Abstract

Using the optical transform intensity and a new inversion formula, we demonstrate the recovery of particulate size distributions. For the optical experiments, chrome masks are fabricated to provide three distributions: the gamma, the log-normal, and the bimodal exponential. Excellent agreement with the theory is obtained.

© 1991 Optical Society of America

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References

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  1. G. Mie, Ann. Phys. 25, 377 (1908).
    [CrossRef]
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  3. K. S. Shifrin, A. Y. Perelman, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967).
  4. A. L. Fymat, K. D. Mease, Appl. Opt. 20, 195 (1981).
    [CrossRef]
  5. W. J. Oliver, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds., Vol. 55 of Applied Mathematics Series (U.S. Govt. Printing Office, Washington, D.C., 1964), Chap. 9.
  6. S. D. Coston, N. George, Appl. Opt. 30, 4786 (1991).
    [CrossRef]
  7. N. George, A. Jain, R. D. S. Melville, Appl. Phys. 7, 157 (1975).
    [CrossRef]
  8. K. S. Shifrin, I. B. Kolmakov, Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 749 (1967).
  9. N. George, J. T. Thomasson, A. Spindel, U.S. patent3,689,772 (September5, 1972).
  10. IMSL, Inc., Houston, Tex.
  11. W. H. Press, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1988).
  12. VAX is a registered trademark of Digital Equipment Corporation, Maynard, Mass.
  13. R. Zisselmar, H. Kellerwessel, Part. Charact. 2, 49 (1985).
    [CrossRef]

1991 (1)

S. D. Coston, N. George, Appl. Opt. 30, 4786 (1991).
[CrossRef]

1985 (1)

R. Zisselmar, H. Kellerwessel, Part. Charact. 2, 49 (1985).
[CrossRef]

1981 (1)

A. L. Fymat, K. D. Mease, Appl. Opt. 20, 195 (1981).
[CrossRef]

1975 (1)

N. George, A. Jain, R. D. S. Melville, Appl. Phys. 7, 157 (1975).
[CrossRef]

1967 (1)

K. S. Shifrin, I. B. Kolmakov, Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 749 (1967).

1908 (1)

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Coston, S. D.

S. D. Coston, N. George, Appl. Opt. 30, 4786 (1991).
[CrossRef]

Fymat, A. L.

A. L. Fymat, K. D. Mease, Appl. Opt. 20, 195 (1981).
[CrossRef]

George, N.

S. D. Coston, N. George, Appl. Opt. 30, 4786 (1991).
[CrossRef]

N. George, A. Jain, R. D. S. Melville, Appl. Phys. 7, 157 (1975).
[CrossRef]

N. George, J. T. Thomasson, A. Spindel, U.S. patent3,689,772 (September5, 1972).

Jain, A.

N. George, A. Jain, R. D. S. Melville, Appl. Phys. 7, 157 (1975).
[CrossRef]

Kellerwessel, H.

R. Zisselmar, H. Kellerwessel, Part. Charact. 2, 49 (1985).
[CrossRef]

Kolmakov, I. B.

K. S. Shifrin, I. B. Kolmakov, Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 749 (1967).

Mease, K. D.

A. L. Fymat, K. D. Mease, Appl. Opt. 20, 195 (1981).
[CrossRef]

Melville, R. D. S.

N. George, A. Jain, R. D. S. Melville, Appl. Phys. 7, 157 (1975).
[CrossRef]

Mie, G.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Oliver, W. J.

W. J. Oliver, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds., Vol. 55 of Applied Mathematics Series (U.S. Govt. Printing Office, Washington, D.C., 1964), Chap. 9.

Perelman, A. Y.

K. S. Shifrin, A. Y. Perelman, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967).

Press, W. H.

W. H. Press, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Shifrin, K. S.

K. S. Shifrin, I. B. Kolmakov, Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 749 (1967).

K. S. Shifrin, A. Y. Perelman, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967).

Spindel, A.

N. George, J. T. Thomasson, A. Spindel, U.S. patent3,689,772 (September5, 1972).

Thomasson, J. T.

N. George, J. T. Thomasson, A. Spindel, U.S. patent3,689,772 (September5, 1972).

van de Hulst, H. C.

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

Zisselmar, R.

R. Zisselmar, H. Kellerwessel, Part. Charact. 2, 49 (1985).
[CrossRef]

Ann. Phys. (1)

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Appl. Opt. (2)

A. L. Fymat, K. D. Mease, Appl. Opt. 20, 195 (1981).
[CrossRef]

S. D. Coston, N. George, Appl. Opt. 30, 4786 (1991).
[CrossRef]

Appl. Phys. (1)

N. George, A. Jain, R. D. S. Melville, Appl. Phys. 7, 157 (1975).
[CrossRef]

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (1)

K. S. Shifrin, I. B. Kolmakov, Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 3, 749 (1967).

Part. Charact. (1)

R. Zisselmar, H. Kellerwessel, Part. Charact. 2, 49 (1985).
[CrossRef]

Other (7)

N. George, J. T. Thomasson, A. Spindel, U.S. patent3,689,772 (September5, 1972).

IMSL, Inc., Houston, Tex.

W. H. Press, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

VAX is a registered trademark of Digital Equipment Corporation, Maynard, Mass.

W. J. Oliver, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds., Vol. 55 of Applied Mathematics Series (U.S. Govt. Printing Office, Washington, D.C., 1964), Chap. 9.

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

K. S. Shifrin, A. Y. Perelman, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1967).

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Figures (5)

Fig. 1
Fig. 1

Far-zone scattering intensity for a 10-μm-radius sphere with an index of refraction of 1.5 calculated with Eq. (1) (the solid curve) and Mie theory (the dotted curve). The values of λ = 0.6328 μm and F = 1 m were chosen.

Fig. 2
Fig. 2

Geometry for recording the far-zone optical transform intensity I(θ) from a distribution of circular apertures f(a), where a is the particle radius. The focal length of the transform lens is F.

Fig. 3
Fig. 3

Particle density f(a′) versus radius a′ for the gamma distribution: recovered density using optical transform data (F = 200 mm and θMAX ~ 0.04) and Eq. (3) (the solid curve), histogram of particle sizes with a 5-μm bin size (the dashed curve), and the gamma distribution (the dotted curve).

Fig. 4
Fig. 4

Particle density versus radius for the log-normal distribution recovered from optical transform data (F = 200 mm and θMAX ~ 0.055) (the solid curve) compared with a histogram from the actual mask using a 5-μm bin size (the dashed curve).

Fig. 5
Fig. 5

Bimodal exponential distribution showing the density recovered from optical transform data (F = 200 mm and θMAX ~ 0.045) (the solid curve) compared with a histogram from the actual mask with a 5-μm bin size (the dashed curve).

Equations (8)

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I ( θ ) = I 0 a 2 J 1 2 ( k a θ ) θ 2 F 2 ,
I ( θ ) = I 0 0 f ( a ) a 2 J 1 2 ( k a θ ) θ 2 F 2 d a .
f ( a ) = 2 π F 2 k 2 a 0 θ 3 I ( θ ) I 0 d d a { a J 1 ( k a θ ) Y 1 ( k a θ ) } d θ ,
d d a { a J 1 ( k a θ ) Y 1 ( k a θ ) } - 1 π sin ( 2 k a θ ) ,
θ MAX 8 λ a .
f ( a ) = N Γ ( 1 b - 2 ) ( t b ) 1 b - 2 ( a - R 0 ) 1 b - 3 × exp [ - ( a - R 0 b t ) ]
f ( a ) = N 2 π σ ( a - R 0 ) exp { - [ ln ( a - R 0 A 0 ) 2 σ ] 2 } ,
f ( a ) = N n a 0 ( a - R 0 a 0 ) n - 1 exp [ - ( a - R 0 a 0 ) n ] ,

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