Abstract

Experimental evidence of transfer of orbital angular momentum of multiringed beams to dielectric particles has been reported recently [e.g., J. Opt. B 4, S82 (2002); Phys. Rev. Lett. 91, 093602 (2003)]. Here we present a detailed theoretical examination of the forces involved in trapping and transferring orbital angular momentum to microparticles due to a multiringed light beam, particularly a Bessel beam. Our investigation gathers, in a more general way, the trapping forces for high-index and low-index dielectric transparent particles, as well as for reflective metallic particles, as a function of particle size and position relative to the dimensions of the rings of the beam. We find that particles can be trapped in different regions of the beam intensity profile according to their size and that an azimuthal force component opposite to the beam helicity may appear under certain circumstances, depending on the relative size and radial equilibrium position with respect to the beam for high-index spheres.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2003 (2)

V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multi-ringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature (London) 426, 421–424 (2003).
[CrossRef]

2002 (3)

K. P. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

V. Garcés-Chávez, K. Volke-Sepúlveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

2001 (2)

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A 63, 063602 (2001).
[CrossRef]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

2000 (2)

J. Courtial and M. J. Padgett, “Limit to the orbital angular momentum per unit energy in a light beam that can be focused onto a small particle,” Opt. Commun. 173, 269–274 (2000).
[CrossRef]

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, and S. Chavez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

1999 (2)

M. Mansuripur, “Geometric-optical rays, Poynting’s vector and field momenta,” Opt. Photonics News 10 (3), 53–56 (1999).
[CrossRef]

S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

1998 (2)

M. V. Berry, “Much ado about nothing: optical distortion lines (phase singularities, zeros, and vortices),” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE 3487, 6 (1998).
[CrossRef]

S. Nemoto and H. Togo, “Axial force acting on a dielectric sphere in a focused laser beam,” Appl. Opt. 37, 6386–6394 (1998).
[CrossRef]

1997 (2)

1996 (1)

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

1995 (2)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

R. C. Gauthier and S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of the force equations,” J. Opt. Soc. Am. B 12, 1680–1686 (1995).
[CrossRef]

1992 (3)

R. Gussgard, T. Lindmo, and I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B 9, 1922–1930 (1992).
[CrossRef]

A. Ashkin, “Forces of a single-beam gradient trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

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, 8185–8189 (1992).
[CrossRef] [PubMed]

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

1986 (1)

1983 (1)

1976 (1)

G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6–8 (1976).
[CrossRef]

1974 (1)

A. Ashkin and J. M. Dziedzic, “Stability of optical levitation by radiation pressure,” Appl. Phys. Lett. 24 (12), 586–588 (1974).
[CrossRef]

Allen, L.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[CrossRef] [PubMed]

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, 8185–8189 (1992).
[CrossRef] [PubMed]

Arlt, J.

K. P. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A 63, 063602 (2001).
[CrossRef]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
[CrossRef] [PubMed]

A. Ashkin and J. M. Dziedzic, “Stability of optical levitation by radiation pressure,” Appl. Phys. Lett. 24 (12), 586–588 (1974).
[CrossRef]

Bajer, J.

R. Horak, Z. Bouchal, and J. Bajer, “Non-diffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[CrossRef]

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, 8185–8189 (1992).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, “Much ado about nothing: optical distortion lines (phase singularities, zeros, and vortices),” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE 3487, 6 (1998).
[CrossRef]

Bjorkholm, J.

Bouchal, Z.

R. Horak, Z. Bouchal, and J. Bajer, “Non-diffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[CrossRef]

Brevik, I.

Chavez-Cerda, S.

Chávez-Cerda, S.

K. P. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

V. Garcés-Chávez, K. Volke-Sepúlveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

Chu, S.

Courtial, J.

J. Courtial and M. J. Padgett, “Limit to the orbital angular momentum per unit energy in a light beam that can be focused onto a small particle,” Opt. Commun. 173, 269–274 (2000).
[CrossRef]

Dholakia, K.

V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multi-ringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature (London) 426, 421–424 (2003).
[CrossRef]

V. Garcés-Chávez, K. Volke-Sepúlveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

K. P. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A 63, 063602 (2001).
[CrossRef]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[CrossRef] [PubMed]

Dultz, W.

V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multi-ringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Dziedzic, J. M.

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Enger, J.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

Friese, M. E. J.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Garcés-Chávez, V.

V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multi-ringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

V. Garcés-Chávez, K. Volke-Sepúlveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

K. P. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

Gauthier, R. C.

Gussgard, R.

Gutierrez-Vega, J. C.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Horak, R.

R. Horak, Z. Bouchal, and J. Bajer, “Non-diffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[CrossRef]

Imbert, C.

G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6–8 (1976).
[CrossRef]

Iturbe-Castillo, M. D.

Kim, J. S.

Lee, S. S.

Lindmo, T.

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature (London) 426, 421–424 (2003).
[CrossRef]

MacVicar, I.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Mansuripur, M.

M. Mansuripur, “Geometric-optical rays, Poynting’s vector and field momenta,” Opt. Photonics News 10 (3), 53–56 (1999).
[CrossRef]

McGloin, D.

V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multi-ringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Nemoto, S.

O’Neil, A. T.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Padgett, M. J.

V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multi-ringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

J. Courtial and M. J. Padgett, “Limit to the orbital angular momentum per unit energy in a light beam that can be focused onto a small particle,” Opt. Commun. 173, 269–274 (2000).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[CrossRef] [PubMed]

Roosen, G.

G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6–8 (1976).
[CrossRef]

Rubinsztein-Dunlop, H.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Schmitzer, H.

V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multi-ringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

Sibbett, W.

V. Garcés-Chávez, K. Volke-Sepúlveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[CrossRef]

Simpson, N. B.

Soneson, J.

J. Arlt, K. Dholakia, J. Soneson, and E. M. Wright, “Optical dipole traps and atomic waveguides based on Bessel light beams,” Phys. Rev. A 63, 063602 (2001).
[CrossRef]

Spalding, G. C.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature (London) 426, 421–424 (2003).
[CrossRef]

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, 8185–8189 (1992).
[CrossRef] [PubMed]

Togo, H.

Volke-Sepúlveda, K.

V. Garcés-Chávez, K. Volke-Sepúlveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Orbital angular momentum transfer to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Volke-Sepúlveda, K. P.

K. P. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

Wallace, S.

Woerdman, J. P.

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Figures (8)

Fig. 1
Fig. 1

Comparison of the axial optical forces exerted on a silica sphere immersed in water (n=1.087) by a Bessel beam and by a Gaussian beam. The parameters for the beams are the following: the radius of the central bright spot of the Bessel beam is ρB=0.9 w0, its maximum propagation distance zmax=40 zR, and its total power P=4 P0, where w0 is the waist spot size of the Gaussian beam, zR is its Rayleigh range in the medium, and P0=1 mW is its total power. The radius of the sphere is R=1.5w0. These curves are exactly the same regardless of the specific value of w0 and the wavelength of the laser light, provided the ratios between the parameters of both beams and with the size of the particle are preserved.

Fig. 2
Fig. 2

Radial optical force per unit power exerted by a Bessel beam with helicity l=2 and zmax=100 µm on three different kinds of spheres as function of the dimensionless parameter r=ρ0/ρ1, where ρ0 is the distance from the beam center and ρ1 is the radius of the first intensity maximum of the beam. The solid black curve corresponds to a high index dielectric sphere (a silica particle in water, n=1.087), the dashed curve is for a low-index sphere (a hollow sphere in water n=0.7502), and the dash-dotted curve is for a reflective metallic sphere. All of them have the same radius, R0=0.6 ρ1. The beam intensity profile is also depicted with a dotted curve as a reference. The parameter zmax determines the magnitude of the forces per unit power, since the larger the value of zmax, the lower the magnitude of the forces. However, the shape of the curves is completely independent of the specific value of ρ1.

Fig. 3
Fig. 3

Percentage of displacement of the radial equilibrium positions with respect to the first intensity maximum in the case of the high-index sphere (solid curve) and with respect to the first dark ring in the cases of low-index (dashed curve) and metallic (dash-dotted curve) spheres, as a function of the size parameter of the particles α=R0/ρ1.

Fig. 4
Fig. 4

Radial optical force exerted by a BB with helicity l=2 and zmax=100 µm on high-index spheres (n=1.087) whose radii are defined by R0=α ρ1, with α=1.1 (solid curve), α=1.15 (dashed curve), and α=1.21 (dash-dotted curve). The dotted curve represents the corresponding intensity profile.

Fig. 5
Fig. 5

Azimuthal optical force as a function of r=ρ0/ρ1 for the three kinds of spheres: high-index (black solid curve), low-index (dashed curve), and metallic particles (dash-dotted curve), for the same parameters as in Fig. 2. The dotted curve represents the corresponding beam profile.

Fig. 6
Fig. 6

Intensity profile (dotted curve) and azimuthal optical force per unit power against r=ρ0/ρ1 for spheres with radius R0=0.6 ρ1 with relative refractive indices n=1.1 (solid curve), n=1.5 (dashed curve), and n=1.8 (dash-dotted curve), and the same beam parameters as in Fig. 2.

Fig. 7
Fig. 7

Vector diagrams of the azimuthal optical force for the same parameters as in Fig. 2: (a) Intensity profile of the BB. Cases of (b) high-index dielectric sphere, (c) low-index dielectric sphere, and (d) metallic sphere.

Fig. 8
Fig. 8

Rotation rates against the total incident power for the silica (solid curve), hollow (dashed curve), and silver (dash-dotted curve) spheres immersed in water. We consider silica, hollow, and silver spheres immersed in water with radii R0=0.6 ρ1, and for the Bessel beam we have l=2 and zmax=100 µm. We assume a viscosity coefficient for water of η=1.0×10-3 N s m-2, which corresponds to a temperature of 20 °C.

Equations (11)

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F=1vm SI(r)cos αi(r)uˆi(r)-R(r)urˆ(r)-T(r)2k=1NR(r)k-1uˆtk(r)dA.
Il(ρ, φ, z)=2l+2l! ktPwc×zzmax2l+1Jl2(ktρ)exp(-2z2/zmax2),
g(r)Il(r)=fρ(r)ρˆ+fφ(r)φˆ+fz(r)zˆ,
fρ(r)=0,fφ(r)=lkρ,fz(r)=cos γ,
ρ=(ρ02+R02 sin2 θ+2ρ0R0 sin θ cos ϕ)1/2,
z=z0+R0 cos θ,
Fx(φ)=R02nmc π/2ππ2π Il(r)ρ0 sin ϕ sin2 θρfφ(r)×sin θT R0ρ sin ϕ-2R ρ0 sin ϕ sin θ cos ϕρ+T2k=1NRk-1(uˆtk)x(φ)dϕdθ,
Fy(φ)=-R02nmc π/2ππ2π Il(r)ρ0 sin ϕ sin2 θρfφ(r)×Tρ0+R0 sin θ cos ϕρ+2R ρ0 sin2 θ sin2 ϕρ-T2k=1NRk-1(uˆtk)y(φ)dϕdθ,
Fx(z)=R02nm2c π/2π02πIl(r)fz(r)-2R cos θ cos ϕ+T2k=1NRk-1(uˆtk)x(z)sin 2θdϕdθ.
Fz(z)=R02nm2c π/2π02πIl(r)fz(r)-T+2R cos2 θ+T2k=1NRk-1(uˆtk)z(z)sin 2θdϕdθ.
vp=Fφ(ρ0)12π2ηρ0R0.

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