Optical theorem for acoustic non-diffracting beams and application to radiation force and torque

Acoustical and optical non-diffracting beams are potentially useful for manipulating particles and larger objects. An extended optical theorem for a non-diffracting beam was given recently in the context of acoustics. The theorem relates the extinction by an object to the scattering at the forward direction of the beam’s plane wave components. Here we use this theorem to examine the extinction cross section of a sphere centered on the axis of the beam, with a non-diffracting Bessel beam as an example. The results are applied to recover the axial radiation force and torque on the sphere by the Bessel beam. © 2013 Optical Society of America OCIS codes: (350.4855) Optical tweezers or optical manipulation; (290.2200) Extinction; (290.4020) Mie theory; (290.5850) Scattering, particles. References and links 1. J. Durnin, “Exact solutions for nondiffraction beams. I. The scalar theory,” J. Opt. Soc. Am 4, 651–654 (1987). 2. P. L. Marston, “Scattering of a Bessel beam by a sphere,” J. Acoust. Soc. Am 121, 753 (2007). 3. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000). 4. A. Belafhal, A. Chafiq, and Z. Hricha, “Scattering of Mathieu beams by a rigid sphere,” Opt. Commun. 284, 3030–3035 (2003). 5. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33(15), 1678–1680 (2008). 6. P. L. Marston, “Axial radiation force of a Bessel beam on a sphere and direction reversal of the force,” J. Acoust. Soc. Am 120, 3518–3524 (2006). 7. P. L. Marston, “Negative axial radiation forces on solid spheres and shells in a Bessel beam,” J. Acoust. Soc. Am 122, 3162–3165 (2007). 8. P. L. Marston, “Radiation force of a helicoidal Bessel beam on a sphere,” J. Acoust. Soc. Am 125, 3539–3547 (2009). 9. F. G. Mitri, “Negative axial radiation force on a fluid and elastic spheres illuminated by a high-order Bessel beam of progressive waves,” J. Phys. A: Math. Theor. 42, 245202 (2009). 10. L. K. Zhang and P. L. Marston, “Geometrical interpretation of negative radiation forces of acoustical Bessel beams on spheres,” Phys. Rev. E 84, 035601(R) (2011). 11. J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011). 12. A. Novitsky, C.-W. Qiu, and H. F. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107, 203601 (2011). 13. N. Wang, J. Chen, S. Y. Liu, and Z. F. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013). 14. L. K. Zhang and P. L. Marston, “Axial radiation force exerted by general non-diffracting beams,” J. Acoust. Soc. Am. 131, EL329–EL335 (2012). #192413 $15.00 USD Received 17 Jun 2013; revised 1 Aug 2013; accepted 2 Aug 2013; published 9 Aug 2013 (C) 2013 OSA 1 September 2013 | Vol. 4, No. 9 | DOI:10.1364/BOE.4.001610 | BIOMEDICAL OPTICS EXPRESS 1610 15. S. Sukhov and A. Dogariu, “Negative nonconservative forces: Optical tractor beams for arbitrary objects,” Phys. Rev. Lett. 107, 203602 (2011). 16. O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, and P. Zemánek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ’tractor beam’,” Nat. Photonics 7, 123–127 (2013). 17. P. L. Marston and L. K. Zhang, “Radiation torques and forces in scattering from spheres and acoustical analogues,” in Optical Trapping Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), p. paper OMB5. 18. Y. Choe, J. W. Kim, K. K. Shung, and E. S. Kim, “Microparticle trapping in an ultrasonic Bessel beam,” Appl. Phys. Lett. 99, 233704 (2011). 19. S. Xu, C. Qiu, and Z. Liu, “Transversally stable acoustic pulling force produced by two crossed plane waves,” Europhys. Lett. 99, 44003 (2012). 20. D. Baresch, J.-L. Thomas, and R. Marchiano, “Three-dimensional acoustic radiation force on an arbitrarily located elastic sphere,” J. Acoust. Soc. Am. 133, 25–36 (2013). 21. C. R. P. Courtney, B. W. Drinkwater, C. E. M. Demore, S. Cochran, A. Grinenko, and P. D. Wilcox, “Dexterous manipulation of microparticles using Bessel-function acoustic pressure fields,” Appl. Phys. Lett. 102, 123508 (2013). 22. D. Baresch, J.-L. Thomas, and R. Marchiano, “Spherical vortex beams of high radial degree for enhanced singlebeam tweezers,” J. Appl. Phys. 113, 184901 (2013). 23. B. T. Hefner and P. L. Marston, “An acoustical helicoidal wave transducer with applications for the alignment of ultrasonic and underwater systems,” J. Acoust. Soc. Am. 106, 3313 (1999). 24. L. K. Zhang and P. L. Marston, “Angular momentum flux of nonparaxial acoustic vortex beams and torques on axisymmetric objects,” Phys. Rev. E 84, 065601(R) (2011). 25. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992). 26. P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly polarized electromagnetic wave,” Phys. Rev. A 30, 2508 – 2516 (1984). 27. C. E. M. Demore, Z. Yang, A. Volovick, S. Cochran, M. P. MacDonald, and G. C. Spalding, “Mechanical evidence of the orbital angular momentum to energy ratio of vortex beams,” Phys. Rev. Lett. 108, 194301 (2012). 28. R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976). 29. J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. A 12, 2708–2715 (1995). 30. P. L. Marston, “Generalized optical theorem for scatterers having inversion symmetry: applications to acoustic backscattering,” J. Acoust. Soc. Am. 109, 1291–1295 (2001). 31. F. G. Mitri, “Equivalence of expressions for the acoustic scattering of a progressive high-order Bessel beam by an elastic sphere,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 56, 1100–1103 (2009). 32. L. K. Zhang and P. L. Marston, “Radiation torque on a sphere centered on an acoustic helicoidal (vortex) Bessel beam.” J. Acoust. Soc. Am. 125, 2552–2552 (2009). 33. L. K. Zhang and P. L. Marston, “Radiation torque on solid spheres and drops centered on an acoustic helicoidal Bessel beam,” J. Acoust. Soc. Am. 129, 2381–2381 (2011). 34. L. K. Zhang and P. L. Marston, “Acoustic radiation torque and the conservation of angular momentum,” J. Acoust. Soc. Am. 129, 1679–1680 (2011). 35. F. G. Mitri, T. P. Lobo, and G. T. Silva, “Axial acoustic radiation torque of a Bessel vortex beam on spherical shells,” Phys. Rev. E 85, 026602 (2012). 36. G. T. Silva, T. P. Lobo, and F. G. Mitri, “Radiation torque produced by an arbitrary acoustic wave,” Europhys. Lett. 97, 54003 (2012). 37. G. T. Silva, “Off-axis scattering of an ultrasound Bessel beam by a sphere,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58, 298–304 (2011). 38. W. L. Nyborg, “Acoustic streaming,” in Nonlinear Acoustics, edited by M. F. Hamilton and D. T. Blackstock (Academic Press, CA, 1998), pp. Chap. 7, pp. 207–231. 39. T. Hasegawa and K. Yosioka, “Acoustic radiation force on fused silica spheres, and intensity determination,” J. Acoust. Soc. Am. 58, 581–585 (1975). 40. X. C. Chen and R. E. Apfel, “Radiation force on a spherical object in an axisymmetric wave field and its application to the calibration of high-frequency transducers,” J. Acoust. Soc. Am. 99, 713–724 (1996). 41. P. L. Marston, “Viscous contributions to low-frequency scattering, power absorption, radiation force, and radiation torque for spheres in acoustic beams,” Proceedings of Meetings on Acoustics (POMA) 19, 045005 (2013).


Introduction
An idealized non-diffracting (optical or acoustic) beam is a beam whose transverse intensity pattern has the feature of propagation-invariance [1][2][3][4][5].Beams which are locally approximately non-diffracting are potentially useful for particle manipulation.
One application is the possibility of generating pulling forces.Situations giving pulling forces for spheres in non-diffracting Bessel beams have been computed in acoustics [6][7][8][9][10] and in optics [11][12][13].It was noticed early [6][7][8] that acoustical situations predicting to give negative forces corresponded to a significant reduction of the far-field scattering into the backward hemisphere relative to the forward hemisphere.An analysis of momentum projection and conservation associated with optical far-field scattering [11] motivated an analogous analysis in the acoustical case [10,14] which shows the relationship between the asymmetry in the scattering and the direction of the radiation force.There have been related theoretical discussions of momentum projection in optics [15] followed by a demonstration of negative optical forces in a beam closely resembling intersecting plane waves [16].
There has been significant recent interest in broader applications of acoustical radiation forces and torques for the manipulation of objects of various sizes [17][18][19][20][21][22].Acoustic [23,24] and optical [25] beams with an extra azimuthal phase dependence exp(imφ ), called vortex beams, have a helicoidal wavefront and carry orbital angular momentum.This feature of angular momentum transport allows the beam to exert a torque to rotate an object.Acoustic vortex beams were analyzed beyond the paraxial approximation in [24] to clarify an analogy with optical vortex beams; the radiation torque on a symmetric object centered on the beam's axis was related to the absorption of power.
It has long been beneficial to consider possible areas of overlap between some related issues arising in acoustical and optical fields of research [17,[23][24][25][26][27].In this paper we illustrate the application of an extended optical theorem on acoustic radiation forces and torques associated with a non-diffracting beam.The optical theorem for an incident plane wave is known as one of the central theorems in scattering theory [28][29][30]; the theorem relates the extinction section to the complex scattering amplitude at the forward direction.An extended theorem for a non-diffracting beam was given recently in the context of acoustics [14].Here we use this extended optical theorem to examine the extinction cross section on a sphere centered on the axis of a non-diffracting beam, in particular, to examine the extinction for a sphere centered on a Bessel beam as an example.The results, together with a prior result of an asymmetry factor of scattering, are then applied to recover the axial radiation force and torque given in [10,24].

Optical theorem of a non-diffracting beam: Review
For a non-diffracting beam (with a speed c 0 in the medium, a frequency ω, and a wavenumber k = ω/c 0 ) propagating along the z axis with an axial wavenumber κ = k cos β , and scattered by an object centered at the origin (refer to Fig. 1), resulting in a scattered far field with a complex amplitude A s (θ , φ ), an extended optical theorem for the extinction cross section, denoted by σ ext , was derived in the context of acoustics by [14] as an azimuthal angle integration where Im denotes the imaginary part, and the asterisk denotes the complex conjugate.This theorem (3) connects the extinction to the scattering at the direction n(β , φ ) by using an angular function g(φ ) of the beam.This angular function g(φ ) relates to the beam's profile through a reduced Whittaker integral This integral (4) represents the beam (1) as a superposition of plane wave components, whose relative amplitude and phase are given by the function g(φ ), and whose wave-vectors have a tilted conical angle β relative to the beam's axis (refer to Fig. 1).The scattering in the theorem (3), A s (n(β , φ )), is the scattering at the forward direction of the beam's plane wave component at the azimuthal angle φ .The extinction results from both scattering and absorption.One can write σ ext = σ sca + σ abs , with σ sca and σ abs denoting the cross sections of scattering and absorption, respectively.The scattering cross section, σ sca , is given by the integral of the scattering coefficient over the whole solid angle element dΩ = sin θ dθ dφ as, On taking the limits, β = 0 and g(φ ) = 1, our extended theorem (3) reduces to the wellknown theorem for a plane wave [28][29][30], where A s (0, 0) is the forward scattering of the incident plane wave ψ i = ψ 0 exp(ikz − iωt).

Application to extinction by a sphere
Here we use the extended theorem (3) to examine the extinction cross section of a sphere centered on an idealized Bessel beam of arbitrary order m, where ρ = x 2 + y 2 , and m is an integer with m = 0 as a special case giving the ordinary Bessel beam.The angular function of this beam is [10,14] g(φ ) = exp(imφ ).
For a sphere of radius a centered on the axis of the beam ( 8), the far-field scattering was given in [10] in terms of a partial wave expansion as where the functions P m n are associated Legendre functions (see also [31]).The functions s n in the partial wave coefficients (s n − 1)/2 are the same as that for a plane wave scattering [2].These coefficients are known for different types of objects, as determined by boundary conditions.Notice that |s n | ≤ 1 and only for non-absorptive scattering are all coefficients |s n | equal to unity.
By using ( 9) and ( 10), and letting θ = β , it immediately follows from the optical theorem (3) that where Re denotes the real part.Given that |s n | ≤ 1, one always has Re[2(1 − s n )] ≥ 0, and hence σ ext ≥ 0 as required; only in the limit of no scattering, s n = 1, does no extinction σ ext = 0 occur.The cross section has the dimension of area.For an object being a sphere of radius a, the dimensionless efficiency factor becomes, which recovers the result in [10] derived from an analytical integration of energy flux (where the scattering coefficients were written as (s n − 1)/2 = α n + iβ n , with α n and β n being the real and imaginary parts).
The separation of the extinction (12) into the scattering and the absorption is straightforward.One may rewrite the coefficients Re The first part, |s n − 1| 2 , associated with the modula of scattering coefficients, corresponds to the contribution from the scattering.The second part, 1 − |s n | 2 , associated with the difference of |s n | 2 from unity, corresponds to the contribution from the absorption.Hence it follows from ( 12) and ( 13) that which again recovers the results in [10] derived from an analytical integration of energy flux of corresponding fields.
In the special case of non-absorptive scattering (|s n | = 1), Q abs = σ abs /(πa 2 ) = 0.In this case using the phase shifts δ n with both the extinction in (12) and the scattering in ( 14) relate to the phase shifts as Lastly, the corresponding powers are given as where I 0 , a function of the amplitude ψ 0 in (8), characterizes the intensity of the incident beam.
In the context of acoustics, if ψ represents the velocity potential field, one has the acoustic intensity

Application to the axial radiation force on a sphere
By the axial projection of momentum, the axial radiation force exerted by the general nondiffracting beam (4) on an object of arbitrary shape and location was given in [14] as or in terms of a dimensionless force function Y p as where the factors, cos β and cos θ s , correspond to the axial projections of extracted momentum and scattered momentum [10,11,14], respectively, and The factor cos θ s is an asymmetry parameter of the scattering: cos θ s is positive or negative when the scattering at the forward hemisphere is stronger or weaker relative to the scattering at the backward hemisphere.An inspection of (20) shows why even in the idealized case of negligible absorption, conditions to achieve negative forces usually require that the conic angle β be large: from the form of (20) the asymmetry must lie between −1 and 1.For a Bessel beam with m = 0 the plane wave limit is recovered by taking β = 0.It follows from the form of (20) that the force must be non-negative in that limit with or without absorption.By taking (20) associated with absorption, Q abs cos β , degrades the negative force [10].
The cross section and efficiency factor associated with the scattering asymmetry factor cos θ s in ( 21) for a sphere of radius a are When a sphere is on the axis of the Bessel beam (8), the efficiency of the scattering asymmetry, Q asym.sca , can be read from [10] (Q asym.sca= −Y 1 therein) as Together the extinction ( 12) from the optical theorem and the asymmetry (23) give the dimensionless force function in (20) as which recovers the result in [10] derived from the analytical integration of the axial projection of stress.Equation ( 24) was derived to associate with the momentum projection [10].Its analytical equivalence with prior results given in [6][7][8][9] was noted in [10].

Application to the radiation torque on a sphere
In an axisymmetric field (traveling or standing waves) with an azimuthal phase dependence exp(imφ ), where the integer m is the topological charge, the axial radiation torque on an axisymmetric object centered on the axis of the field was revealed in [24] to be associated with the absorption of energy as or in terms of a dimensionless torque efficiency factor Q T for a sphere of a radius a as, When a sphere is placed on the axis of the vortex Bessel beam ( 8) with a non-zero integer m, using the absorption (15) from the optical theorem, it has which were published in [10,24] (and presented even earlier in [32,33]), where the absorption was derived from an analytical integration of energy flux.
In [24], Eq. ( 25) was generalized to the torque on any axisymmetric object centered in any vortex wave field (traveling or standing waves; where the derivation started from the conservation of of angular momentum [34]).Experiments in [27] confirm (25) for a disk shaped object in a vortex wave field having an adjustable topological charge m.For most cases series expansions giving the absorption, like our (15), are not currently available.While some related work has appeared [35,36], the reader is cautioned the left side of Eq. ( 18) in [35] is incorrect.Other corrections to [35] were given in an erratum published by Mitri et al. late in 2012.Notice also that since the analysis in [24] included the standing wave case, (25) applies to the standing wave case examined in [35].

Conclusions and discussion
The applications of the extended acoustic extinction theorem (3) illustrated here concern a sphere placed on the axis of an idealized Bessel beam of arbitrary integer order m.In that case the resulting predictions for the extinction cross section and associated efficiency factor (12) agree with results derived a different way in [10].Also the normalized radiation force (24) agrees with the result from analytical integration of the axial radiation stress projection in [10] (equivalent to prior results in [6][7][8][9]).The torque and absorption efficiencies in (27) agree with the results from analytical integration of angular momentum and energy flux in [10,24].
The extinction theorem (3) and the associated axial stress projection relation (19) involving the scattering asymmetry (21) neither require the object to be a sphere nor to be centered on the axis [14].In that case, however, more complicated approaches are needed for evaluating the scattering amplitude.In the case of a sphere some computational approaches have been described by other researchers for various choices of beam types [20,22,37].
There is an additional complication in the case of small spheres in thermal viscous fluids in that highly intense sound waves can establish a steady flow pattern commonly referred to as acoustic streaming [38] not allowed for in the present analysis since all of the flow induced by the acoustic wave is assumed to oscillate at the frequency ω according to (1) and ( 2).The magnitude and flow pattern of acoustic streaming tends to be somewhat dependent on the transducer geometry and apparatus boundary conditions [38].Methods have been introduced to reduce the effects of streaming yielding agreement between measured and computed forces for objects in traveling waves [39,40].
An additional complication resulting from thermal-viscous dissipation in the surrounding fluid are the contributions to the effective absorption efficiency (15) and correspondingly the torque efficiency (27) from certain of the partial waves.For example, in the case of a small solid sphere when ka 0.5 the viscous correction to the dipole scattering can become important when the average density of the sphere differs from that of the surrounding fluid.In the usual case in which the thickness of the oscillating viscous boundary layer is small relative to the radius of the sphere, a simple approximation is available for estimating the viscous correction to s 1 [41].For solid spheres having small intrinsic absorption, this correction tends to decrease in significance the larger the radius of the sphere [41].
A situation where absorption (either by the sphere or in the adjacent boundary layer) can be beneficial concerns the induced rotation of a sphere on the axis of an acoustic vortex Bessel beam.That is because the radiation torque applied to the sphere is proportional to the absorbed power [23,24] as in the case of a sphere made of a lossy dielectric placed in circularly polarized light [26].If it is desirable to activate the absorption associated with the aforementioned viscous correction to the dipole term s 1 , it is necessary to use a beam having m = 1 [41].When the effect of the absorption of angular momentum by the surrounding fluid is negligible, the rotation rate for the sphere may be estimated by balancing the radiation torque with the viscous drag torque [41] as in the electromagnetic case [26].

Fig. 1 .
Fig. 1.The radiation force and/or torque on an object centered on the axis of an idealized non-diffracting beam relates to the extinction by the object via scattering and/or absorption.The beam, propagating along the z axis, is characterized by an angular function g(φ ) (refer to text) and by a conical angle β determining the direction of wave vectors k(β , φ ) = kn(β , φ ) of the beam's plane wave components.The scattering and/or absorption are characterized by a far-field scattering complex amplitude A s (n(θ , φ )).