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References

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  1. J. H. Altman, Appl. Opt. 3, 35 (1964).
    [Crossref]
  2. L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 36, 203 (1946).
    [Crossref]
  3. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals and Products (Academic Press Inc., New York, 1965), pp. 538 and 541.
  4. H. T. Davis, Tables of the Higher Mathematical Functions (The Principia Press, Inc., Bloomington, Indiana, 1935).
  5. E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publ. Co., Inc., Reading, Mass., 1963), p. 115.

1964 (1)

1946 (1)

Altman, J. H.

Davis, H. T.

H. T. Davis, Tables of the Higher Mathematical Functions (The Principia Press, Inc., Bloomington, Indiana, 1935).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals and Products (Academic Press Inc., New York, 1965), pp. 538 and 541.

Higgins, G. C.

Jones, L. A.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publ. Co., Inc., Reading, Mass., 1963), p. 115.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals and Products (Academic Press Inc., New York, 1965), pp. 538 and 541.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Other (3)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals and Products (Academic Press Inc., New York, 1965), pp. 538 and 541.

H. T. Davis, Tables of the Higher Mathematical Functions (The Principia Press, Inc., Bloomington, Indiana, 1935).

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publ. Co., Inc., Reading, Mass., 1963), p. 115.

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Figures (1)

Fig. 1
Fig. 1

Percent error E in measurements of rms density fluctuations plotted vs relative transmittance fluctuations σ(T)/〈Tav for various values of average transmittance 〈Tav. The error is due to linearization of the relation D = −logT when the transmittance fluctuations can be adequately described by a beta distribution. The abscissa corresponding to the intersection of the dashed curve and a particular solid curve gives a reasonable upper bound on σ(T)/〈Tav for each 〈Tav.

Equations (15)

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σ A ( D ) = 0.434 [ σ ( T ) / T av ]
D = - log T .
σ E 2 ( D ) = D 2 av - D av 2 ,
D av = 0 1 log T p ( T ) d T
D 2 av = 0 1 log 2 T p ( T ) d T .
p ( T ) = Γ ( p + q ) Γ ( p ) Γ ( q ) T p - 1 ( 1 - T ) q - 1 ,
T av = p / p + q
σ 2 ( T ) = p q / ( p + q ) 2 ( p + q + 1 ) .
C = σ ( T ) / T av
p = [ ( 1 - T av ) / C 2 ] - T av ,
q = p ( 1 - T av ) / T av .
σ ( D ) = log 10 e [ ψ ( p ) - ψ ( p + q ) ] 1 2 ,
ψ ( z ) = k = 0 1 ( z + k ) 2 .
E = 100 [ σ A ( D ) - σ E ( D ) ] / σ A ( D )
( σ ( T ) / T av ) max = ( l / L ) [ ( T av ) - 1 - 1 ] 1 2 ,