Abstract

We formulate the Chin–Shifrin integral transform inversion for calculating the particle-size distribution from an observed forward-scattering pattern in a way that does not require knowledge of the derivative of the data. The resulting equations are suitable for use with a photodiode array of the type commonly used to record forward-scattering signatures. By attention to the range of integration in the inversion, and by suitable apodization of the input signal, we are able to reduce noise in the inversion to an acceptable level. Some sample analyses are presented.

© 1991 Optical Society of America

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  1. J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Chapman & Hall, London, 1957).
  3. Y. C. Agrawal, J. B. Riley, “Optical particle sizing for hydrodynamics based on near forward scattering,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 68–76 (1984).
  4. E. D. Hirleman, “Particle sizing by optical nonimaging techniques,” in Liquid Particle Size Measurement Techniques, ASTM STP 848, J. M. Tishkoff, R. D. Ingebo, J. B. Kennedy, eds. (American Society for Testing Materials, Philadelphia, 1984).
  5. E. C. Titchmarsh, “Extension of Fourier's integral formula to formulae involving Bessel functions,” Proc. London Math. Soc. Ser. 2 23(7), XXII–XXIV (1925).
  6. J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 841–844 (1955).
    [CrossRef]
  7. L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
  8. A. L. Fymat, K. D. Mease, “Reconstructing the size distribution of spherical particles from angular forward scattering data,” in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, eds. (Elsevier, Amsterdam, 1978).
  9. A. L. Fymat, “Analytical inversions in remote sensing of particle size distribution. 2: Angular and spectral scattering in diffraction approximations,” Appl. Opt. 17, 1677–1678 (1978).
    [CrossRef] [PubMed]
  10. J. Koo, E. D. Hirleman, “A synthesis of integral transform techniques for reconstruction of particle size distributions from the forward-scattered light,” Appl. Opt.
  11. W. Steele, Interferometry (Cambridge U. Press, London, 1967).
  12. N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. Patent3,689,772 (5Sept.1972).
  13. E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response characteristics of laser diffraction particle size analyzers: optical sample volume extent and lens effects,”; Opt. Eng. 23, 610–619 (1984).
    [CrossRef]
  14. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, London, 1922), p. 136.
  15. J. H. Koo, “Particle size analysis using integral transform techniques on Fraunhofer diffraction patterns,” Ph.D. dissertation, (Arizona State University and George Washington University, Washington, D.C.1987).
  16. Malvern Instruments, Ltd., Malvern, Worcs., England.
  17. Seragen Diagnostics, Inc., Indianapolis, Ind.

1984

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response characteristics of laser diffraction particle size analyzers: optical sample volume extent and lens effects,”; Opt. Eng. 23, 610–619 (1984).
[CrossRef]

1978

1977

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

1955

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

1925

E. C. Titchmarsh, “Extension of Fourier's integral formula to formulae involving Bessel functions,” Proc. London Math. Soc. Ser. 2 23(7), XXII–XXIV (1925).

Abbot, D.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Agrawal, Y. C.

Y. C. Agrawal, J. B. Riley, “Optical particle sizing for hydrodynamics based on near forward scattering,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 68–76 (1984).

Bayvel, L. P.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).

Beer, J. M.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Chigier, N. A.

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response characteristics of laser diffraction particle size analyzers: optical sample volume extent and lens effects,”; Opt. Eng. 23, 610–619 (1984).
[CrossRef]

Chin, J. H.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

Fymat, A. L.

A. L. Fymat, “Analytical inversions in remote sensing of particle size distribution. 2: Angular and spectral scattering in diffraction approximations,” Appl. Opt. 17, 1677–1678 (1978).
[CrossRef] [PubMed]

A. L. Fymat, K. D. Mease, “Reconstructing the size distribution of spherical particles from angular forward scattering data,” in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, eds. (Elsevier, Amsterdam, 1978).

George, N.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. Patent3,689,772 (5Sept.1972).

Hirleman, E. D.

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response characteristics of laser diffraction particle size analyzers: optical sample volume extent and lens effects,”; Opt. Eng. 23, 610–619 (1984).
[CrossRef]

J. Koo, E. D. Hirleman, “A synthesis of integral transform techniques for reconstruction of particle size distributions from the forward-scattered light,” Appl. Opt.

E. D. Hirleman, “Particle sizing by optical nonimaging techniques,” in Liquid Particle Size Measurement Techniques, ASTM STP 848, J. M. Tishkoff, R. D. Ingebo, J. B. Kennedy, eds. (American Society for Testing Materials, Philadelphia, 1984).

Jones, A. R.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).

Koo, J.

J. Koo, E. D. Hirleman, “A synthesis of integral transform techniques for reconstruction of particle size distributions from the forward-scattered light,” Appl. Opt.

Koo, J. H.

J. H. Koo, “Particle size analysis using integral transform techniques on Fraunhofer diffraction patterns,” Ph.D. dissertation, (Arizona State University and George Washington University, Washington, D.C.1987).

McCreath, G. C.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Mease, K. D.

A. L. Fymat, K. D. Mease, “Reconstructing the size distribution of spherical particles from angular forward scattering data,” in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, eds. (Elsevier, Amsterdam, 1978).

Oechsle, V.

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response characteristics of laser diffraction particle size analyzers: optical sample volume extent and lens effects,”; Opt. Eng. 23, 610–619 (1984).
[CrossRef]

Riley, J. B.

Y. C. Agrawal, J. B. Riley, “Optical particle sizing for hydrodynamics based on near forward scattering,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 68–76 (1984).

Sliepcevich, C. M.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

Spindel, A.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. Patent3,689,772 (5Sept.1972).

Steele, W.

W. Steele, Interferometry (Cambridge U. Press, London, 1967).

Swithenbank, J.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Taylor, D. S.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Thomasson, J. T.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. Patent3,689,772 (5Sept.1972).

Titchmarsh, E. C.

E. C. Titchmarsh, “Extension of Fourier's integral formula to formulae involving Bessel functions,” Proc. London Math. Soc. Ser. 2 23(7), XXII–XXIV (1925).

Tribus, M.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Chapman & Hall, London, 1957).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, London, 1922), p. 136.

Appl. Opt.

J. Phys. Chem.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

Opt. Eng.

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response characteristics of laser diffraction particle size analyzers: optical sample volume extent and lens effects,”; Opt. Eng. 23, 610–619 (1984).
[CrossRef]

Proc. London Math. Soc. Ser. 2

E. C. Titchmarsh, “Extension of Fourier's integral formula to formulae involving Bessel functions,” Proc. London Math. Soc. Ser. 2 23(7), XXII–XXIV (1925).

Prog. Astronaut. Aeronaut.

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot, G. C. McCreath, “A laser diagnostic technique for the measurement of droplet and particle size distribution,” Prog. Astronaut. Aeronaut. 53, 421–447 (1977).

Other

H. C. van de Hulst, Light Scattering by Small Particles (Chapman & Hall, London, 1957).

Y. C. Agrawal, J. B. Riley, “Optical particle sizing for hydrodynamics based on near forward scattering,” in Ocean Optics VII, M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 68–76 (1984).

E. D. Hirleman, “Particle sizing by optical nonimaging techniques,” in Liquid Particle Size Measurement Techniques, ASTM STP 848, J. M. Tishkoff, R. D. Ingebo, J. B. Kennedy, eds. (American Society for Testing Materials, Philadelphia, 1984).

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).

A. L. Fymat, K. D. Mease, “Reconstructing the size distribution of spherical particles from angular forward scattering data,” in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, eds. (Elsevier, Amsterdam, 1978).

J. Koo, E. D. Hirleman, “A synthesis of integral transform techniques for reconstruction of particle size distributions from the forward-scattered light,” Appl. Opt.

W. Steele, Interferometry (Cambridge U. Press, London, 1967).

N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. Patent3,689,772 (5Sept.1972).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, London, 1922), p. 136.

J. H. Koo, “Particle size analysis using integral transform techniques on Fraunhofer diffraction patterns,” Ph.D. dissertation, (Arizona State University and George Washington University, Washington, D.C.1987).

Malvern Instruments, Ltd., Malvern, Worcs., England.

Seragen Diagnostics, Inc., Indianapolis, Ind.

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