Abstract

The localized model of the beam-shape coefficients for Gaussian beam-scattering theory by a spherical particle provides a great simplification in the numerical implementation of the theory. We derive an alternative form for the localized coefficients that is more convenient for computer computations and that provides physical insight into the details of the scattering process. We construct a fortran program for Gaussian beam scattering with the localized model and compare its computer run time on a personal computer with that of a traditional Mie scattering program and with three other published methods for computing Gaussian beam scattering. We show that the analytical form of the beam-shape coefficients makes evident the fact that the excitation rate of morphology-dependent resonances is greatly enhanced for far off-axis incidence of the Gaussian beam.

© 1995 Optical Society of America

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  1. Lord Rayleigh, “The incidence of light upon a transparent sphere of dimensions comparable with the wave length,” Proc. R. Soc. London Ser. A84, 25–46 (1910);Scientific Papers by Lord Rayleigh, J. N. Howard, ed. (Dover, New York, 1964), Vol. 5, paper 344, pp. 547–568.
    [CrossRef]
  2. J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
    [CrossRef] [PubMed]
  3. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.
  5. P. W. Barber, D.-S. Y. Wang, M. B. Long, “Scattering calculations using a microcomputer,” Appl. Opt. 20, 1121–1123 (1981).
    [CrossRef] [PubMed]
  6. G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
    [CrossRef]
  7. G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc.Am.A 5, 1427–1443 (1988).
    [CrossRef]
  8. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  9. C. Yeh, S. Colak, P. Barber, “Scattering of sharply focused beams by arbitrarily shaped dielectric particles: an exact solution,” Appl. Opt. 21, 4426–4433 (1982).
    [CrossRef] [PubMed]
  10. E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
    [CrossRef] [PubMed]
  11. E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
    [CrossRef]
  12. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  13. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  14. J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I: On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  15. G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II: Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  16. G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef] [PubMed]
  17. G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gmn in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  18. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
    [CrossRef]
  19. K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: faster algorithm for computations of beam shape coefficients, gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
    [CrossRef]
  20. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), Eq. (11.5).
  21. Ref. 20, Eq. (11.114).
  22. Ref. 20, Table 11.2
  23. G. B. Thomas, Calculus and Analytic Geometry, 3rd ed. (Addison-Wesley, Reading, Mass., 1964), Section 16-9.
  24. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Tables 9.8–9.11.
  25. Ref. 24, Eq. (9.6.32).
  26. Ref. 20, Eqs. (11.129) and (11.133).
  27. Ref. 24, Tables 9.1–9.4.
  28. Ref. 24, Eq. (9.1.40).
  29. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), p. 155.
  30. F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
    [CrossRef] [PubMed]
  31. F. Guilloteau, G. Gréhan, G. Gouesbet, “Optical levitation experiments to assess the validity of the generalized Lorenz– Mie theory,” Appl. Opt. 31, 2942–2951 (1992).
    [CrossRef] [PubMed]
  32. J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
    [CrossRef]
  33. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
    [CrossRef]
  34. J. P. Barton, D. R. Alexander, 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]
  35. E. E. M. Khaled, S. C. Hill, P. W. Barber, “Internal electric energy in a spherical particle illuminated with a plane wave or off-axis Gaussian beam,” Appl. Opt. 33, 524–532 (1994).
    [CrossRef] [PubMed]
  36. E. A. Hovenac, J. A. Lock, “Calibration of the forward-scattering spectrometer probe: modeling scattering from a multimode laser beam,” J. Atmos. Oceanic Technol. 10, 518–525 (1993).
    [CrossRef]
  37. S. A. Schaub, Mountain Technical Center, Schuller International, Littleton, Col. 80127 (personal communication, March1992).
  38. T. Bear, “Continuous-wave laser oscillation in a Nd:YAG sphere,” Opt. Lett. 12, 392–394 (1987).
    [CrossRef]
  39. J.-Z. Zhang, D. H. Leach, R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13, 270–272 (1988).
    [CrossRef] [PubMed]
  40. J.-Z. Zhang, G. Chen, R. K. Chang, “Pumping of stimulated Raman scattering by stimulated Brillouin scattering within a single liquid droplet: input laser line width effects,” J. Opt. Soc. Am. B 7, 108–115 (1990).
    [CrossRef]
  41. A. Messiah, Quantum Mechanics (Wiley, New York, 1968), Vol. 1, appendix B.II.6.
  42. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sect. 12.31.
  43. B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
    [CrossRef]
  44. C. C. Lam, P. T. Leung, K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585–1592 (1992).
    [CrossRef]
  45. S. Schiller, “Asymptotic expansion of morphological resonance frequencies in Mie scattering,” Appl. Opt. 32, 2181–2185 (1993).
    [CrossRef] [PubMed]
  46. Ref. 24, Sect. 10.4.

1994 (3)

1993 (5)

E. A. Hovenac, J. A. Lock, “Calibration of the forward-scattering spectrometer probe: modeling scattering from a multimode laser beam,” J. Atmos. Oceanic Technol. 10, 518–525 (1993).
[CrossRef]

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

S. Schiller, “Asymptotic expansion of morphological resonance frequencies in Mie scattering,” Appl. Opt. 32, 2181–2185 (1993).
[CrossRef] [PubMed]

J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
[CrossRef]

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

1992 (4)

1990 (2)

1989 (4)

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, 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]

1988 (3)

J.-Z. Zhang, D. H. Leach, R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13, 270–272 (1988).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc.Am.A 5, 1427–1443 (1988).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

1987 (1)

1986 (1)

1985 (1)

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

1984 (1)

1982 (1)

1981 (1)

1980 (1)

1979 (1)

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

1969 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Tables 9.8–9.11.

Alexander, D. R.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, 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]

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Allano, D.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), Eq. (11.5).

Barber, P.

Barber, P. W.

Barton, J. P.

J. P. Barton, D. R. Alexander, 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]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Bear, T.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.

Chang, R. K.

Chen, G.

Chowdhury, D. Q.

Colak, S.

Dave, J. V.

Davis, L. W.

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

Gouesbet, G.

J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I: On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II: Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
[CrossRef]

F. Guilloteau, G. Gréhan, G. Gouesbet, “Optical levitation experiments to assess the validity of the generalized Lorenz– Mie theory,” Appl. Opt. 31, 2942–2951 (1992).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: faster algorithm for computations of beam shape coefficients, gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gmn in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc.Am.A 5, 1427–1443 (1988).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

Gréhan, G.

F. Guilloteau, G. Gréhan, G. Gouesbet, “Optical levitation experiments to assess the validity of the generalized Lorenz– Mie theory,” Appl. Opt. 31, 2942–2951 (1992).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: faster algorithm for computations of beam shape coefficients, gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gmn in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc.Am.A 5, 1427–1443 (1988).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

Guilloteau, F.

Hill, S. C.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Internal electric energy in a spherical particle illuminated with a plane wave or off-axis Gaussian beam,” Appl. Opt. 33, 524–532 (1994).
[CrossRef] [PubMed]

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
[CrossRef] [PubMed]

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), p. 155.

Hovenac, E. A.

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

E. A. Hovenac, J. A. Lock, “Calibration of the forward-scattering spectrometer probe: modeling scattering from a multimode laser beam,” J. Atmos. Oceanic Technol. 10, 518–525 (1993).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.

Khaled, E. E. M.

Lam, C. C.

Leach, D. H.

Leung, P. T.

Lock, J. A.

Long, M. B.

Maheu, B.

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gmn in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc.Am.A 5, 1427–1443 (1988).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

Messiah, A.

A. Messiah, Quantum Mechanics (Wiley, New York, 1968), Vol. 1, appendix B.II.6.

Rayleigh, Lord

Lord Rayleigh, “The incidence of light upon a transparent sphere of dimensions comparable with the wave length,” Proc. R. Soc. London Ser. A84, 25–46 (1910);Scientific Papers by Lord Rayleigh, J. N. Howard, ed. (Dover, New York, 1964), Vol. 5, paper 344, pp. 547–568.
[CrossRef]

Ren, K. F.

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: faster algorithm for computations of beam shape coefficients, gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, 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]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

S. A. Schaub, Mountain Technical Center, Schuller International, Littleton, Col. 80127 (personal communication, March1992).

Schiller, S.

Slimani, F.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Tables 9.8–9.11.

Thomas, G. B.

G. B. Thomas, Calculus and Analytic Geometry, 3rd ed. (Addison-Wesley, Reading, Mass., 1964), Section 16-9.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sect. 12.31.

Wang, D.-S. Y.

Wiscombe, W. J.

Yeh, C.

Young, K.

Zhang, J.-Z.

Am. J. Phys. (1)

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

Appl. Opt. (10)

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

F. Guilloteau, G. Gréhan, G. Gouesbet, “Optical levitation experiments to assess the validity of the generalized Lorenz– Mie theory,” Appl. Opt. 31, 2942–2951 (1992).
[CrossRef] [PubMed]

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Internal electric energy in a spherical particle illuminated with a plane wave or off-axis Gaussian beam,” Appl. Opt. 33, 524–532 (1994).
[CrossRef] [PubMed]

J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
[CrossRef] [PubMed]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

P. W. Barber, D.-S. Y. Wang, M. B. Long, “Scattering calculations using a microcomputer,” Appl. Opt. 20, 1121–1123 (1981).
[CrossRef] [PubMed]

C. Yeh, S. Colak, P. Barber, “Scattering of sharply focused beams by arbitrarily shaped dielectric particles: an exact solution,” Appl. Opt. 21, 4426–4433 (1982).
[CrossRef] [PubMed]

E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
[CrossRef] [PubMed]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

S. Schiller, “Asymptotic expansion of morphological resonance frequencies in Mie scattering,” Appl. Opt. 32, 2181–2185 (1993).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag. (1)

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

J. Appl. Phys. (4)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, 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]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. Atmos. Oceanic Technol. (1)

E. A. Hovenac, J. A. Lock, “Calibration of the forward-scattering spectrometer probe: modeling scattering from a multimode laser beam,” J. Atmos. Oceanic Technol. 10, 518–525 (1993).
[CrossRef]

J. Opt. (Paris) (1)

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

J. Opt. Soc. Am. A (4)

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

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

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc.Am.A 5, 1427–1443 (1988).
[CrossRef]

Opt. Commun. (1)

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

Opt. Lett. (2)

Part. Part. Syst. Charact. (1)

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of generalized Lorenz-Mie theory: faster algorithm for computations of beam shape coefficients, gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

Phys. Rev. A (1)

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

Other (16)

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), Eq. (11.5).

Ref. 20, Eq. (11.114).

Ref. 20, Table 11.2

G. B. Thomas, Calculus and Analytic Geometry, 3rd ed. (Addison-Wesley, Reading, Mass., 1964), Section 16-9.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), Tables 9.8–9.11.

Ref. 24, Eq. (9.6.32).

Ref. 20, Eqs. (11.129) and (11.133).

Ref. 24, Tables 9.1–9.4.

Ref. 24, Eq. (9.1.40).

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), p. 155.

S. A. Schaub, Mountain Technical Center, Schuller International, Littleton, Col. 80127 (personal communication, March1992).

A. Messiah, Quantum Mechanics (Wiley, New York, 1968), Vol. 1, appendix B.II.6.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sect. 12.31.

Lord Rayleigh, “The incidence of light upon a transparent sphere of dimensions comparable with the wave length,” Proc. R. Soc. London Ser. A84, 25–46 (1910);Scientific Papers by Lord Rayleigh, J. N. Howard, ed. (Dover, New York, 1964), Vol. 5, paper 344, pp. 547–568.
[CrossRef]

Ref. 24, Sect. 10.4.

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Figures (5)

Fig. 1
Fig. 1

Focused Gaussian beam that is incident upon a spherical particle. The center of the particle is at the origin of the coordinates, and the center of the beam's focal waist is at (a) xf, yf, zf; (b) xf ≠ 0, yf = 0; and (c) xf = 0, yf ≠ 0.

Fig. 2
Fig. 2

(a) Complex Q plane as defined in Eq. (22). (b) Focused Gaussian beam that is propagating from left to right. The points labeled A through E in the off-axis beam are the positions of the spherical particle within the beam's focal waist. They also correspond to the indicated locations in the complex Q plane.

Fig. 3
Fig. 3

(a) Complex P plane as defined in Eq. (16). (b) Focused Gaussian beam that is propagating from left to right. The points labeled A through D in the off-axis beam are the positions of the spherical particle outside the beam's focal waist. A–D also correspond to the indicated locations in the complex P plane.

Fig. 4
Fig. 4

Weighted beam-shape coefficients |Almloc τlm| (filled circles) and |Blmloc τlm| (open circles) as a function of |m| for l = 430 and an off-axis Gaussian beam with λ = 0.6328.

Fig. 5
Fig. 5

Weighted beam-shape coefficient |Almlocτlm | as a function of xf for l = 430 and an off-axis Gaussian beam with λ = 0.6328 μm and yf = zf = 0. The short dashed curve is the weighted beam-shape coefficient for |m| = 1. This is the only coefficient that remains nonzero in the on-axis limit and is proportional to the MDR excitation rate by an incident plane wave.

Tables (1)

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Table 1 Parameters in Published Gaussian Beam-Scattering Calculations

Equations (63)

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I ( r , θ , ϕ ) = 1 2 μ 0 c k 2 r 2 [ | S 1 ( θ , ϕ ) | 2 + | S 2 ( θ , ϕ ) | 2 ] ,
S 1 ( θ , ϕ ) = l = 1 m = l l 2 l + 1 2 l ( l + 1 ) [ im α lm π l | m | ( θ ) + β lm τ l | m | ( θ ) ] exp ( im ϕ ) , S 2 ( θ , ϕ ) = l = 1 m = l l 2 l + 1 2 l ( l + 1 ) [ im β lm π l | m | ( θ ) + α lm τ l | m | ( θ ) ] exp ( im ϕ ) .
k = 2 π λ
α lm = A lm a l , β lm = B lm b l ,
A lm = ( i ) l 1 2 π kr j l ( kr ) ( l | m | ) ! ( l + | m | ) ! 0 π sin θ d θ × 0 2 π d ϕ P l | m | ( cos θ ) exp ( im ϕ ) E inc rad ( r , θ , ϕ ) , B lm = ( i ) l 1 2 π kr j l ( kr ) ( l | m | ) ! ( l + | m | ) ! 0 π sin θ d θ × 0 2 π d ϕ P l | m | ( cos θ ) exp ( im ϕ ) c B inc rad ( r , θ , ϕ ) ,
π l | m | ( θ ) = 1 sin θ P l | m | ( cos θ ) , τ l | m | ( θ ) = d d θ P l | m | ( cos θ ) ,
A lm loc = K lm 2 F l j = 0 p = 0 j ( Ψ ¯ jp δ j 2 p + 1 , m + Ψ ¯ jp δ j 2 p 1 , m ) , B lm loc = K lm 2 i F l j = 0 p = 0 j ( Ψ ¯ jp δ j 2 p + 1 , m Ψ ¯ jp δ j 2 p 1 , m ) ,
F l = D exp [ D ( x f 2 + y f 2 w 0 2 ) ] exp [ = D s 2 ( l + 1 2 ) 2 ] × exp ( i s z f w 0 ) ,
s = 1 k w 0 ,
D = ( 1 2 is z f w 0 ) 1 ,
K lm = { 2 il ( l + 1 ) l + 1 2 if m = 0 2 ( i l + 1 2 ) | m | 1 if m 0 ,
Ψ ¯ jp = [ s ( l + 1 2 ) D ] j ( x f i y f w 0 ) j p ( j p ) ! ( x f + i y f w 0 ) p p ! .
A lm loc = { A lm + loc if m > 0 A lm loc if m < 0 A l 0 loc if m = 0 ,
A lm ± loc = F l ( r f l + 1 2 ) m 1 [ J m 1 ( P ) ( r f ) 2 J m + 1 ( P ) ] , A l 0 loc = F l 2 l ( l + 1 ) ( l + 1 2 ) x f ( x f 2 + y f 2 ) 1 / 2 J 1 ( P ) ,
r f ± = x f ± i y f ( x f 2 + y f 2 ) 1 / 2 ,
P = ( l + 1 2 ) z f / w 0 ( x f 2 + y f 2 w 0 2 ) 1 / 2 ( 1 + i w 0 2 s z f ) 1 .
B lm loc = { B lm + loc if m > 0 B lm loc if m < 0 B l 0 loc if m = 0 ,
B lm ± loc = ± F l i ( r f l + 1 2 ) m 1 [ J m 1 ( P ) + ( r f ) 2 J m + 1 ( P ) ] , B l 0 loc = F l 2 l ( l + 1 ) ( l + 1 2 ) y f ( x f 2 + y f 2 ) 1 / 2 J 1 ( P ) ,
J n ( x ) = i n I n ( ix ) ,
A lm ± loc = F l ( i r f l + 1 2 ) m 1 [ I m 1 ( Q ) + ( r f ) 2 I m + 1 ( Q ) ] , A l 0 loc = F l 2 il ( l + 1 ) ( l + 1 2 ) x f ( x f 2 + y f 2 ) 1 / 2 I 1 ( Q ) ,
B lm ± loc = ± F l i ( i r f l + 1 2 ) m 1 [ I m 1 ( Q ) ( r f ) 2 I m + 1 ( Q ) ] , B l 0 loc = F l 2 il ( l + 1 ) ( l + 1 2 ) x f ( x f 2 + y f 2 ) 1 / 2 I 1 ( Q ) ,
Q = 2 s ( l + 1 2 ) ( x f 2 + y f 2 w 0 2 ) 1 / 2 ( 1 2 is z f w 0 ) 1 = iP .
A lm ± loc = F l ( i l + 1 2 x f | x f | ) m 1 [ I m 1 ( Q ) + I m + 1 ( Q ) ] , A l 0 loc = F l 2 il ( l + 1 ) ( l + 1 2 ) x f | x f | I 1 ( Q ) , B lm ± loc = ± F l i ( i l + 1 2 x f | x f | ) m 1 [ I m 1 ( Q ) I m + 1 ( Q ) ] , B l 0 loc = 0 ,
A lm ± loc = F l ( 1 l + 1 2 y f | y f | ) m 1 [ I m 1 ( Q ) I m + 1 ( Q ) ] , A l 0 loc = 0 , B lm ± loc = ± F l i ( 1 l + 1 2 y f | y f | ) m 1 [ I m 1 ( Q ) + I m + 1 ( Q ) ] , B l 0 loc = F l 2 il ( l + 1 ) ( l + 1 2 ) y f | y f | I 1 ( Q ) .
I n ( Q ) = ( Q 2 ) n k = 0 ( Q 2 / 4 ) k ( n + k ) !
I n ( Q ) = exp ( Q ) ( 2 π Q ) 1 / 2 { 1 + k = 1 ( 1 ) k k ! ( 8 Q ) k ( 4 n 2 1 ) × ( 4 n 2 9 ) [ 4 n 2 ( 2 k 1 ) 2 ] }
k max = Re Q + 7 + 0.5 | Im Q | .
k max = n + 12 .
I n ( Q * ) = I n * ( Q ) .
| z f | w 0 2 s = L 2 ,
J n ( P ) = ( P 2 ) n k = 0 ( 1 ) k ( P 2 / 4 ) k k ! ( n + k ) !
k max = Re P + 12 + 0.5 Im P
| z f | w 0 2 s = L 2
J n ( P ) = ( 2 π P ) 1 / 2 { [ 1 ( 4 n 2 1 ) ( 4 n 2 9 ) 2 ! ( 8 P ) 2 + ( 4 n 2 1 ) ( 4 n 2 9 ) ( 4 n 2 25 ) ( 4 n 2 49 ) 4 ! ( 8 P ) 4 ] × cos [ P ( n + 1 2 ) π 2 ] + [ 0 ( 4 n 2 1 ) 1 ! ( 8 P ) + ( 4 n 2 1 ) ( 4 n 2 9 ) ( 4 n 2 25 ) 3 ! ( 8 P ) 3 ] × sin [ P ( n + 1 2 ) π 2 ] }
k max = n + 9 .
cos [ P ( n + 1 2 ) π 2 ] = cos [ Re P ( n + 1 2 ) π 2 ] cosh ( Im P ) i sin [ Re P ( n + 1 2 ) π 2 ] sinh ( Im P ) , sin [ P ( n + 1 2 ) π 2 ] = sin [ Re P ( n + 1 2 ) π 2 ] cosh ( Im P ) + i cos [ Re P ( n + 1 2 ) π 2 ] sinh ( Im P ) .
P = r e i θ ,
P 1 / 2 = r 1 / 2 exp ( i θ / 2 ) .
( P ) z f > 0 = ( P * ) z f < 0 .
J n [ ( P ) z f > 0 ] = ( 1 ) n J n * [ ( P ) z f < 0 ] .
π l + 1 m ( θ ) = 2 l + 1 l + 1 m cos θ π l m ( θ ) l + m l + 1 m π l 1 m ( θ ) ,
τ l m ( θ ) = l cos θ π l m ( θ ) ( l + m ) π l 1 m ( θ ) ,
π l 1 l ( θ ) = 0 , π l l ( θ ) = ( 2 l 1 ) ! ! sin l 1 θ
S = cos θ Π l m ( θ ) ,
T = S Π l 1 m ( θ ) ,
τ l m ( θ ) = l m T Π l 1 m ( θ ) ,
Π l + 1 m ( θ ) = S + l + m l + 1 m T ,
Π l m ( θ ) = m π l m ( θ ) .
S 1 ( θ , ϕ ) = l = 1 l max 2 l + 1 2 l ( l + 1 ) B l 0 b l τ l 0 ( θ ) + m = 1 m max l = m l max 2 l + 1 2 l ( l + 1 ) i a l Π l m ( θ ) × [ A lm + exp ( im ϕ ) + A lm exp ( im ϕ ) ] + m = 1 m max l = m l max 2 l + 1 2 l ( l + 1 ) b l τ l m ( θ ) × [ B lm + exp ( im ϕ ) + B lm exp ( im ϕ ) ] S 2 ( θ , ϕ ) = l = 1 l max 2 l + 1 2 l ( l + 1 ) A l 0 a l τ l 0 ( θ ) + m = 1 m max l = m l max 2 l + 1 2 l ( l + 1 ) a l τ l m ( θ ) × [ A lm + exp ( im ϕ ) + A lm exp ( im ϕ ) ] + m = 1 m max l = m l max 2 l + 1 2 l ( l + 1 ) i b l Π l m ( θ ) × [ B lm + exp ( im ϕ ) B lm exp ( im ϕ ) ] ,
l max = 2 + X + 4.3 X 1 / 3 ,
X = 2 π a λ ,
| A lm ± loc | exp [ s 2 ( l + 1 2 ) 2 ] exp ( x f 2 / w 0 2 ) ( l + 1 2 ) m 1 × ( 2 π Q ) 1 / 2 exp ( Q ) ( 2 3 4 Q m 2 Q ) , | B lm ± loc | exp [ s 2 ( l + 1 2 ) 2 ] exp ( x f 2 / w 0 2 ) ( l + 1 2 ) m 1 × ( 2 π Q ) 1 / 2 exp ( Q ) ( 2 m Q ) .
Π l m ( θ ) ( m l ) ( 2 π ) 1 / 2 l m + 1 / 2 ( sin θ ) 3 / 2 × cos [ ( l + 1 2 ) θ + m π 2 π 4 ] , τ l m ( θ ) ( 2 π ) 1 / 2 l m + 1 / 2 ( sin θ ) 1 / 2 × sin [ ( l + 1 2 ) θ + m π 2 π 4 ]
| τ l m ( θ ) | l m | Π l m ( θ ) | l m + 1 / 2 .
| A lm ± loc | | B lm ± loc | exp [ s 2 ( l + 1 2 ) 2 ] exp ( x f 2 / w 0 2 ) ( l + 1 2 ) m 1 × ( Q / 2 ) m 1 ( m 1 ) ! exp ( Q 2 / 4 m ) .
| A lm ± loc τ l m | m max | A lm ± loc τ l m | small m = 2 ( Q 2 ) m max 1 ( m max 1 ) ! exp ( Q 2 / 4 m max ) × ( 2 π Q ) 1 / 2 exp ( Q ) 10 8 .
m max = 6.5 Q 1 / 2 for 6 Q 40 .
m max = ( 6.5 + 2.0 | Im Q | Re Q ) ( Re Q ) 1 / 2 for | Im Q | Re Q .
S 1 total ( θ , ϕ ) = { S 1 scattered ( θ , ϕ ) for θ > 10 s S 1 scattered ( θ , ϕ ) sin ϕ S incident ( θ , ϕ ) for θ 10 s , S 2 total ( θ , ϕ ) = { S 2 scattered ( θ , ϕ ) for θ > 10 s S 2 scattered ( θ , ϕ ) cos ϕ S incident ( θ , ϕ ) for θ 10 s , S incident ( θ , ϕ ) = 1 2 s 2 exp ( θ 2 / 4 s 2 ) × exp [ i θ s ( x f w 0 cos ϕ + y f w 0 sin ϕ ) ] × exp ( i z f / s w 0 ) exp ( i θ 2 z f / 2 s w 0 ) .
| A lm ± loc τ l m | l ( 4 π s | x f w 0 | ) 1 / 2 exp { [ s ( l + 1 2 ) | x f w 0 | ] 2 } × ( 2 3 4 Q m 2 Q ) .
l + 1 2 | x f | w 0 s = 2 π λ | x f | .
n x = ( l + 1 2 ) + α i ( l + 1 2 ) 1 / 3 2 1 / 3 V ( n 2 1 ) 1 / 2 , V = { n for a TE resonance 1 n for a TE resonance ,
y f a n α i ( n 2 ) 1 / 3 1 X 2 / 3 + n ( n 2 1 ) 1 / 2 X + 1 6 ( 2 n ) 1 / 3 α i 2 X 4 / 3 for a TE resonance x f a n α i ( n 2 ) 1 / 3 1 X 2 / 3 + n n ( n 2 1 ) 1 / 2 X + 1 6 ( 2 n ) 1 / 3 α i 2 X 4 / 3 for a TM resonance ,

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