Abstract

We characterize the shape of a large nonspherical particle by means of the two-dimensional Fourier transformation of its diffraction pattern, called the S function. The main properties of S functions are considered. Some ways in which to retrieve the geometric parameters of a particle by use of its S function are discussed. In particular, the parameter of nonsphericity of a particle is defined by means of the S function.

© 2000 Optical Society of America

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References

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  1. T. Allen, Particle Size Measurement (Chapman & Hall, London, 1990).
    [CrossRef]
  2. See 7th European Symposium on Particle Characterization, preprints I, II (NurnbergMesse, Nurnberg, Germany, 1998).
  3. D. H. Tycko, M. H. Metz, E. A. Epstein, A. Grinbaum, “Flow-cytometric light scattering measurements of red blood cell volume and hemoglobin concentration,” Appl. Opt. 24, 1355–1365 (1985).
    [CrossRef]
  4. E. D. Hirleman, C. F. Bohren, eds., feature on particle sizing, Appl. Opt. 30, 4685–4986 (1991).
  5. J. A. Davies, D. L. Collins, “Comparison of the size distribution of boron powders as measured by Malvern diffractometer and Coulter counter,” Part. Part. Syst. Charact. 5, 116–121 (1988).
    [CrossRef]
  6. C. M. G. Heffels, D. Heitzman, E. D. Hirleman, B. Scarlett, “The use of azimuthal intensity variations in diffraction patterns for particle shape characterization,” Part. Part. Syst. Charact. 11, 194–199 (1994).
    [CrossRef]
  7. P. Kaye, E. Hirst, Z. Whang-Thomas, “Neural-network-based spatial light-scattering instrument for hazardous airborne fiber detection,” Appl. Opt. 36, 6149–6156 (1997).
    [CrossRef] [PubMed]
  8. J. List, R. Weichert, “Detection of fibers by light diffraction,” in 7th European Symposium on Particle Characterization, preprints (NurnbergMesse, Nurnberg, Germany, 1998), pp. 705–713.
  9. W. Loffler, G. Lindenthal, “Measurements of particle size distribution with a white light Fraunhofer diffraction instrument for controlling of grinding process,” in 1st European Symposium on Process Technology in Pharmaceutical and Nutritional Sciences, preprints (NurnbergMesse, Nurnberg, Germany, 1998), pp. 141–150.
  10. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  11. A. G. Borovoi, “Straight-ray approximation in problems of scattering and propagation of waves in random media,” Opt. Atmosferi. 1,(7), 17–21 (1988), in Russian.
  12. A. G. Borovoi, E. I. Naats, U. G. Oppel, “Scattering of light by a red blood cell,” J. Biomed. Opt. 3, 364–372 (1998).
    [CrossRef] [PubMed]

1998 (1)

A. G. Borovoi, E. I. Naats, U. G. Oppel, “Scattering of light by a red blood cell,” J. Biomed. Opt. 3, 364–372 (1998).
[CrossRef] [PubMed]

1997 (1)

1994 (1)

C. M. G. Heffels, D. Heitzman, E. D. Hirleman, B. Scarlett, “The use of azimuthal intensity variations in diffraction patterns for particle shape characterization,” Part. Part. Syst. Charact. 11, 194–199 (1994).
[CrossRef]

1991 (1)

1988 (2)

J. A. Davies, D. L. Collins, “Comparison of the size distribution of boron powders as measured by Malvern diffractometer and Coulter counter,” Part. Part. Syst. Charact. 5, 116–121 (1988).
[CrossRef]

A. G. Borovoi, “Straight-ray approximation in problems of scattering and propagation of waves in random media,” Opt. Atmosferi. 1,(7), 17–21 (1988), in Russian.

1985 (1)

Allen, T.

T. Allen, Particle Size Measurement (Chapman & Hall, London, 1990).
[CrossRef]

Borovoi, A. G.

A. G. Borovoi, E. I. Naats, U. G. Oppel, “Scattering of light by a red blood cell,” J. Biomed. Opt. 3, 364–372 (1998).
[CrossRef] [PubMed]

A. G. Borovoi, “Straight-ray approximation in problems of scattering and propagation of waves in random media,” Opt. Atmosferi. 1,(7), 17–21 (1988), in Russian.

Collins, D. L.

J. A. Davies, D. L. Collins, “Comparison of the size distribution of boron powders as measured by Malvern diffractometer and Coulter counter,” Part. Part. Syst. Charact. 5, 116–121 (1988).
[CrossRef]

Davies, J. A.

J. A. Davies, D. L. Collins, “Comparison of the size distribution of boron powders as measured by Malvern diffractometer and Coulter counter,” Part. Part. Syst. Charact. 5, 116–121 (1988).
[CrossRef]

Epstein, E. A.

Grinbaum, A.

Heffels, C. M. G.

C. M. G. Heffels, D. Heitzman, E. D. Hirleman, B. Scarlett, “The use of azimuthal intensity variations in diffraction patterns for particle shape characterization,” Part. Part. Syst. Charact. 11, 194–199 (1994).
[CrossRef]

Heitzman, D.

C. M. G. Heffels, D. Heitzman, E. D. Hirleman, B. Scarlett, “The use of azimuthal intensity variations in diffraction patterns for particle shape characterization,” Part. Part. Syst. Charact. 11, 194–199 (1994).
[CrossRef]

Hirleman, E. D.

C. M. G. Heffels, D. Heitzman, E. D. Hirleman, B. Scarlett, “The use of azimuthal intensity variations in diffraction patterns for particle shape characterization,” Part. Part. Syst. Charact. 11, 194–199 (1994).
[CrossRef]

Hirst, E.

Kaye, P.

Lindenthal, G.

W. Loffler, G. Lindenthal, “Measurements of particle size distribution with a white light Fraunhofer diffraction instrument for controlling of grinding process,” in 1st European Symposium on Process Technology in Pharmaceutical and Nutritional Sciences, preprints (NurnbergMesse, Nurnberg, Germany, 1998), pp. 141–150.

List, J.

J. List, R. Weichert, “Detection of fibers by light diffraction,” in 7th European Symposium on Particle Characterization, preprints (NurnbergMesse, Nurnberg, Germany, 1998), pp. 705–713.

Loffler, W.

W. Loffler, G. Lindenthal, “Measurements of particle size distribution with a white light Fraunhofer diffraction instrument for controlling of grinding process,” in 1st European Symposium on Process Technology in Pharmaceutical and Nutritional Sciences, preprints (NurnbergMesse, Nurnberg, Germany, 1998), pp. 141–150.

Metz, M. H.

Naats, E. I.

A. G. Borovoi, E. I. Naats, U. G. Oppel, “Scattering of light by a red blood cell,” J. Biomed. Opt. 3, 364–372 (1998).
[CrossRef] [PubMed]

Oppel, U. G.

A. G. Borovoi, E. I. Naats, U. G. Oppel, “Scattering of light by a red blood cell,” J. Biomed. Opt. 3, 364–372 (1998).
[CrossRef] [PubMed]

Scarlett, B.

C. M. G. Heffels, D. Heitzman, E. D. Hirleman, B. Scarlett, “The use of azimuthal intensity variations in diffraction patterns for particle shape characterization,” Part. Part. Syst. Charact. 11, 194–199 (1994).
[CrossRef]

Tycko, D. H.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

Weichert, R.

J. List, R. Weichert, “Detection of fibers by light diffraction,” in 7th European Symposium on Particle Characterization, preprints (NurnbergMesse, Nurnberg, Germany, 1998), pp. 705–713.

Whang-Thomas, Z.

Appl. Opt. (3)

J. Biomed. Opt. (1)

A. G. Borovoi, E. I. Naats, U. G. Oppel, “Scattering of light by a red blood cell,” J. Biomed. Opt. 3, 364–372 (1998).
[CrossRef] [PubMed]

Opt. Atmosferi. (1)

A. G. Borovoi, “Straight-ray approximation in problems of scattering and propagation of waves in random media,” Opt. Atmosferi. 1,(7), 17–21 (1988), in Russian.

Part. Part. Syst. Charact. (2)

J. A. Davies, D. L. Collins, “Comparison of the size distribution of boron powders as measured by Malvern diffractometer and Coulter counter,” Part. Part. Syst. Charact. 5, 116–121 (1988).
[CrossRef]

C. M. G. Heffels, D. Heitzman, E. D. Hirleman, B. Scarlett, “The use of azimuthal intensity variations in diffraction patterns for particle shape characterization,” Part. Part. Syst. Charact. 11, 194–199 (1994).
[CrossRef]

Other (5)

J. List, R. Weichert, “Detection of fibers by light diffraction,” in 7th European Symposium on Particle Characterization, preprints (NurnbergMesse, Nurnberg, Germany, 1998), pp. 705–713.

W. Loffler, G. Lindenthal, “Measurements of particle size distribution with a white light Fraunhofer diffraction instrument for controlling of grinding process,” in 1st European Symposium on Process Technology in Pharmaceutical and Nutritional Sciences, preprints (NurnbergMesse, Nurnberg, Germany, 1998), pp. 141–150.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

T. Allen, Particle Size Measurement (Chapman & Hall, London, 1990).
[CrossRef]

See 7th European Symposium on Particle Characterization, preprints I, II (NurnbergMesse, Nurnberg, Germany, 1998).

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Figures (9)

Fig. 1
Fig. 1

Three experimental schemes for viewing the Fraunhofer diffraction patterns.

Fig. 2
Fig. 2

(a) Projection of a model particle, (b) its phase function, (c) its S function.

Fig. 3
Fig. 3

Sections of the S function at various heights: (a) particle from Fig. 2, (b) ellipse, (c) rectangle, (d) triangle.

Fig. 4
Fig. 4

Red-blood cell with an absorbing inclusion: (a) particle shape, (b) real part of the S function, (c) imaginary part of the S function, (d) phase function.

Fig. 5
Fig. 5

Geometrical proof of Eq. (20).

Fig. 6
Fig. 6

(a) Projection of a model particle, (b) its diffraction pattern, (c) its S function.

Fig. 7
Fig. 7

S function of Fig. 6 retrieved from (a) the central spot of the diffraction pattern, (b) the central plus first-order spots.

Fig. 8
Fig. 8

S function of Fig. 6 retrieved from the diffraction pattern distorted by quantization of the detected intensity. The numbers of detection levels are (a) 128, (b) 64, and (c) 32.

Fig. 9
Fig. 9

Sections of the S function of Fig. 6 for various amplitudes A of the noise in the detected intensity: (a) A = 0.25, (b) A = 0.75, (c) A = 1.

Equations (26)

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Φρ=k -mρ, z-1dz,
sρ=expiΦρ-1ηρ.
ηρ=1inside the particle shadow projection0outside the shadow.
fn=k2πi  exp-iknρsρdρ,
sρ=-ηρ.
In=|fn|2.
 Indn=σ.
Sρ= Inexpiknρdn,
In=k2π2  Sρexp-iknρdρ.
Sρ= sρs*ρ-ρdρ.
Sx, y=ab-b|x|-a|y|+|xy||x|<a, |y|<b0|x|>a, |y|>b.
Sρ=S*-ρ.
Sρ=S1ρ+iS2ρ,
S1ρ=S1-ρ,
S2ρ=-S2-ρ.
sρ-ρ0=s-ρ+ρ0.
S2ρ= s2ρs1ρ+ρdρ- s1ρs2ρ+ρdρ=0,
dρ=hρ.
S0=σ.
dS0dl=Pm.
dS0dl=i NiPim
dS0dl=pπ.
α=4π σp2,
Sr= Sr, φdφ/2π,
Sr=k2π2  J0k|n|rIndn,
dS0/dlS0-Sr0/r0.

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