Abstract

An expression for a circularly polarized Gaussian laser beam is obtained from the electromagnetic vector potential A, of which the scalar part becomes the complex-source-point spherical wave. Based on the theory of laser-beam scattering by a stationary homogeneous sphere, the analytical formulas are derived for the optical torque components exerted on the levitated sphere in the circularly polarized focused laser beam. The optical torque is numerically calculated, and interpretations of the results are presented. For a sphere with size parameters ρ = (2πα/λ) = 10π and complex refractive index N = 1.47 + i0.000001, levitated at the center of a 1-W Ar+ laser beam, the z component of angular velocity in air ωrot,z is found to be about 4.0 rad/sec.

© 1985 Optical Society of America

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References

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  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [Crossref]
  2. J. S. Kim and S. S. Lee, “Radiation pressure on a dielectric sphere in a Gaussian laser beam,” Opt. Acta 29, 801–806 (1982).
    [Crossref]
  3. A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283–285 (1971).
    [Crossref]
  4. A. Ashkin, “The pressure of laser light,” Sci. Am. 226, 63–71 (1972).
    [Crossref]
  5. A. Ashkin and J. M. Dziedzic, “Stability of optical levitation by radiation pressure,” Appl. Phys. Lett. 24, 586–588 (1974).
    [Crossref]
  6. A. Ashkin, “Optical levitation of liquid drops by radiation pressure,” Science 187, 1073–1075 (1975).
    [Crossref] [PubMed]
  7. A. Ashkin and J. M. Dziedzic, “Optical levitation in high vacuum,” Appl. Phys. Lett. 28, 333–335 (1976).
    [Crossref]
  8. G. Roosen and C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. 59A, 6–8 (1976).
  9. G. Roosen, B. Delaunay, and C. Imbert, “Radiation pressure exerted by a light beam on refractive spheres: theoretical and experimental study,” J. Opt. 8, 181–187 (1977).
    [Crossref]
  10. G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
    [Crossref]
  11. G. Roosen and C. Imbert, “The TEM01* mode laser beam—a powerful tool for optical levitation of various types of spheres,” Opt. Commun. 26, 432–436 (1978).
    [Crossref]
  12. G. Roosen and S. Slansky, “Influence of the beam divergence on the exerted force on a sphere by a laser beam and required conditions for stable optical levitation,” Opt. Commun. 29, 341–346 (1979).
    [Crossref]
  13. G. Roosen, “La levitation optique de spheres,” Can. J. Phys. 57, 1260–1279 (1979).
    [Crossref]
  14. A. Ashkin, “Applications of laser radiation pressure,” Science 210, 1081–1088 (1980).
    [Crossref] [PubMed]
  15. A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
    [Crossref]
  16. G. Grehan and G. Gouesbet, “Optical levitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
    [Crossref] [PubMed]
  17. A. Ashkin and J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19, 660–668 (1980).
    [Crossref] [PubMed]
  18. A. Ashkin and J. M. Dziedzic, “Observation of a new nonlinear photoelectric effect using optical levitation,” Phys. Rev. Lett. 36, 267–270 (1976).
    [Crossref]
  19. A. Ashkin and J. M. Dziedzic, “Feedback stabilization of optically levitated particles,” Appl. Phys. Lett. 30, 202–204 (1977).
    [Crossref]
  20. 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); “Radiation torque on a sphere illuminated with circularly polarized light,” J. Opt. Soc. Am. B 1, 528–529 (1984); “Radiation torque on a sphere illuminated with circularly-polarized light and the angular momentum of the scattered radiation,” presented at the 1984 CRDC Conference on Obscuration Science and Aerosol Research, 1984.
    [Crossref]
  21. J. S. Kim and S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73, 303–312 (1983).
    [Crossref]
  22. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [Crossref] [PubMed]
  23. G. A. Descamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [Crossref]
  24. L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
    [Crossref]
  25. S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
    [Crossref]
  26. L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19, 1177–1179 (1979).
    [Crossref]
  27. M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
    [Crossref]
  28. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 414–420, 485–486.
  29. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 429.
  30. E. J. Konopinski, Electromagnetic Fields and Relativistic Particles (McGraw-Hill, New York, 1981), pp. 160–170.
  31. P. Chylek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross-section,” J. Opt. Soc. Am. 66, 285–287 (1976).
    [Crossref]
  32. P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
    [Crossref]
  33. C. F. Bohren, “How can a particle absorb more than the light incident on it?” Am. J. Phys. 51, 323–327 (1983).
    [Crossref]
  34. T. R. Lettieri, W. D. Jenkins, and D. A. Swyt, “Sizing of individual optically levitated evaporating droplets by measurement of resonances in the polarization ratio,” Appl. Opt. 20, 2799–2805 (1981).
    [Crossref] [PubMed]
  35. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, London, 1959), pp. 63–69.
  36. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970), pp. 524–525, 560–561.

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); “Radiation torque on a sphere illuminated with circularly polarized light,” J. Opt. Soc. Am. B 1, 528–529 (1984); “Radiation torque on a sphere illuminated with circularly-polarized light and the angular momentum of the scattered radiation,” presented at the 1984 CRDC Conference on Obscuration Science and Aerosol Research, 1984.
[Crossref]

1983 (2)

C. F. Bohren, “How can a particle absorb more than the light incident on it?” Am. J. Phys. 51, 323–327 (1983).
[Crossref]

J. S. Kim and S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73, 303–312 (1983).
[Crossref]

1982 (1)

J. S. Kim and S. S. Lee, “Radiation pressure on a dielectric sphere in a Gaussian laser beam,” Opt. Acta 29, 801–806 (1982).
[Crossref]

1981 (2)

1980 (3)

1979 (3)

G. Roosen and S. Slansky, “Influence of the beam divergence on the exerted force on a sphere by a laser beam and required conditions for stable optical levitation,” Opt. Commun. 29, 341–346 (1979).
[Crossref]

G. Roosen, “La levitation optique de spheres,” Can. J. Phys. 57, 1260–1279 (1979).
[Crossref]

L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

1978 (2)

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[Crossref]

G. Roosen and C. Imbert, “The TEM01* mode laser beam—a powerful tool for optical levitation of various types of spheres,” Opt. Commun. 26, 432–436 (1978).
[Crossref]

1977 (5)

A. Ashkin and J. M. Dziedzic, “Feedback stabilization of optically levitated particles,” Appl. Phys. Lett. 30, 202–204 (1977).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

G. Roosen, B. Delaunay, and C. Imbert, “Radiation pressure exerted by a light beam on refractive spheres: theoretical and experimental study,” J. Opt. 8, 181–187 (1977).
[Crossref]

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
[Crossref]

S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
[Crossref]

1976 (5)

P. Chylek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross-section,” J. Opt. Soc. Am. 66, 285–287 (1976).
[Crossref]

L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Observation of a new nonlinear photoelectric effect using optical levitation,” Phys. Rev. Lett. 36, 267–270 (1976).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Optical levitation in high vacuum,” Appl. Phys. Lett. 28, 333–335 (1976).
[Crossref]

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

1975 (1)

A. Ashkin, “Optical levitation of liquid drops by radiation pressure,” Science 187, 1073–1075 (1975).
[Crossref] [PubMed]

1974 (1)

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

1972 (1)

A. Ashkin, “The pressure of laser light,” Sci. Am. 226, 63–71 (1972).
[Crossref]

1971 (2)

A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283–285 (1971).
[Crossref]

G. A. Descamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[Crossref]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[Crossref]

1966 (1)

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970), pp. 524–525, 560–561.

Ashkin, A.

A. Ashkin, “Applications of laser radiation pressure,” Science 210, 1081–1088 (1980).
[Crossref] [PubMed]

A. Ashkin and J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19, 660–668 (1980).
[Crossref] [PubMed]

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Feedback stabilization of optically levitated particles,” Appl. Phys. Lett. 30, 202–204 (1977).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Observation of a new nonlinear photoelectric effect using optical levitation,” Phys. Rev. Lett. 36, 267–270 (1976).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Optical levitation in high vacuum,” Appl. Phys. Lett. 28, 333–335 (1976).
[Crossref]

A. Ashkin, “Optical levitation of liquid drops by radiation pressure,” Science 187, 1073–1075 (1975).
[Crossref] [PubMed]

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

A. Ashkin, “The pressure of laser light,” Sci. Am. 226, 63–71 (1972).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283–285 (1971).
[Crossref]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[Crossref]

Belanger, P. A.

M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[Crossref]

Bohren, C. F.

C. F. Bohren, “How can a particle absorb more than the light incident on it?” Am. J. Phys. 51, 323–327 (1983).
[Crossref]

Chylek, P.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[Crossref]

P. Chylek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross-section,” J. Opt. Soc. Am. 66, 285–287 (1976).
[Crossref]

Couture, M.

M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[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); “Radiation torque on a sphere illuminated with circularly polarized light,” J. Opt. Soc. Am. B 1, 528–529 (1984); “Radiation torque on a sphere illuminated with circularly-polarized light and the angular momentum of the scattered radiation,” presented at the 1984 CRDC Conference on Obscuration Science and Aerosol Research, 1984.
[Crossref]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

Delaunay, B.

G. Roosen, B. Delaunay, and C. Imbert, “Radiation pressure exerted by a light beam on refractive spheres: theoretical and experimental study,” J. Opt. 8, 181–187 (1977).
[Crossref]

Descamps, G. A.

G. A. Descamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[Crossref]

Dziedzic, J. M.

A. Ashkin and J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19, 660–668 (1980).
[Crossref] [PubMed]

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Feedback stabilization of optically levitated particles,” Appl. Phys. Lett. 30, 202–204 (1977).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Observation of a new nonlinear photoelectric effect using optical levitation,” Phys. Rev. Lett. 36, 267–270 (1976).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Optical levitation in high vacuum,” Appl. Phys. Lett. 28, 333–335 (1976).
[Crossref]

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

A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283–285 (1971).
[Crossref]

Felsen, L. B.

Gouesbet, G.

Grehan, G.

Imbert, C.

G. Roosen and C. Imbert, “The TEM01* mode laser beam—a powerful tool for optical levitation of various types of spheres,” Opt. Commun. 26, 432–436 (1978).
[Crossref]

G. Roosen, B. Delaunay, and C. Imbert, “Radiation pressure exerted by a light beam on refractive spheres: theoretical and experimental study,” J. Opt. 8, 181–187 (1977).
[Crossref]

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 429.

Jenkins, W. D.

Kiehl, J. T.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[Crossref]

Kim, J. S.

J. S. Kim and S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73, 303–312 (1983).
[Crossref]

J. S. Kim and S. S. Lee, “Radiation pressure on a dielectric sphere in a Gaussian laser beam,” Opt. Acta 29, 801–806 (1982).
[Crossref]

Ko, M. K. W.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[Crossref]

Kogelnik, H.

Konopinski, E. J.

E. J. Konopinski, Electromagnetic Fields and Relativistic Particles (McGraw-Hill, New York, 1981), pp. 160–170.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, London, 1959), pp. 63–69.

Lee, S. S.

J. S. Kim and S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73, 303–312 (1983).
[Crossref]

J. S. Kim and S. S. Lee, “Radiation pressure on a dielectric sphere in a Gaussian laser beam,” Opt. Acta 29, 801–806 (1982).
[Crossref]

Lettieri, T. R.

Li, T.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, London, 1959), pp. 63–69.

Marston, P. L.

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); “Radiation torque on a sphere illuminated with circularly polarized light,” J. Opt. Soc. Am. B 1, 528–529 (1984); “Radiation torque on a sphere illuminated with circularly-polarized light and the angular momentum of the scattered radiation,” presented at the 1984 CRDC Conference on Obscuration Science and Aerosol Research, 1984.
[Crossref]

Roosen, G.

G. Roosen, “La levitation optique de spheres,” Can. J. Phys. 57, 1260–1279 (1979).
[Crossref]

G. Roosen and S. Slansky, “Influence of the beam divergence on the exerted force on a sphere by a laser beam and required conditions for stable optical levitation,” Opt. Commun. 29, 341–346 (1979).
[Crossref]

G. Roosen and C. Imbert, “The TEM01* mode laser beam—a powerful tool for optical levitation of various types of spheres,” Opt. Commun. 26, 432–436 (1978).
[Crossref]

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
[Crossref]

G. Roosen, B. Delaunay, and C. Imbert, “Radiation pressure exerted by a light beam on refractive spheres: theoretical and experimental study,” J. Opt. 8, 181–187 (1977).
[Crossref]

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

Shin, S. Y.

Slansky, S.

G. Roosen and S. Slansky, “Influence of the beam divergence on the exerted force on a sphere by a laser beam and required conditions for stable optical levitation,” Opt. Commun. 29, 341–346 (1979).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 414–420, 485–486.

Swyt, D. A.

Am. J. Phys. (1)

C. F. Bohren, “How can a particle absorb more than the light incident on it?” Am. J. Phys. 51, 323–327 (1983).
[Crossref]

Appl. Opt. (4)

Appl. Phys. Lett. (4)

A. Ashkin and J. M. Dziedzic, “Feedback stabilization of optically levitated particles,” Appl. Phys. Lett. 30, 202–204 (1977).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283–285 (1971).
[Crossref]

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

A. Ashkin and J. M. Dziedzic, “Optical levitation in high vacuum,” Appl. Phys. Lett. 28, 333–335 (1976).
[Crossref]

Can. J. Phys. (1)

G. Roosen, “La levitation optique de spheres,” Can. J. Phys. 57, 1260–1279 (1979).
[Crossref]

Electron. Lett. (1)

G. A. Descamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[Crossref]

J. Opt. (1)

G. Roosen, B. Delaunay, and C. Imbert, “Radiation pressure exerted by a light beam on refractive spheres: theoretical and experimental study,” J. Opt. 8, 181–187 (1977).
[Crossref]

J. Opt. Soc. Am. (4)

Opt. Acta (1)

J. S. Kim and S. S. Lee, “Radiation pressure on a dielectric sphere in a Gaussian laser beam,” Opt. Acta 29, 801–806 (1982).
[Crossref]

Opt. Commun. (3)

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
[Crossref]

G. Roosen and C. Imbert, “The TEM01* mode laser beam—a powerful tool for optical levitation of various types of spheres,” Opt. Commun. 26, 432–436 (1978).
[Crossref]

G. Roosen and S. Slansky, “Influence of the beam divergence on the exerted force on a sphere by a laser beam and required conditions for stable optical levitation,” Opt. Commun. 29, 341–346 (1979).
[Crossref]

Phys. Lett. (1)

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

Phys. Rev. A (4)

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[Crossref]

L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[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); “Radiation torque on a sphere illuminated with circularly polarized light,” J. Opt. Soc. Am. B 1, 528–529 (1984); “Radiation torque on a sphere illuminated with circularly-polarized light and the angular momentum of the scattered radiation,” presented at the 1984 CRDC Conference on Obscuration Science and Aerosol Research, 1984.
[Crossref]

Phys. Rev. Lett. (3)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Observation of a new nonlinear photoelectric effect using optical levitation,” Phys. Rev. Lett. 36, 267–270 (1976).
[Crossref]

Sci. Am. (1)

A. Ashkin, “The pressure of laser light,” Sci. Am. 226, 63–71 (1972).
[Crossref]

Science (2)

A. Ashkin, “Optical levitation of liquid drops by radiation pressure,” Science 187, 1073–1075 (1975).
[Crossref] [PubMed]

A. Ashkin, “Applications of laser radiation pressure,” Science 210, 1081–1088 (1980).
[Crossref] [PubMed]

Other (5)

L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, London, 1959), pp. 63–69.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970), pp. 524–525, 560–561.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 414–420, 485–486.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 429.

E. J. Konopinski, Electromagnetic Fields and Relativistic Particles (McGraw-Hill, New York, 1981), pp. 160–170.

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Figures (7)

Fig. 1
Fig. 1

The circularly polarized Gaussian laser beam propagating along the z axis. The origin of the coordinate system is at the center of the sphere of radius a, and the beam center is at (x0, y0, z0).

Fig. 2
Fig. 2

z component of the optical torque of the sphere levitated at the center of the laser beam as the index of extinction κ changes from 0 to 1 × 10−6.

Fig. 3
Fig. 3

z component of the torque of the sphere levitated at the beam center as the size parameter ρ is varied. The computation is made for Δρ = 0.01. The two types of resonance peaks (⇣ and ↓) are shown.

Fig. 4
Fig. 4

Variation of the z component of the torque as the sphere is displaced from the beam center along the laser-beam axis.

Fig. 5
Fig. 5

The optical torque components, Trot,y, and Trot,z, when the sphere is gradually displaced from the beam center to x0 = −8λ, y0 = z0 = 0.

Fig. 6
Fig. 6

The optical torque components, Trot,y and Trot,z, when the sphere is gradually displaced from the beam center to x0 = −10λ, y0 = z0 = 0.

Fig. 7
Fig. 7

The z component of the torque Trev,z for revolution around the laser-beam axis as the sphere moves from the beam center to x0 = −8λ, y0 = z0 = 0.

Equations (30)

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E i = ( x ^ ± i y ^ ) - i b z - i b exp [ i k z + i k ( x 2 + y 2 2 ( z - i b ) ] ,
ψ = C exp ( i k R ) i k R , R = [ ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 - i b ) 2 ] 1 / 2 ,
A = ( x ^ ± i y ^ ) ψ , E i = i k × × A , H i = × A ,
ψ = { C l , m H ( 1 ) ( l , m ) F l , m ( 0 ) r < R 0 exp ( ± i Δ ) , C l , m H ( 0 ) ( l , m ) F l , m ( 1 ) R 0 < r exp ( ± i Δ ) , 0 θ < π / 2 , C l , m H ( 0 ) ( l , m ) F l , m ( 2 ) R 0 < r exp ( ± i Δ ) , π / 2 < θ π ,
Δ = cos - 1 [ x x 0 + y y 0 + z ( z 0 + i b ) r R 0 ] , F l , m ( σ ) = Z l ( σ ) ( k r ) P l m ( cos θ ) exp ( i m ϕ )             ( σ = 0 , 1 , 2 ) , H ( η ) ( l , m ) = ( 2 l + 1 ) ( l - m ) ! ( l + m ) ! Z l ( η ) ( k R 0 ) P l m ( cos θ 0 ) × exp ( - i m ϕ 0 )             ( η = 0 , 1 ) ,
R 0 = [ ( x 0 2 + y 0 2 + ( z 0 + i b ) 2 ] 1 / 2 , cos θ 0 = z 0 + i b R 0 , ϕ 0 = tan - 1 ( y 0 x 0 ) .
A = ( C k ) l , m [ α ( η ) ( l , m ) M l , m ( σ ) + β ( η ) ( l , m ) N l , m ( σ ) + γ ( η ) ( l , m ) L l , m ( σ ) ] ,
E i = i C l , m [ α ( η ) ( l , m ) M l , m ( σ ) + β ( η ) ( l , m ) N l , m ( σ ) ] , H i = C l , m [ β ( η ) ( l , m ) M l , m ( σ ) + α ( η ) ( l , m ) N l , m ( σ ) ] .
α ( η ) ( l , m ) = ( - i k ) H ( η ) ( l , m - 1 ) l ( l + 1 ) , β ( η ) ( l , m ) = ( - k ) [ H ( η ) ( l + 1 , m - 1 ) ( 2 l + 3 ) ( l + 1 ) - H ( η ) ( l - 1 , m - 1 ) ( 2 l - 1 ) l ]
α ( η ) ( l , m ) = ( - i k ) ( l - m ) ( l + m + 1 ) l ( l + 1 ) H ( η ) ( l , m + 1 ) , β ( η ) ( l , m ) = k [ ( l + m + 1 ) ( l + m + 2 ) ( 2 l + 3 ) ( l + 1 ) H ( η ) ( l + 1 , m + 1 ) - ( l - m ) ( l - m - 1 ) ( 2 l - 1 ) l H ( η ) ( l - 1 , m + 1 ) ]
E s = i C l , m [ a l α ( 1 ) ( l , m ) M l , m ( 1 ) + b l β ( 1 ) ( l , m ) N l , m ( 1 ) ] , H s = C l , m [ a l α ( 1 ) ( l , m ) N l , m ( 1 ) + b l β ( 1 ) ( l , m ) M l , m ( 1 ) ] , E t = i C l , m [ c l α ( 1 ) ( l , m ) M ˜ l , m ( 0 ) + d l β ( 1 ) ( l , m ) N ˜ l , m ( 0 ) ] , H t = C l , m [ c l α ( 1 ) ( l , m ) N ˜ l , m ( 0 ) + d l β ( 1 ) ( l , m ) M ˜ l , m ( 0 ) ] ,
a l = - j l ( N ρ ) [ ρ j l ( ρ ) ] - [ N ρ j l ( N ρ ) ] j l ( ρ ) j l ( N ρ ) [ ρ h l ( 1 ) ( ρ ) ] - [ N ρ j l ( N ρ ) ] h l ( 1 ) ( ρ ) , b l = - N 2 j l ( N ρ ) [ ρ j l ( ρ ) ] - [ N ρ j l ( N ρ ) ] j l ( ρ ) N 2 j l ( N ρ ) [ ρ h l ( 1 ) ( ρ ) ] - [ N ρ j l ( N ρ ) ] h l ( 1 ) ( ρ ) , c l = j l ( ρ ) [ ρ h l ( 1 ) ( ρ ) ] - [ ρ j l ( ρ ) ] h l ( 1 ) ( ρ ) j l ( N ρ ) [ ρ h l ( 1 ) ( ρ ) ] - [ N ρ j l ( N ρ ) ] h l ( 1 ) ( ρ ) , d l = N j l ( ρ ) [ ρ h l ( 1 ) ( ρ ) ] - [ ρ j l ( ρ ) ] h l ( 1 ) ( ρ ) N 2 j l ( N ρ ) [ ρ h l ( 1 ) ( ρ ) ] - [ N ρ j l ( N ρ ) ] h l ( 1 ) ( ρ ) ,
M = 1 8 π Re [ ( E out E out * + H out H out * ) - 1 2 ( E out 2 + H out 2 ) I ] ,
T rot = - d s · M × r ,
T rot , x = C 2 4 k 3 Re l , m l ( l + 1 ) 2 l + 1 ( l + m ) ! ( l - m ) ! ( l + m + 1 ) ( l - m ) × [ β ( 1 ) ( l , m ) β ( 1 ) * ( l , m + 1 ) ( b l + b l * + 2 b l 2 ) + α ( 1 ) ( l , m ) α ( 1 ) * ( l , m + 1 ) ( a l + a l * + 2 a l 2 ) ] , T rot , y = C 2 4 k 3 Im l , m l ( l + 1 ) 2 l + 1 ( l + m ) ! ( l - m ) ! ( l + m + 1 ) ( l - m ) × [ β ( 1 ) ( l , m ) β ( 1 ) * ( l , m + 1 ) ( b l + b l * + 2 b l 2 ) + α ( 1 ) ( l , m ) α ( 1 ) * ( l , m + 1 ) ( a l + a l * + 2 a l 2 ) ] , T rot , z = - C 2 2 k 3 Re l , m l ( l + 1 ) 2 l + 1 ( l + m ) ! ( l - m ) ! m [ α ( 1 ) ( l , m ) 2 × ( a l 2 + a l ) + β ( 1 ) ( l , m ) 2 ( b l 2 + b l ) ] .
α ( 1 ) ( l , 1 ) = β ( 1 ) ( l , 1 ) = k exp ( k b ) k b 2 l + 1 l ( l + 1 ) i l - 1
α ( 1 ) ( l , - 1 ) = - β ( 1 ) ( l , - 1 ) = k exp ( k b ) k b ( 2 l + 1 ) i l - 1
T rot , x = T rot , y = 0 T rot , z = ± C ˜ 2 2 k Re l ( 2 l + 1 ) ( a l 2 + b l 2 + a l + b l ) ,
T rev , z = ( d s · M × r 0 ) z = - ( r 0 × F ) z = - x 0 F y + y 0 F x ,
F x = C 2 4 k 2 Re l , m ( l + m ) ! ( 2 l + 1 ) ( l + m ) ! × { - i l ( l + 2 ) ( l + m + 1 ) ( l + m + 2 ) ( 2 l + 3 ) × [ M i a α ( 1 ) ( l , m ) α ( 1 ) * ( l + 1 , m + 1 ) + M i b β ( 1 ) ( l , m ) β ( 1 ) * ( l + 1 , m + 1 ) ] + l ( l + 2 ) 2 l + 3 [ M i a α ( 1 ) ( l , m ) α ( 1 ) * ( l + 1 , m - 1 ) + M i b β ( 1 ) ( l , m ) β ( 1 ) * ( l + 1 , m - 1 ) ] + ( l - m ) ( l + m + 1 ) M i c α ( 1 ) ( l , m ) β ( 1 ) * ( l , m + 1 ) + M i c α ( 1 ) ( l , m ) β ( 1 ) * ( l , m - 1 ) } , F y = C 2 4 k 2 Im l , m ( l + m ) ! ( 2 l + 1 ) ( l - m ) ! × { - i l ( l + 2 ) ( l + m + 1 ) ( l + m + 2 ) 2 l + 3 × [ M i a α ( 1 ) ( l , m ) α ( 1 ) * ( l + 1 , m + 1 ) + M i b β ( 1 ) ( l , m ) β ( 1 ) * ( l + 1 , m + 1 ) ] - i l ( l + 2 ) 2 l + 3 [ M i a α ( 1 ) ( l , m ) α ( 1 ) * ( l + 1 , m - 1 ) + M i b β ( 1 ) ( l , m ) β ( 1 ) * ( l + 1 , m - 1 ) ] + ( l - m ) ( l + m + 1 ) M i c α ( 1 ) ( l , m ) β ( 1 ) * ( l , m + 1 ) - M i c α ( 1 ) ( l , m ) α ( 1 ) * ( l , m - 1 ) } ,
T drag , j = - 8 π η a 3 ω j , F drag , j = - 6 π η a v j ,             j = x , y , z ,
× A = × [ ( x ^ ± i y ^ ) C l , m H ( η ) ( l , m ) F l , m ( σ ) ] = C l , m [ α ( η ) ( l , m ) N l , m ( σ ) + β ( η ) ( l , m ) M l , m ( σ ) ] ,
l , m α ( η ) ( l , m ) [ l ( l + 1 ) k r Z l ( η ) P l m exp ( i m ϕ ) ] = l , m i H ( η ) ( l , m ) 1 r Z l ( σ ) ( ± d P l m d θ - m cos θ sin θ P l m ) exp [ i ( m ± 1 ) ϕ ] = { l , m - i H ( η ) ( l , m ) 1 r Z l ( σ ) P l m + 1 exp [ i ( m + 1 ) ϕ ] : ( + ) l , m - i H ( η ) ( l , m ) ( l + m ) ( l - m + 1 ) r Z l ( σ ) P l m - 1 exp [ i ( m - 1 ) ϕ ] : ( - ) = { l , m [ - i k H ( η ) ( l , m - 1 ) 1 l ( l + 1 ) ] [ l ( l + 1 ) k r Z l ( σ ) P l m exp ( i m ϕ ) ] : ( + ) l , m [ - i k H ( η ) ( l , m + 1 ) ( l - m ) ( l + m + 1 ) l ( l + 1 ) ] [ l ( l + 1 ) k r Z l ( σ ) P l m exp ( i m ϕ ) ] : ( - ) .
W in = S i n · d s = ( c / 8 π ) Re ( E i × H i * ) · d s = 1 × 10 7 ( erg / sec ) ,
W in = 2 c k 2 C 2 l , l l ( l + 1 ) { Im [ α ( 1 ) ( l , 1 ) α ( 1 ) * ( l , 1 ) ] × d P l 1 ( 0 ) d θ P l 1 ( 0 ) sin [ ( l - l ) π / 2 ] ( l - l ) ( l + l + 1 ) + Re [ α ( 1 ) ( l , 1 ) α ( 1 ) * ( l , 1 ) ] × P l 1 ( 0 ) P l 1 ( 0 ) ( l + 1 ) sin [ ( l - l + 1 ) π / 2 ] ( 2 l + 1 ) ( l + l ) ( l - l + 1 ) - l sin [ ( l - l - 1 ) π / 2 ] ( 2 l + 1 ) ( l - l - 1 ) ( l + l + 2 ) } = C 2 Q .
T rot = - 1 8 π Re d s [ r ^ · ( E i * + E s * ) ( E i + E s ) × r + r ^ · ( B i * + B s * ) ( B i + B s ) × r ] ,
T rot = C 2 8 π Re l , m l , m 0 π d θ 0 2 π d ϕ exp [ i ( m - m ) ϕ ] × { ϕ ^ [ P l m P l m ( i m ) U + sin θ P l m d P l m d θ V ] + θ ^ [ sin θ P m d P l m l d θ U - P l m P l m ( i m ) V ] } ,
U = l ( l + 1 ) k r r 3 { j l j l [ β ( 1 ) * ( l , m ) α ( 1 ) ( l , m ) + α ( 1 ) * ( l , m ) β ( 1 ) ( l , m ) ] + j l h l ( 1 ) [ β ( 1 ) * ( l , m ) α ( 1 ) ( l , m ) a l + α ( 1 ) * ( l , m ) β ( 1 ) ( l , m ) b l ] + h l ( 1 ) * j l [ β ( 1 ) * ( l , m ) α ( 1 ) ( l , m ) b l * + α ( 1 ) * ( l , m ) β ( 1 ) ( l , m ) a l * ] + h l ( 1 ) * h l ( 1 ) [ β ( 1 ) * ( l , m ) α ( 1 ) ( l , m ) b l * a l + α ( 1 ) * ( l , m ) β ( 1 ) ( l , m ) a l * b l ] } , V = l ( l + 1 ) ( k r ) 2 r 3 { j l d d r ( r j l ) ( β ( 1 ) * ( l , m ) β ( 1 ) ( l , m ) + α ( 1 ) * ( l , m ) α ( 1 ) ( l , m ) ] + j l d d r ( r h l ( 1 ) ) [ β ( 1 ) * ( l , m ) β ( 1 ) ( l , m ) b l + α ( 1 ) * ( l , m ) α ( 1 ) ( l , m ) a l ] + h l ( 1 ) * d d r ( r j l ) × [ β ( 1 ) * ( l , m ) β ( 1 ) ( l , m ) b l * + α ( 1 ) * ( l , m ) α ( 1 ) ( l , m ) a l * ] + h l ( 1 ) * d d r ( r h l ( 1 ) ) [ β ( 1 ) * ( l , m ) β ( 1 ) ( l , m ) b l * b l + α ( 1 ) * ( l , m ) α ( 1 ) ( l , m ) a l * a l ] } ,
m P l m cos θ - d P l m d θ sin θ = P l m + 1 sin θ , m P l m cos θ + d P l m d θ sin θ = ( l + m ) ( l - m + 1 ) P l m - 1 sin θ , m P l m sin θ d P l m d θ cos θ = 1 2 l + 1 [ l P l + 1 m + 1 + ( l + 1 ) P l - 1 m + 1 ] , m P l m sin θ + d P l m d θ cos θ = 1 2 l + 1 [ ( l + m ) ( l + m - 1 ) ( l + 1 ) × P l - 1 m - 1 + l ( l - m + 2 ) ( l - m + 1 ) P l + 1 m - 1 ] , 0 π sin θ P l m P l m d θ = 2 2 l + 1 ( l + m ) ! ( l - m ) ! δ l , l , 0 π sin 2 θ P l m d P l m d θ d θ = 2 2 l + 1 ( l + m ) ! ( l - m ) ! [ ( l - 1 ) ( l - m ) 2 l - 1 δ l , l - 1 - ( l + 2 ) ( l + m + 1 ) 2 l + 3 δ l , l + 1 ] .
( 2 l + 1 ) d Z l ( σ ) d ( k r ) = l Z l - 1 ( σ ) - ( l + 1 ) Z l + 1 ( σ ) , ( 2 l + 1 ) Z l ( σ ) k r = Z l - 1 ( σ ) + Z l + 1 ( σ ) , h l ( 1 ) ( k r ) ~ ( - i ) l exp ( i k r ) i k r , j l ( k r ) ~ 1 k r sin ( k r - l 2 π ) .

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