Abstract

The forward scattering of a Gaussian laser beam by a spherical particle located along the beam axis is analyzed with the generalized Lorenz–Mie theory (GLMT) and with diffraction theory. Forward-scattering and near-forward-scattering profiles from electrodynamically levitated droplets, 51.6 μm in diameter, are also presented and compared with GLMT-based predictions. The total intensity in the forward direction, formed by the superposition of the incident and the scattered fields, is found to correlate with the particle-extinction cross section, the particle diameter, and the beam width. Based on comparison with the GLMT, the diffraction solution is accurate when beam widths that are approximately greater than or equal to the particle diameter are considered and when large particles that have an extinction efficiency near the asymptotic value of 2 are considered. However, diffraction fails to describe the forward intensity for more tightly focused beams. The experimental observations, which are in good agreement with GLMT-based predictions, reveal that the total intensity profile about the forward direction is quite sensitive to particle axial position within a Gaussian beam. These finite beam effects are significant when the ratio of the beam to the particle diameter is less than approximately 5:1. For larger beam-to-particle-diameter ratios, the total field in the forward direction is dominated by the incident beam.

© 1995 Optical Society of America

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  9. 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]
  10. 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).
    [CrossRef]
  11. G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formalism,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [CrossRef]
  12. 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]
  13. K. F. Ren, G. Gréhan, G. Gouesbet, “Laser sheet scattering from spherical particles,” Part. Part. Syst. Char. 10, 146–151 (1993).
    [CrossRef]
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    [CrossRef] [PubMed]
  20. G. Gréhan, F. Guilloteau, G. Gouesbet, “GLMT-validations: the off-axis case,” Part. Part. Syst. Char. 7, 248–249 (1990).
    [CrossRef]
  21. F. Guilloteau, G. Gréhan, G. Gouesbet, “Optical levitation experiments to assess the validity of the GLMT,” Appl. Opt. 31, 2942–2951 (1992).
    [CrossRef] [PubMed]
  22. J. T. Hodges, G. Gréhan, C. Presser, H. G. Semerjian, “Elastic scattering from spheres under non plane-wave illumination,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed. Proc. Soc. Photo-Opt. Instrum. Eng.1862, 294–308 (1993).
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    [CrossRef]
  25. 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]
  26. J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  29. 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]
  30. G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representation of laser beams for use in light-scattering calculations,” Appl. Opt. (to be published).
  31. 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]
  32. G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
    [CrossRef] [PubMed]
  33. 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]
  34. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 3, p. 57.
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    [CrossRef]
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    [CrossRef]
  37. E. J. Davis, E. Chorbajian, “The measurement of evaporation rates of submicron aerosol droplets,” Indust. Eng. Chem. Fundam. 13, 272–277 (1974).
    [CrossRef]
  38. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 13, p. 231.
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1994 (1)

1993 (3)

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

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

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]

1992 (3)

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

E. J. Davis, “Microchemical engineering: the physics and chemistry of the microparticle,” Adv. Chem. Eng. 18, 1–95 (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]

1991 (1)

1990 (2)

1989 (1)

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

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]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formalism,” 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]

1985 (3)

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]

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).
[CrossRef]

E. J. Davis, “Electrodynamic balance stability characteristics and applications to the study of aerocolloidal particles,” Langmuir 1, 379–387 (1985).
[CrossRef]

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]

1981 (1)

1980 (1)

1979 (1)

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

1978 (1)

1977 (1)

1975 (1)

1974 (1)

E. J. Davis, E. Chorbajian, “The measurement of evaporation rates of submicron aerosol droplets,” Indust. Eng. Chem. Fundam. 13, 272–277 (1974).
[CrossRef]

1968 (1)

N. Morita, T. Tanaka, T. Yamasaki, Y. Nahanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

1890 (1)

L. Lorenz, “Lysbevaegelsen i og uden for en half plane lysbölger belyst Kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

Alexander, D. R.

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.

Barton, J. P.

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]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small ParticlesWiley, New York, 1983), Chap. 4, p. 118.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 3, p. 57.

Casperson, L. W.

Chen, S. 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]

J. P. Chevaillier, J. Fabre, G. Gréhan, G. Gouesbet, “Comparison of diffraction theory and generalized Lorenz–Mie theory for a sphere located on the axis of a laser beam,” Appl. Opt. 29, 1293–1298 (1990).
[CrossRef] [PubMed]

Chorbajian, E.

E. J. Davis, E. Chorbajian, “The measurement of evaporation rates of submicron aerosol droplets,” Indust. Eng. Chem. Fundam. 13, 272–277 (1974).
[CrossRef]

Corriveau, R.

Davis, E. J.

E. J. Davis, “Microchemical engineering: the physics and chemistry of the microparticle,” Adv. Chem. Eng. 18, 1–95 (1992).
[CrossRef]

E. J. Davis, “Electrodynamic balance stability characteristics and applications to the study of aerocolloidal particles,” Langmuir 1, 379–387 (1985).
[CrossRef]

E. J. Davis, E. Chorbajian, “The measurement of evaporation rates of submicron aerosol droplets,” Indust. Eng. Chem. Fundam. 13, 272–277 (1974).
[CrossRef]

Davis, L. W.

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

Fabre, J.

Glantschnig, W. J.

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]

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

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

J. P. Chevaillier, J. Fabre, G. Gréhan, G. Gouesbet, “Comparison of diffraction theory and generalized Lorenz–Mie theory for a sphere located on the axis of a laser beam,” Appl. Opt. 29, 1293–1298 (1990).
[CrossRef] [PubMed]

G. Gréhan, F. Guilloteau, G. Gouesbet, “GLMT-validations: the off-axis case,” Part. Part. Syst. Char. 7, 248–249 (1990).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients 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 formalism,” 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]

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]

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).
[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]

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, “Optical levitation of a single particle to study the theory of quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
[CrossRef] [PubMed]

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representation of laser beams for use in light-scattering calculations,” Appl. Opt. (to be published).

Gréhan, G.

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

F. Guilloteau, G. Gréhan, G. Gouesbet, “Optical levitation experiments to assess the validity of the GLMT,” 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]

J. P. Chevaillier, J. Fabre, G. Gréhan, G. Gouesbet, “Comparison of diffraction theory and generalized Lorenz–Mie theory for a sphere located on the axis of a laser beam,” Appl. Opt. 29, 1293–1298 (1990).
[CrossRef] [PubMed]

G. Gréhan, F. Guilloteau, G. Gouesbet, “GLMT-validations: the off-axis case,” Part. Part. Syst. Char. 7, 248–249 (1990).
[CrossRef]

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

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 formalism,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

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).
[CrossRef]

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]

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, “Optical levitation of a single particle to study the theory of quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
[CrossRef] [PubMed]

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representation of laser beams for use in light-scattering calculations,” Appl. Opt. (to be published).

J. T. Hodges, G. Gréhan, C. Presser, H. G. Semerjian, “Elastic scattering from spheres under non plane-wave illumination,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed. Proc. Soc. Photo-Opt. Instrum. Eng.1862, 294–308 (1993).

Guilloteau, F.

F. Guilloteau, G. Gréhan, G. Gouesbet, “Optical levitation experiments to assess the validity of the GLMT,” 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]

G. Gréhan, F. Guilloteau, G. Gouesbet, “GLMT-validations: the off-axis case,” Part. Part. Syst. Char. 7, 248–249 (1990).
[CrossRef]

Hamelin, P.

P. Hamelin, “Application de la diffusion lumineuse à la métrologie des particules en écoulement diphasique dispersé,” Ph.D. dissertation (l’Institut National Polytechnique de Toulouse, Toulouse, France, 1986).

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), Chap. 11, p. 397.

Hodges, J. T.

J. T. Hodges, G. Gréhan, C. Presser, H. G. Semerjian, “Elastic scattering from spheres under non plane-wave illumination,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed. Proc. Soc. Photo-Opt. Instrum. Eng.1862, 294–308 (1993).

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]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small ParticlesWiley, New York, 1983), Chap. 4, p. 118.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 3, p. 57.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, San Diego, Calif., 1969), Chap. 3, p. 27.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Lock, J. A.

Lorenz, L.

L. Lorenz, “Lysbevaegelsen i og uden for en half plane lysbölger belyst Kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

Maheu, B.

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 formalism,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

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

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).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Morita, N.

N. Morita, T. Tanaka, T. Yamasaki, Y. Nahanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

Nahanishi, Y.

N. Morita, T. Tanaka, T. Yamasaki, Y. Nahanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

Pogorzelski, R. J.

Presser, C.

J. T. Hodges, G. Gréhan, C. Presser, H. G. Semerjian, “Elastic scattering from spheres under non plane-wave illumination,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed. Proc. Soc. Photo-Opt. Instrum. Eng.1862, 294–308 (1993).

Ren, K. F.

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

Schaub, S. A.

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Figures (10)

Fig. 1
Fig. 1

Coordinate systems and scattering geometry: OG(u, v, w) is the primary coordinate system, with an origin at the beam-waist center, and Op(x, y, z) is the particle-referenced system. The conjugate axes of the two coordinate systems are parallel, and the scattering particle is located at OG(x0, y0, z0).

Fig. 2
Fig. 2

GLMT calculations for a particle located at the Gaussian beam-waist center: θ = 0, m = 1.5; λ = 0.5145 μm; d = 5, 25, 50, 100 μm. These data illustrate the correlation between the extinction cross section (normalized by the incident power in the beam per unit incident beam intensity πω02/2), with the normalized total forward intensity It0/Ii0. The nonmonotonic portion of each curve (at fixed particle diameter) corresponds to the size regime 2ω0/d ≲ 1.

Fig. 3
Fig. 3

Calculated extinction efficiency Qext for a particle located at the beam-waist center versus 2 ω0/d. Input parameters: m = 1.5; λ = 0.5145 μm; d = 5, 25, 50, 100 μm. Qext is defined as the scattered power divided by the power that is geometrically incident upon the particle. The results were determined with the GLMT-based extinction cross section given by Eq. (10).

Fig. 4
Fig. 4

GLMT and diffraction calculations of the normalized total field intensity in the forward direction, It0/Ii0, versus 2 ω0/d for a particle located at the beam-waist center. GLMT calculations: m = 1.5; λ = 0.5145 μm; d = 5, 25, 50, 100 μm. The symbols ▲ correspond to the diffraction solution given by Eq. (24).

Fig. 5
Fig. 5

Schematic of the experimental apparatus: 1, electrodynamic balance; 2, three-axis translation stage with 2-μm resolution; 3, CCD video camera for detection of scattered light; 4, motorized rail; 5, 8-bit frame grabber and 386 computer; 6, long-distance microscope; 7, CCD video camera for droplet imaging; 8, videocassette recorder; 9, video monitor; 10, Ar+ laser; 11, laser-focusing singlet, f = 100 mm; 12, power supply for electrodynamic balance.

Fig. 6
Fig. 6

Measured and computed Gaussian beam profiles (λ = 0.5145 μm) in the far field at the axial location w = 146 mm. □, experimental profile obtained along the axis given by OG(0, v, 146 mm). The Gaussian beam computations are represented by the dashed curve for ω0 = 18 μm and by the solid curve for ω0 = 24 μm. Note that the local beam width that corresponds to the smaller beam-waist radius, ω0 = 18 μm, is larger than the beam width that corresponds to the ω0 = 24 μm beam because of the larger beam-confinement factor of the more tightly focused beam.

Fig. 7
Fig. 7

Variation of It0/Ii0 with droplet axial position z0 for a droplet located along the beam axis. □, experimental data for a 51.6-μm-diameter DOP droplet (cases 1–3). The GLMT calculations are represented by the dashed and the solid curves. The input data to the GLMT are the detector at OG(0, 0, 146 mm), d = 51.6 μm, m = 1.4845; dashed curve, ω0 = 18 μm; solid curve, ω0 = 24 μm. The diffraction-based calculations (represented by ▲) were obtained from Eq. (24) with the local beam-waist radius ωw(z0, ω0 = 24 μm) and d = 51.6 μm. For clarity, this expression is displayed only for z0 < 0 because it is symmetric about z0 = 0.

Fig. 8
Fig. 8

Ratio of particle diameter to local beam-waist diameter, (2 ωw/d)−1, versus the normalized total forward intensity It0/Ii0. The experimental results for the levitated DOP droplet (cases 1–3) are given by □. The three ▲ symbols are the GLMT calculations of It0/Ii0 with the parameters in Table 1 for cases 1–3. The solid curve represents the diffraction solution, Eq. (24), assuming that ω0 = 24 μm.

Fig. 9
Fig. 9

Normalized total field intensity about the forward direction measured along the axis given by OG(0, v, 146 mm). The subtended angles Δθ are given by ±0.813°, ±0.897°, ±0.988° for cases 1, 2, and 3, respectively. The experimental data are represented by the discrete symbols □, and the GLMT calculations are represented by the dashed curves for ω0 = 18 μm, and the solid curves for ω0 = 24 μm. (a), (b) □, case 1 data; z0 = 5 mm. (c), (d) □, case 2 data; z0 = 18.2 mm. (e), (f) □, case 3 data; z0 = 30.0 mm.

Fig. 10
Fig. 10

Near-forward-scattering data for cases 1, 2, and 3 obtained experimentally (□) and GLMT calculations (solid curves). These measurements were made along the axis given by OG(0, v, 146 mm).

Tables (2)

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Table 1 Experimental Parameters

Tables Icon

Table 2 Local Beam Radii Based on ω0 = 24 μm

Equations (24)

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E u , i = ω 0 ω w E 0 exp [ ( u 2 + v 2 ) ω w 2 ] exp [ i ( k w Φ w Φ u v ) ] ,
E w , i = 2 i + 2 w s ω 0 u E u , i ,
ω w 2 ω 0 2 = 1 + ( 2 s w ω 0 ) 2 ,
lim w ω 0 / 2 s ω w = 2 s w .
E s = E θ, s θ ˆ + E ϕ , s ϕ ˆ ,
E ϕ , s = E 0 k r i exp ( i k r ) sin ( ϕ ) S 1 ,
E θ , s = E 0 k r i exp ( i k r ) cos ( ϕ ) S 2 ,
S 1 = n = 1 2 n + 1 n ( n + 1 ) g n ( s , z 0 ) × [ a n ( m , α ) π n ( μ ) + b n ( m , α ) τ n ( μ ) ] ,
S 2 = n = 1 2 n + 1 n ( n + 1 ) g n ( s , z 0 ) × [ a n ( m , α ) τ n ( μ ) + b n ( m , α ) π n ( μ ) ] ,
C ext = C sca + C abs = λ 2 2 π Re n = 1 ( 2 n + 1 ) | g n ( s , z 0 ) | 2 × [ a n ( m , α ) + b n ( m , α ) ] ,
Q ext = I 0 ( C sca + C abs ) P inc = I 0 C ext P inc ,
P inc = I 0 2 π 0 d / 2 exp [ ( 2 a 2 / ω 0 2 ) ] a d a = I 0 πω 0 2 2 { 1 exp [ ( d 2 / 2 ω 0 2 ) ] } ,
Q ext = C ext πω 0 2 2 { 1 exp [ ( d 2 2 ω 0 2 ) ] } .
E t = E s + E i .
I t = 1 2 | E 0 | 2 Re [ ( E θ , t H ϕ , t * ) ( E ϕ , t H θ , t * ) ] ,
I t = 1 | E 0 | 2 ( E θ , t E θ , t * + E ϕ , t E ϕ , t * ) .
I i = 1 | E 0 | 2 ( E θ , i E θ , i * + E ϕ , i E ϕ , i * ) .
C ext πω 0 2 ( 1 I t 0 I i 0 )
Q ext = C ext π ω 0 2 2 [ 1 exp ( d 2 2 ω 0 2 ) ] 2 πω 0 2 πω 0 2 ( 1 I t 0 / I i 0 ) { 1 exp [ ( d 2 2 ω 0 2 ) ] } ,
d 2 ω 0 1 2 [ 2 ln ( Q ext Q ext 2 + 2 I t 0 / I i 0 ) ] 1 / 2 ,
d 2 ω 0 1 2 [ 2 ln ( I i 0 / I t 0 ) ] 1 / 2 ,
E t ( θ ) = A { ( k ω 0 ) 2 2 exp [ ( k ω 0 θ ) 2 2 ] 0 k d / 2 u d u J 0 ( u θ ) exp [ u 2 ( k ω 0 ) 2 ] } ,
E t 0 = A ( k ω 0 ) 2 2 exp ( d 2 4 ω 0 2 ) .
I t 0 I i 0 = exp ( d 2 2 ω 0 2 ) .

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