Abstract

An approach to small particle counting and classification is proposed in which the size and shape distribution is determined from the far-field pattern of light scattered by the particles. By presuming that the scattering function for each particle species is known from auxiliary experiments or calculations, a general result applicable to irregular particles as well as simply shaped particles is obtained. Inversion formulas, by which the particle distribution is determined from the far-field data, are presented in forms suitable for digital or optical processing. An experimental approach to the design of a real-time optical counting and classifying instrument is suggested.

© 1971 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. J. Wyatt, Appl. Opt. 7, 1879 (1968).
    [Crossref] [PubMed]
  2. L. C. Roess, J. Chem. Phys. 14, 695 (1946).
    [Crossref]
  3. J. Riseman, Acta Crystallogr. 5, 193 (1952).
    [Crossref]
  4. J. H. Letcher, P. W. Schmidt, J. Appl. Phys. 37, 649 (1966).
    [Crossref]
  5. V. Luzzati, Acta Crystallogr. 10, 33 (1957).
    [Crossref]
  6. K. S. Shifrin, A. Y. Perelman, in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967).
  7. R. S. Powell, R. R. Circle, D. C. Vogel, P. D. Woodson, B. Donn, Planet. Space Sci. 15, 1641 (1967).
    [Crossref]
  8. S. Twomey, H. B. Howell, Appl. Opt. 6, 2125 (1967).
    [Crossref] [PubMed]
  9. J. H. Ward, J. Opt. Soc. Amer. 58, 1566 (1968).
  10. B. J. Thompson, SPIE J. 2, 43 (1963).
  11. G. B. Parrent, B. J. Thompson, Opt. Acta 11, 183 (1964).
    [Crossref]
  12. B. A. Silverman, B. J. Thompson, J. H. Ward, J. Appl. Meteorol. 3, 792 (1964).
    [Crossref]
  13. B. J. Thompson, Japan. J. Appl. Phys. 4, Suppl. 1, 302 (1965).
    [Crossref]
  14. J. R. Hodkinson, Appl. Opt. 5, 839 (1966).
    [Crossref] [PubMed]
  15. L. J. Cutrona, in Optical and Electro-Optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965).

1968 (2)

P. J. Wyatt, Appl. Opt. 7, 1879 (1968).
[Crossref] [PubMed]

J. H. Ward, J. Opt. Soc. Amer. 58, 1566 (1968).

1967 (2)

R. S. Powell, R. R. Circle, D. C. Vogel, P. D. Woodson, B. Donn, Planet. Space Sci. 15, 1641 (1967).
[Crossref]

S. Twomey, H. B. Howell, Appl. Opt. 6, 2125 (1967).
[Crossref] [PubMed]

1966 (2)

J. H. Letcher, P. W. Schmidt, J. Appl. Phys. 37, 649 (1966).
[Crossref]

J. R. Hodkinson, Appl. Opt. 5, 839 (1966).
[Crossref] [PubMed]

1965 (1)

B. J. Thompson, Japan. J. Appl. Phys. 4, Suppl. 1, 302 (1965).
[Crossref]

1964 (2)

G. B. Parrent, B. J. Thompson, Opt. Acta 11, 183 (1964).
[Crossref]

B. A. Silverman, B. J. Thompson, J. H. Ward, J. Appl. Meteorol. 3, 792 (1964).
[Crossref]

1963 (1)

B. J. Thompson, SPIE J. 2, 43 (1963).

1957 (1)

V. Luzzati, Acta Crystallogr. 10, 33 (1957).
[Crossref]

1952 (1)

J. Riseman, Acta Crystallogr. 5, 193 (1952).
[Crossref]

1946 (1)

L. C. Roess, J. Chem. Phys. 14, 695 (1946).
[Crossref]

Circle, R. R.

R. S. Powell, R. R. Circle, D. C. Vogel, P. D. Woodson, B. Donn, Planet. Space Sci. 15, 1641 (1967).
[Crossref]

Cutrona, L. J.

L. J. Cutrona, in Optical and Electro-Optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965).

Donn, B.

R. S. Powell, R. R. Circle, D. C. Vogel, P. D. Woodson, B. Donn, Planet. Space Sci. 15, 1641 (1967).
[Crossref]

Hodkinson, J. R.

Howell, H. B.

Letcher, J. H.

J. H. Letcher, P. W. Schmidt, J. Appl. Phys. 37, 649 (1966).
[Crossref]

Luzzati, V.

V. Luzzati, Acta Crystallogr. 10, 33 (1957).
[Crossref]

Parrent, G. B.

G. B. Parrent, B. J. Thompson, Opt. Acta 11, 183 (1964).
[Crossref]

Perelman, A. Y.

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

Powell, R. S.

R. S. Powell, R. R. Circle, D. C. Vogel, P. D. Woodson, B. Donn, Planet. Space Sci. 15, 1641 (1967).
[Crossref]

Riseman, J.

J. Riseman, Acta Crystallogr. 5, 193 (1952).
[Crossref]

Roess, L. C.

L. C. Roess, J. Chem. Phys. 14, 695 (1946).
[Crossref]

Schmidt, P. W.

J. H. Letcher, P. W. Schmidt, J. Appl. Phys. 37, 649 (1966).
[Crossref]

Shifrin, K. S.

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

Silverman, B. A.

B. A. Silverman, B. J. Thompson, J. H. Ward, J. Appl. Meteorol. 3, 792 (1964).
[Crossref]

Thompson, B. J.

B. J. Thompson, Japan. J. Appl. Phys. 4, Suppl. 1, 302 (1965).
[Crossref]

B. A. Silverman, B. J. Thompson, J. H. Ward, J. Appl. Meteorol. 3, 792 (1964).
[Crossref]

G. B. Parrent, B. J. Thompson, Opt. Acta 11, 183 (1964).
[Crossref]

B. J. Thompson, SPIE J. 2, 43 (1963).

Twomey, S.

Vogel, D. C.

R. S. Powell, R. R. Circle, D. C. Vogel, P. D. Woodson, B. Donn, Planet. Space Sci. 15, 1641 (1967).
[Crossref]

Ward, J. H.

J. H. Ward, J. Opt. Soc. Amer. 58, 1566 (1968).

B. A. Silverman, B. J. Thompson, J. H. Ward, J. Appl. Meteorol. 3, 792 (1964).
[Crossref]

Woodson, P. D.

R. S. Powell, R. R. Circle, D. C. Vogel, P. D. Woodson, B. Donn, Planet. Space Sci. 15, 1641 (1967).
[Crossref]

Wyatt, P. J.

Acta Crystallogr. (2)

J. Riseman, Acta Crystallogr. 5, 193 (1952).
[Crossref]

V. Luzzati, Acta Crystallogr. 10, 33 (1957).
[Crossref]

Appl. Opt. (3)

J. Appl. Meteorol. (1)

B. A. Silverman, B. J. Thompson, J. H. Ward, J. Appl. Meteorol. 3, 792 (1964).
[Crossref]

J. Appl. Phys. (1)

J. H. Letcher, P. W. Schmidt, J. Appl. Phys. 37, 649 (1966).
[Crossref]

J. Chem. Phys. (1)

L. C. Roess, J. Chem. Phys. 14, 695 (1946).
[Crossref]

J. Opt. Soc. Amer. (1)

J. H. Ward, J. Opt. Soc. Amer. 58, 1566 (1968).

Japan. J. Appl. Phys. (1)

B. J. Thompson, Japan. J. Appl. Phys. 4, Suppl. 1, 302 (1965).
[Crossref]

Opt. Acta (1)

G. B. Parrent, B. J. Thompson, Opt. Acta 11, 183 (1964).
[Crossref]

Planet. Space Sci. (1)

R. S. Powell, R. R. Circle, D. C. Vogel, P. D. Woodson, B. Donn, Planet. Space Sci. 15, 1641 (1967).
[Crossref]

SPIE J. (1)

B. J. Thompson, SPIE J. 2, 43 (1963).

Other (2)

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

L. J. Cutrona, in Optical and Electro-Optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (38)

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

F ( ρ , ϕ ) = 0 r d r 0 2 π f ( r , θ ) exp [ - i r ρ cos ( θ - ϕ ) ] d θ .
f ( r , θ ) = m = - f m ( r ) e i m θ ,
F ( ρ , ϕ ) = m = - F m ( ρ ) exp [ i m ( ϕ - π / 2 ) ] ,
F m ( ρ ) = 0 f m ( r ) J m ( r ρ ) r d r ,
f ( r , θ ) F . T . F ( ρ , ϕ ) ,
f ( r , θ - α ) F . T . F ( ρ , ϕ - α )
F ( ρ , ϕ - α ) exp [ - i ρ r 0 cos ( θ 0 - ϕ ) ] .
F N ( ρ , ϕ ) = k = 1 N F ( ρ , ϕ - α k ) exp [ - i ρ r k cos ( θ k - ϕ ) ] ,
F N ( ρ , ϕ ) = m = - F m ( ρ ) k = 1 N exp × { i [ m ( ϕ - α k - π / 2 ) - ρ r k cos ( θ k - ϕ ) ] } .
I N ( ρ ) = F N ( ρ , ϕ ) 2 = N m = - F m ( ρ ) 2 ,
I N ( ρ ) = N 0 2 π [ F ( ρ , ϕ ) ] 2 d ϕ ,
N = j = 1 M N j ,
F N ( ρ , ϕ ) = j = 1 M m = - F j m ( ρ ) k = 1 N j exp × { i [ m ( ϕ - α k - π / 2 ) - ρ r k cos ( θ k - ϕ ) ] } ,
F N ( ρ , ϕ ) 2 = j = 1 M N j m = - F j m ( ρ ) 2 .
I ( ρ ) = j = 1 M G j ( ρ ) N j
G j ( ρ ) = m = - F j m ( ρ ) 2 = 0 2 π [ F j ( ρ , ϕ ) ] 2 d ϕ .
I i = I ( ρ i ) ,
G i j = G j ( ρ i ) ,
I i = j = 1 M G i j N j ,
I = G N .
I ( ρ ) = v 1 v 2 n ( v ) G ( ρ , v ) d v ,
( I - J ) T ( I - J ) = i = 1 P ( I i - J i ) 2 ,
I i - J i = I ( ρ i ) - J ( ρ i ) .
N = ( G T G ) - 1 G T J
ϕ ϕ j + 1 d ϕ ρ i ρ i + 1 ρ d ρ = Δ ϕ j ρ i ρ i + 1 ρ d ρ .
K i = ( 2 π / M ) ρ i ρ + 1 J ( ρ ) ρ d ρ ,
H i j = ( 2 π / M ) ρ i ρ i + 1 G j ( ρ ) ρ d ρ ,
N = ( H T H ) - 1 H T K
{ ( H T H ) - 1 H T } j i ,
L ( ρ ) d ρ = ( 2 π / M ) J ( ρ ) ρ d ρ ,
U ( ρ ) = Q ( ρ ) N ,
U ( ρ ) = ( 2 π / M ) ρ I ( ρ )
Q ( ρ ) = ( 2 π / M ) ρ G ( ρ ) .
ρ 0 ρ M [ U ( ρ ) - L ( ρ ) ] 2 d ρ = ρ 0 ρ M [ N T Q T ( ρ ) Q ( ρ ) N - 2 Q ( ρ ) N L ( ρ ) + L 2 ( ρ ) ] d ρ ,
U 2 ( ρ ) = [ Q ( ρ ) N ] T Q ( ρ ) N = N T Q T ( ρ ) Q ( ρ ) N .
N = [ ρ 0 ρ M Q T ( ζ ) Q ( ζ ) d ζ ] - 1 ρ 0 ρ M Q T ( ρ ) L ( ρ ) d ρ .
N j = ( 2 π / M ) ρ 0 ρ M V j ( ρ ) J ( ρ ) ρ d ρ
V j ( ρ ) = k = 1 M { [ ρ 0 ρ M Q T ( ζ ) Q ( ζ ) d ζ ] - 1 } j k Q k ( ρ ) .

Metrics