Abstract

The probability density function and average contrast of the sum of n uncorrelated, partially developed speckle patterns have been theoretically investigated. A new form of the probability density function is given for the sum of n uncorrelated, partially developed speckle patterns under the assumption that the individual speckle fields to be added follow the circular statistics.

© 1977 Optical Society of America

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References

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  1. J. M. Burch, in Optical Instruments and Techniques, J. Home Dickson, ed. (Oriel, Newcastle upon Tyne, 1970), p. 213.
  2. R. Barakat, Opt. Commun. 8, 14 (1973).
    [CrossRef]
  3. J. W. Goodman, Opt. Commun. 13, 244 (1975).
    [CrossRef]
  4. J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), p. 9.
    [CrossRef]
  5. J. Ohtsubo, T. Asakura, Appl. Phys. (to be published).
  6. J. W. Goodman, Proc. IEEE 53, 1688 (1965).
    [CrossRef]
  7. J. C. Dainty, Opt. Acta 17, 761 (1970).
    [CrossRef]
  8. J. C. Dainty, Opt. Acta 18, 327 (1971).
    [CrossRef]
  9. J. C. Dainty, J. Opt. Soc. Am. 62, 595 (1972).
    [CrossRef]
  10. J. Ohtsubo, T. Asakura, Optik 45, 65 (1976).

1976 (1)

J. Ohtsubo, T. Asakura, Optik 45, 65 (1976).

1975 (1)

J. W. Goodman, Opt. Commun. 13, 244 (1975).
[CrossRef]

1973 (1)

R. Barakat, Opt. Commun. 8, 14 (1973).
[CrossRef]

1972 (1)

1971 (1)

J. C. Dainty, Opt. Acta 18, 327 (1971).
[CrossRef]

1970 (1)

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

1965 (1)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

Asakura, T.

J. Ohtsubo, T. Asakura, Optik 45, 65 (1976).

J. Ohtsubo, T. Asakura, Appl. Phys. (to be published).

Barakat, R.

R. Barakat, Opt. Commun. 8, 14 (1973).
[CrossRef]

Burch, J. M.

J. M. Burch, in Optical Instruments and Techniques, J. Home Dickson, ed. (Oriel, Newcastle upon Tyne, 1970), p. 213.

Dainty, J. C.

J. C. Dainty, J. Opt. Soc. Am. 62, 595 (1972).
[CrossRef]

J. C. Dainty, Opt. Acta 18, 327 (1971).
[CrossRef]

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Opt. Commun. 13, 244 (1975).
[CrossRef]

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), p. 9.
[CrossRef]

Ohtsubo, J.

J. Ohtsubo, T. Asakura, Optik 45, 65 (1976).

J. Ohtsubo, T. Asakura, Appl. Phys. (to be published).

J. Opt. Soc. Am. (1)

Opt. Acta (2)

J. C. Dainty, Opt. Acta 17, 761 (1970).
[CrossRef]

J. C. Dainty, Opt. Acta 18, 327 (1971).
[CrossRef]

Opt. Commun. (2)

R. Barakat, Opt. Commun. 8, 14 (1973).
[CrossRef]

J. W. Goodman, Opt. Commun. 13, 244 (1975).
[CrossRef]

Optik (1)

J. Ohtsubo, T. Asakura, Optik 45, 65 (1976).

Proc. IEEE (1)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

Other (3)

J. M. Burch, in Optical Instruments and Techniques, J. Home Dickson, ed. (Oriel, Newcastle upon Tyne, 1970), p. 213.

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), p. 9.
[CrossRef]

J. Ohtsubo, T. Asakura, Appl. Phys. (to be published).

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Figures (3)

Fig. 1
Fig. 1

Probability density functions of the sum of partially developed speckle patterns with equal intensity as a function of the number n of added speckle patterns. In this case, the intensity ratio IS/〈IN〉 between the diffuse and specular components of speckle patterns is fixed as r = 1 and the average contrasts V corresponding to each of the probability density functions AE are also given.

Fig. 2
Fig. 2

Probability density functions of the sum of five partially developed speckle patterns (n = 5) as a function of the intensity ratio r between the diffuse and specular components in speckle patterns. The average contrasts V corresponding to the probability density functions AE are also given.

Fig. 3
Fig. 3

Average contrast of the sum of partially developed speckle patterns with intensity ratios of r = 0, 1, 10 as a function of the number n of added speckle patterns shown in a logarithmic scale.

Equations (12)

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I = k = 1 n I k = k = 1 n | A k | 2 = k = 1 n ( A r k 2 + A i k 2 ) ,
P A ( A r k , A i k ) = 1 2 π σ k × exp [ ( A rk I S k ) 2 + A i k 2 2 σ k 2 ] ,
σ k 2 = 1 2 I N k ,
P I ( I ) = 1 2 π exp ( i z I ) exp ( i z I ) d z .
exp ( i z I ) = k = 1 n exp [ i z ( A r k 2 + A i k 2 ) ] = 1 ( 1 i z I N ) n × exp [ n I S I N + n I S I N ( 1 i z I N ) ] = exp ( n I S I N ) l = 0 ( n I S / I N ) l ( 1 i z I N ) n + l ,
P I ( I ) = ( 2 I ) n 1 I N × exp ( I + n I S I N ) M n 1 ( 2 n I S I / I N ) ( 2 n I S I / I N ) n 1 ,
P I ( I / I ) = n n ( 1 + r ) n ( 2 I ) n 1 × exp { n [ ( 1 + r ) I + r ] } M n 1 ( 2 n r ( 1 + r ) I ) ( 2 n r ( 1 + r ) I ) n 1 ,
P I ( I / I ) = ( 1 + r ) × exp { [ ( 1 + r ) I + r ] } M 0 ( 2 r ( 1 + r ) I ) .
P I ( I / I ) = n n I n 1 exp ( n I ) Γ ( n ) ;
V = ( I 2 I 2 ) 1 / 2 I = ( k = 1 n λ k 2 V k 2 ) 1 / 2 ,
λ k = I k I , V k = ( I k 2 I k 2 ) 1 / 2 I k .
V = 1 n V 0 = 1 n ( 2 I S I N + I N 2 ) 1 / 2 I S + I N = 1 n ( 1 + 2 r ) 1 + r 1 / 2 ,

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