Abstract

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

Corrections

Likun Zhang and Philip L. Marston, "Optical theorem for acoustic non-diffracting beams and application to radiation force and torque: erratum," Biomed. Opt. Express 4, 2988-2988 (2013)
https://www.osapublishing.org/boe/abstract.cfm?uri=boe-4-12-2988

1. Introduction

An idealized non-diffracting (optical or acoustic) beam is a beam whose transverse intensity pattern has the feature of propagation-invariance [15]. 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 [610] and in optics [1113]. It was noticed early [68] 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 [1722]. Acoustic [23, 24] and optical [25] beams with an extra 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, 2327]. 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 [2830]; 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].

2. Optical theorem of a non-diffracting beam: Review

For a non-diffracting beam (with a speed c0 in the medium, a frequency ω, and a wavenumber k = ω/c0) propagating along the z axis with an axial wavenumber κ = k cos β,

ψi=ψi0(x,y;β)exp(iκziωt),
and scattered by an object centered at the origin (refer to Fig. 1), resulting in a scattered far field with a complex amplitude As(θ, ϕ),
ψs=ψ0As(n)exp(ikriωt)/r,
an extended optical theorem for the extinction cross section, denoted by σext, was derived in the context of acoustics by [14] as an azimuthal angle integration
σext=4πkIm[12π02πg*(ϕ)As(n(β,ϕ))dϕ],
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
ψi0(x,y;β)=ψ02π02πg(ϕ)exp[iμ(xcosϕ+ysinϕ)]dϕ,withμ=ksinβ.
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
k(μcosϕ,μsinϕ,κ)=k(sinβcosϕ,sinβsinϕ,cosβ)=kn(β,ϕ),
have a tilted conical angle β relative to the beam’s axis (refer to Fig. 1). The scattering in the theorem (3), As(n(β, ϕ)), is the scattering at the forward direction of the beam’s plane wave component at the azimuthal angle ϕ.

 

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 As(n(θ, ϕ)).

Download Full Size | PPT Slide | PDF

The extinction results from both scattering and absorption. One can write σext = σsca + σabs, with σsca and σabs denoting the cross sections of scattering and absorption, respectively. The scattering cross section, σsca, is given by the integral of the scattering coefficient over the whole solid angle element dΩ = sinθdθdϕ as,

σsca=|As(n)|2dΩ.

On taking the limits, β = 0 and g(ϕ) = 1, our extended theorem (3) reduces to the well-known theorem for a plane wave [2830],

σext=4πkIm[As(0,0)],
where As(0, 0) is the forward scattering of the incident plane wave ψi = ψ0 exp(ikziωt).

3. 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,

ψi=ψ0imJm(μρ)exp(iκz+imϕ),
where ρ=x2+y2, 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

As(n(θ,ϕ))=exp(imϕ)ikn=|m|(sn1)2(2n+1)(nm)!(n+m)!Pnm(cosβ)Pnm(cosθ),
where the functions Pnm are associated Legendre functions (see also [31]). The functions sn in the partial wave coefficients (sn − 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 |sn| ≤ 1 and only for non-absorptive scattering are all coefficients |sn| equal to unity.

By using (9) and (10), and letting θ = β, it immediately follows from the optical theorem (3) that

σext=πk2n=|m|(2n+1)[Pnm(cosβ)]2Re[2(1sn)],
where Re denotes the real part. Given that |sn| ≤ 1, one always has Re[2(1 − sn)] ≥ 0, and hence σext ≥ 0 as required; only in the limit of no scattering, sn = 1, does no extinction σext = 0 occur. The cross section has the dimension of area. For an object being a sphere of radius a, the dimensionless efficiency factor becomes,
Qextσext/(πa2)=1(ka)2n=|m|(2n+1)(nm)!(n+m)![Pnm(cosβ)]2Re[2(1sn)],
which recovers the result in [10] derived from an analytical integration of energy flux (where the scattering coefficients were written as (sn − 1)/2 = αn + n, with αn and βn 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[2(1 − sn)] as

Re[2(1sn)]=(|sn1|2)+(1|sn|2).
The first part, |sn − 1|2, associated with the modula of scattering coefficients, corresponds to the contribution from the scattering. The second part, 1 − |sn|2, associated with the difference of |sn|2 from unity, corresponds to the contribution from the absorption. Hence it follows from (12) and (13) that
Qscaσsca/(πa2)=1(ka)2n=|m|(2n+1)[Pnm(cosβ)]2(|sn1|2),
Qabsσabs/(πa2)=1(ka)2n=|m|(2n+1)[Pnm(cosβ)]2(1|sn|2).
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 (|sn| = 1), Qabs = σabs/(πa2) = 0. In this case using the phase shifts δn with

sn=exp(i2δn),
both the extinction in (12) and the scattering in (14) relate to the phase shifts as
Qext,sca=σext,sca/(πa2)=4(ka)2n=|m|(2n+1)[Pnm(cosβ)]2sin2(δn).

Lastly, the corresponding powers are given as

Pext,sca,abs=I0σext,sca,abs=I0(πa2)Qext,sca,abs,
where I0, a function of the amplitude ψ0 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 I0 = (ρoco/2)(0)2.

4. 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

Fz=c01(PextcosβPscacosθs)
or in terms of a dimensionless force function Yp as
Fz=πa2c01I0Yp,Yp=QextcosβQscacosθs,
where the factors, cos β and 〈cos θs, correspond to the axial projections of extracted momentum and scattered momentum [10, 11, 14], respectively, and
cosθs=cosθ|As(n)|2dΩ|As(n)|2dΩ.

The factor 〈cos θs is an asymmetry parameter of the scattering: 〈cos θs 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 Qext = Qsca + Qabs, an implicit term in (20) associated with absorption, Qabs cos β, degrades the negative force [10].

The cross section and efficiency factor associated with the scattering asymmetry factor 〈cos θs in (21) for a sphere of radius a are

σasym.scacosθ|As(n)|2dΩ,Qasym.scaσasym.sca/(πa2).
When a sphere is on the axis of the Bessel beam (8), the efficiency of the scattering asymmetry, Qasym.sca, can be read from [10] (Qasym.sca = −Y1 therein) as
Qasym.sca=σasym.scaπa2=(2ka)2n=|m|[2(αnαn+1+βnβn+1)](nm+1)!(n+m)!Pnm(b)Pn+1m(b).
Together the extinction (12) from the optical theorem and the asymmetry (23) give the dimensionless force function in (20) as
Yp=cosβ(ka2)n=|m|(4αn)(2n+1)(nm)!(n+m)![Pnm(cosβ)]2(2ka)2n=|m|[2(αnαn+1+βnβn+1)](nm+1)!(n+m)!Pnm(cosβ)Pn+1m(cosβ),
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 [69] was noted in [10].

5. 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

Tz=mωPabs,
or in terms of a dimensionless torque efficiency factor QT for a sphere of a radius a as,
Tz=πa2I0QT/ω,QT=mQabs.
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
QT=mQabs=m(ka)2n=|m|(2n+1)(nm)!(n+m)![Pnm(cosβ)]2(1|sn|2),
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].

6. 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 [69]). 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 s1[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 s1, 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].

Acknowledgments

Zhang acknowledges the support from NASA and ONR, and Marston acknowledges support from ONR.

References and links

1. J. Durnin, “Exact solutions for nondiffraction beams. I. The scalar theory,” J. Opt. Soc. Am. A. 4, 651–654 (1987). [CrossRef]  

2. P. L. Marston, “Scattering of a Bessel beam by a sphere,” J. Acoust. Soc. Am 121, 753 (2007). [CrossRef]   [PubMed]  

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). [CrossRef]  

4. A. Belafhal, A. Chafiq, and Z. Hricha, “Scattering of Mathieu beams by a rigid sphere,” Opt. Commun. 284, 3030–3035 (2003). [CrossRef]  

5. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33(15), 1678–1680 (2008). [CrossRef]   [PubMed]  

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). [CrossRef]  

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). [CrossRef]  

8. P. L. Marston, “Radiation force of a helicoidal Bessel beam on a sphere,” J. Acoust. Soc. Am 125, 3539–3547 (2009). [CrossRef]   [PubMed]  

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). [CrossRef]  

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). [CrossRef]  

11. J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011). [CrossRef]  

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). [CrossRef]   [PubMed]  

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). [CrossRef]  

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). [CrossRef]   [PubMed]  

15. S. Sukhov and A. Dogariu, “Negative nonconservative forces: Optical tractor beams for arbitrary objects,” Phys. Rev. Lett. 107, 203602 (2011). [CrossRef]  

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). [CrossRef]  

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). [CrossRef]  

19. S. Xu, C. Qiu, and Z. Liu, “Transversally stable acoustic pulling force produced by two crossed plane waves,” Europhys. Lett. 99, 44003 (2012). [CrossRef]  

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). [CrossRef]   [PubMed]  

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). [CrossRef]  

22. D. Baresch, J.-L. Thomas, and R. Marchiano, “Spherical vortex beams of high radial degree for enhanced single-beam 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). [CrossRef]  

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). [CrossRef]  

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). [CrossRef]   [PubMed]  

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). [CrossRef]  

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). [CrossRef]   [PubMed]  

28. R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976). [CrossRef]  

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). [CrossRef]  

30. P. L. Marston, “Generalized optical theorem for scatterers having inversion symmetry: applications to acoustic backscattering,” J. Acoust. Soc. Am. 109, 1291–1295 (2001). [CrossRef]   [PubMed]  

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). [CrossRef]   [PubMed]  

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). [CrossRef]  

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). [CrossRef]   [PubMed]  

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). [CrossRef]  

36. G. T. Silva, T. P. Lobo, and F. G. Mitri, “Radiation torque produced by an arbitrary acoustic wave,” Europhys. Lett. 97, 54003 (2012). [CrossRef]  

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). [CrossRef]   [PubMed]  

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). [CrossRef]  

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). [CrossRef]   [PubMed]  

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). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. J. Durnin, “Exact solutions for nondiffraction beams. I. The scalar theory,” J. Opt. Soc. Am. A. 4, 651–654 (1987).
    [Crossref]
  2. P. L. Marston, “Scattering of a Bessel beam by a sphere,” J. Acoust. Soc. Am 121, 753 (2007).
    [Crossref] [PubMed]
  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).
    [Crossref]
  4. A. Belafhal, A. Chafiq, and Z. Hricha, “Scattering of Mathieu beams by a rigid sphere,” Opt. Commun. 284, 3030–3035 (2003).
    [Crossref]
  5. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33(15), 1678–1680 (2008).
    [Crossref] [PubMed]
  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).
    [Crossref]
  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).
    [Crossref]
  8. P. L. Marston, “Radiation force of a helicoidal Bessel beam on a sphere,” J. Acoust. Soc. Am 125, 3539–3547 (2009).
    [Crossref] [PubMed]
  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).
    [Crossref]
  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).
    [Crossref]
  11. J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
    [Crossref]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref]
  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).
    [Crossref] [PubMed]
  15. S. Sukhov and A. Dogariu, “Negative nonconservative forces: Optical tractor beams for arbitrary objects,” Phys. Rev. Lett. 107, 203602 (2011).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  19. S. Xu, C. Qiu, and Z. Liu, “Transversally stable acoustic pulling force produced by two crossed plane waves,” Europhys. Lett. 99, 44003 (2012).
    [Crossref]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref]
  22. D. Baresch, J.-L. Thomas, and R. Marchiano, “Spherical vortex beams of high radial degree for enhanced single-beam 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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref]
  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).
    [Crossref] [PubMed]
  28. R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976).
    [Crossref]
  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).
    [Crossref]
  30. P. L. Marston, “Generalized optical theorem for scatterers having inversion symmetry: applications to acoustic backscattering,” J. Acoust. Soc. Am. 109, 1291–1295 (2001).
    [Crossref] [PubMed]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref]
  36. G. T. Silva, T. P. Lobo, and F. G. Mitri, “Radiation torque produced by an arbitrary acoustic wave,” Europhys. Lett. 97, 54003 (2012).
    [Crossref]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref]
  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).
    [Crossref] [PubMed]
  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).
    [Crossref]

2013 (5)

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).
[Crossref]

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).
[Crossref]

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).
[Crossref] [PubMed]

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).
[Crossref]

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).
[Crossref]

2012 (5)

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).
[Crossref] [PubMed]

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).
[Crossref]

G. T. Silva, T. P. Lobo, and F. G. Mitri, “Radiation torque produced by an arbitrary acoustic wave,” Europhys. Lett. 97, 54003 (2012).
[Crossref]

S. Xu, C. Qiu, and Z. Liu, “Transversally stable acoustic pulling force produced by two crossed plane waves,” Europhys. Lett. 99, 44003 (2012).
[Crossref]

L. K. Zhang and P. L. Marston, “Axial radiation force exerted by general non-diffracting beams,” J. Acoust. Soc. Am. 131, EL329–EL335 (2012).
[Crossref] [PubMed]

2011 (9)

S. Sukhov and A. Dogariu, “Negative nonconservative forces: Optical tractor beams for arbitrary objects,” Phys. Rev. Lett. 107, 203602 (2011).
[Crossref]

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).
[Crossref]

J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

A. Novitsky, C.-W. Qiu, and H. F. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107, 203601 (2011).
[Crossref] [PubMed]

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).
[Crossref]

G. T. Silva, “Off-axis scattering of an ultrasound Bessel beam by a sphere,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58, 298–304 (2011).
[Crossref] [PubMed]

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).
[Crossref]

L. K. Zhang and P. L. Marston, “Acoustic radiation torque and the conservation of angular momentum,” J. Acoust. Soc. Am. 129, 1679–1680 (2011).
[Crossref] [PubMed]

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).
[Crossref]

2009 (4)

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).
[Crossref] [PubMed]

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).

P. L. Marston, “Radiation force of a helicoidal Bessel beam on a sphere,” J. Acoust. Soc. Am 125, 3539–3547 (2009).
[Crossref] [PubMed]

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).
[Crossref]

2008 (1)

2007 (2)

P. L. Marston, “Scattering of a Bessel beam by a sphere,” J. Acoust. Soc. Am 121, 753 (2007).
[Crossref] [PubMed]

P. L. Marston, “Negative axial radiation forces on solid spheres and shells in a Bessel beam,” J. Acoust. Soc. Am 122, 3162–3165 (2007).
[Crossref]

2006 (1)

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).
[Crossref]

2003 (1)

A. Belafhal, A. Chafiq, and Z. Hricha, “Scattering of Mathieu beams by a rigid sphere,” Opt. Commun. 284, 3030–3035 (2003).
[Crossref]

2001 (1)

P. L. Marston, “Generalized optical theorem for scatterers having inversion symmetry: applications to acoustic backscattering,” J. Acoust. Soc. Am. 109, 1291–1295 (2001).
[Crossref] [PubMed]

2000 (1)

1999 (1)

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).
[Crossref]

1996 (1)

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).
[Crossref] [PubMed]

1995 (1)

1992 (1)

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).
[Crossref] [PubMed]

1987 (1)

J. Durnin, “Exact solutions for nondiffraction beams. I. The scalar theory,” J. Opt. Soc. Am. A. 4, 651–654 (1987).
[Crossref]

1984 (1)

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).
[Crossref]

1976 (1)

R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976).
[Crossref]

1975 (1)

T. Hasegawa and K. Yosioka, “Acoustic radiation force on fused silica spheres, and intensity determination,” J. Acoust. Soc. Am. 58, 581–585 (1975).
[Crossref]

1849 (1)

D. Baresch, J.-L. Thomas, and R. Marchiano, “Spherical vortex beams of high radial degree for enhanced single-beam tweezers,” J. Appl. Phys. 113, 184901 (2013).

Allen, L.

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).
[Crossref] [PubMed]

Apfel, R. E.

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).
[Crossref] [PubMed]

Bandres, M. A.

Baresch, D.

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).
[Crossref] [PubMed]

D. Baresch, J.-L. Thomas, and R. Marchiano, “Spherical vortex beams of high radial degree for enhanced single-beam tweezers,” J. Appl. Phys. 113, 184901 (2013).

Beijersbergen, M. W.

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).
[Crossref] [PubMed]

Belafhal, A.

A. Belafhal, A. Chafiq, and Z. Hricha, “Scattering of Mathieu beams by a rigid sphere,” Opt. Commun. 284, 3030–3035 (2003).
[Crossref]

Brzobohatý, O.

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).
[Crossref]

Chafiq, A.

A. Belafhal, A. Chafiq, and Z. Hricha, “Scattering of Mathieu beams by a rigid sphere,” Opt. Commun. 284, 3030–3035 (2003).
[Crossref]

Chan, C. T.

J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

Chávez-Cerda, S.

Chen, J.

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).
[Crossref]

J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

Chen, X. C.

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).
[Crossref] [PubMed]

Choe, Y.

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).
[Crossref]

Chvátal, L.

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).
[Crossref]

Cižmár, T.

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).
[Crossref]

Cochran, S.

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).
[Crossref]

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).
[Crossref] [PubMed]

Courtney, C. R. P.

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).
[Crossref]

Crichton, J. H.

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).
[Crossref]

Demore, C. E. M.

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).
[Crossref]

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).
[Crossref] [PubMed]

Dogariu, A.

S. Sukhov and A. Dogariu, “Negative nonconservative forces: Optical tractor beams for arbitrary objects,” Phys. Rev. Lett. 107, 203602 (2011).
[Crossref]

Drinkwater, B. W.

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).
[Crossref]

Durnin, J.

J. Durnin, “Exact solutions for nondiffraction beams. I. The scalar theory,” J. Opt. Soc. Am. A. 4, 651–654 (1987).
[Crossref]

Gouesbet, G.

Grinenko, A.

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).
[Crossref]

Gutiérrez-Vega, J. C.

Hasegawa, T.

T. Hasegawa and K. Yosioka, “Acoustic radiation force on fused silica spheres, and intensity determination,” J. Acoust. Soc. Am. 58, 581–585 (1975).
[Crossref]

Hefner, B. T.

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).
[Crossref]

Hodges, J. T.

Hricha, Z.

A. Belafhal, A. Chafiq, and Z. Hricha, “Scattering of Mathieu beams by a rigid sphere,” Opt. Commun. 284, 3030–3035 (2003).
[Crossref]

Iturbe-Castillo, M. D.

Karásek, V.

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).
[Crossref]

Kim, E. S.

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).
[Crossref]

Kim, J. W.

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).
[Crossref]

Lin, Z. F.

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).
[Crossref]

J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

Liu, S. Y.

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).
[Crossref]

Liu, Z.

S. Xu, C. Qiu, and Z. Liu, “Transversally stable acoustic pulling force produced by two crossed plane waves,” Europhys. Lett. 99, 44003 (2012).
[Crossref]

Lobo, T. P.

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).
[Crossref]

G. T. Silva, T. P. Lobo, and F. G. Mitri, “Radiation torque produced by an arbitrary acoustic wave,” Europhys. Lett. 97, 54003 (2012).
[Crossref]

Lock, J. A.

MacDonald, M. P.

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).
[Crossref] [PubMed]

Marchiano, R.

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).
[Crossref] [PubMed]

D. Baresch, J.-L. Thomas, and R. Marchiano, “Spherical vortex beams of high radial degree for enhanced single-beam tweezers,” J. Appl. Phys. 113, 184901 (2013).

Marston, P. L.

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).
[Crossref]

L. K. Zhang and P. L. Marston, “Axial radiation force exerted by general non-diffracting beams,” J. Acoust. Soc. Am. 131, EL329–EL335 (2012).
[Crossref] [PubMed]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

L. K. Zhang and P. L. Marston, “Acoustic radiation torque and the conservation of angular momentum,” J. Acoust. Soc. Am. 129, 1679–1680 (2011).
[Crossref] [PubMed]

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).

P. L. Marston, “Radiation force of a helicoidal Bessel beam on a sphere,” J. Acoust. Soc. Am 125, 3539–3547 (2009).
[Crossref] [PubMed]

P. L. Marston, “Negative axial radiation forces on solid spheres and shells in a Bessel beam,” J. Acoust. Soc. Am 122, 3162–3165 (2007).
[Crossref]

P. L. Marston, “Scattering of a Bessel beam by a sphere,” J. Acoust. Soc. Am 121, 753 (2007).
[Crossref] [PubMed]

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).
[Crossref]

P. L. Marston, “Generalized optical theorem for scatterers having inversion symmetry: applications to acoustic backscattering,” J. Acoust. Soc. Am. 109, 1291–1295 (2001).
[Crossref] [PubMed]

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).
[Crossref]

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).
[Crossref]

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.

Mitri, F. G.

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).
[Crossref]

G. T. Silva, T. P. Lobo, and F. G. Mitri, “Radiation torque produced by an arbitrary acoustic wave,” Europhys. Lett. 97, 54003 (2012).
[Crossref]

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).
[Crossref] [PubMed]

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).
[Crossref]

Newton, R. G.

R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976).
[Crossref]

Ng, J.

J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

Novitsky, A.

A. Novitsky, C.-W. Qiu, and H. F. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107, 203601 (2011).
[Crossref] [PubMed]

Nyborg, W. L.

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.

Qiu, C.

S. Xu, C. Qiu, and Z. Liu, “Transversally stable acoustic pulling force produced by two crossed plane waves,” Europhys. Lett. 99, 44003 (2012).
[Crossref]

Qiu, C.-W.

A. Novitsky, C.-W. Qiu, and H. F. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107, 203601 (2011).
[Crossref] [PubMed]

Shung, K. K.

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).
[Crossref]

Šiler, M.

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).
[Crossref]

Silva, G. T.

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).
[Crossref]

G. T. Silva, T. P. Lobo, and F. G. Mitri, “Radiation torque produced by an arbitrary acoustic wave,” Europhys. Lett. 97, 54003 (2012).
[Crossref]

G. T. Silva, “Off-axis scattering of an ultrasound Bessel beam by a sphere,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58, 298–304 (2011).
[Crossref] [PubMed]

Spalding, G. C.

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).
[Crossref] [PubMed]

Spreeuw, R. J. C.

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).
[Crossref] [PubMed]

Sukhov, S.

S. Sukhov and A. Dogariu, “Negative nonconservative forces: Optical tractor beams for arbitrary objects,” Phys. Rev. Lett. 107, 203602 (2011).
[Crossref]

Thomas, J.-L.

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).
[Crossref] [PubMed]

D. Baresch, J.-L. Thomas, and R. Marchiano, “Spherical vortex beams of high radial degree for enhanced single-beam tweezers,” J. Appl. Phys. 113, 184901 (2013).

Volovick, A.

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).
[Crossref] [PubMed]

Wang, H. F.

A. Novitsky, C.-W. Qiu, and H. F. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107, 203601 (2011).
[Crossref] [PubMed]

Wang, N.

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).
[Crossref]

Wilcox, P. D.

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).
[Crossref]

Woerdman, J. P.

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).
[Crossref] [PubMed]

Xu, S.

S. Xu, C. Qiu, and Z. Liu, “Transversally stable acoustic pulling force produced by two crossed plane waves,” Europhys. Lett. 99, 44003 (2012).
[Crossref]

Yang, Z.

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).
[Crossref] [PubMed]

Yosioka, K.

T. Hasegawa and K. Yosioka, “Acoustic radiation force on fused silica spheres, and intensity determination,” J. Acoust. Soc. Am. 58, 581–585 (1975).
[Crossref]

Zemánek, P.

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).
[Crossref]

Zhang, L. K.

L. K. Zhang and P. L. Marston, “Axial radiation force exerted by general non-diffracting beams,” J. Acoust. Soc. Am. 131, EL329–EL335 (2012).
[Crossref] [PubMed]

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).
[Crossref]

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).
[Crossref]

L. K. Zhang and P. L. Marston, “Acoustic radiation torque and the conservation of angular momentum,” J. Acoust. Soc. Am. 129, 1679–1680 (2011).
[Crossref] [PubMed]

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).
[Crossref]

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).

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.

Am. J. Phys. (1)

R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976).
[Crossref]

Appl. Phys. Lett. (2)

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).
[Crossref]

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).
[Crossref]

Europhys. Lett. (2)

S. Xu, C. Qiu, and Z. Liu, “Transversally stable acoustic pulling force produced by two crossed plane waves,” Europhys. Lett. 99, 44003 (2012).
[Crossref]

G. T. Silva, T. P. Lobo, and F. G. Mitri, “Radiation torque produced by an arbitrary acoustic wave,” Europhys. Lett. 97, 54003 (2012).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (2)

G. T. Silva, “Off-axis scattering of an ultrasound Bessel beam by a sphere,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58, 298–304 (2011).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

J. Acoust. Soc. Am (4)

P. L. Marston, “Scattering of a Bessel beam by a sphere,” J. Acoust. Soc. Am 121, 753 (2007).
[Crossref] [PubMed]

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).
[Crossref]

P. L. Marston, “Negative axial radiation forces on solid spheres and shells in a Bessel beam,” J. Acoust. Soc. Am 122, 3162–3165 (2007).
[Crossref]

P. L. Marston, “Radiation force of a helicoidal Bessel beam on a sphere,” J. Acoust. Soc. Am 125, 3539–3547 (2009).
[Crossref] [PubMed]

J. Acoust. Soc. Am. (9)

L. K. Zhang and P. L. Marston, “Axial radiation force exerted by general non-diffracting beams,” J. Acoust. Soc. Am. 131, EL329–EL335 (2012).
[Crossref] [PubMed]

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).

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).
[Crossref]

L. K. Zhang and P. L. Marston, “Acoustic radiation torque and the conservation of angular momentum,” J. Acoust. Soc. Am. 129, 1679–1680 (2011).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

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).
[Crossref]

T. Hasegawa and K. Yosioka, “Acoustic radiation force on fused silica spheres, and intensity determination,” J. Acoust. Soc. Am. 58, 581–585 (1975).
[Crossref]

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).
[Crossref] [PubMed]

P. L. Marston, “Generalized optical theorem for scatterers having inversion symmetry: applications to acoustic backscattering,” J. Acoust. Soc. Am. 109, 1291–1295 (2001).
[Crossref] [PubMed]

J. Appl. Phys. (1)

D. Baresch, J.-L. Thomas, and R. Marchiano, “Spherical vortex beams of high radial degree for enhanced single-beam tweezers,” J. Appl. Phys. 113, 184901 (2013).

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (1)

J. Durnin, “Exact solutions for nondiffraction beams. I. The scalar theory,” J. Opt. Soc. Am. A. 4, 651–654 (1987).
[Crossref]

J. Phys. A: Math. Theor. (1)

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).
[Crossref]

Nat. Photonics (2)

J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011).
[Crossref]

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).
[Crossref]

Opt. Commun. (1)

A. Belafhal, A. Chafiq, and Z. Hricha, “Scattering of Mathieu beams by a rigid sphere,” Opt. Commun. 284, 3030–3035 (2003).
[Crossref]

Opt. Lett. (2)

Phys. Rev. A (3)

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).
[Crossref]

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).
[Crossref] [PubMed]

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).
[Crossref]

Phys. Rev. E (3)

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Phys. Rev. Lett. (3)

A. Novitsky, C.-W. Qiu, and H. F. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. 107, 203601 (2011).
[Crossref] [PubMed]

S. Sukhov and A. Dogariu, “Negative nonconservative forces: Optical tractor beams for arbitrary objects,” Phys. Rev. Lett. 107, 203602 (2011).
[Crossref]

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).
[Crossref] [PubMed]

Proceedings of Meetings on Acoustics (POMA) (1)

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).
[Crossref]

Other (2)

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.

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.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

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 As(n(θ, ϕ)).

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

ψ i = ψ i 0 ( x , y ; β ) exp ( i κ z i ω t ) ,
ψ s = ψ 0 A s ( n ) exp ( i k r i ω t ) / r ,
σ ext = 4 π k Im [ 1 2 π 0 2 π g * ( ϕ ) A s ( n ( β , ϕ ) ) d ϕ ] ,
ψ i 0 ( x , y ; β ) = ψ 0 2 π 0 2 π g ( ϕ ) exp [ i μ ( x cos ϕ + y sin ϕ ) ] d ϕ , with μ = k sin β .
k ( μ cos ϕ , μ sin ϕ , κ ) = k ( sin β cos ϕ , sin β sin ϕ , cos β ) = k n ( β , ϕ ) ,
σ sca = | A s ( n ) | 2 d Ω .
σ ext = 4 π k Im [ A s ( 0 , 0 ) ] ,
ψ i = ψ 0 i m J m ( μ ρ ) exp ( i κ z + i m ϕ ) ,
g ( ϕ ) = exp ( i m ϕ ) .
A s ( n ( θ , ϕ ) ) = exp ( i m ϕ ) i k n = | m | ( s n 1 ) 2 ( 2 n + 1 ) ( n m ) ! ( n + m ) ! P n m ( cos β ) P n m ( cos θ ) ,
σ ext = π k 2 n = | m | ( 2 n + 1 ) [ P n m ( cos β ) ] 2 Re [ 2 ( 1 s n ) ] ,
Q ext σ ext / ( π a 2 ) = 1 ( k a ) 2 n = | m | ( 2 n + 1 ) ( n m ) ! ( n + m ) ! [ P n m ( cos β ) ] 2 Re [ 2 ( 1 s n ) ] ,
Re [ 2 ( 1 s n ) ] = ( | s n 1 | 2 ) + ( 1 | s n | 2 ) .
Q sca σ sca / ( π a 2 ) = 1 ( k a ) 2 n = | m | ( 2 n + 1 ) [ P n m ( cos β ) ] 2 ( | s n 1 | 2 ) ,
Q abs σ abs / ( π a 2 ) = 1 ( k a ) 2 n = | m | ( 2 n + 1 ) [ P n m ( cos β ) ] 2 ( 1 | s n | 2 ) .
s n = exp ( i 2 δ n ) ,
Q ext , sca = σ ext , sca / ( π a 2 ) = 4 ( k a ) 2 n = | m | ( 2 n + 1 ) [ P n m ( cos β ) ] 2 sin 2 ( δ n ) .
P ext , sca , abs = I 0 σ ext , sca , abs = I 0 ( π a 2 ) Q ext , sca , abs ,
F z = c 0 1 ( P ext cos β P sca cos θ s )
F z = π a 2 c 0 1 I 0 Y p , Y p = Q ext cos β Q sca cos θ s ,
cos θ s = cos θ | A s ( n ) | 2 d Ω | A s ( n ) | 2 d Ω .
σ asym . sca cos θ | A s ( n ) | 2 d Ω , Q asym . sca σ asym . sca / ( π a 2 ) .
Q asym . sca = σ asym . sca π a 2 = ( 2 k a ) 2 n = | m | [ 2 ( α n α n + 1 + β n β n + 1 ) ] ( n m + 1 ) ! ( n + m ) ! P n m ( b ) P n + 1 m ( b ) .
Y p = cos β ( k a 2 ) n = | m | ( 4 α n ) ( 2 n + 1 ) ( n m ) ! ( n + m ) ! [ P n m ( cos β ) ] 2 ( 2 k a ) 2 n = | m | [ 2 ( α n α n + 1 + β n β n + 1 ) ] ( n m + 1 ) ! ( n + m ) ! P n m ( cos β ) P n + 1 m ( cos β ) ,
T z = m ω P abs ,
T z = π a 2 I 0 Q T / ω , Q T = m Q abs .
Q T = m Q abs = m ( k a ) 2 n = | m | ( 2 n + 1 ) ( n m ) ! ( n + m ) ! [ P n m ( cos β ) ] 2 ( 1 | s n | 2 ) ,

Metrics