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, S. S. Lee, “Radiation pressure on a dielectric sphere in a Gaussian laser beam,” Opt. Acta 29, 801–806 (1982).
    [CrossRef]
  3. A. Ashkin, 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, 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, J. M. Dziedzic, “Optical levitation in high vacuum,” Appl. Phys. Lett. 28, 333–335 (1976).
    [CrossRef]
  8. G. Roosen, 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, 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, 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, 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, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
    [CrossRef]
  16. G. Grehan, 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, J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19, 660–668 (1980).
    [CrossRef] [PubMed]
  18. A. Ashkin, J. M. Dziedzic, “Observation of a new nonlinear photoelectric effect using optical levitation,” Phys. Rev. Lett. 36, 267–270 (1976).
    [CrossRef]
  19. A. Ashkin, J. M. Dziedzic, “Feedback stabilization of optically levitated particles,” Appl. Phys. Lett. 30, 202–204 (1977).
    [CrossRef]
  20. P. L. Marston, 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, 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, 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, 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, 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, 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, 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, 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, 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)

J. S. Kim, 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]

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

1982 (1)

J. S. Kim, 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, 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)

G. Roosen, 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]

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

1977 (5)

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

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

G. Roosen, B. Delaunay, 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]

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

1976 (5)

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

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

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

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

1975 (1)

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

1974 (1)

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

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

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

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

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

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

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

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

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

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

A. Ashkin, 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, 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, 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, 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, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Kim, J. S.

J. S. Kim, 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, 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, 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, E. M. Lifshitz, Fluid Mechanics (Pergamon, London, 1959), pp. 63–69.

Lee, S. S.

J. S. Kim, 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, 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, E. M. Lifshitz, Fluid Mechanics (Pergamon, London, 1959), pp. 63–69.

Marston, P. L.

P. L. Marston, 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, 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, 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, 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, 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, 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, J. M. Dziedzic, “Feedback stabilization of optically levitated particles,” Appl. Phys. Lett. 30, 202–204 (1977).
[CrossRef]

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

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

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

P. L. Marston, 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, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[CrossRef]

A. Ashkin, 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, 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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