Abstract

In the framework of generalized Lorenz–Mie theory, laser beams are described by sets of beam-shape coefficients. The modified localized approximation to evaluate these coefficients for a focused Gaussian beam is presented. A new description of Gaussian beams, called standard beams, is introduced. A comparison is made between the values of the beam-shape coefficients in the framework of the localized approximation and the beam-shape coefficients of standard beams. This comparison leads to new insights concerning the electromagnetic description of laser beams. The relevance of our discussion is enhanced by a demonstration that the localized approximation provides a very satisfactory description of top-hat beams as well.

© 1995 Optical Society of America

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  1. G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase-Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
    [CrossRef]
  2. G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase-Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–144 (1994).
    [CrossRef]
  3. S. A. Schaub, D. R. Alexander, J. P. Barton, “Theoretical analysis of the effects of particle trajectory and structural resonances on the performance of a phase-Doppler particle analyzer,” Appl. Opt. 33, 473–483 (1994).
    [CrossRef] [PubMed]
  4. K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering by spherical particles,” Part. Part. Syst. Charact. 10, 146–151 (1993).
    [CrossRef]
  5. G. Gréhan, K. F. Ren, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a particle sizing technique based on laser sheets,” Part. Part. Syst. Charact. 11, 101–106 (1994).
    [CrossRef]
  6. K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser-sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
    [CrossRef]
  7. D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a top-hat laser beam technique,” J. Phys. D 17, 43–58 (1984).
    [CrossRef]
  8. G. Gréhan, G. Gouesbet, “Simultaneous measurements of velocities and sizes of particles in flows using a combined system incorporating a top-hat beam technique,” Appl. Opt. 25, 3527–3538 (1986).
    [CrossRef] [PubMed]
  9. M. Maeda, K. Hishida, “Application of top-hat laser beam to particle sizing in LDV system,” in Proceedings of the First International Symposium on Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988).
  10. F. Corbin, G. Gréhan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
    [CrossRef]
  11. G. Gréhan, 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]
  12. 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]
  13. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  14. G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in N. Chigier ed., Combustion Measurements (Hemisphere, New York, 1991), pp. 339–384.
  15. G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).
    [CrossRef]
  16. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  17. 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]
  18. G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985); republished in Selected Papers On Light Scattering, M. Kerker, ed., Vol. 951 of SPIE Milestone Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1988), Part 1, pp. 352–360.
    [CrossRef]
  19. 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]
  20. 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]
  21. 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]
  22. B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
    [CrossRef]
  23. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sects. 12.31, 12.33.
  24. G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory, using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
    [CrossRef]
  25. G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the coefficients gn in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
    [CrossRef] [PubMed]
  26. G. Gouesbet, G. Gréhan, B. Maheu, “A localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  27. 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]
  28. S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficients for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
    [CrossRef]
  29. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), Chap. 15.

1994 (7)

1993 (3)

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]

K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering by spherical particles,” Part. Part. Syst. Charact. 10, 146–151 (1993).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase-Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

1991 (1)

F. Corbin, G. Gréhan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

1990 (1)

1989 (3)

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficients for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (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, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

1988 (4)

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 illuminated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory, using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the coefficients gn in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

1986 (2)

1985 (1)

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985); republished in Selected Papers On Light Scattering, M. Kerker, ed., Vol. 951 of SPIE Milestone Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1988), Part 1, pp. 352–360.
[CrossRef]

1984 (1)

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a top-hat laser beam technique,” J. Phys. D 17, 43–58 (1984).
[CrossRef]

1980 (1)

1979 (1)

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

Alexander, D. R.

S. A. Schaub, D. R. Alexander, J. P. Barton, “Theoretical analysis of the effects of particle trajectory and structural resonances on the performance of a phase-Doppler particle analyzer,” Appl. Opt. 33, 473–483 (1994).
[CrossRef] [PubMed]

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]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficients for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

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

Allano, D.

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a top-hat laser beam technique,” J. Phys. D 17, 43–58 (1984).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), Chap. 15.

Barton, J. P.

S. A. Schaub, D. R. Alexander, J. P. Barton, “Theoretical analysis of the effects of particle trajectory and structural resonances on the performance of a phase-Doppler particle analyzer,” Appl. Opt. 33, 473–483 (1994).
[CrossRef] [PubMed]

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]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficients for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

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

Corbin, F.

F. Corbin, G. Gréhan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

Davis, L. W.

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

Durst, F.

G. Gréhan, K. F. Ren, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a particle sizing technique based on laser sheets,” Part. Part. Syst. Charact. 11, 101–106 (1994).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase-Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–144 (1994).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase-Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

Gouesbet, G.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase-Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–144 (1994).
[CrossRef]

G. Gréhan, K. F. Ren, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a particle sizing technique based on laser sheets,” Part. Part. Syst. Charact. 11, 101–106 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser-sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (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]

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]

K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering by spherical particles,” Part. Part. Syst. Charact. 10, 146–151 (1993).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase-Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

F. Corbin, G. Gréhan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “A localized interpretation to compute all the coefficients gnm 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, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory, using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the coefficients gn in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (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]

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. Gréhan, G. Gouesbet, “Simultaneous measurements of velocities and sizes of particles in flows using a combined system incorporating a top-hat beam technique,” Appl. Opt. 25, 3527–3538 (1986).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985); republished in Selected Papers On Light Scattering, M. Kerker, ed., Vol. 951 of SPIE Milestone Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1988), Part 1, pp. 352–360.
[CrossRef]

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a top-hat laser beam technique,” J. Phys. D 17, 43–58 (1984).
[CrossRef]

G. Gréhan, 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]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in N. Chigier ed., Combustion Measurements (Hemisphere, New York, 1991), pp. 339–384.

Gréhan, G.

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser-sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase-Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–144 (1994).
[CrossRef]

G. Gréhan, K. F. Ren, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a particle sizing technique based on laser sheets,” Part. Part. Syst. Charact. 11, 101–106 (1994).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase-Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering by spherical particles,” Part. Part. Syst. Charact. 10, 146–151 (1993).
[CrossRef]

F. Corbin, G. Gréhan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “A localized interpretation to compute all the coefficients gnm 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, G. Gréhan, B. Maheu, “Computations of the coefficients gn in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory, using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[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. Gréhan, G. Gouesbet, “Simultaneous measurements of velocities and sizes of particles in flows using a combined system incorporating a top-hat beam technique,” Appl. Opt. 25, 3527–3538 (1986).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985); republished in Selected Papers On Light Scattering, M. Kerker, ed., Vol. 951 of SPIE Milestone Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1988), Part 1, pp. 352–360.
[CrossRef]

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a top-hat laser beam technique,” J. Phys. D 17, 43–58 (1984).
[CrossRef]

G. Gréhan, 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]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in N. Chigier ed., Combustion Measurements (Hemisphere, New York, 1991), pp. 339–384.

Hishida, K.

M. Maeda, K. Hishida, “Application of top-hat laser beam to particle sizing in LDV system,” in Proceedings of the First International Symposium on Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988).

Lisiecki, D.

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a top-hat laser beam technique,” J. Phys. D 17, 43–58 (1984).
[CrossRef]

Lock, J. A.

Maeda, M.

M. Maeda, K. Hishida, “Application of top-hat laser beam to particle sizing in LDV system,” in Proceedings of the First International Symposium on Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988).

Maheu, B.

G. Gouesbet, G. Gréhan, B. Maheu, “A localized interpretation to compute all the coefficients gnm 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, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory, using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the coefficients gn in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (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]

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, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985); republished in Selected Papers On Light Scattering, M. Kerker, ed., Vol. 951 of SPIE Milestone Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1988), Part 1, pp. 352–360.
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in N. Chigier ed., Combustion Measurements (Hemisphere, New York, 1991), pp. 339–384.

Naqwi, A.

G. Gréhan, K. F. Ren, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a particle sizing technique based on laser sheets,” Part. Part. Syst. Charact. 11, 101–106 (1994).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase-Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–144 (1994).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase-Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

Ren, K. F.

G. Gréhan, K. F. Ren, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a particle sizing technique based on laser sheets,” Part. Part. Syst. Charact. 11, 101–106 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser-sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering by spherical particles,” Part. Part. Syst. Charact. 10, 146–151 (1993).
[CrossRef]

Schaub, S. A.

S. A. Schaub, D. R. Alexander, J. P. Barton, “Theoretical analysis of the effects of particle trajectory and structural resonances on the performance of a phase-Doppler particle analyzer,” Appl. Opt. 33, 473–483 (1994).
[CrossRef] [PubMed]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficients for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

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

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sects. 12.31, 12.33.

Appl. Opt. (5)

Appl. Phys. Lett. (1)

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficients for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

J. Appl. Phys. (2)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle illuminated 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. Opt. (Paris) (2)

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985); republished in Selected Papers On Light Scattering, M. Kerker, ed., Vol. 951 of SPIE Milestone Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1988), Part 1, pp. 352–360.
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory, using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

J. Opt. Soc. Am. A (6)

J. Phys. D (1)

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a top-hat laser beam technique,” J. Phys. D 17, 43–58 (1984).
[CrossRef]

Opt. Commun. (1)

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

Part. Part. Syst. Charact. (6)

F. Corbin, G. Gréhan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering by spherical particles,” Part. Part. Syst. Charact. 10, 146–151 (1993).
[CrossRef]

G. Gréhan, K. F. Ren, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a particle sizing technique based on laser sheets,” Part. Part. Syst. Charact. 11, 101–106 (1994).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Particle trajectory effects in phase-Doppler systems: computations and experiments,” Part. Part. Syst. Charact. 10, 332–338 (1993).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase-Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–144 (1994).
[CrossRef]

G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).
[CrossRef]

Phys. Rev. A (1)

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

Other (4)

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in N. Chigier ed., Combustion Measurements (Hemisphere, New York, 1991), pp. 339–384.

M. Maeda, K. Hishida, “Application of top-hat laser beam to particle sizing in LDV system,” in Proceedings of the First International Symposium on Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sects. 12.31, 12.33.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), Chap. 15.

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Figures (6)

Fig. 1
Fig. 1

Two coordinate systems that describe a focused Gaussian beam that is propagating along the z axis. The origin of the xyz′ coordinate system is at the center of the beam waist, and the partial-wave expansion is carried out with respect to the x, y, z coordinate system.

Fig. 2
Fig. 2

Value K of the kth-order standard beam required for convergence of the BSC g1 as a function of z0.

Fig. 3
Fig. 3

Value K of the kth-order standard beam required for the convergence of the BSC g1 as a function of z0/l. The individual curves from Fig. 2 now coincide.

Fig. 4
Fig. 4

Value K of the kth-order standard beam required for the convergence of the BSC gn as a function of z0/l and D = s(n − 1)1/2(n + 2)1/2 for λ = 0.5 μm, w0 = 5 μm, and s = 0.016.

Fig. 5
Fig. 5

Dominant beam-shape function F1 for the localized top-hat beam of Eq. (45) with (a) w0 = 25 μm, (b) w0 = 7.5 μm, and (c) w0 = 2.5 μm as a function of the distance ρ from the z axis.

Fig. 6
Fig. 6

Dominant beam-shape function F1 for the localized top-hat beam of Eq. (50), which possesses a smooth roll-off of width ∊. The curves are for w0 = 25 μm and (a) ∊ = 0.05w0 and (b) ∊ = 0.10w0. The smoothing of the edge of the beam profile dramatically decreases the oscillatory ringing in F1.

Tables (3)

Tables Icon

Table 1 Coefficients α, β, and γ of Eqs. (34), (35), and (39), Respectively, as a Function of Partial Wave

Tables Icon

Table 2 BSC’s as a Function of Partial Wave for s = 0.001 for the Localized Approximation (LA); the Modified Localized Approximation (MLA); the First- (D1), Third- (D3), and Fifth-order (D5) approximations to the Standard Beam; and the Standard Beam (S)a

Tables Icon

Table 3 BSC’s as a Function of Partial Wave for s = 0.16 for the Localized Approximation (LA); the Modified Localized Approximation (MLA); the First- (D1), Third- (D3), and Fifth-Order (D5) Approximations to the Standard Beam; and the Standard Beam (S)a

Equations (51)

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A = ( A x , 0 , 0 ) .
A x = ψ ( x , y , z ) exp ( i k z ) .
ξ = x w 0 , η = y w 0 , ζ = z l .
2 A + k 2 A = 0 ,
( 2 ξ 2 + 2 η 2 ) ψ 2 i ∂ψ ∂ζ + s 2 2 ψ ζ 2 = 0 .
s = w 0 / l = 1 / k w 0 .
ψ = ψ 0 + s 2 ψ 2 + s 4 ψ 4 + .
ψ 0 = i Q exp [ i Q ( ξ 2 + η 2 ) ] ,
Q = 1 i + 2 ζ .
( 2 ξ 2 + 2 η 2 2 i ∂ζ ) ψ 2 n + 2 = 2 ζ 2 ψ 2 n , n 0 .
E = i c k ( · A ) i ω A ,
H = ( × A ) / μ ,
E x = E 0 [ ψ 0 + s 2 ( ψ 2 + 2 ψ 0 ξ 2 ) + ] exp ( i k z ) ,
E y = E 0 [ s 2 2 ψ 0 ∂ξ∂η + s 4 2 ψ 2 ∂ξ∂η + ] exp ( i k z ) ,
E z = E 0 [ i s ψ 0 ∂ξ i s 3 ( ψ 2 ∂ξ + i 2 ψ 0 ∂ξ∂ζ ) + ] exp ( i k z ) ,
H x = 0 ,
H y = H 0 [ ψ 0 + s 2 ( ψ 2 + i ψ 0 ∂ζ ) + ] exp ( i k z ) ,
H z = H 0 [ i s ψ 0 ∂η i s 3 ψ 2 ∂η + ] exp ( i k z ) .
g n = 1 2 i n 1 R j n ( R ) 1 n ( n + 1 ) 0 π sin 2 θ d θ f ( R , θ ) × exp ( i R cos θ ) P n 1 ( cos θ ) ,
( E r / E 0 H r / H 0 ) = exp ( i R cos θ ) f ( R , θ ) sin θ ( cos ϕ sin ϕ ) .
g n k = l = 0 k ( 1 ) l s 2 l l ! ( n 1 ) ! ( n 1 l ) ! ( n + 1 + l ) ! ( n + 1 ) ! ,
g n 1 = 1 ( n 1 ) ( n + 2 ) s 2 ,
g n 3 = g n 1 + 1 2 ( n 2 ) ( n 1 ) ( n + 2 ) ( n + 3 ) s 4 1 6 ( n 3 ) ( n 2 ) ( n 1 ) ( n + 2 ) ( n + 3 ) × ( n + 4 ) s 6 ,
g n 5 = g n 3 + 1 24 ( n 4 ) ( n 3 ) ( n 2 ) ( n 1 ) ( n + 2 ) × ( n + 3 ) ( n + 4 ) ( n + 5 ) s 8 1 120 ( n 5 ) × ( n 1 ) ( n + 2 ) ( n + 6 ) s 10 .
g n k = j = 0 j + 2 l = l = 0 2 k + 1 ( 2 i s z 0 w 0 ) j ( 1 ) l s 2 l ( l + j ) ! l ! j ! 1 l ! × ( n 1 ) ! ( n 1 l ) ! ( n + 1 + l ) ! ( n + 1 ) ! exp ( i k z 0 ) .
g n = j = 0 l = 0 ( 2 i s z 0 w 0 ) j ( 1 ) l s 2 l ( l + j ) ! l ! j ! 1 l ! × ( n 1 ) ! ( n 1 l ) ! ( n + 1 + l ) ! ( n + 1 ) ! exp ( i k z 0 ) ,
g n = l = 0 ( 1 ) l s 2 l l ! ( n 1 ) ! ( n 1 l ) ! ( n + 1 + l ) ! ( n + 1 ) !
E r = E 0 exp ( i k z ) sin θ cos ϕ f ( k r , θ ) ,
f ( k r , θ ) = i Q exp ( i Q r 2 sin 2 θ w 0 2 ) ( 1 2 Q s r cos θ / w 0 ) .
L ˆ f ( R , θ ) = f ( n + 1 2 , π / 2 ) ,
g n ¯ = exp [ s 2 ( n + 1 2 ) 2 ] .
g n exp [ s 2 ( n 1 ) ( n + 2 ) ] .
exp [ s 2 ( n 1 ) ( n + 2 ) ] = g n 1 + 1 2 α ( n 2 ) ( n 1 ) ( n + 2 ) ( n + 3 ) s 4 1 6 β ( n 3 ) ( n 2 ) ( n 1 ) ( n + 2 ) ( n + 3 ) × ( n + 4 ) s 6 + ,
α = ( n 1 ) ( n + 2 ) ( n 2 ) ( n + 3 ) ,
β = ( n 1 ) 2 ( n + 2 ) 2 ( n 3 ) ( n 2 ) ( n + 3 ) ( n + 4 ) .
L ˆ mod f ( R , θ ) = f [ ( n 1 ) 1 / 2 ( n + 2 ) 1 / 2 , π / 2 ] ,
g n , mod ¯ = exp [ s 2 ( n 1 ) ( n + 2 ) ] .
g n , mod ¯ = exp [ s 2 γ ( n + 1 2 ) 2 ] ,
γ = ( n 1 ) ( n + 2 ) ( n + 1 2 ) 2 .
g n ¯ = ( 1 + 2 i s z 0 w 0 ) 1 exp ( i k z 0 ) × exp [ s 2 ( n + 1 2 ) 2 1 + 2 i s z 0 / w 0 ] ,
g n , mod ¯ = ( 1 + 2 i s z 0 w 0 ) 1 exp ( i k z 0 ) × exp [ s 2 ( n 1 ) ( n + 2 ) 1 + 2 i s z 0 / w 0 ] .
ρ n = ( n + 1 2 ) λ 2 π
D = ( n 1 ) 1 / 2 ( n + 2 ) 1 / 2 s ,
ρ n = ( n 1 ) 1 / 2 ( n + 2 ) 1 / 2 λ 2 π .
f ( k r , θ ) = { 1 if r sin θ w 0 0 if r sin θ > w 0 ,
g ¯ n = { 1 if n k w 0 1 2 0 if n > k w 0 1 2 .
E x ( k r , π / 2 , ϕ ) = F 1 ( k r ) F 2 ( k r ) sin 2 ϕ ,
E y ( k r , π / 2 , ϕ ) = F 2 ( k r ) sin ϕ cos ϕ ,
E z ( k r , π / 2 , ϕ ) = F 3 ( k r ) cos ϕ ,
f ( k r , θ ) = { 1 if r sin θ w 0 exp [ ( r sin θ w 0 ) 2 / 2 ] if r sin θ > w 0 ,
g ¯ n = { 1 if n k w 0 1 2 exp [ ( n + 1 2 k w 0 ) 2 / k 2 2 ] if n > k w 0 1 2 .

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