Abstract

A generalized theoretical model for the response of a phase–Doppler particle analyzer (PDPA) to homogeneous, spherical particles passing at arbitrary locations through a crossed beam measurement volume is presented. The model is based on the arbitrary beam theory [J. Appl. Phys. 64, 1632 (1988)] and is valid for arbitrary particle size and complex refractive index. In contrast to classical Lorenz–Mie theory, the arbitrary beam approach has the added capability of accounting for effects that are due to the presence of the finite-size crossed incident beams that are used in the PDPA measurement technique.

The theoretical model is used to compute phase shift as a function of both the particle position within the measurement volume and particle diameter (1.0 μm < diameter water droplets < 10.0 μm for both resonant and nonresonant sizes) for 30° off-axis receiver configuration. Results indicate that trajectory effects are most pronounced for particle trajectories through the edge of the crossed beam measurement volume on the side opposite the detector. Trajectories through the center of the probe volume gave phase shifts that are nearly identical to those obtained with Lorenz–Mie plane-wave theory. Phase shifts calculated for particle diameters corresponding to electric-wave resonances showed the largest deviation from the corresponding nonresonance diameter phase shifts. Phase shifts for droplets at magnetic wave resonance conditions showed smaller effects, closely following the behavior of nonresonant particle sizes. The major influence of aerosol trajectory on actual particle size determination (for both resonant and nonresonant particle sizes) is that the measured aerosol size distributions will appear broader than the actual size distribution that exists within a spray.

© 1994 Optical Society of America

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  2. K. D. Ahlers, D. R. Alexander, “Microcomputer based digital image processing system developed to count and size laser-generated small particle images,” Opt. Eng. 24, 1060–1065 (1985).
  3. G. P. Bertollini, L. M. Oberdier, Y. H. Lee, “Image processing system to analyze droplet distributions in sprays,” Opt. Eng. 24, 464–469 (1985).
  4. B. A. Weiss, P. Derov, D. DeBiase, H. C. Simmons, “Fluid particle sizing using a fully automated optical imaging system,” Opt. Eng. 23, 561–566 (1984).
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    [CrossRef] [PubMed]
  7. W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
  8. W. D. Bachalo, “Analysis and testing of a new method for drop size measurements using laser light scattering interferometry,” Rep. NAS3-23684 (NASA, Washington, D.C., 1983), pp. 1–65.
  9. M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” presented at the Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, 1984.
  10. L. V. Lorenz, “Upon the light reflected and refracted by a transparent sphere,” Vidensk. Selsk. Shrifter, 6, 1–62 (1890).
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    [CrossRef]
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    [CrossRef]
  14. C. R. Negus, L. E. Drain, “Mie calculations of the scattered light from a spherical particle traversing a fringe pattern produced by two intersecting laser beams,” J. Phys. D 15, 375–402 (1982).
    [CrossRef]
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    [CrossRef] [PubMed]
  17. A. Ungut, G. Grehan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
    [CrossRef] [PubMed]
  18. S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. D. P. Taylor, “Calculation of calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 611–622.
  19. S. V. Sankar, W. D. Bachalo, “Response characteristics of the phase Doppler particle analyzer for sizing spherical particles larger than the light wavelength,” Appl. Opt. 30, 1487–1496 (1991).
    [CrossRef] [PubMed]
  20. D. R. Alexander, K. J. Wiles, S. A. Schaub, M. P. Seeman, “Effects of non-spherical drops on a phase Doppler spray analyzer,” in Particle Sizing and Spray Analysis, N. Chigier, G. W. Stewart, eds., Proc. Soc. Photo-Opt. Instrum. Eng.573, 67–72 (1985).
  21. K. D. Marx, C. F. Edwards, W. K. Chin, “Effects of the single particle constraint on the measurement of droplet distribution functions,” presented at the Fifth Annual Conference on Liquid Atomization and Spray Systems, San Ramon, Calif., 18–20 May 1992.
  22. R. W. Sellens, “Alignment errors in phase Doppler receiving optics,” Part. Part. Syst. Characterization 7, 116–120 (1990).
    [CrossRef]
  23. A. Naqwi, F. Durst, “Contributions to the optical design of the phase/Doppler system,” in Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State U. Press, Tempe, Ariz., 1990), pp. 521–530.
  24. G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “On elimination of trajectory effects in phase Doppler systems,” presented at the European Symposium on Particle Characterization, Nürnberg, Germany, 24–26 March, 1992.
  25. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  26. 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]
  27. J. P. Barton, W. Ma, S. A. Schaub, D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706–4715 (1991).
    [CrossRef] [PubMed]
  28. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic field for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  29. I. Gonda, “Aerosols for delivery of therapeutic and diagnostic agents to the respiratory tract,” Crit. Rev. Ther. Drug Deliv. Syst. 6, 273–313 (1990).
  30. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 3, p. 99.
  31. S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a 5th-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989); Appl. Phys. Lett. 59, 1798 (1991).
    [CrossRef]
  32. G. M. Hale, M. R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength range,” Appl. Opt. 12, 555–563 (1973).
    [CrossRef] [PubMed]
  33. S. V. Sankar, B. J. Weber, D. Y. Damemoto, W. D. Bachalo, “Sizing fine particles with the phase doppler interferometric technique,” Appl. Opt. 30, 4914–4920 (1991).
    [CrossRef] [PubMed]
  34. 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]
  35. A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
    [CrossRef]
  36. A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814 (1981).
    [CrossRef] [PubMed]
  37. S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Aerosol Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.
  38. J. A. Lock, “Interference enhancement of the internal fields at structural scattering resonance of a coated sphere,” Appl. Opt. 29, 3180–3187 (1990).
    [CrossRef] [PubMed]
  39. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 11, pp.300–305.
  40. M. Born, E. Wolf, eds., Principles of Optics, 6th ed. (Pergamon, Elmsford, N.Y., 1987), Chap. 13, pp. 633–664.
  41. G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a phase Doppler system using generalized Lorenz–Mie theory,” presented at the International Conference on Multiphase Flows ’91, Tsukuba, Japan, 1991.

1991 (3)

1990 (3)

J. A. Lock, “Interference enhancement of the internal fields at structural scattering resonance of a coated sphere,” Appl. Opt. 29, 3180–3187 (1990).
[CrossRef] [PubMed]

R. W. Sellens, “Alignment errors in phase Doppler receiving optics,” Part. Part. Syst. Characterization 7, 116–120 (1990).
[CrossRef]

I. Gonda, “Aerosols for delivery of therapeutic and diagnostic agents to the respiratory tract,” Crit. Rev. Ther. Drug Deliv. Syst. 6, 273–313 (1990).

1989 (3)

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a 5th-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989); Appl. Phys. Lett. 59, 1798 (1991).
[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]

1988 (1)

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

1985 (2)

K. D. Ahlers, D. R. Alexander, “Microcomputer based digital image processing system developed to count and size laser-generated small particle images,” Opt. Eng. 24, 1060–1065 (1985).

G. P. Bertollini, L. M. Oberdier, Y. H. Lee, “Image processing system to analyze droplet distributions in sprays,” Opt. Eng. 24, 464–469 (1985).

1984 (2)

B. A. Weiss, P. Derov, D. DeBiase, H. C. Simmons, “Fluid particle sizing using a fully automated optical imaging system,” Opt. Eng. 23, 561–566 (1984).

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).

1982 (1)

C. R. Negus, L. E. Drain, “Mie calculations of the scattered light from a spherical particle traversing a fringe pattern produced by two intersecting laser beams,” J. Phys. D 15, 375–402 (1982).
[CrossRef]

1981 (3)

1979 (1)

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

1977 (1)

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

1974 (1)

A. R. Jones, “Light scattering by a sphere situated in an interference pattern, with relevance to fringe anemometry and particle sizing,” J. Phys. D 7, 1369–1376 (1974).
[CrossRef]

1973 (1)

1972 (1)

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

1908 (1)

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

1890 (1)

L. V. Lorenz, “Upon the light reflected and refracted by a transparent sphere,” Vidensk. Selsk. Shrifter, 6, 1–62 (1890).

Ahlers, K. D.

K. D. Ahlers, D. R. Alexander, “Microcomputer based digital image processing system developed to count and size laser-generated small particle images,” Opt. Eng. 24, 1060–1065 (1985).

Al-Chalabi, S. A. M.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. D. P. Taylor, “Calculation of calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 611–622.

Alexander, D. R.

J. P. Barton, W. Ma, S. A. Schaub, D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706–4715 (1991).
[CrossRef] [PubMed]

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]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a 5th-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989); Appl. Phys. Lett. 59, 1798 (1991).
[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 field for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

K. D. Ahlers, D. R. Alexander, “Microcomputer based digital image processing system developed to count and size laser-generated small particle images,” Opt. Eng. 24, 1060–1065 (1985).

D. R. Alexander, K. J. Wiles, S. A. Schaub, M. P. Seeman, “Effects of non-spherical drops on a phase Doppler spray analyzer,” in Particle Sizing and Spray Analysis, N. Chigier, G. W. Stewart, eds., Proc. Soc. Photo-Opt. Instrum. Eng.573, 67–72 (1985).

Ashkin, A.

A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814 (1981).
[CrossRef] [PubMed]

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

Bachalo, W. D.

S. V. Sankar, W. D. Bachalo, “Response characteristics of the phase Doppler particle analyzer for sizing spherical particles larger than the light wavelength,” Appl. Opt. 30, 1487–1496 (1991).
[CrossRef] [PubMed]

S. V. Sankar, B. J. Weber, D. Y. Damemoto, W. D. Bachalo, “Sizing fine particles with the phase doppler interferometric technique,” Appl. Opt. 30, 4914–4920 (1991).
[CrossRef] [PubMed]

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).

W. D. Bachalo, “Analysis and testing of a new method for drop size measurements using laser light scattering interferometry,” Rep. NAS3-23684 (NASA, Washington, D.C., 1983), pp. 1–65.

Barton, J. P.

J. P. Barton, W. Ma, S. A. Schaub, D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706–4715 (1991).
[CrossRef] [PubMed]

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]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a 5th-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989); Appl. Phys. Lett. 59, 1798 (1991).
[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 field for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Benner, R. E.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Aerosol Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

Bertollini, G. P.

G. P. Bertollini, L. M. Oberdier, Y. H. Lee, “Image processing system to analyze droplet distributions in sprays,” Opt. Eng. 24, 464–469 (1985).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 11, pp.300–305.

Buchhave, P.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” presented at the Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, 1984.

Chen, S.-H.

Chin, W. K.

K. D. Marx, C. F. Edwards, W. K. Chin, “Effects of the single particle constraint on the measurement of droplet distribution functions,” presented at the Fifth Annual Conference on Liquid Atomization and Spray Systems, San Ramon, Calif., 18–20 May 1992.

Damemoto, D. Y.

Davis, L. W.

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

DeBiase, D.

B. A. Weiss, P. Derov, D. DeBiase, H. C. Simmons, “Fluid particle sizing using a fully automated optical imaging system,” Opt. Eng. 23, 561–566 (1984).

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

Derov, P.

B. A. Weiss, P. Derov, D. DeBiase, H. C. Simmons, “Fluid particle sizing using a fully automated optical imaging system,” Opt. Eng. 23, 561–566 (1984).

Drain, L. E.

C. R. Negus, L. E. Drain, “Mie calculations of the scattered light from a spherical particle traversing a fringe pattern produced by two intersecting laser beams,” J. Phys. D 15, 375–402 (1982).
[CrossRef]

Durst, F.

A. Naqwi, F. Durst, “Contributions to the optical design of the phase/Doppler system,” in Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State U. Press, Tempe, Ariz., 1990), pp. 521–530.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a phase Doppler system using generalized Lorenz–Mie theory,” presented at the International Conference on Multiphase Flows ’91, Tsukuba, Japan, 1991.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “On elimination of trajectory effects in phase Doppler systems,” presented at the European Symposium on Particle Characterization, Nürnberg, Germany, 24–26 March, 1992.

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814 (1981).
[CrossRef] [PubMed]

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

Edwards, C. F.

K. D. Marx, C. F. Edwards, W. K. Chin, “Effects of the single particle constraint on the measurement of droplet distribution functions,” presented at the Fifth Annual Conference on Liquid Atomization and Spray Systems, San Ramon, Calif., 18–20 May 1992.

Farmer, W. M.

Glantschnig, W. J.

Gonda, I.

I. Gonda, “Aerosols for delivery of therapeutic and diagnostic agents to the respiratory tract,” Crit. Rev. Ther. Drug Deliv. Syst. 6, 273–313 (1990).

Gouesbet, G.

A. Ungut, G. Grehan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a phase Doppler system using generalized Lorenz–Mie theory,” presented at the International Conference on Multiphase Flows ’91, Tsukuba, Japan, 1991.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “On elimination of trajectory effects in phase Doppler systems,” presented at the European Symposium on Particle Characterization, Nürnberg, Germany, 24–26 March, 1992.

Grehan, G.

A. Ungut, G. Grehan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “On elimination of trajectory effects in phase Doppler systems,” presented at the European Symposium on Particle Characterization, Nürnberg, Germany, 24–26 March, 1992.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a phase Doppler system using generalized Lorenz–Mie theory,” presented at the International Conference on Multiphase Flows ’91, Tsukuba, Japan, 1991.

Hale, G. M.

Hardalupas, Y.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. D. P. Taylor, “Calculation of calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 611–622.

Hill, S. C.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Aerosol Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

Houser, M. J.

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 11, pp.300–305.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 3, p. 99.

Jones, A. R.

A. R. Jones, “Light scattering by a sphere situated in an interference pattern, with relevance to fringe anemometry and particle sizing,” J. Phys. D 7, 1369–1376 (1974).
[CrossRef]

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. D. P. Taylor, “Calculation of calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 611–622.

Lee, Y. H.

G. P. Bertollini, L. M. Oberdier, Y. H. Lee, “Image processing system to analyze droplet distributions in sprays,” Opt. Eng. 24, 464–469 (1985).

Lock, J. A.

Lorenz, L. V.

L. V. Lorenz, “Upon the light reflected and refracted by a transparent sphere,” Vidensk. Selsk. Shrifter, 6, 1–62 (1890).

Ma, W.

Magnus, D. E.

D. S. Mahler, D. E. Magnus, “Hot-wire technique for droplet measurement,” ASTM Special Tech. Pub. 848 (American Society for Testing and Materials, Philadelphia, Pa., 1984), pp. 153–165.

Mahler, D. S.

D. S. Mahler, D. E. Magnus, “Hot-wire technique for droplet measurement,” ASTM Special Tech. Pub. 848 (American Society for Testing and Materials, Philadelphia, Pa., 1984), pp. 153–165.

Marx, K. D.

K. D. Marx, C. F. Edwards, W. K. Chin, “Effects of the single particle constraint on the measurement of droplet distribution functions,” presented at the Fifth Annual Conference on Liquid Atomization and Spray Systems, San Ramon, Calif., 18–20 May 1992.

Mie, G.

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

Naqwi, A.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a phase Doppler system using generalized Lorenz–Mie theory,” presented at the International Conference on Multiphase Flows ’91, Tsukuba, Japan, 1991.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “On elimination of trajectory effects in phase Doppler systems,” presented at the European Symposium on Particle Characterization, Nürnberg, Germany, 24–26 March, 1992.

A. Naqwi, F. Durst, “Contributions to the optical design of the phase/Doppler system,” in Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State U. Press, Tempe, Ariz., 1990), pp. 521–530.

Negus, C. R.

C. R. Negus, L. E. Drain, “Mie calculations of the scattered light from a spherical particle traversing a fringe pattern produced by two intersecting laser beams,” J. Phys. D 15, 375–402 (1982).
[CrossRef]

Oberdier, L. M.

G. P. Bertollini, L. M. Oberdier, Y. H. Lee, “Image processing system to analyze droplet distributions in sprays,” Opt. Eng. 24, 464–469 (1985).

Querry, M. R.

Saffman, M.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” presented at the Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, 1984.

Sankar, S. V.

Schaub, S. A.

J. P. Barton, W. Ma, S. A. Schaub, D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706–4715 (1991).
[CrossRef] [PubMed]

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]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a 5th-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989); Appl. Phys. Lett. 59, 1798 (1991).
[CrossRef]

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

D. R. Alexander, K. J. Wiles, S. A. Schaub, M. P. Seeman, “Effects of non-spherical drops on a phase Doppler spray analyzer,” in Particle Sizing and Spray Analysis, N. Chigier, G. W. Stewart, eds., Proc. Soc. Photo-Opt. Instrum. Eng.573, 67–72 (1985).

Seeman, M. P.

D. R. Alexander, K. J. Wiles, S. A. Schaub, M. P. Seeman, “Effects of non-spherical drops on a phase Doppler spray analyzer,” in Particle Sizing and Spray Analysis, N. Chigier, G. W. Stewart, eds., Proc. Soc. Photo-Opt. Instrum. Eng.573, 67–72 (1985).

Sellens, R. W.

R. W. Sellens, “Alignment errors in phase Doppler receiving optics,” Part. Part. Syst. Characterization 7, 116–120 (1990).
[CrossRef]

Simmons, H. C.

B. A. Weiss, P. Derov, D. DeBiase, H. C. Simmons, “Fluid particle sizing using a fully automated optical imaging system,” Opt. Eng. 23, 561–566 (1984).

Tanger, H.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” presented at the Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, 1984.

Taylor, A. M. D. P.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. D. P. Taylor, “Calculation of calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 611–622.

Ungut, A.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 12, p. 200, 212.

Weber, B. J.

Weiner, B. B.

B. B. Weiner, “Particle and droplet sizing using Fraunhofer diffraction,” Modern Methods of Particle Size Analysis, H. G. Barth, ed. (Wiley Interscience, New York, 1984), 135–172.

Weiss, B. A.

B. A. Weiss, P. Derov, D. DeBiase, H. C. Simmons, “Fluid particle sizing using a fully automated optical imaging system,” Opt. Eng. 23, 561–566 (1984).

Wiles, K. J.

D. R. Alexander, K. J. Wiles, S. A. Schaub, M. P. Seeman, “Effects of non-spherical drops on a phase Doppler spray analyzer,” in Particle Sizing and Spray Analysis, N. Chigier, G. W. Stewart, eds., Proc. Soc. Photo-Opt. Instrum. Eng.573, 67–72 (1985).

Ann. Phys. (2)

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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

S. V. Sankar, W. D. Bachalo, “Response characteristics of the phase Doppler particle analyzer for sizing spherical particles larger than the light wavelength,” Appl. Opt. 30, 1487–1496 (1991).
[CrossRef] [PubMed]

J. P. Barton, W. Ma, S. A. Schaub, D. R. Alexander, “Electromagnetic field for a beam incident on two adjacent spherical particles,” Appl. Opt. 30, 4706–4715 (1991).
[CrossRef] [PubMed]

S. V. Sankar, B. J. Weber, D. Y. Damemoto, W. D. Bachalo, “Sizing fine particles with the phase doppler interferometric technique,” Appl. Opt. 30, 4914–4920 (1991).
[CrossRef] [PubMed]

A. Ungut, G. Grehan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a 5th-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989); Appl. Phys. Lett. 59, 1798 (1991).
[CrossRef]

Crit. Rev. Ther. Drug Deliv. Syst. (1)

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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 field for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

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A. R. Jones, “Light scattering by a sphere situated in an interference pattern, with relevance to fringe anemometry and particle sizing,” J. Phys. D 7, 1369–1376 (1974).
[CrossRef]

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[CrossRef]

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W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).

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G. P. Bertollini, L. M. Oberdier, Y. H. Lee, “Image processing system to analyze droplet distributions in sprays,” Opt. Eng. 24, 464–469 (1985).

B. A. Weiss, P. Derov, D. DeBiase, H. C. Simmons, “Fluid particle sizing using a fully automated optical imaging system,” Opt. Eng. 23, 561–566 (1984).

Part. Part. Syst. Characterization (1)

R. W. Sellens, “Alignment errors in phase Doppler receiving optics,” Part. Part. Syst. Characterization 7, 116–120 (1990).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Other (15)

D. S. Mahler, D. E. Magnus, “Hot-wire technique for droplet measurement,” ASTM Special Tech. Pub. 848 (American Society for Testing and Materials, Philadelphia, Pa., 1984), pp. 153–165.

B. B. Weiner, “Particle and droplet sizing using Fraunhofer diffraction,” Modern Methods of Particle Size Analysis, H. G. Barth, ed. (Wiley Interscience, New York, 1984), 135–172.

A. Naqwi, F. Durst, “Contributions to the optical design of the phase/Doppler system,” in Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State U. Press, Tempe, Ariz., 1990), pp. 521–530.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “On elimination of trajectory effects in phase Doppler systems,” presented at the European Symposium on Particle Characterization, Nürnberg, Germany, 24–26 March, 1992.

W. D. Bachalo, “Analysis and testing of a new method for drop size measurements using laser light scattering interferometry,” Rep. NAS3-23684 (NASA, Washington, D.C., 1983), pp. 1–65.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” presented at the Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon, 1984.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Chap. 12, p. 200, 212.

D. R. Alexander, K. J. Wiles, S. A. Schaub, M. P. Seeman, “Effects of non-spherical drops on a phase Doppler spray analyzer,” in Particle Sizing and Spray Analysis, N. Chigier, G. W. Stewart, eds., Proc. Soc. Photo-Opt. Instrum. Eng.573, 67–72 (1985).

K. D. Marx, C. F. Edwards, W. K. Chin, “Effects of the single particle constraint on the measurement of droplet distribution functions,” presented at the Fifth Annual Conference on Liquid Atomization and Spray Systems, San Ramon, Calif., 18–20 May 1992.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Aerosol Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 11, pp.300–305.

M. Born, E. Wolf, eds., Principles of Optics, 6th ed. (Pergamon, Elmsford, N.Y., 1987), Chap. 13, pp. 633–664.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of a phase Doppler system using generalized Lorenz–Mie theory,” presented at the International Conference on Multiphase Flows ’91, Tsukuba, Japan, 1991.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 3, p. 99.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. D. P. Taylor, “Calculation of calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988), pp. 611–622.

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Figures (9)

Fig. 1
Fig. 1

Simplified schematic showing the primary components of a typical phase–Doppler system. PMT, photomultiplier tube.

Fig. 2
Fig. 2

Schematic of the geometry for crossed beam–aerosol interaction calculations.

Fig. 3
Fig. 3

Computer-generated Doppler half-burst signal for an 8-μm-diameter water droplet passing through an 80-μm-diameter beam waist with X = 0 μm 2θ b = 1.8°, θ d = 30°, and δ ≈ 20 μm. The receiver is located at θ d =30° with focal length f rec = 495 mm and diameter d rec = 105 mm.

Fig. 4
Fig. 4

Phase shift as a function of aerosol diameter and particle trajectory as calculated with both Lorenz–Mie plane-wave theory and the arbitrary beam theory. (2w 0 = 80 μm, 2θ b = 1.8°, δ ≈ 20 μm, θ d = 30°, f rec = 495 mm, d rec = 105 mm.)

Fig. 5
Fig. 5

Absorption efficiency as a function of droplet size ( n ¯ = 1.33 + 10−8 i).

Fig. 6
Fig. 6

Calculated phase shift as a function of particle trajectory for both nonresonance and a(l, 1) resonance conditions. (2w 0 = 80μm, 2θ b = 1.8°, δ ≈ 20 μm, θ d = 30°, f rec = 495 mm, d rec = 105 mm.)

Fig. 7
Fig. 7

Calculated phase shift as a function of particle trajectory for both nonresonance and b(l, 1) resonance conditions. (2w 0 = 80 μm, 2θ b = 1.8°, δ ≈ 20 μm, θ d = 30°, f rec = 495 mm, d rec = 105 mm.)

Fig. 8
Fig. 8

Calculated phase shift as a function of particle trajectory for both nonresonance and a(l, 2) resonance conditions. (2w 0 = 80 μm, 2θ b = 1.8°, δ ≈ 20 μm, θ d = 30°, f rec = 495 mm, d rec = 105 mm.)

Fig. 9
Fig. 9

Calculated phase shift as a function of particle trajectory for both nonresonance and b(l, 2) resonance conditions. (2w 0 = 80 μm, 2θ b = 1.8°, δ ≈ 20 μm, θ d = 30°, f rec = 495 mm, d rec = 105 mm.)

Tables (1)

Tables Icon

Table 1 Actual Aerosol Diameter Along With the Corresponding Measured Diameter Range for Theoretical Models

Equations (29)

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A lm = 1 l ( l + 1 ) ψ l ( α ) 0 2 π 0 π sin θ p × E ˜ r p ( inc ) ( r ˜ p = 1 , θ p , ϕ p ) Y lm * ( θ p , ϕ p ) d θ p d ϕ p ,
B lm = 1 l ( l + 1 ) ψ l ( α ) 0 2 π 0 π sin θ p × H ˜ r p ( inc ) ( r ˜ p = 1 , θ p , ϕ p ) Y lm * ( θ p , ϕ p ) d θ p d ϕ p .
x ˜ p = sin θ p cos ϕ p ,
y ˜ p = sin θ p sin ϕ p ,
z ˜ p = cos θ p ,
x ˜ 0 = x ˜ p + X ˜ ,
y ˜ 0 = y ˜ p + Y ˜ ,
z ˜ 0 = z ˜ p + Z ˜ .
x ˜ 1 = x ˜ 0 ,
y ˜ 1 = y ˜ 0 cos θ b + z ˜ 0 sin θ b ,
z ˜ 1 = z ˜ 0 cos θ b - y ˜ 0 sin θ b ,
x ˜ 2 = x ˜ 0 ,
y ˜ 2 = y ˜ 0 cos θ b - z ˜ 0 sin θ b ,
z ˜ 2 = z ˜ 0 cos θ b + y ˜ 0 sin θ b .
E ˜ x 0 ( inc ) = [ E ˜ x 1 ( inc ) + E ˜ x 2 ( inc ) ] / 2 = E ˜ x p ( inc ) ,
E ˜ y 0 ( inc ) = { [ E ˜ y 1 ( inc ) + E ˜ y 2 ( inc ) ] cos θ b - [ E ˜ z 1 ( inc ) - E ˜ z 2 ( inc ) ] sin θ b } / 2 = E ˜ y p ( inc ) ,
E ˜ z 0 ( inc ) = { [ E ˜ z 1 ( inc ) + E ˜ z 2 ( inc ) ] cos θ b + [ E ˜ y 1 ( inc ) - E ˜ y 2 ( inc ) ] sin θ b } / 2 = E ˜ z p ( inc ) .
E ˜ r p ( inc ) = E ˜ x p ( inc ) sin θ p cos ϕ p + E ˜ y p ( inc ) sin θ p sin ϕ p + E ˜ z p ( inc ) cos θ p .
δ = λ 2 sin θ b ,
P ˜ n ( R , z ˜ d ) = A n S ˜ r ( R , r d ) d x ˜ d d y ˜ d ,             n = 1 , 2 , 3 ,
= A n R ( E ˜ θ H ˜ ϕ * - E ˜ ϕ H ˜ θ * ) d x ˜ d d y ˜ d ,
ϕ 13 ( degrees ) = 360 τ 13 τ D ,
P n ( Y ) = C 1 Y 2 + C 2 Y + C 3 ,
Y min = - C 2 / 2 C 1 .
ϕ 13 ( degrees ) = 1.698 d ( μ m ) .
m ˜ = f rec ϕ 13 δ 360 s d ,
E θ ( sca ) ( r ˜ , θ , ϕ ) = cos ϕ α r ˜ exp ( i α r ˜ ) l = 1 ( - i ) l [ a l τ l ( cos θ ) + b l π l ( cos θ ) ] ,
π l ( cos θ ) = P l ( 1 ) ( cos θ ) / sin θ ,
τ l ( cos θ ) = d d θ P l ( 1 ) ( cos θ ) .

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