Abstract

We discuss some experimental results concerning the statistical properties of a light beam scattered by a rotating ground glass with large surface inhomogeneities (average size 20 μm), when the illuminated area of the scattering surface contains only a few scattering centers. Photon-count distribution measurements indicate that, to a high degree of accuracy, the field amplitude of the scattered light fluctuates as a log-normal variate. The irradiance correlation function is a gaussian function of time with a half-width at half-height that varies inversely with the angular speed of the ground glass but is largely independent of the angle of scattering. Identical functional dependence was previously found, using a ground glass with much smaller surface inhomogeneities (average size 1 μm). Finally, we report some experimental tests of a recent calculation by Mitchell concerning what has been defined as the permanence of the log-normal distribution. We find that, even when the detecting surface of the photomultiplier is illuminated by several coherence areas of the scattered field during a single counting interval, the field-amplitude distribution remains log-normal to an excellent approximation.

© 1972 Optical Society of America

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References

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  1. Ch. Bendjaballah, Phys. Letters 31A, 471 (1970).
  2. L. E. Estes, L. M. Narducci, and R. A. Tuft, J. Opt. Soc. Am. 61, 1301 (1971).
    [Crossref]
  3. M. Rousseau, J. Opt. Soc. Am. 61, 1307 (1971) and references therein, for previous work on rotating ground glass.
    [Crossref]
  4. An extensive discussion of gaussian optical fields can be found in B. Picinbono and M. Rousseau, Phys. Rev. A 1, 635 (1970).See also L. Mandel, Phys. Rev. 181, 75 (1969).
    [Crossref]
  5. R. L. Mitchell, J. Opt. Soc. Am. 58, 1267 (1968).
    [Crossref]
  6. Comprehensive discussions of the digital correlation technique as well as a comparison with other methods used in irradiance-fluctuation spectroscopy can be found in E. Jakeman, E. R. Pike, and S. Swain, J. Phys. A 4, 517 (1971);V. Degiorgio and J. B. Lastovka, Phys. Rev. A 4, 2033 (1971);and H. C. Kelly, IEEE J. QE-7, 541 (1971).
    [Crossref]
  7. From the acknowledgments of Ref. 5 we learn that the title“permanence of the log-normal distribution” was suggested to Mitchell by D. L. Fried.
  8. D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
    [Crossref]
  9. Z. Kopal, Numerical Analysis (Chapman & Hall, London, 1955), Ch. 7.
  10. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, Natl. Bur. Std. (U. S.) Appl. Math. Ser. (U. S. Government Printing Office, Washington, D. C., 1964;Handbook of Mathematical FunctionsDover, New York, 1965), p. 923.

1971 (3)

L. E. Estes, L. M. Narducci, and R. A. Tuft, J. Opt. Soc. Am. 61, 1301 (1971).
[Crossref]

M. Rousseau, J. Opt. Soc. Am. 61, 1307 (1971) and references therein, for previous work on rotating ground glass.
[Crossref]

Comprehensive discussions of the digital correlation technique as well as a comparison with other methods used in irradiance-fluctuation spectroscopy can be found in E. Jakeman, E. R. Pike, and S. Swain, J. Phys. A 4, 517 (1971);V. Degiorgio and J. B. Lastovka, Phys. Rev. A 4, 2033 (1971);and H. C. Kelly, IEEE J. QE-7, 541 (1971).
[Crossref]

1970 (2)

Ch. Bendjaballah, Phys. Letters 31A, 471 (1970).

An extensive discussion of gaussian optical fields can be found in B. Picinbono and M. Rousseau, Phys. Rev. A 1, 635 (1970).See also L. Mandel, Phys. Rev. 181, 75 (1969).
[Crossref]

1968 (1)

1967 (1)

Bendjaballah, Ch.

Ch. Bendjaballah, Phys. Letters 31A, 471 (1970).

Estes, L. E.

Fried, D. L.

D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
[Crossref]

From the acknowledgments of Ref. 5 we learn that the title“permanence of the log-normal distribution” was suggested to Mitchell by D. L. Fried.

Jakeman, E.

Comprehensive discussions of the digital correlation technique as well as a comparison with other methods used in irradiance-fluctuation spectroscopy can be found in E. Jakeman, E. R. Pike, and S. Swain, J. Phys. A 4, 517 (1971);V. Degiorgio and J. B. Lastovka, Phys. Rev. A 4, 2033 (1971);and H. C. Kelly, IEEE J. QE-7, 541 (1971).
[Crossref]

Keister, M. P.

Kopal, Z.

Z. Kopal, Numerical Analysis (Chapman & Hall, London, 1955), Ch. 7.

Mevers, G. E.

Mitchell,

From the acknowledgments of Ref. 5 we learn that the title“permanence of the log-normal distribution” was suggested to Mitchell by D. L. Fried.

Mitchell, R. L.

Narducci, L. M.

Picinbono, B.

An extensive discussion of gaussian optical fields can be found in B. Picinbono and M. Rousseau, Phys. Rev. A 1, 635 (1970).See also L. Mandel, Phys. Rev. 181, 75 (1969).
[Crossref]

Pike, E. R.

Comprehensive discussions of the digital correlation technique as well as a comparison with other methods used in irradiance-fluctuation spectroscopy can be found in E. Jakeman, E. R. Pike, and S. Swain, J. Phys. A 4, 517 (1971);V. Degiorgio and J. B. Lastovka, Phys. Rev. A 4, 2033 (1971);and H. C. Kelly, IEEE J. QE-7, 541 (1971).
[Crossref]

Rousseau, M.

M. Rousseau, J. Opt. Soc. Am. 61, 1307 (1971) and references therein, for previous work on rotating ground glass.
[Crossref]

An extensive discussion of gaussian optical fields can be found in B. Picinbono and M. Rousseau, Phys. Rev. A 1, 635 (1970).See also L. Mandel, Phys. Rev. 181, 75 (1969).
[Crossref]

Swain, S.

Comprehensive discussions of the digital correlation technique as well as a comparison with other methods used in irradiance-fluctuation spectroscopy can be found in E. Jakeman, E. R. Pike, and S. Swain, J. Phys. A 4, 517 (1971);V. Degiorgio and J. B. Lastovka, Phys. Rev. A 4, 2033 (1971);and H. C. Kelly, IEEE J. QE-7, 541 (1971).
[Crossref]

Tuft, R. A.

J. Opt. Soc. Am. (4)

J. Phys. A (1)

Comprehensive discussions of the digital correlation technique as well as a comparison with other methods used in irradiance-fluctuation spectroscopy can be found in E. Jakeman, E. R. Pike, and S. Swain, J. Phys. A 4, 517 (1971);V. Degiorgio and J. B. Lastovka, Phys. Rev. A 4, 2033 (1971);and H. C. Kelly, IEEE J. QE-7, 541 (1971).
[Crossref]

Phys. Letters (1)

Ch. Bendjaballah, Phys. Letters 31A, 471 (1970).

Phys. Rev. A (1)

An extensive discussion of gaussian optical fields can be found in B. Picinbono and M. Rousseau, Phys. Rev. A 1, 635 (1970).See also L. Mandel, Phys. Rev. 181, 75 (1969).
[Crossref]

Other (3)

From the acknowledgments of Ref. 5 we learn that the title“permanence of the log-normal distribution” was suggested to Mitchell by D. L. Fried.

Z. Kopal, Numerical Analysis (Chapman & Hall, London, 1955), Ch. 7.

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, Natl. Bur. Std. (U. S.) Appl. Math. Ser. (U. S. Government Printing Office, Washington, D. C., 1964;Handbook of Mathematical FunctionsDover, New York, 1965), p. 923.

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Figures (8)

F. 1
F. 1

Experimental probability distribution P(n) of recording n photoelectron pulses in a counting time T = 10 μs. The rotating coarse-ground glass is the scattering medium. The solid curve through the points represents the computer-generated theoretical fit using the first and second moments of the experimental distribution.

F. 2
F. 2

Experimental probability distribution P(n) of recording n photoelectron pulses in a counting time T = 10 μs. A rotating ground glass with 1-μm average inhomogeneities is the scattering medium. The straight line through the experimental points represents the theoretical Bose–Einstein distribution corresponding to the observed mean number of photoelectrons.

F. 3
F. 3

Schematic of the sampling scheme used in our correlation measurements. See also Ref. 6 for further details.

F. 4
F. 4

A typical normalized irradiance autocorrelation function defined in such a way that the value at the origin is 1 and the asymptotic value for τ → ∞ is zero. The solid curve represents the best gaussian fit to the experimental points.

F. 5
F. 5

Linear dependence of the correlation time τc on the period of rotation of the ground glass. In all cases, the illuminated area on the surface of the ground glass was 9.4 cm away from the center of the axis of rotation.

F. 6
F. 6

Photon-count distribution obtained with a fine-ground glass. A large number of coherence areas are focused on the collecting aperture of the detector. The straight line is the theoretical Bose–Einstein distribution corresponding to the observed mean number.

F. 7
F. 7

Photon-count distribution obtained with a coarse-ground glass. The geometry of the experiment is the same as in Fig. 6. The solid curve is obtained by numerical integration of Eq. (1) using the experimental values of the first and second moments of the distribution.

F. 8
F. 8

Photon-count distribution with a coarse-ground glass when the illuminated pinhole is imaged on the collecting aperture of the detector. The solid curve is obtained by numerical integration of Eq. (1) using the experimental values of the first and second moments of the distribution.

Equations (7)

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P ( n ) = 0 d I I n n ! e I P ( I ) ,
P ( I ) = 1 2 I 1 ( 2 π σ 2 ) 1 2 exp ( ln 2 ( I / I 0 ) 4 σ 2 )
n 1 ( t ) n 2 ( t + τ ) = η 2 T 2 G ( 2 ) ( τ ) ,
G ( 2 ) ( τ ) = E ( ) ( t ) E ( ) ( t + τ ) E ( + ) ( t + τ ) E ( + ) ( t )
0 e I f ( I ) d I = j = 1 m H j f ( a j ) ,
E ( m ! ) 2 ( 2 m ) ! f ( 2 m ) ( η ) ,
f ( I ) = 1 ( 2 π σ 2 ) 1 2 1 2 I I n n ! exp ( ln 2 ( I / I 0 ) 4 σ 2 ) .