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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References

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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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]
  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 (1)

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 (1)

1977 (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).

1955 (1)

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 (1)

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).
[CrossRef]

Bayvel, L. P.

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

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).
[CrossRef]

Jones, A. R.

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

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).
[CrossRef]

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. (1)

J. Phys. Chem. (1)

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. (1)

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 (1)

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. (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).

Other (12)

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).
[CrossRef]

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).
[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.

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).

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

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).

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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Figures (6)

Fig. 1
Fig. 1

Effect of apodization on the size spectrum resulting from the Chin–Shifrin inversion of a theoretical Fraunhofer-diffraction pattern calculated over a limited angular range. The input signal (a), a delta function at a = 20 μm, was used to compute a diffraction signature for 200 equispaced points over the angular range of 0.085–4.1 deg. This scattering pattern was then inverted by the use of Eq. (3) without (b) and with (c) apodization of the data.

Fig. 2
Fig. 2

Delta function at a = 20 μm used to calculate a theoretical scattering pattern for a 30-element log-spaced detector array with radii as reported by Hirleman et al.13 The scattering pattern was inverted, with the value for Ndel in Eq. (8) and for θmax in Eq. (5) being determined by the number of detector elements in the array (b) and by the use of condition (9) applied at each value of a (c).

Fig. 3
Fig. 3

Measurement of a monodisperse sample of nominal size of 20 μm from a scattering pattern recorded using the f = 100-mm lens. The mass frequency distributions resulting from the analytical inversion (solid line) and the Malvern proprietary software (broken line) are plotted against the particle size.

Fig. 4
Fig. 4

Measurement of a bimodal sample. The nominal sizes of the spheres in the sample were 15 and 90 μm as marked on the top axis.

Fig. 5
Fig. 5

Measurement of a sample displaying three modes nominally at 45, 90, and 173 μm. The scattering pattern was recorded using the f = 300-mm lens.

Fig. 6
Fig. 6

Measurement of a sample containing four modes at nominal sizes of 15, 20, 45, and 90 μm.

Equations (12)

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I ( θ ) = I inc [ a J 1 ( x θ ) θ ] 2 ,
I ( θ ) = I inc 0 f ( a ) [ a J 1 ( x θ ) θ ] 2 d a .
a 2 f ( a ) = 4 π 2 λ 0 K ( x θ ) d d θ ( θ 3 I θ ) d θ ,
K ( x θ ) = ( x θ ) J 1 ( x θ ) Y 1 ( x θ ) ,
A ( θ ) = ( 1 θ 2 θ max 2 ) 2 ,
E = C 0 a 2 f ( a ) [ 1 J 0 2 ( 2 π a θ λ ) J 1 2 ( 2 π a θ λ ) ] d a ,
E n = C 0 a 2 f ( a ) [ J 0 2 ( 2 π a θ in n λ ) + J 1 2 ( 2 π a θ in n λ ) J 0 2 ( 2 π a θ out n λ ) J 1 2 ( 2 π θ out n λ ) ] d a ,
a 2 f ( a ) = 4 π 2 λ ( n = 1 N det I n θ in n θ out n K ( x θ ) d d θ [ A ( θ ) θ 3 ] d θ + n = 1 N det 1 θ out n δ θ in n + 1 + δ K ( x θ ) { d d θ [ A ( θ ) θ 3 ] I n } d θ )
n = 1 N det I n θ in n θ out n P ( z ) J 1 ( z ) Y 1 ( z ) d z ,
P ( z ) = 3 z 3 x 3 10 z 5 x 5 0 max 2 + 7 z 7 x 7 0 max 4
n = 1 N det 1 { [ K ( x θ ) θ 3 A ( θ ) I n ] θ out n θ in n + 1 θ out n θ in n + 1 θ 3 I n A ( θ ) d [ K ( x θ ) ] } ,
θ 15.47 / x .

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