Abstract

The use of laser sheets is of growing interest in many measurement techniques relying on light scattering. Device performances must often be analyzed with generalized Lorenz–Mie theory, which requires the evaluation of beam shape coefficients describing the incident beams. In the case of Gaussian beams these coefficients are most efficiently computed by a localized approximation. The localized approximation procedure for Gaussian beams is here successfully generalized to the case of laser sheets.

© 1994 Optical Society of America

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References

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  1. A. A. Naqwi, X.-Z. Liu, F. Durst, “Dual-cylindrical wave method for particle sizing,” Part. Part. Syst. Charact. 7, 45–53 (1990).
    [CrossRef]
  2. A. A. Naqwi, X.-Z. Liu, F. Durst, “Evaluation of the dual-cylindrical wave laser technique for sizing of liquid droplets, Part. Part. Syst. Charact. 9, 44–51 (1992).
    [CrossRef]
  3. R. J. Adrian, “Scattering particle characteristics and their effect on pulsed laser measurements of fluid flow: speckle velocimetry vs particle image velocimetry,” Appl. Opt. 23, 1960–1961 (1984).
    [CrossRef]
  4. R. J. Adrian, “The role of particle image velocimetry in fluid mechanics,” in Proceedings of Optical Methods and Data Processing in Heat and Fluid Flow, (Institution of Mechanical Engineers, London, 1992), pp. 1–6.
  5. 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]
  6. B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
    [CrossRef]
  7. G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), Chap. 10, pp. 339–384.
  8. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
    [CrossRef]
  9. 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]
  10. G. Gréhan, G. Gouesbet, F. Guilloteau, J. P. Chevaillier, “Comparison of the diffraction theory and the generalized Lorenz–Mie theory for a sphere arbitrarily located into a laser beam,” Opt. Commun. 90, 1–6 (1992).
    [CrossRef]
  11. G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “On elimination of the trajectory effects in phase-Doppler systems,” in Proceedings of the 5th European Symposium on Particle Characterization, Nürnberg, Germany, March1992, pp. 309–318.
  12. G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase-Doppler systems: use of polarizers and additional detectors to suppress the effect,” in Proceedings of the 6th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July1992.
  13. K. F. Ren, G. Gréhan, G. Gouesbet, “Electromagnetic field expression of laser sheet and the order of approximation,” submitted to J. Opt. (Paris).
  14. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focussed laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  15. 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]
  16. 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]
  17. 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]
  18. B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–26 (1987).
    [CrossRef] [PubMed]
  19. B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
    [CrossRef]
  20. G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempts to design a localized approximation to the computation of the coefficients,” J. Opt. (Paris) 20, 31–43 (1989).
    [CrossRef]
  21. G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  22. K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of generalized Lorenz–Mie theory: fast algorithm for computations of beam shape coefficients gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
    [CrossRef]
  23. J. A. Lock, “The 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]
  24. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
    [CrossRef]

1993 (1)

1992 (4)

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

A. A. Naqwi, X.-Z. Liu, F. Durst, “Evaluation of the dual-cylindrical wave laser technique for sizing of liquid droplets, Part. Part. Syst. Charact. 9, 44–51 (1992).
[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]

G. Gréhan, G. Gouesbet, F. Guilloteau, J. P. Chevaillier, “Comparison of the diffraction theory and the generalized Lorenz–Mie theory for a sphere arbitrarily located into a laser beam,” Opt. Commun. 90, 1–6 (1992).
[CrossRef]

1990 (2)

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

A. A. Naqwi, X.-Z. Liu, F. Durst, “Dual-cylindrical wave method for particle sizing,” Part. Part. Syst. Charact. 7, 45–53 (1990).
[CrossRef]

1989 (2)

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

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempts to design a localized approximation to the computation of the coefficients,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

1988 (5)

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

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

1987 (1)

1986 (1)

1984 (1)

1982 (1)

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

1972 (1)

Adrian, R. J.

R. J. Adrian, “Scattering particle characteristics and their effect on pulsed laser measurements of fluid flow: speckle velocimetry vs particle image velocimetry,” Appl. Opt. 23, 1960–1961 (1984).
[CrossRef]

R. J. Adrian, “The role of particle image velocimetry in fluid mechanics,” in Proceedings of Optical Methods and Data Processing in Heat and Fluid Flow, (Institution of Mechanical Engineers, London, 1992), pp. 1–6.

Alexander, D. R.

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

Barton, J. P.

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

Carter, W. H.

Chevaillier, J. P.

G. Gréhan, G. Gouesbet, F. Guilloteau, J. P. Chevaillier, “Comparison of the diffraction theory and the generalized Lorenz–Mie theory for a sphere arbitrarily located into a laser beam,” Opt. Commun. 90, 1–6 (1992).
[CrossRef]

Durst, F.

A. A. Naqwi, X.-Z. Liu, F. Durst, “Evaluation of the dual-cylindrical wave laser technique for sizing of liquid droplets, Part. Part. Syst. Charact. 9, 44–51 (1992).
[CrossRef]

A. A. Naqwi, X.-Z. Liu, F. Durst, “Dual-cylindrical wave method for particle sizing,” Part. Part. Syst. Charact. 7, 45–53 (1990).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “On elimination of the trajectory effects in phase-Doppler systems,” in Proceedings of the 5th European Symposium on Particle Characterization, Nürnberg, Germany, March1992, pp. 309–318.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase-Doppler systems: use of polarizers and additional detectors to suppress the effect,” in Proceedings of the 6th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July1992.

Gouesbet, G.

G. Gréhan, G. Gouesbet, F. Guilloteau, J. P. Chevaillier, “Comparison of the diffraction theory and the generalized Lorenz–Mie theory for a sphere arbitrarily located into a laser beam,” Opt. Commun. 90, 1–6 (1992).
[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: fast 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 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, “On the generalized Lorenz–Mie theory: first attempts to design a localized approximation to the computation of the coefficients,” J. Opt. (Paris) 20, 31–43 (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]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–26 (1987).
[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]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase-Doppler systems: use of polarizers and additional detectors to suppress the effect,” in Proceedings of the 6th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July1992.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “On elimination of the trajectory effects in phase-Doppler systems,” in Proceedings of the 5th European Symposium on Particle Characterization, Nürnberg, Germany, March1992, pp. 309–318.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), Chap. 10, pp. 339–384.

K. F. Ren, G. Gréhan, G. Gouesbet, “Electromagnetic field expression of laser sheet and the order of approximation,” submitted to J. Opt. (Paris).

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: fast algorithm for computations of beam shape coefficients gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

G. Gréhan, G. Gouesbet, F. Guilloteau, J. P. Chevaillier, “Comparison of the diffraction theory and the generalized Lorenz–Mie theory for a sphere arbitrarily located into a laser beam,” Opt. Commun. 90, 1–6 (1992).
[CrossRef]

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

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempts to design a localized approximation to the computation of the coefficients,” J. Opt. (Paris) 20, 31–43 (1989).
[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]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (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]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–26 (1987).
[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]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “On elimination of the trajectory effects in phase-Doppler systems,” in Proceedings of the 5th European Symposium on Particle Characterization, Nürnberg, Germany, March1992, pp. 309–318.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase-Doppler systems: use of polarizers and additional detectors to suppress the effect,” in Proceedings of the 6th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July1992.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), Chap. 10, pp. 339–384.

K. F. Ren, G. Gréhan, G. Gouesbet, “Electromagnetic field expression of laser sheet and the order of approximation,” submitted to J. Opt. (Paris).

Guilloteau, F.

G. Gréhan, G. Gouesbet, F. Guilloteau, J. P. Chevaillier, “Comparison of the diffraction theory and the generalized Lorenz–Mie theory for a sphere arbitrarily located into a laser beam,” Opt. Commun. 90, 1–6 (1992).
[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]

Liu, X.-Z.

A. A. Naqwi, X.-Z. Liu, F. Durst, “Evaluation of the dual-cylindrical wave laser technique for sizing of liquid droplets, Part. Part. Syst. Charact. 9, 44–51 (1992).
[CrossRef]

A. A. Naqwi, X.-Z. Liu, F. Durst, “Dual-cylindrical wave method for particle sizing,” Part. Part. Syst. Charact. 7, 45–53 (1990).
[CrossRef]

Lock, J. A.

Maheu, B.

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

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempts to design a localized approximation to the computation of the coefficients,” J. Opt. (Paris) 20, 31–43 (1989).
[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]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–26 (1987).
[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]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), Chap. 10, pp. 339–384.

Naqwi, A.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “On elimination of the trajectory effects in phase-Doppler systems,” in Proceedings of the 5th European Symposium on Particle Characterization, Nürnberg, Germany, March1992, pp. 309–318.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase-Doppler systems: use of polarizers and additional detectors to suppress the effect,” in Proceedings of the 6th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July1992.

Naqwi, A. A.

A. A. Naqwi, X.-Z. Liu, F. Durst, “Evaluation of the dual-cylindrical wave laser technique for sizing of liquid droplets, Part. Part. Syst. Charact. 9, 44–51 (1992).
[CrossRef]

A. A. Naqwi, X.-Z. Liu, F. Durst, “Dual-cylindrical wave method for particle sizing,” Part. Part. Syst. Charact. 7, 45–53 (1990).
[CrossRef]

Ren, K. F.

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

K. F. Ren, G. Gréhan, G. Gouesbet, “Electromagnetic field expression of laser sheet and the order of approximation,” submitted to J. Opt. (Paris).

Schaub, S. A.

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

Appl. Opt. (5)

J. Appl. Phys. (1)

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

J. Opt. (Paris) (4)

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempts to design a localized approximation to the computation of the coefficients,” J. Opt. (Paris) 20, 31–43 (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]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

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

G. Gréhan, G. Gouesbet, F. Guilloteau, J. P. Chevaillier, “Comparison of the diffraction theory and the generalized Lorenz–Mie theory for a sphere arbitrarily located into a laser beam,” Opt. Commun. 90, 1–6 (1992).
[CrossRef]

Part. Part. Syst. Charact. (3)

A. A. Naqwi, X.-Z. Liu, F. Durst, “Dual-cylindrical wave method for particle sizing,” Part. Part. Syst. Charact. 7, 45–53 (1990).
[CrossRef]

A. A. Naqwi, X.-Z. Liu, F. Durst, “Evaluation of the dual-cylindrical wave laser technique for sizing of liquid droplets, Part. Part. Syst. Charact. 9, 44–51 (1992).
[CrossRef]

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

Other (5)

R. J. Adrian, “The role of particle image velocimetry in fluid mechanics,” in Proceedings of Optical Methods and Data Processing in Heat and Fluid Flow, (Institution of Mechanical Engineers, London, 1992), pp. 1–6.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), Chap. 10, pp. 339–384.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “On elimination of the trajectory effects in phase-Doppler systems,” in Proceedings of the 5th European Symposium on Particle Characterization, Nürnberg, Germany, March1992, pp. 309–318.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase-Doppler systems: use of polarizers and additional detectors to suppress the effect,” in Proceedings of the 6th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July1992.

K. F. Ren, G. Gréhan, G. Gouesbet, “Electromagnetic field expression of laser sheet and the order of approximation,” submitted to J. Opt. (Paris).

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Figures (3)

Fig. 1
Fig. 1

Geometry of the scattering problem.

Fig. 2
Fig. 2

Definition of long and short axes of laser sheet.

Fig. 3
Fig. 3

Accuracy of g n m computation: relative error between quadrature and localized approximation.

Tables (4)

Tables Icon

Table 1 Comparison of Beam Shape Coefficient g n m Values Computed by Localized Approximation or Quadraturea

Tables Icon

Table 2 Comparison of Beam Shape Coefficient g n m Values Computed by Localized Approximation or Quadraturea

Tables Icon

Table 3 Comparison of Beam Shape Coefficient g n m Values Computed by Localized Approximation or Quadraturea

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Table 4 Comparison of Computational Time (in Seconds) by Localized Approximation or Quadraturea

Equations (60)

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s = 1 k min ( w 0 x , w 0 y ) ,
E r ( r , θ , ϕ ) = E 0 ψ 0 sh [ ( cos ϕ ) ( sin θ ) ( 1 2 Q x l x r cos θ ) + 2 Q x l x x 0 cos θ ] exp ( K ) ,
H r ( r , θ , ϕ ) = E 0 ψ 0 sh [ ( sin ϕ ) ( sin θ ) ( 1 2 Q y l y r cos θ ) + 2 Q y l y y 0 cos θ ] exp ( K ) ,
K = i k ( r cos θ z 0 ) ,
ψ 0 sh = ( Q x Q y ) 1 / 2 exp [ i Q x w 0 x 2 ( r cos ϕ sin θ x 0 ) 2 i Q y w 0 y 2 ( r sin ϕ sin θ y 0 ) 2 ] ,
Q x = 1 i + ( 2 / l x ) ( r cos θ z 0 ) , Q y = 1 i + ( 2 / l y ) ( r cos θ z 0 ) ,
l x = k w 0 x 2 , l y = k w 0 y 2 .
( E r H r ) = m = m = + ( E r m H r m ) ,
( E r m H r m ) = [ ( E 0 H 0 ) exp ( ikr cos θ ) exp ( i m ϕ ) sin θ ] × ( r m r m ) ,
ψ 0 sh = ψ 0 0 , sh ψ 0 ϕ , sh ,
ψ 0 0 , sh = ( Q x Q y ) 1 / 2 exp [ i 2 ( Q x w 0 x 2 + Q y w 0 y 2 ) r 2 sin 2 θ ] × exp [ i ( Q x w 0 x 2 x 0 2 + Q y w 0 y 2 y 0 2 ) ] ,
ψ 0 ϕ , sh = exp { A [ exp ( 2 i ϕ ) + exp ( 2 i ϕ ) ] + B exp ( i ϕ ) + C exp ( i ϕ ) } ,
A = r 2 sin 2 θ 4 ( i Q x w 0 x 2 i Q y w 0 y 2 ) ,
B = ( r sin θ ) ( i Q x w 0 x 2 x 0 + Q y w 0 y 2 y 0 ) ,
C = ( r sin θ ) ( i Q x w 0 x 2 x 0 Q y w 0 y 2 y 0 ) ,
ψ 0 ϕ , sh = pqtj Ψ pqtj exp [ i ϕ ( 2 p 2 q + t j ) ] ,
pqtj = p = 0 q = 0 t = 0 j = 0 ,
Ψ pqtj = A p + q B t C j p ! q ! t ! j ! .
E r = E 0 F x 2 [ pqtj Ψ pqtj exp ( i p + ϕ ) + pqtj Ψ pqtj exp ( i p ϕ ) ] + E 0 x 0 G x pqtj Ψ pqtj exp ( i p 0 ϕ ) ,
H r = H 0 F y 2 i [ pqtj Ψ pqtj exp ( i p + ϕ ) pqtj Ψ pqtj exp ( i p ϕ ) ] + H 0 y 0 G y pqtj Ψ pqtj exp ( i p 0 ϕ ) ,
F i = ψ 0 0 , sh ( sin θ ) ( 1 2 Q i l i r cos θ ) exp ( K ) ,
G i = ψ 0 0 , sh 2 Q i l i ( cos θ ) exp ( K ) ,
p + = 2 p 2 q + t j + 1 = p 0 + 1 ,
p = 2 p 2 q + t j 1 = p 0 1 .
r m = ψ 0 0 , sh exp ( i k z 0 ) [ 1 2 ( 1 2 Q x l x r cos θ ) ( p + = m pqtj Ψ pqtj + p = m pqtj Ψ pqtj ) + x 0 2 Q x l x cos θ sin θ p 0 = m pqtj Ψ pqtj ] ,
r m = ψ 0 0 , sh exp ( i k z 0 ) [ 1 2 i ( 1 2 Q y l y r cos θ ) ( p + = m pqtj Ψ pqtj p = m pqtj Ψ pqtj ) + y 0 2 Q y l y cos θ sin θ p 0 = m pqtj Ψ pqtj ] ,
( n , TM m n , TE m ) = Z n m F ̂ ( r m r m ) ,
Z n 0 = 2 n ( n + 1 ) 2 n + 1 i ,
Z n m = ( 2 i 2 n + 1 ) | m | 1 , m 0 ,
F ̂ A ( k r , θ ) = A ( n + 1 / 2 , π / 2 ) .
( n , TM m n , TE m ) = 1 2 Z n m ( 1 i ) exp ( i k z 0 ) ( F ̂ ψ 0 0 , sh ) × [ p + = m pqtj ( F ̂ Ψ pqtj ) + p = m pqtj ( F ̂ Ψ pqtj ) ] ,
( n , TM 0 n , TE 0 ) = 1 2 Z n 0 ( 1 i ) exp ( i k z 0 ) ( F ̂ ψ 0 0 , sh ) × j = 0 q = 0 p = 0 q + [ ( j + 1 ) / 2 ] F ̂ ( A p + q B j C j ) p ! q ! j ! ( 2 q 2 p + j + 1 ) ! × [ F ̂ C 2 q 2 p + 1 + F ̂ ( B 2 q 2 p + 1 B 2 q 2 p + 1 ) ] ,
( n , TM m n , TE m ) = 1 2 Z n m ( 1 i ) exp ( i k z 0 ) ( F ̂ ψ 0 0 , sh ) × { j = 0 q = 0 p = 0 q + [ ( m + j 1 ) / 2 ] F ̂ ( A p + q B 2 q 2 p + j + m 1 C j ) p ! q ! j ! ( 2 q 2 p + j + m 1 ) ! + j = 0 q = 0 p = 0 q + [ ( m + j + 1 ) / 2 ] F ̂ ( A p + q B 2 q 2 p + j + m + 1 C j ) p ! q ! j ! ( 2 q 2 p + j + m + 1 ) ! } .
( n , TM | m | n , TE | m | ) = 1 2 Z n m ( 1 i ) exp ( i k z 0 ) ( F ̂ ψ 0 0 , sh ) × { j = 0 q = 0 p = 0 q + [ ( | m | + j 1 ) / 2 ] F ̂ ( A p + q B j C 2 q 2 p + j + | m | 1 ) p ! q ! j ! ( 2 q 2 p + j + | m | 1 ) ! + j = 0 q = 0 p = 0 q + [ ( | m | + j + 1 ) / 2 ] F ̂ ( A p + q B j C 2 q 2 p + j + | m | + 1 ) p ! q ! j ! ( 2 q 2 p + j + | m | + 1 ) ! } .
( g n , TM m g n , TE m ) ( A , B , C ) = ( g n , TM m g n , TE m ) ( A , C , B )
( n , TM m n , TE m ) ( | m | , F ̂ A , F ̂ B , F ̂ C ) = ( n , TM m n , TE m ) ( | m | , F ̂ A , F ̂ C , F ̂ B ) ,
F ̂ B = F ̂ C = 0.
( n , TM m n , TE m ) = 1 2 Z n m ( 1 i ) exp ( i k z 0 ) ( F ̂ 0 0 , sh ) x 0 = y 0 = 0 × ( p + = m p q F ̂ A p + q p ! q ! + p = m p q F ̂ A p + q p ! q ! ) ,
p q = p = 0 q = 0 , p + = 2 p 2 q + 1 , p = 2 p 2 q 1 ,
n , TM 2 p = n , TE 2 p = 0.
( n , TM 2 j + 1 n , TE 2 j + 1 ) = 1 2 Z n 2 j + 1 ( 1 i ) exp ( i k z 0 ) ( F ̂ ψ 0 0 , sh ) x 0 = y 0 = 0 × ( F ̂ A j ) p = 0 F ̂ A 2 p p ! ( p + j ) ! ( 1 + F ̂ A p + j + 1 ) .
( n , TM m n , TE m ) = ( + n , TM m n , TE m ) ,
( n , TM 0 n , TE 0 ) = 1 2 Z n 0 ( 1 i ) × exp ( i k z 0 ) ( F ̂ ψ 0 0 , sh ) w 0 x = w 0 y ( F ̂ B + F ̂ C ) × j = 0 F ̂ ( B j C j ) j ! ( j + 1 ) !
( n , TM m n , TE m ) = 1 2 Z n m ( 1 i ) exp ( i k z 0 ) F ̂ ψ 0 0 , sh w 0 x = w 0 y × j = 0 F ̂ ( B j + m 1 C j ) j ! ( j + m 1 ) ! × [ 1 + F ̂ B 2 ( j + m ) ( j + m + 1 ) ] ,
( g n , TM m g n , TE m ) = i n + 1 4 π ( n | m | ) ! ( n + | m | ) ! k r ψ n ( 1 ) ( k r ) × 0 π 0 2 π ( E r / E 0 H r / H 0 ) P n | m | ( cos θ ) × exp ( i m ϕ ) sin θ d θ d ϕ ,
g n , TM m ( x 0 , y 0 , z 0 ) = ( 1 ) m 1 g n , TM m ( x 0 , y 0 , z 0 ) ,
g n , TM m ( x 0 , y 0 , z 0 ) = g n , TM m ( x 0 , y 0 , z 0 ) ,
g n , TM m ( x 0 , y 0 , z 0 ) = ( 1 ) m 1 g n , TM m * ( x 0 , y 0 , z 0 ) ,
g n , TE m ( x 0 , y 0 , z 0 ) = ( 1 ) m g n , TE m ( x 0 , y 0 , z 0 ) ,
g n , TE m ( x 0 , y 0 , z 0 ) = g n , TE m ( x 0 , y 0 , z 0 ) ,
g n , TE m ( x 0 , y 0 , z 0 ) = ( 1 ) m 1 g n , TE m * ( x 0 , y 0 , z 0 ) .
B ( x 0 , y 0 ) = C ( x 0 , y 0 ) .
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