Abstract

Dual-beam laser measuring techniques are now being used, not only for velocimetry, but also for simultaneous measurements of particle size and velocity in particulate two-phase flows. However, certain details of these optical techniques, such as the effect of Gaussian beam profiles on the accuracy of the measurements, need to be further explored. To implement innovative improvements, a general analytic framework is needed in which performances of various dual-beam instruments could be quantitatively studied and compared. For this purpose, the analysis of light scattering in a generalized dual-wave system is presented in this paper. The present simulation model provides a basis for studying effects of nonplanar beam structures of incident waves, taking into account arbitrary modes of polarization. A polarizer is included in the receiving optics as well. The peculiar aspects of numerical integration of scattered light over circular, rectangular, and truncated circular apertures are also considered.

© 1993 Optical Society of America

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  1. R. J. Adrain, W. L. Earley, “Evaluation of LDV performance using Mie scattering theory,” presented at the Symposium on Laser Anemometry, University of Minnesota, Minneapolis, Minn., 1976.
  2. F. Durst, M. Macagno, G. Richter, “Light scattering by small particles: refined numerical computations” Rep. SFB 80/TM/195 (University of Karlsruhe, Karlsruhe, Germany, 1981).
  3. A. R. Jones, “Light scattering by a sphere situated in an interference pattern, with reference to fringe anemometry and particle sizing,” J. Phys. D 7, 1369–1376 (1974).
    [CrossRef]
  4. J. D. Pendleton, “Mie and refraction theory comparison for particle sizing with the laser velocimetry,” Appl. Opt. 21, 684–688 (1982).
    [CrossRef] [PubMed]
  5. F. Durst, M. Zaré, “Laser Doppler measurements in two-phase flows,” presented at the Symposium on the Accuracy of Flow Measurements by the Laser Doppler Method (Copenhagen, 1975).
  6. K. Bauckhage, H. H. Floegel, “Simultaneous measurement of droplet size and velocity in nozzle sprays,” presented at the Second Symposium on Applications in Laser Anemometry in Fluid Mechanics, Lisbon, 1984.
  7. W. D. Bachalo, M. House, “Phase Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
  8. M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics—II, R. Adrian, D. Durão, F. Durst, H. Mishina, J. Whitelaw, eds. (Ladoan—Instituto Superior Técnico, Lisbon, 1984), pp. 85–104.
  9. A. Naqwi, X. Liu, F. Durst, “Dual cylindrical wave method for particle sizing,” Part. Part. Syst. Charact. 7, 45–53 (1990).
    [CrossRef]
  10. 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]
  11. A. Naqwi, F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt. 29, 1780–1785 (1990).
    [CrossRef] [PubMed]
  12. F. Durst, R. Müller, A. Naqwi, “Measurement accuracy of semiconductor LDA systems,” Exp. Fluids 10, 125–137 (1990).
    [CrossRef]
  13. J. Domnick, F. Durst, R. Müiller, A. Naqwi, “Improved optical systems for velocimetry and particle sizing using semiconductor lasers and detectors,” in Applications of Laser Techniques in Fluid Mechanics (Springer-Verlag, Berlin, 1991).
  14. K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements, part 2: optical particle characteristics as a base for the new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
    [CrossRef]
  15. S. V. Sankar, B. J. Weber, W. D. Bachalo, “Sizing fine particles with phase Doppler interferometric technique,” Appl. Opt. 30, 4914–4920 (1991).
    [CrossRef] [PubMed]
  16. S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of the calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 107–120.
  17. A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements, Part 1: theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
    [CrossRef]
  18. J. Gardavský, R. Kleine, “Reflection suppression by polarization in backscattering LDA measurements near walls and in two-phase flows,” Appl. Opt. 20, 4110–4123 (1981).
    [CrossRef] [PubMed]
  19. M. Saffman, “The use of polarized light for optical particle sizing,” in Laser Anemometry in Fluid Mechanics—III, R. Adrian, I. Asanuma, D. Durão, F. Durst, T. Mishina, J. Whitelaw, eds. (Ladoan—Instituto Superior Técnico, Lisbon, Portugal, 1986), pp. 85–104.
  20. A. Naqwi, F. Durst, G. Kraft, “Sizing of submicrometer particles using a phase/Doppler system,” Appl. Opt. 30, 4903–4913 (1991).
    [CrossRef] [PubMed]
  21. F. Durst, A. Naqwi, “Optical methods for studies of multiphase flows,” in Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State U. Press, Tempe, Ariz., 1990), pp. 269–276.
  22. A. Naqwi, F. Durst, X. Liu, “An extended phase Doppler system for characterization of multiphase flows,” Part. Part. Syst. Charact. 8, 16–22 (1991).
    [CrossRef]
  23. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metalllösungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  24. P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. (Leipzig) 30, 57–136 (1909).
  25. G. Gouesbet, B. Maheu, G. Gréhen, “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]
  26. C. F. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), App. A.
  27. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), pp. 25–32.
  28. G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of phase Doppler systems using generalized Lorenz–Mie theory,” in Proceedings of the International Conference on Multiphase Flows (University of Tsukuba, Tsukuba, Japan, 1991), Vol. 2, pp. 291–294.

1991 (4)

A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements, Part 1: theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
[CrossRef]

A. Naqwi, F. Durst, X. Liu, “An extended phase Doppler system for characterization of multiphase flows,” Part. Part. Syst. Charact. 8, 16–22 (1991).
[CrossRef]

A. Naqwi, F. Durst, G. Kraft, “Sizing of submicrometer particles using a phase/Doppler system,” Appl. Opt. 30, 4903–4913 (1991).
[CrossRef] [PubMed]

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

1990 (4)

A. Naqwi, F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt. 29, 1780–1785 (1990).
[CrossRef] [PubMed]

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

F. Durst, R. Müller, A. Naqwi, “Measurement accuracy of semiconductor LDA systems,” Exp. Fluids 10, 125–137 (1990).
[CrossRef]

1988 (2)

K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements, part 2: optical particle characteristics as a base for the new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhen, “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]

1984 (1)

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

1982 (1)

1981 (1)

1974 (1)

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

1909 (1)

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

1908 (1)

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

Adrain, R. J.

R. J. Adrain, W. L. Earley, “Evaluation of LDV performance using Mie scattering theory,” presented at the Symposium on Laser Anemometry, University of Minnesota, Minneapolis, Minn., 1976.

Al-Chalabi, S. A. M.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of the calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 107–120.

Bachalo, W. D.

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

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

Bauckhage, K.

K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements, part 2: optical particle characteristics as a base for the new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
[CrossRef]

K. Bauckhage, H. H. Floegel, “Simultaneous measurement of droplet size and velocity in nozzle sprays,” presented at the Second Symposium on Applications in Laser Anemometry in Fluid Mechanics, Lisbon, 1984.

Bohren, C. F.

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), pp. 25–32.

Buchhave, P.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics—II, R. Adrian, D. Durão, F. Durst, H. Mishina, J. Whitelaw, eds. (Ladoan—Instituto Superior Técnico, Lisbon, 1984), pp. 85–104.

Debye, P.

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

Domnick, J.

J. Domnick, F. Durst, R. Müiller, A. Naqwi, “Improved optical systems for velocimetry and particle sizing using semiconductor lasers and detectors,” in Applications of Laser Techniques in Fluid Mechanics (Springer-Verlag, Berlin, 1991).

Durst, F.

A. Naqwi, F. Durst, G. Kraft, “Sizing of submicrometer particles using a phase/Doppler system,” Appl. Opt. 30, 4903–4913 (1991).
[CrossRef] [PubMed]

A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements, Part 1: theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
[CrossRef]

A. Naqwi, F. Durst, X. Liu, “An extended phase Doppler system for characterization of multiphase flows,” Part. Part. Syst. Charact. 8, 16–22 (1991).
[CrossRef]

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

A. Naqwi, F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt. 29, 1780–1785 (1990).
[CrossRef] [PubMed]

F. Durst, R. Müller, A. Naqwi, “Measurement accuracy of semiconductor LDA systems,” Exp. Fluids 10, 125–137 (1990).
[CrossRef]

F. Durst, A. Naqwi, “Optical methods for studies of multiphase flows,” in Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State U. Press, Tempe, Ariz., 1990), pp. 269–276.

J. Domnick, F. Durst, R. Müiller, A. Naqwi, “Improved optical systems for velocimetry and particle sizing using semiconductor lasers and detectors,” in Applications of Laser Techniques in Fluid Mechanics (Springer-Verlag, Berlin, 1991).

F. Durst, M. Macagno, G. Richter, “Light scattering by small particles: refined numerical computations” Rep. SFB 80/TM/195 (University of Karlsruhe, Karlsruhe, Germany, 1981).

F. Durst, M. Zaré, “Laser Doppler measurements in two-phase flows,” presented at the Symposium on the Accuracy of Flow Measurements by the Laser Doppler Method (Copenhagen, 1975).

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of phase Doppler systems using generalized Lorenz–Mie theory,” in Proceedings of the International Conference on Multiphase Flows (University of Tsukuba, Tsukuba, Japan, 1991), Vol. 2, pp. 291–294.

Earley, W. L.

R. J. Adrain, W. L. Earley, “Evaluation of LDV performance using Mie scattering theory,” presented at the Symposium on Laser Anemometry, University of Minnesota, Minneapolis, Minn., 1976.

Floegel, H. H.

K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements, part 2: optical particle characteristics as a base for the new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
[CrossRef]

K. Bauckhage, H. H. Floegel, “Simultaneous measurement of droplet size and velocity in nozzle sprays,” presented at the Second Symposium on Applications in Laser Anemometry in Fluid Mechanics, Lisbon, 1984.

Fritsching, U.

K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements, part 2: optical particle characteristics as a base for the new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
[CrossRef]

Gardavský, J.

Gouesbet, G.

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

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of phase Doppler systems using generalized Lorenz–Mie theory,” in Proceedings of the International Conference on Multiphase Flows (University of Tsukuba, Tsukuba, Japan, 1991), Vol. 2, pp. 291–294.

Gréhan, G.

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. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of phase Doppler systems using generalized Lorenz–Mie theory,” in Proceedings of the International Conference on Multiphase Flows (University of Tsukuba, Tsukuba, Japan, 1991), Vol. 2, pp. 291–294.

Gréhen, G.

Hardalupas, Y.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of the calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 107–120.

Hiller, R.

K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements, part 2: optical particle characteristics as a base for the new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
[CrossRef]

House, M.

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

Huffman, D.

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

Jones, A. R.

A. R. Jones, “Light scattering by a sphere situated in an interference pattern, with reference 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. K. P. Taylor, “Calculation of the calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 107–120.

Kleine, R.

Kraft, G.

Liu, X.

A. Naqwi, F. Durst, X. Liu, “An extended phase Doppler system for characterization of multiphase flows,” Part. Part. Syst. Charact. 8, 16–22 (1991).
[CrossRef]

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

Macagno, M.

F. Durst, M. Macagno, G. Richter, “Light scattering by small particles: refined numerical computations” Rep. SFB 80/TM/195 (University of Karlsruhe, Karlsruhe, Germany, 1981).

Maheu, B.

Mie, G.

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

Müiller, R.

J. Domnick, F. Durst, R. Müiller, A. Naqwi, “Improved optical systems for velocimetry and particle sizing using semiconductor lasers and detectors,” in Applications of Laser Techniques in Fluid Mechanics (Springer-Verlag, Berlin, 1991).

Müller, R.

F. Durst, R. Müller, A. Naqwi, “Measurement accuracy of semiconductor LDA systems,” Exp. Fluids 10, 125–137 (1990).
[CrossRef]

Naqwi, A.

A. Naqwi, F. Durst, G. Kraft, “Sizing of submicrometer particles using a phase/Doppler system,” Appl. Opt. 30, 4903–4913 (1991).
[CrossRef] [PubMed]

A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements, Part 1: theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
[CrossRef]

A. Naqwi, F. Durst, X. Liu, “An extended phase Doppler system for characterization of multiphase flows,” Part. Part. Syst. Charact. 8, 16–22 (1991).
[CrossRef]

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

A. Naqwi, F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt. 29, 1780–1785 (1990).
[CrossRef] [PubMed]

F. Durst, R. Müller, A. Naqwi, “Measurement accuracy of semiconductor LDA systems,” Exp. Fluids 10, 125–137 (1990).
[CrossRef]

F. Durst, A. Naqwi, “Optical methods for studies of multiphase flows,” in Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State U. Press, Tempe, Ariz., 1990), pp. 269–276.

J. Domnick, F. Durst, R. Müiller, A. Naqwi, “Improved optical systems for velocimetry and particle sizing using semiconductor lasers and detectors,” in Applications of Laser Techniques in Fluid Mechanics (Springer-Verlag, Berlin, 1991).

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of phase Doppler systems using generalized Lorenz–Mie theory,” in Proceedings of the International Conference on Multiphase Flows (University of Tsukuba, Tsukuba, Japan, 1991), Vol. 2, pp. 291–294.

Pendleton, J. D.

Richter, G.

F. Durst, M. Macagno, G. Richter, “Light scattering by small particles: refined numerical computations” Rep. SFB 80/TM/195 (University of Karlsruhe, Karlsruhe, Germany, 1981).

Saffman, M.

M. Saffman, “The use of polarized light for optical particle sizing,” in Laser Anemometry in Fluid Mechanics—III, R. Adrian, I. Asanuma, D. Durão, F. Durst, T. Mishina, J. Whitelaw, eds. (Ladoan—Instituto Superior Técnico, Lisbon, Portugal, 1986), pp. 85–104.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics—II, R. Adrian, D. Durão, F. Durst, H. Mishina, J. Whitelaw, eds. (Ladoan—Instituto Superior Técnico, Lisbon, 1984), pp. 85–104.

Sankar, S. V.

Tanger, H.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics—II, R. Adrian, D. Durão, F. Durst, H. Mishina, J. Whitelaw, eds. (Ladoan—Instituto Superior Técnico, Lisbon, 1984), pp. 85–104.

Taylor, A. M. K. P.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of the calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 107–120.

Weber, B. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), pp. 25–32.

Zaré, M.

F. Durst, M. Zaré, “Laser Doppler measurements in two-phase flows,” presented at the Symposium on the Accuracy of Flow Measurements by the Laser Doppler Method (Copenhagen, 1975).

Ann. Phys. (1)

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

Ann. Phys. (Leipzig) (1)

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

Appl. Opt. (5)

Exp. Fluids (1)

F. Durst, R. Müller, A. Naqwi, “Measurement accuracy of semiconductor LDA systems,” Exp. Fluids 10, 125–137 (1990).
[CrossRef]

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

J. Phys. D (1)

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

Opt. Eng. (1)

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

Part. Part. Syst. Charact. (4)

K. Bauckhage, H. H. Floegel, U. Fritsching, R. Hiller, “The phase Doppler difference method, a new laser Doppler technique for simultaneous size and velocity measurements, part 2: optical particle characteristics as a base for the new diagnostic technique,” Part. Part. Syst. Charact. 5, 66–71 (1988).
[CrossRef]

A. Naqwi, F. Durst, “Light scattering applied to LDA and PDA measurements, Part 1: theory and numerical treatments,” Part. Part. Syst. Charact. 8, 245–258 (1991).
[CrossRef]

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

A. Naqwi, F. Durst, X. Liu, “An extended phase Doppler system for characterization of multiphase flows,” Part. Part. Syst. Charact. 8, 16–22 (1991).
[CrossRef]

Other (12)

M. Saffman, “The use of polarized light for optical particle sizing,” in Laser Anemometry in Fluid Mechanics—III, R. Adrian, I. Asanuma, D. Durão, F. Durst, T. Mishina, J. Whitelaw, eds. (Ladoan—Instituto Superior Técnico, Lisbon, Portugal, 1986), pp. 85–104.

F. Durst, A. Naqwi, “Optical methods for studies of multiphase flows,” in Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State U. Press, Tempe, Ariz., 1990), pp. 269–276.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of the calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 107–120.

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

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1986), pp. 25–32.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of phase Doppler systems using generalized Lorenz–Mie theory,” in Proceedings of the International Conference on Multiphase Flows (University of Tsukuba, Tsukuba, Japan, 1991), Vol. 2, pp. 291–294.

M. Saffman, P. Buchhave, H. Tanger, “Simultaneous measurement of size, concentration and velocity of spherical particles by a laser Doppler method,” in Laser Anemometry in Fluid Mechanics—II, R. Adrian, D. Durão, F. Durst, H. Mishina, J. Whitelaw, eds. (Ladoan—Instituto Superior Técnico, Lisbon, 1984), pp. 85–104.

J. Domnick, F. Durst, R. Müiller, A. Naqwi, “Improved optical systems for velocimetry and particle sizing using semiconductor lasers and detectors,” in Applications of Laser Techniques in Fluid Mechanics (Springer-Verlag, Berlin, 1991).

F. Durst, M. Zaré, “Laser Doppler measurements in two-phase flows,” presented at the Symposium on the Accuracy of Flow Measurements by the Laser Doppler Method (Copenhagen, 1975).

K. Bauckhage, H. H. Floegel, “Simultaneous measurement of droplet size and velocity in nozzle sprays,” presented at the Second Symposium on Applications in Laser Anemometry in Fluid Mechanics, Lisbon, 1984.

R. J. Adrain, W. L. Earley, “Evaluation of LDV performance using Mie scattering theory,” presented at the Symposium on Laser Anemometry, University of Minnesota, Minneapolis, Minn., 1976.

F. Durst, M. Macagno, G. Richter, “Light scattering by small particles: refined numerical computations” Rep. SFB 80/TM/195 (University of Karlsruhe, Karlsruhe, Germany, 1981).

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Figures (8)

Fig. 1
Fig. 1

Coordinate systems for incident beams and the moving particle.

Fig. 2
Fig. 2

Receiver and polarizer geometry.

Fig. 3
Fig. 3

Standard aperture geometries: (a) circular aperture; (b) rectangular aperture; (c) truncated circular aperture.

Fig. 4
Fig. 4

Resolution of the scattered fields parallel and perpendicular to the polarizer axis.

Fig. 5
Fig. 5

Two schemes for numerical integration of scattered light intensity over a truncated circular aperture: (a) single summation, (b) double summation.

Fig. 6
Fig. 6

Convergence of computed results with increasing grid points: comparison between elliptic and parabolic interpolation.

Fig. 7
Fig. 7

Scattered intensity field of a water droplet. Left-hand side, particle on a bright fringe; right-hand side, particle on a dark fringe. (a) ψpa = ψpt = 90°; (b) ψpa = ψpt = 45°; (c) ψpa = ψpt = 45°, polarizer at ωpol = 10°.

Fig. 8
Fig. 8

Phase diameter relations for three different arrangements of polarization.

Equations (95)

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s a = sin α i + cos α k ,
s t = sin α i + cos α k ,
p a = cos ψ p a cos α i + sin ψ p a j cos ψ p a sin α k ,
p t = cos ψ p t cos α i + sin ψ p t j + cos ψ p t sin α k ,
X P = X X 0 V t ,
[ R ] = [ cos ψ cos ω sin ϕ sin ψ cos ω + cos ϕ sin ω cos ϕ sin ψ cos ω + sin ϕ sin ω cos ψ sin ω sin ϕ sin ψ sin ω + cos ϕ cos ω cos ϕ sin ψ sin ω + sin ϕ cos ω sin ψ sin ϕ cos ψ cos ϕ cos ψ ] .
r pol = [ cos ω pol sin ω pol 0 sin ω pol cos ω pol 0 0 0 1 ] r r ,
s a p = [ R pol ] s a , etc .
X pol = [ R pol ] ( X P + X 0 + V t ) .
X P = [ R pol ] T X pol X 0 V t ,
r r = sin θ r cos ϕ r i + sin θ r sin ϕ r j + cos θ r k .
y r cos θ r x + z r sin θ r x = 0 ,
x r cos θ r y z r sin θ r y = 0 ,
r r = cos θ r x sin θ r y i sin θ r x cos θ r y j + cos θ r x cos θ r y k ( 1 sin 2 θ r x sin 2 θ r y ) 1 / 2 .
cos θ r x 0 = cos δ cos θ r y ( cos 2 θ r y cos 2 δ sin 2 θ r y ) 1 / 2 .
d Ω = | r r χ 1 × r r χ 2 | d χ 1 d χ 2 ,
d Ω = sin θ r d θ r d ϕ r .
d Ω = cos θ r x cos θ r y d θ r x d θ r y ( 1 sin 2 θ r x sin 2 θ r y ) 3 / 2 ,
cos θ a = r pol s a p .
cos ϕ a ( s a p × p a p ) ( s a p × r pol ) ,
sin ϕ a p a p ( s a p × r pol ) .
e ϕ a = sin ϕ a p a p + cos ϕ a ( s a p × p a p ) ,
e θ a = e ϕ a × r pol .
r pol = cos θ p x sin θ p y i sin θ p x cos θ p y j + cos θ p x cos θ p y k ( 1 sin 2 θ p x sin 2 θ p y ) 1 / 2 .
u x = r pol / θ y p | r pol / θ y p | = cos θ p y i + sin θ p x sin θ p y cos θ p x j cos 2 θ p x sin θ p y k ( 1 sin 2 θ p x sin 2 θ p y ) 1 / 2 .
u y = sin θ p x sin θ p y cos θ p y i cos θ p x j cos 2 θ p y sin θ p x k ( 1 sin 2 θ p x sin 2 θ p y ) 1 / 2 .
u x u y = sin θ p x sin θ p y .
E a = A 0 a e 0 a ( X ) exp { i [ k ( s a . X ) ω a t ] } ,
E t = A 0 t e 0 t ( X ) exp { i [ k ( s t . X ) ω t t ] } ,
E a P = A 0 a a P exp [ i ( ω a k s a V ) t ] ,
E t P = A 0 t t P exp [ i ( ω t k s t V ) t ] ,
a P = e 0 a ( X P + X 0 + V t ) exp [ i k s a ( X P + X 0 ) ] ,
t P = e 0 t ( X P + X 0 + V t ) exp [ i k s t ( X P + X 0 ) ] .
E asP = A 0 a S ( a P , r P , m , q ) × exp [ i ( ω a k s a V ) t ] exp ( i k X P ) k X P ,
E tsP = A 0 t S ( t P , r P , m , q ) × exp [ i ( ω t k s t V ) t ] exp ( i k X P ) k X P .
X P X P = X pol 2 + ( X 0 + V t ) ( X 0 + V t ) 2 [ R pol ] T X pol ( X 0 + V t ) .
X P = X pol [ R pol ] T r pol ( X 0 + V t ) ,
r P = [ R pol ] T r pol ,
E a s = A 0 a S ( a , r pol , m , q ) × exp { i [ ω a k ( s a + [ R pol ] T r pol ) V ] t } × exp ( i k X pol ) k X pol ,
E t s = A 0 t S ( t , r pol , m , q ) × exp { i [ ω t k ( s t + [ R pol ] T r pol ) V ] t } × exp ( i k X pol ) k X pol ,
X pol = X pol [ R pol ] T r pol X 0 .
S ( a , r pol , m , q ) = i ( S 2 a e θ a S 1 a e ϕ a ) = E asx u x + E asy u y ,
S ( t , r pol , m , q ) = i ( S 2 t e θ t S 1 t e ϕ t ) = E tsx s x u x + E tsy u y .
E asx = i S 2 a [ e θ a u x ( u x u y ) ( e θ a u y ) ] S 1 a [ e ϕ a u x ( u x u y ) ( e ϕ a u y ) ] 1 ( u x u y ) 2 .
E asx = i [ ( e θ a x c y e θ a z s y ) S 2 a ( e ϕ a x c y e ϕ a z s y ) S 1 a ] ,
c y = ( r p z 2 + r p y 2 ) 1 / 2 ,
s y = r p x c y / r p z ,
c x = ( r p z 2 + r p x 2 ) 1 / 2 ,
s x = r p y c x / r p z .
S a D = t x exp ( i ϕ 0 ) E asx u x + t y E asy u y ,
E a s D = A 0 a S a D exp { i [ ω a k ( s a + [ R pol ] T r pol ) V ] t } × exp ( i k X pol ) k X pol ,
E t s D = A 0 t S t D exp { i [ ω t k ( s t + [ R pol ] T r pol ) V ] t } × exp ( i k X pol ) k X pol .
I s ( t ) = 1 k 2 X pol 2 { G ( t ) + [ H ( t ) exp ( i ω s t ) ] } ,
ω s = ω a ω t + k ( s t s a ) V ,
G ( t ) = A 0 a 2 | S a D | 2 + A 0 t 2 | S t D | 2 ,
H ( t ) = 2 A 0 a A 0 t S a D S t D * .
| S a D | 2 = t x 2 | E asx | 2 + t y 2 | E asy | 2 + 2 t x t y ( u x . u y ) [ E asx E asy * exp ( i ϕ 0 ) ] ,
S a D S t D * = t x 2 E asx E tsx * + t y 2 E asy E tsy * + t x t y ( u x . u y ) × [ E asy E tsx * exp ( i ϕ 0 ) + E asx E tsy * exp ( i ϕ 0 ) ] .
u x u y = r p x r p y c x c y ,
P s = Ω I s X pol 2 d Ω .
P s = 1 k 2 ( G ¯ + H ¯ r cos ω s t + H ¯ i sin ω s t ) ,
= P [ 1 + η cos ( ω s t + Φ ) ] ,
P = G ¯ k 2 ,
η = ( H ¯ r 2 + H ¯ i 2 ) 1 / 2 G ¯ ,
Φ = tan 1 ( H ¯ i H ¯ r ) .
Φ P = Φ Φ I .
Φ I = tan 1 ( H I i H I r ) ,
H I = a P t P * at X P = 0 ,
= e 0 a e 0 t * exp [ i k ( s t s a ) X 0 ] .
Φ P = tan 1 ( H P i H P r ) ,
H ¯ P = H ¯ e 0 a * e 0 t exp [ i k ( s t s a ) X 0 ] .
p a I = p t I = j ;
p a II = cos α i + sin α k ,
p t II = cos α i sin α k .
S 1 a I = sin ϕ a I S 1 a ,
S 2 a I = cos ϕ a I S 2 a .
cos ϕ a II = sin ϕ a I ,
sin ϕ a II = cos ϕ a I .
S 1 a II = cos ϕ a I S 1 a ,
S 2 a II = sin ϕ a I S 2 a .
S 1 a = [ a I exp ( i δ pol ) sin ϕ a I + a II cos ϕ a I ] S 1 a ,
S 2 a = [ a I exp ( i δ pol ) cos ϕ a I a II sin ϕ a I ] S 2 a ,
a I = ( sin 2 ψ p a + p 2 cos 2 ψ p a 1 + p 2 ) 1 / 2 ,
a II = ( cos 2 ψ p a + p 2 sin 2 ψ p a 1 + p 2 ) 1 / 2 ,
sin δ pol ( p ) ,
cos δ pol ( 1 p 2 ) cos ψ p a sin ψ p a .
ω s t = 0 , 2 π , , ω s t = π , 3 π , ,
I s = 1 k 2 X pol 2 ( G ± H r ) .
G ¯ = n θ = 1 N θ W n θ sin θ r [ n ϕ = 1 N ϕ G ( θ r x , θ r y ) Δ ϕ r ] Δ θ r ,
G ¯ = 1 N G ( θ r x , θ r y ) Δ Ω ,
G ¯ = n y = 1 N y W n y cos θ r y × [ n x = 1 N x W n x G ( θ r x , θ r y ) cos θ r x ( 1 sin 2 θ r x sin 2 θ r y ) 3 / 2 Δ θ r x ] Δ θ r y .
J ( θ r y ) = ( δ 2 θ r y 2 ) 1 / 2 ( b θ r y + c ) ,
b = 1 Δ θ r y [ J h ( δ 2 θ ryh 2 ) 1 / 2 J l ( δ 2 θ ryl 2 ) 1 / 2 ] ,
c = 1 Δ θ r y [ J l θ ryh ( δ 2 θ ryl 2 ) 1 / 2 J h θ ryl ( δ 2 θ ryh 2 ) 1 / 2 ] ,
Δ G = b 3 [ ( δ 2 θ ryh 2 ) 3 / 2 ( δ 2 θ ryl 2 ) 3 / 2 ] c 2 { δ 2 [ cos 1 ( θ ryh δ ) cos 1 ( θ ryl δ ) ] θ ryh ( δ 2 θ ryh 2 ) 1 / 2 + θ ryl ( δ 2 θ ryl 2 ) 1 / 2 } .

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