Abstract

We show that integral transform inversions for Fraunhofer-diffraction particle sizing possess an important feature that has useful implications. The selection of three angular parameters, Δθ (angular resolution), θmin (minimum scattering angle), and θmax (maximum scattering angle), necessary for reconstructing a given kind of particle size distribution without undergoing mathematical limits that contradict the Fraunhofer theory and exceed practical measurement limitations, depends inverse-linearly on the optical size parameter χ, χ = 2πa/λ (a, particle radius; λ, wavelength). Two series of numerical experiments, in which the Chin–Shifrin inversion is used, are performed to assess the reconstruction of original discontinuous (narrow-type) and continuous (board-type) particle size distributions from simulated Fraunhofer-diffracted patterns, assuming linear- and log-scaled light detector configurations, respectively. New and useful findings regarding the roles of these three key angular parameters in the Chin–Shifrin inversion process, including general criteria relating the χ and the selections of Δθ and θmax for effective size retrieval, were obtained.

© 1997 Optical Society of America

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  1. J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of intensity of forward-scattered light at very small angle,” J. Phys. Chem. 59, 841–844 (1955).
    [CrossRef]
  2. K. S. Shifrin, A. Y. Perleman, “Inversion of light scattering data for the determination of spherical particle spectrum,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967).
  3. K. S. Shifrin, I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 749–753 (1967).
  4. L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium power,” Combust. Explosion Shock Waves 12, 258–262 (1976).
    [CrossRef]
  5. S. D. Coston, N. George, “Particle sizing by inversion of the optical transform pattern,” Appl. Opt. 30, 4785–4794 (1991).
    [CrossRef] [PubMed]
  6. J. H. Koo, E. D. Hirleman, “Synthesis of integral transform solutions for the reconstruction of particle-size distributions from forward-scattered light,” Appl. Opt. 31, 2130–2140 (1992).
    [CrossRef] [PubMed]
  7. L. C. Chow, C. L. Tien, “Inversion techniques for determining the droplet size distribution in clouds: numerical examination,” Appl. Opt. 15, 378–383 (1976).
    [CrossRef] [PubMed]
  8. D. Holve, S. A. Self, “Optical particle sizing for in situ measurements. Parts 1,” Appl. Opt. 18, 1632–1645 (1979).
    [CrossRef] [PubMed]
  9. R. Santer, M. Herman, “Particle size distributions from forward scattered light using the Chahine inversion scheme,” Appl. Opt. 22, 2294–2301 (1983).
    [CrossRef] [PubMed]
  10. E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
    [CrossRef]
  11. K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–253 (1993).
    [CrossRef]
  12. K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 514–518 (1966).
  13. K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).
  14. A. L. Fymat, K. D. Mease, “Reconstruction of the size distribution of spherical particles from angular forward scattering data,” in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. Fymat, V. E. Zuev, eds. (Elsevier, New York, 1978).
  15. J. C. Knight, D. Ball, G. N. Robertson, “Analytical inversion for laser diffraction spectrometry giving improved resolution and accuracy in size distribution,” Appl. Opt. 30, 4795–4799 (1991).
    [CrossRef] [PubMed]
  16. A. N. Tikhonov, V. Y. Arsenin, Solution on Ill-Posed Problems (Winston, Washington D.C., 1977).
  17. E. C. Titchmarch, “Extension of Fourier’s integral formula to formulae involving Bessel functions,” Proc. London Math. Soc. Ser. 2 23, 22–24 (1924).
  18. H. Bateman, “On the inversion of a definite integral,” Proc. London Math. Soc. Ser. 4, 461–498 (1906).
  19. J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. Part II. Sunlight reflected by terrestrial water clouds,” J. Atmos. Sci. 28, 1400–1426 (1972).
    [CrossRef]
  20. Malvern Instruments Ltd., Malvern, Worcsester, England.
  21. 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]
  22. K. S. Shifrin, V. A. Punina, “Light scattering indicatrix in the region of small angles,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 4, 450–453 (1968).
  23. A. L. Fymat, K. D. Mease, “Mie forward scattering: improved semiempirical approximation with application to particle size distribution inversion,” Appl. Opt. 20, 194–198 (1981).
    [CrossRef] [PubMed]

1993

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–253 (1993).
[CrossRef]

1992

1991

1987

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
[CrossRef]

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]

1983

1981

1979

1976

L. C. Chow, C. L. Tien, “Inversion techniques for determining the droplet size distribution in clouds: numerical examination,” Appl. Opt. 15, 378–383 (1976).
[CrossRef] [PubMed]

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium power,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

1972

J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. Part II. Sunlight reflected by terrestrial water clouds,” J. Atmos. Sci. 28, 1400–1426 (1972).
[CrossRef]

1968

K. S. Shifrin, V. A. Punina, “Light scattering indicatrix in the region of small angles,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 4, 450–453 (1968).

1967

K. S. Shifrin, I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 749–753 (1967).

1966

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 514–518 (1966).

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).

1955

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

1924

E. C. Titchmarch, “Extension of Fourier’s integral formula to formulae involving Bessel functions,” Proc. London Math. Soc. Ser. 2 23, 22–24 (1924).

1906

H. Bateman, “On the inversion of a definite integral,” Proc. London Math. Soc. Ser. 4, 461–498 (1906).

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solution on Ill-Posed Problems (Winston, Washington D.C., 1977).

Ball, D.

Bateman, H.

H. Bateman, “On the inversion of a definite integral,” Proc. London Math. Soc. Ser. 4, 461–498 (1906).

Boitova, L. N.

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium power,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

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 angle,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

Chow, L. C.

Coston, S. D.

Fymat, A. L.

A. L. Fymat, K. D. Mease, “Mie forward scattering: improved semiempirical approximation with application to particle size distribution inversion,” Appl. Opt. 20, 194–198 (1981).
[CrossRef] [PubMed]

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

George, N.

Hansen, J. E.

J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. Part II. Sunlight reflected by terrestrial water clouds,” J. Atmos. Sci. 28, 1400–1426 (1972).
[CrossRef]

Herman, M.

Hirleman, E. D.

J. H. Koo, E. D. Hirleman, “Synthesis of integral transform solutions for the reconstruction of particle-size distributions from forward-scattered light,” Appl. Opt. 31, 2130–2140 (1992).
[CrossRef] [PubMed]

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
[CrossRef]

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]

Holve, D.

Knight, J. C.

Kolmakov, I. B.

K. S. Shifrin, I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 749–753 (1967).

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 514–518 (1966).

Koo, J. H.

Kudryavitskii, F. A.

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium power,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

Mease, K. D.

A. L. Fymat, K. D. Mease, “Mie forward scattering: improved semiempirical approximation with application to particle size distribution inversion,” Appl. Opt. 20, 194–198 (1981).
[CrossRef] [PubMed]

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

Perleman, A. Y.

K. S. Shifrin, A. Y. Perleman, “Inversion of light scattering data for the determination of spherical particle spectrum,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967).

Petrov, G. D.

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium power,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

Punina, V. A.

K. S. Shifrin, V. A. Punina, “Light scattering indicatrix in the region of small angles,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 4, 450–453 (1968).

Robertson, G. N.

Robinskii, V. L.

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium power,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

Santer, R.

Self, S. A.

Shifrin, K. S.

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–253 (1993).
[CrossRef]

K. S. Shifrin, V. A. Punina, “Light scattering indicatrix in the region of small angles,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 4, 450–453 (1968).

K. S. Shifrin, I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 749–753 (1967).

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 514–518 (1966).

K. S. Shifrin, A. Y. Perleman, “Inversion of light scattering data for the determination of spherical particle spectrum,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967).

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 angle,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

Sokolov, R. N.

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium power,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

Tien, C. L.

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solution on Ill-Posed Problems (Winston, Washington D.C., 1977).

Titchmarch, E. C.

E. C. Titchmarch, “Extension of Fourier’s integral formula to formulae involving Bessel functions,” Proc. London Math. Soc. Ser. 2 23, 22–24 (1924).

Tonna, G.

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–253 (1993).
[CrossRef]

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 angle,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

Adv. Geophys.

K. S. Shifrin, G. Tonna, “Inverse problems related to light scattering in the atmosphere and ocean,” Adv. Geophys. 34, 175–253 (1993).
[CrossRef]

Appl. Opt.

Combust. Explosion Shock Waves

L. N. Boitova, F. A. Kudryavitskii, G. D. Petrov, V. L. Robinskii, R. N. Sokolov, “Particle size distribution for the flame from a mixture containing magnesium power,” Combust. Explosion Shock Waves 12, 258–262 (1976).
[CrossRef]

Izv. Acad. Sci. USSR Atmos. Oceanic Phys.

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 514–518 (1966).

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 2, 559–561 (1966).

K. S. Shifrin, I. B. Kolmakov, “Calculation of particle size spectrum from direct and integral values of the indicatrix in the small angle region,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 749–753 (1967).

K. S. Shifrin, V. A. Punina, “Light scattering indicatrix in the region of small angles,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 4, 450–453 (1968).

J. Atmos. Sci.

J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. Part II. Sunlight reflected by terrestrial water clouds,” J. Atmos. Sci. 28, 1400–1426 (1972).
[CrossRef]

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 angle,” 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]

Part. Charact.

E. D. Hirleman, “Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature,” Part. Charact. 4, 128–133 (1987).
[CrossRef]

Proc. London Math. Soc. Ser.

H. Bateman, “On the inversion of a definite integral,” Proc. London Math. Soc. Ser. 4, 461–498 (1906).

Proc. London Math. Soc. Ser. 2

E. C. Titchmarch, “Extension of Fourier’s integral formula to formulae involving Bessel functions,” Proc. London Math. Soc. Ser. 2 23, 22–24 (1924).

Other

Malvern Instruments Ltd., Malvern, Worcsester, England.

K. S. Shifrin, A. Y. Perleman, “Inversion of light scattering data for the determination of spherical particle spectrum,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967).

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

A. N. Tikhonov, V. Y. Arsenin, Solution on Ill-Posed Problems (Winston, Washington D.C., 1977).

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Figures (12)

Fig. 1
Fig. 1

Identical numerically resolved reconstructed particle-size distributions from theoretical Fraunhofer-diffraction patterns when the Chin–Shifrin inversion for the assumed rectangular-pulse-size distributions are used.

Fig. 2
Fig. 2

Simulated, normalized intensity profiles for the two rectangular-pulse-size distributions.

Fig. 3
Fig. 3

Theoretical (I/I 03 profile for the rectangular-pulse size distribution with the radius a = 3000 µm, λ = 0.6328 µm.

Fig. 4
Fig. 4

Theoretical (I/I 03 profile for the rectangular-pulse size distribution with the radius a = 0.3 µm, λ = 0.6328 µm.

Fig. 5
Fig. 5

Theoretical (I/I 03 profile for the rectangular-pulse-size distribution with the radius a = 30 µm, λ = 0.6328 µm.

Fig. 6
Fig. 6

Theoretical (I/I 03 profile for the trigamma-type particle-size distribution.

Fig. 7
Fig. 7

Effects of Δθ, θmax, and θmin on the size reconstruction of assumed single-rectangular-pulse and trigamma-type particle-size distributions.

Fig. 8
Fig. 8

Inverted, trirectangular pulse size distribution from theoretical Fraunhofer-diffraction patterns using the Chin–Shifrin inversion.

Fig. 9
Fig. 9

Inverted, single-gamma-type size distribution from theoretical Fraunhofer-diffraction patterns where the Chin–Shifrin inversion is used.

Fig. 10
Fig. 10

Size reconstructions by the Chin–Shifrin inversion for the same rectangular pulse and trigamma size distributions from theoretical Fraunhofer-diffraction signatures simulating a commercial20 log-scaled detector array configurations with the ring radii as reported by Hirleman et al. 21; transform lenses of focal lengths F = 63, 300 mm and λ = 0.6328 µm are used. (a) Effect of θmax on the size reconstruction of the rectangular pulse size distribution. (b) Effect of θmax on the size reconstruction of the trigamma size distribution. (c) Effect of Δθ on the size reconstruction of the trigamma size distribution. (d) Effect of Δθ on the size reconstruction of the rectangular pulse size distribution. (e) Effect of θmin on the size reconstruction of the rectangular pulse size distribution.

Fig. 11
Fig. 11

Comparison of inversions between theoretical Fraunhofer-diffraction patterns simulating linear- and log-scaled21 detector configurations for the trigamma size function.

Fig. 12
Fig. 12

Comparison of inversions between theoretical Fraunhofer-diffraction patterns simulating linear- and log-scaled21 detector configurations for the rectangular pulse size function.

Tables (1)

Tables Icon

Table 1 Near-Forward-Scattering Angles of a Commercial Malvern Log-scaled Photodiode Detector Array21

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Iθ=I0aJ1xθθ2.
Iθ=I0ofaaJ1xθθ2da.
fbx=-4π2λ2I00 HixθddθEiθdθ,
a2fa=-2πλI00 2πxθJ1xθY1xθddθIθ3dθ.
Hcsxθ=2πxθJ1xθY1xθ,  ddθEcsθ=ddθIθ3,
fa=ci=13a-Ri1/b-3 exp-a-RibtaRi0a<Ri.

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