Abstract

It is well known that optical vortex beams carry orbital as well as spin angular momentum. This optical angular momentum can manifest itself mechanically, for example in tightly focused Laguerre–Gaussian beams, where trapped, weakly absorbing spheres rotate at a rate proportional to the total angular momentum carried by the beam. In the present paper we subject this system to a rigorous analysis involving expansions in vector spherical wave functions that culminates in a simple expression for the torque on the sphere. It is seen that, for large weakly absorbing spheres, the induced torque per unit power is independent of the detailed structure of the incident field, being a simple function of two indices that describe the helicity and polarization state of the beam, the relative refractive indices of the sphere and ambient medium, the absorption index of the sphere, and its radius. A number of relationships between the coefficients of these expansions are also developed.

© 2008 Optical Society of America

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  1. J. H. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London, Ser. A 82, 560-567 (1909).
    [CrossRef]
  2. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115-125 (1936).
    [CrossRef]
  3. 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]
  4. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
    [CrossRef]
  5. 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]
  6. 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]
  7. 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]
  8. A. T. O'Neil and M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185, 139-143 (2000).
    [CrossRef]
  9. J. E. Molloy and M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. 43, 241-258 (2002).
    [CrossRef]
  10. M. J. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. 50, 1555-1562 (2003).
    [CrossRef]
  11. R. Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A 68, 013806 (2003).
    [CrossRef]
  12. S. Chang and S. S. Lee, “Optical torque exerted on a homogeneous sphere levitated in the circularly polarized fundamental-mode of a laser beam,” J. Opt. Soc. Am. B 2, 1853-1860 (1985).
    [CrossRef]
  13. H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, “Optical trapping of absorbing particles,” Adv. Quantum Chem. 30, 469-492 (1998).
    [CrossRef]
  14. A. S. van de Nes and P. Török, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express 15, 13360-13374 (2007).
    [CrossRef] [PubMed]
  15. S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110, 670-678 (1994).
    [CrossRef]
  16. T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
    [CrossRef]
  17. T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
    [CrossRef]
  18. S. H. Simpson and S. Hanna, “Numerical calculation of inter-particle forces arising in association with holographic assembly,” J. Opt. Soc. Am. A 23, 1419-1431 (2006).
    [CrossRef]
  19. S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430-443 (2007).
    [CrossRef]
  20. S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
    [CrossRef]
  21. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).
  22. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594-4602 (1989).
    [CrossRef]
  23. G. Gouesbet, J. A. Lock, and G. Grehan, “Partial wave representations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133-2143 (1995).
    [CrossRef] [PubMed]
  24. A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
    [CrossRef]
  25. O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B 22, 1620-1631 (2005).
    [CrossRef]
  26. J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electronic J. Differ. Equations, Conf. 04, 37-50 (2000).
  27. Y. Zhao, J. S. Edgar, G. D. M. Jefferies, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
    [CrossRef] [PubMed]
  28. A. O'Neil, I. MacVicar, L. Allen, and M. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
    [CrossRef] [PubMed]
  29. L. Allen and M. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67-71 (2000).
    [CrossRef]
  30. S. H. Simpson and S. Hanna, in preparation, to be called “Axial trapping in Laguerre-Gaussian beams.”

2007 (4)

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

Y. Zhao, J. S. Edgar, G. D. M. Jefferies, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430-443 (2007).
[CrossRef]

A. S. van de Nes and P. Török, “Rigorous analysis of spheres in Gauss-Laguerre beams,” Opt. Express 15, 13360-13374 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (1)

2003 (3)

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
[CrossRef]

M. J. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. 50, 1555-1562 (2003).
[CrossRef]

R. Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A 68, 013806 (2003).
[CrossRef]

2002 (2)

A. O'Neil, I. MacVicar, L. Allen, and M. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

J. E. Molloy and M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. 43, 241-258 (2002).
[CrossRef]

2001 (2)

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

2000 (3)

L. Allen and M. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67-71 (2000).
[CrossRef]

A. T. O'Neil and M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185, 139-143 (2000).
[CrossRef]

J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electronic J. Differ. Equations, Conf. 04, 37-50 (2000).

1998 (1)

H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, “Optical trapping of absorbing particles,” Adv. Quantum Chem. 30, 469-492 (1998).
[CrossRef]

1997 (1)

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]

G. Gouesbet, J. A. Lock, and G. Grehan, “Partial wave representations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133-2143 (1995).
[CrossRef] [PubMed]

1994 (1)

S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110, 670-678 (1994).
[CrossRef]

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

1989 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

1985 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

1936 (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115-125 (1936).
[CrossRef]

1909 (1)

J. H. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London, Ser. A 82, 560-567 (1909).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Allen, L.

A. O'Neil, I. MacVicar, L. Allen, and M. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

L. Allen and M. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67-71 (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]

S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110, 670-678 (1994).
[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, 8185-8189 (1992).
[CrossRef] [PubMed]

Barnett, S. M.

M. J. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. 50, 1555-1562 (2003).
[CrossRef]

S. M. Barnett and L. Allen, “Orbital angular-momentum and nonparaxial light-beams,” Opt. Commun. 110, 670-678 (1994).
[CrossRef]

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594-4602 (1989).
[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]

Benito, D. C.

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115-125 (1936).
[CrossRef]

Bishop, A. I.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Chang, S.

Chiu, D. T.

Y. Zhao, J. S. Edgar, G. D. M. Jefferies, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Crichton, J. H.

J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electronic J. Differ. Equations, Conf. 04, 37-50 (2000).

Dholakia, K.

Edgar, J. S.

Y. Zhao, J. S. Edgar, G. D. M. Jefferies, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[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.

H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, “Optical trapping of absorbing particles,” Adv. Quantum Chem. 30, 469-492 (1998).
[CrossRef]

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]

Gouesbet, G.

Grehan, G.

Hanna, S.

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430-443 (2007).
[CrossRef]

S. H. Simpson and S. Hanna, “Numerical calculation of inter-particle forces arising in association with holographic assembly,” J. Opt. Soc. Am. A 23, 1419-1431 (2006).
[CrossRef]

S. H. Simpson and S. Hanna, in preparation, to be called “Axial trapping in Laguerre-Gaussian beams.”

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.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, “Optical trapping of absorbing particles,” Adv. Quantum Chem. 30, 469-492 (1998).
[CrossRef]

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]

Jefferies, G. D. M.

Y. Zhao, J. S. Edgar, G. D. M. Jefferies, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Lee, S. S.

Lock, J. A.

Loudon, R.

M. J. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. 50, 1555-1562 (2003).
[CrossRef]

R. Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A 68, 013806 (2003).
[CrossRef]

MacVicar, I.

A. O'Neil, I. MacVicar, L. Allen, and M. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Marston, P. L.

J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electronic J. Differ. Equations, Conf. 04, 37-50 (2000).

McGloin, D.

Y. Zhao, J. S. Edgar, G. D. M. Jefferies, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Moine, O.

Molloy, J. E.

J. E. Molloy and M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. 43, 241-258 (2002).
[CrossRef]

Nieminen, T. A.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, “Optical trapping of absorbing particles,” Adv. Quantum Chem. 30, 469-492 (1998).
[CrossRef]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

O'Neil, A.

A. O'Neil, I. MacVicar, L. Allen, and M. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

O'Neil, A. T.

A. T. O'Neil and M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185, 139-143 (2000).
[CrossRef]

Padgett, M.

A. O'Neil, I. MacVicar, L. Allen, and M. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

L. Allen and M. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67-71 (2000).
[CrossRef]

Padgett, M. J.

M. J. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Opt. 50, 1555-1562 (2003).
[CrossRef]

J. E. Molloy and M. J. Padgett, “Lights, action: optical tweezers,” Contemp. Phys. 43, 241-258 (2002).
[CrossRef]

A. T. O'Neil and M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185, 139-143 (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]

Poynting, J. H.

J. H. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London, Ser. A 82, 560-567 (1909).
[CrossRef]

Rubinsztein-Dunlop, H.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical application and measurement of torque on microparticles of isotropic nonabsorbing material,” Phys. Rev. A 68, 033802 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transf. 70, 627-637 (2001).
[CrossRef]

H. Rubinsztein-Dunlop, T. A. Nieminen, M. E. J. Friese, and N. R. Heckenberg, “Optical trapping of absorbing particles,” Adv. Quantum Chem. 30, 469-492 (1998).
[CrossRef]

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]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

Simpson, N. B.

Simpson, S. H.

S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430-443 (2007).
[CrossRef]

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

S. H. Simpson and S. Hanna, “Numerical calculation of inter-particle forces arising in association with holographic assembly,” J. Opt. Soc. Am. A 23, 1419-1431 (2006).
[CrossRef]

S. H. Simpson and S. Hanna, in preparation, to be called “Axial trapping in Laguerre-Gaussian beams.”

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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).
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M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

van de Nes, A. S.

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

Zhao, Y.

Y. Zhao, J. S. Edgar, G. D. M. Jefferies, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

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Appl. Opt. (1)

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[CrossRef]

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J. Opt. Soc. Am. B (2)

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[CrossRef]

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[CrossRef]

Phys. Rev. A (5)

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]

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[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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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]

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[CrossRef] [PubMed]

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[CrossRef]

Other (2)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

S. H. Simpson and S. Hanna, in preparation, to be called “Axial trapping in Laguerre-Gaussian beams.”

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Figures (5)

Fig. 1
Fig. 1

Schematic showing the division of the incident field into “incoming” and “outgoing” parts.

Fig. 2
Fig. 2

Far-field intensities (normalized to one) of the incoming and outgoing parts of a fifth-order Davis beam.

Fig. 3
Fig. 3

Odd and even components of a fifth-order Davis beam plotted as a function of polar angle θ.

Fig. 4
Fig. 4

Integration surfaces appearing in Eq. (15).

Fig. 5
Fig. 5

Fractional error in the large-sphere approximation for n = 3 and 4 as a function of the sphere size factor x = k 1 a . The error is defined by { [ α e , o 2 Re ( α e , o ) ] [ α n 2 Re ( α n ) ] } [ α n 2 Re ( α n ) ] .

Equations (86)

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Rg M n m ( k r ) = 1 2 [ M n m ( 1 ) ( k r ) + M n m ( 2 ) ( k r ) ] ,
Rg N n m ( k r ) = 1 2 [ N n m ( 1 ) ( k r ) + N n m ( 2 ) ( k r ) ] .
E inc = n = 1 m = n n [ a n m ( 1 ) Rg M n m ( k 1 r ) + a n m ( 2 ) Rg N n m ( k 1 r ) ] ,
E scat = n = 1 m = n n [ p n m ( 1 ) M n m ( 1 ) ( k 1 r ) + p n m ( 2 ) N n m ( 1 ) ( k 1 r ) ] ,
M n m ( 1 ) ( k r ) ( i ) ( n + 1 ) e i k r k r γ n m C n m ( θ , ϕ ) ,
N n m ( 1 ) ( k r ) ( i ) n e i k r k r γ n m B n m ( θ , ϕ ) ,
M n m ( 2 ) ( k r ) i ( n + 1 ) e i k r k r γ n m C n m ( θ , ϕ ) ,
N n m ( 2 ) ( k r ) i n e i k r k r γ n m B n m ( θ , ϕ ) ,
E inc ( k 1 r ) = a ̃ Rg Ψ ( k 1 r ) ,
E scat ( k 1 r ) = p ̃ Ψ ( 1 ) ( k 1 r ) ,
p ̃ = T a ̃ .
Γ z = ϵ 1 2 π k 1 3 k = 1 2 n = 1 m = n n m { p n m ( k ) 2 + Re [ p n m ( k ) a n m * ( k ) ] } ,
lim k r f in ( ψ ) = 0 : θ π 2 ,
lim k r f out ( ψ ) = 0 : θ π 2 ,
ψ = f even ( ψ ) + f odd ( ψ ) ,
P n m ( cos θ ) = ( 1 ) n + m P n m ( cos θ ) ,
d P n m ( cos θ ) d θ = ( 1 ) n + m + 1 d P n m ( cos θ ) d θ .
a ̃ even = { a n m ( 1 ) : ( n + m ) even , a n m ( 2 ) : ( n + m ) odd } ,
a ̃ odd = { a n m ( 1 ) : ( n + m ) odd , a n m ( 2 ) : ( n + m ) even } .
f out ( ψ ) = f out [ f even ( ψ ) + f odd ( ψ ) ] = f out [ f even ( ψ ) ] + f out [ f odd ( ψ ) ] .
lim ( k r ) f out [ f even ( ψ ) ] = lim ( k r ) f out [ f odd ( ψ ) ] : θ π 2 .
ρ / t = j = s ,
t V ρ d v = j d σ = V s d v ,
j d σ = C j k ̂ d x d y + H j d σ = 0 .
C j ( ψ ) k ̂ d x d y = lim R H j [ f out ( ψ ) ] d σ = lim R S j [ f out ( ψ ) ] d σ .
E inc = e i k 1 r k 1 r F ( θ , ϕ ) ,
F ( θ , ϕ ) = n , m γ n m 2 ( i ) n [ i a n m ( 1 ) C n m ( θ , ϕ ) + a n m ( 2 ) B n m ( θ , ϕ ) ] .
H inc = ϵ μ e i k 1 r k 1 r [ r ̂ × F ( θ , ϕ ) ] .
S ( r ) = r ̂ 1 2 r 2 ϵ μ F F * .
P = S ( r ) k ̂ d x d y = 0 2 π d ϕ 0 π d θ sin ( θ ) S ( r ) r ̂ = 1 8 k 1 2 ϵ μ n , m a n m ( 1 ) 2 + a n m ( 2 ) 2 = 1 8 k 1 2 ϵ μ a ̃ 2 .
a ̃ even 2 = a ̃ odd 2 .
J z = d x d y k ̂ [ T M ( r ) × r ] = ϵ 1 8 k 1 3 n , m m ( a n m ( 1 ) 2 + a n m ( 2 ) 2 ) .
E = α E ( ρ , z ) e i l ϕ i ̂ + β E ( ρ , z ) e i l ϕ j ̂ + E z ( ρ , ϕ , z ) k ̂ .
z E z = x ( α E e i l ϕ ) y ( β E e i l ϕ )
E z = cos ϕ e i l ϕ [ α E z ( 1 ) ( ρ , z ) + β i l ρ E z ( 2 ) ( ρ , z ) ] + sin ϕ e i l ϕ [ α i l ρ E z ( 2 ) ( ρ , z ) β E z ( 1 ) ( ρ , z ) ] .
σ z = i ( α β * β α * ) ,
a ̃ + = { g n + : m = l + 1 0 : otherwise } ; a ̃ = { g n : m = l 1 0 : otherwise } ,
a ̃ + 2 = a ̃ 2 ,
a ̃ x = 1 2 ( a ̃ + + a ̃ ) ; a ̃ y = i 2 ( a ̃ + a ̃ ) ,
n min = l + σ z
a ̃ + 2 = a ̃ 2 = a ̃ x 2 = a ̃ y 2 = A ,
J z ( σ z ) = ϵ 1 8 k 1 3 ( l + σ z ) A .
J z ( α z ) P = ( l + σ z ) ω ,
T m n m n 11 = δ m m δ n n α n , T m n m n 22 = δ m m δ n n β n ,
α n = η Ψ n ( x ) Ψ n ( η x ) Ψ n ( η x ) Ψ n ( x ) η ζ n ( x ) Ψ n ( η x ) Ψ n ( η x ) ζ n ( x ) ,
β n = η Ψ n ( η x ) Ψ n ( x ) Ψ n ( x ) Ψ n ( η x ) η ζ n ( x ) Ψ n ( η x ) Ψ n ( η x ) ζ n ( x ) ,
Γ z + = ϵ 1 2 k 1 3 ( l + 1 ) n = 1 { a n , l + 1 ( 1 ) 2 [ α n 2 Re ( α n ) ] + a n , l + 1 ( 2 ) 2 [ β n 2 Re ( β n ) ] } ,
Γ z = ϵ 1 2 k 1 3 ( l 1 ) n = 1 { a n , l 1 ( 1 ) 2 [ α n 2 Re ( α n ) ] + a n , l 1 ( 2 ) 2 [ β n 2 Re ( β n ) ] } ,
Γ z x = 1 2 ( Γ z + + Γ z ) ,
α e = [ cos ( x ) sin ( η x ) η sin ( x ) cos ( η x ) ] e i x [ i η cos ( η x ) + sin ( η x ) ] ,
α o = [ η cos ( x ) sin ( η x ) sin ( x ) cos ( η x ) ] e i x [ η sin ( η x ) + i cos ( η x ) ] ;
β e = α o ,
β o = α e ;
Γ z + ϵ 1 2 k 3 ( l + 1 ) { a ̃ + , even 2 [ α e 2 Re ( α e ) ] + a ̃ + , odd 2 [ α o 2 Re ( α o ) ] } = ϵ 1 2 k 3 A 2 ( l + 1 ) [ α e 2 + α o 2 Re ( α e + α o ) ] ,
Γ z ( σ z ) P 2 ( l + σ z ) [ α e 2 + α o 2 Re ( α e + α o ) ] ω .
n 2 n + i n ,
α e = z 1 z 1 + i z 2 ,
z 1 = cos ( x ) sin ( η x ) η sin ( x ) cos ( η x ) ,
z 2 = sin ( x ) sin ( η x ) + η cos ( x ) cos ( η x ) .
α e 2 Re ( α ) = i Im ( z 1 z 2 ) z 1 + i z 2 2 .
z 1 = cos ( x ) [ sin ( η x ) cosh ( δ x ) + i sinh ( δ x ) cos ( η x ) ] ( η + i δ ) sin ( x ) [ cos ( η x ) cosh ( δ x ) i sin ( η x ) sinh ( δ x ) ] ,
z 2 = sin ( x ) [ sin ( η x ) cosh ( δ x ) + i sinh ( δ x ) cos ( η x ) ] + ( η + i δ ) cos ( x ) [ cos ( η x ) cosh ( δ x ) i sin ( η x ) sinh ( δ x ) ] .
z 1 a 0 + i a 1 δ + a 2 δ 2 + ,
z 2 b 0 + i b 1 δ + b 2 δ 2 + ,
a 0 = cos ( x ) sin ( η x ) η sin ( x ) cos ( η x ) ,
a 1 = x [ cos ( x ) cos ( η x ) + η sin ( x ) sin ( η x ) ] sin ( x ) cos ( η x ) ,
a 2 = 1 2 x 2 [ cos ( x ) sin ( η x ) η sin ( x ) cos ( η x ) ] x sin ( x ) cos ( η x ) ,
b 0 = sin ( x ) sin ( η x ) + η cos ( x ) cos ( η x ) ,
b 1 = x [ sin ( x ) cos ( η x ) η cos ( x ) sin ( η x ) ] + cos ( x ) cos ( η x ) ,
b 2 = 1 2 x 2 [ sin ( x ) sin ( η x ) + η cos ( x ) cos ( η x ) ] + x cos ( x ) sin ( η x ) .
i Im ( z 1 z 2 ) z 1 + i z 2 2 ( a 0 b 1 a 1 b 0 ) a 0 2 + b 0 2 δ + 2 ( a 0 b 1 a 1 b 0 ) 2 ( a 0 2 + b 0 2 ) 2 δ 2 .
a 0 b 1 a 1 b 0 a 0 2 + b 0 2 = [ η x ( 1 2 ) sin ( 2 η x ) 1 + ( η 2 1 ) cos 2 ( η x ) ] C e .
a 0 b 1 a 1 b 0 a 0 2 + b 0 2 = [ η x + ( 1 2 ) sin ( 2 η x ) 1 + ( η 2 1 ) sin 2 ( η x ) ] C o .
α e 2 + α o 2 Re ( α e + α o ) ( C e + C o ) δ + 2 ( C e 2 + C o 2 ) δ 2 ,
C e + C o = [ η x ( η 2 + 1 ) ( 1 4 ) sin ( 4 η x ) ( η 2 1 ) η 2 + ( 1 4 ) ( η 2 1 ) 2 sin 2 ( 2 η x ) ] ,
C e 2 + C o 2 = ( C e + C o ) 2 2 C e C o = [ η x ( η 2 + 1 ) ( 1 4 ) sin ( 4 η x ) ( η 2 1 ) η 2 + ( 1 4 ) ( η 2 1 ) 2 sin 2 ( 2 η x ) ] 2 [ 2 ( η x ) 2 ( 1 2 ) sin 2 ( 2 η x ) η 2 + ( 1 4 ) ( η 2 1 ) 2 sin ( 2 η x ) ] .
η 2 ( 1 4 ) ( η 2 1 ) 2 .
α e 2 + α o 2 Re ( α e + α o ) ( η + η 1 ) ( x δ ) + 2 ( η 2 + η 2 ) ( x δ ) 2 + .
Γ z ( σ z ) P 2 x δ ω ( l + σ z ) ( η + η 1 ) = 2 a n c ( l + σ z ) ( n n 1 + n 1 n ) ,
( F n m , G n m ) = 0 2 π d ϕ 0 π d θ sin ( θ ) F n m ( θ , ϕ ) G n m * ( θ , ϕ ) ,
( B n m , C n m ) = ( B n m , P n m ) = ( C n m , P n m ) = 0 ,
( B n m , B n m ) = ( C n m , C n m ) = 1 ( γ n m ) 2 δ n n δ m m ,
( P n m , P n m ) = 1 ( γ n m ) 2 δ n n δ m m ,
0 2 π d ϕ e i m ϕ e i m ϕ = 2 π δ m m .
( E inc , C n m ) = a n m ( 1 ) j n ( k 1 r ) δ n n δ m m γ n m ,
( E inc , B n m ) = a n m ( 2 ) γ n m 1 k 1 r d [ k 1 r j n ( k 1 r ) ] d ( k 1 r ) δ n n δ m m .

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