 1).

N] PWE µ 2000 PWE (absorption)

1000 1 60

axial force [ 0 50 −10 −5 0 5 10 0.5 z - axis [mm] 40 N

Fig. 5. Axial radiation force versus the droplet’s position along 0 µ z direction for the resonant scattering regime (a) ka = 1 and (b) 30 ka = 5. -axis [mm] y −0.5 20

10 pushes the droplet to the forward scattering direction. −1 0 In Fig. 6, the transverse radiation force versus the −1 −0.5 0 0.5 1 droplet’s position along x direction is displayed for the x -axis [mm] resonant scattering regime (ka = 1 and 5). It is clearly shown that the transverse trapping is still possible in Fig. 7. Vector field of the radiation force in the transducer focal plane this scattering regime. This happens because the incident produced on the silicone oil droplet in the resoant regime ka = 1. beam is tightly focused in the transverse direction. More- The vector field is plot on top of the axial radiation force. over, the transverse radiation force remains practically the same regardless of ultrasound absorption within the droplet. V. SUMMARY AND CONCLUSIONS The vector field of the transverse radiation force on the A method to compute the axial and the transverse silicone-oil droplet placed in the transducer focal plane is acoustic radiation force on a sphere based on the transla- displayed in Fig. 7. The background map represents the tional addition theorem for spherical wave functions was 7 presented. The method relies on the fact that once the The leading term of the separation matrix in (5) is BSCs are known with respect to a coordinate system, related to the spherical Bessel function of smallest order, they can be calculated in a new translated coordinate i.e. q = n ν . Therefore, the truncation error can be system. The radiation force generated by an ultrasound approximated| − to| focused beam in the paraxial approximation and exerted ǫ (4π)3/2√2N 2ν + 3 (ν,µ; N + 1, 0; N ν + 1) on a silicone-oil droplet placed anywhere in the host ≃ − G µ − medium was calculated using the proposed method. Both j N−ν+1(kd)YN−ν+1(θd, ϕd) . × (27) axial and transverse radiation forces were computed in the Rayleigh and resonant scattering regimes. In the After integrating over (θd, ϕd), we obtain Rayleigh regime, the obtained results were compared to Gorkov’s radiation force theory. Good agreement was ǫ jN−ν+1(kd) . (28) ≃ | | found between these methods when ultrasound absorp- When the order of the spherical Bessel function is larger tion inside the droplet was neglected. Nevertheless, a than its argument, the error can be approximated to [39] significant deviation between the results takes place in the axial radiation force when the ultrasound absorption 1 1 3 ǫ exp f(x,L) L + is considered. It was showm that the transducer under ≃ 2 xf(x,L) − 2 s    investigation, with driving frequency of 3.1 MHz and an 3 f(x,L) F-number of 1.6, can operate as a single-beam acoustical ln L + + , (29) 2 x tweezer in 3D for the Rayleigh scattering regime. In   contrast, only transverse trapping of the silicone-oil where L = N ν + 1, x = kd, and f( x,L) = − droplet was achieved for the resonant scattering regime. (L + 3/2)2 x2. We define L+3/2= x(1+δ). Since In conclusion, the translational addition theorem of the spherical Bessel− function decreases rapidly as the p spherical wave functions was used in combination with order becomes larger than the argument, the parameter δ the partial-wave expansion to compute the axial and the is assumed to be much smaller than the unit. Therefore, transverse acoustic radiation force on a sphere. This the error can be expressed as method may become a useful tool in the design and 3/2 ǫ (2δ)−1/4e−x(2δ) /3 (30) evaluation of acoustical tweezers in particle manipulation ≃ applications. The second term in this equation is much smaller than the first and then dominates the error. Thus, taking the ACKNOWLEDGEMENTS logarithm on both sides of this equation, we can estimate the truncation order in (6) as This work was supported by grants CAPES 2163/2009-AUX-PE-PNPD, CNPq 306697/2010-6, N = ν + kd + 1.8(log ǫ−1)2/3(kd)1/3. (31) and CNPq 481284/2012-5 (Brazilian agencies). The term log ǫ−1 is closely related to the number of precision digits, which is given by the nearest integer APPENDIX to (log ǫ−1 + 1.0 log 2). − Assume the series in (6) is truncated in n = N. Thus, the truncation error is given by REFERENCES ∞ n m µm d [1] J. Wu, “Acoustical tweezers,” J. Acoust. Soc. Am., vol. 89, pp. ǫ = an Sνn (k ) . (25) 2140–2143, 1991.

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