Abstract

The deterioration that is due to saturation of a revealing apparatus on high-order moments of intensity is investigated in the case of Furutsu’s distribution. This is a two-parameter distribution and seems suitable for describing intensity fluctuations that are due to atmospheric turbulence in the case of intermediate and strong scintillation. The effect of saturation was shown to depend on only one parameter, αDS, defined as the ratio between the saturation intensity of the system and the mean intensity to be measured, a result already found in the case of a log-normal distribution. An expression of the deteriorated moments is given in terms of αDS and of the two parameters of the distribution. The theoretical results are used to give a qualitative interpretation of measured moments after laser propagation along a 1800-m path above the city of Florence.

© 1986 Optical Society of America

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References

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  1. A. K. Majumdar, “Higher-order statistics in laser-irradiance fluctuations due to turbulence,” J. Opt. Soc. Am. A 1, 1067–1074 (1984); see also A. K. Majumdar, “Uniqueness of statistics derived from moments of irradiance fluctuations in atmospheric optical propagation,” Opt. Commun. 50, 1–7 (1984).
    [CrossRef]
  2. A. Consortini, G. Conforti, “Detector saturation effect on higher-order moments of intensity fluctuations in atmospheric laser propagation measurement,” J. Opt. Soc. Am. A 1, 1075–1077 (1984).
    [CrossRef]
  3. G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,”J. Opt. Soc. Am. 69, 796–798 (1979).
    [CrossRef]
  4. K. Furutsu, “Theory of irradiance distribution function in turbulent media—cluster approximation,”J. Math. Phys. 17, 1252–1263 (1976).
    [CrossRef]
  5. S. Ito, K. Furutsu, “Theoretical analysis of the high-order irradiance moments of light waves observed in turbulent air,”J. Opt. Soc. Am. 72, 760–764 (1982).
    [CrossRef]
  6. R. L. Phillips, L. C. Andrews, “Universal statistical model for irradiance fluctuations in a turbulent medium,”J. Opt. Soc. Am. 72, 864–870 (1982).
    [CrossRef]
  7. P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Effect of recording process in evaluating higher-order propagation statistics,”J. Opt. Soc. Am. 68, 1350–1352 (1978).
    [CrossRef]
  8. P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Detection effects on the statistics of intensity fluctuations of an atmospherically degraded laser beam,” Nuovo Cimento B 48, 63–73 (1978).
    [CrossRef]
  9. P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Evidence of log-normal distributed intensity fluctuations of an atmospherically degraded laser beam,” Opt. Acta 26, 253–259 (1979).
    [CrossRef]
  10. G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1958).
  11. R. L. Phillips, L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,”J. Opt. Soc. Am. 71, 1440–1445 (1981).
    [CrossRef]
  12. D. A. de Wolf, “Strong irradiance fluctuations in turbulent air: plane waves,”J. Opt. Soc. Am. 63, 171–179 (1973).
    [CrossRef]
  13. L. C. Andrews, R. L. Phillips, “I–K distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985).
    [CrossRef]

1985 (1)

1984 (2)

1982 (2)

1981 (1)

1979 (2)

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Evidence of log-normal distributed intensity fluctuations of an atmospherically degraded laser beam,” Opt. Acta 26, 253–259 (1979).
[CrossRef]

G. Parry, P. N. Pusey, “K distributions in atmospheric propagation of laser light,”J. Opt. Soc. Am. 69, 796–798 (1979).
[CrossRef]

1978 (2)

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Effect of recording process in evaluating higher-order propagation statistics,”J. Opt. Soc. Am. 68, 1350–1352 (1978).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Detection effects on the statistics of intensity fluctuations of an atmospherically degraded laser beam,” Nuovo Cimento B 48, 63–73 (1978).
[CrossRef]

1976 (1)

K. Furutsu, “Theory of irradiance distribution function in turbulent media—cluster approximation,”J. Math. Phys. 17, 1252–1263 (1976).
[CrossRef]

1973 (1)

Andrews, L. C.

Ciolli, P.

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Evidence of log-normal distributed intensity fluctuations of an atmospherically degraded laser beam,” Opt. Acta 26, 253–259 (1979).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Effect of recording process in evaluating higher-order propagation statistics,”J. Opt. Soc. Am. 68, 1350–1352 (1978).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Detection effects on the statistics of intensity fluctuations of an atmospherically degraded laser beam,” Nuovo Cimento B 48, 63–73 (1978).
[CrossRef]

Conforti, G.

Consortini, A.

A. Consortini, G. Conforti, “Detector saturation effect on higher-order moments of intensity fluctuations in atmospheric laser propagation measurement,” J. Opt. Soc. Am. A 1, 1075–1077 (1984).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Evidence of log-normal distributed intensity fluctuations of an atmospherically degraded laser beam,” Opt. Acta 26, 253–259 (1979).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Effect of recording process in evaluating higher-order propagation statistics,”J. Opt. Soc. Am. 68, 1350–1352 (1978).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Detection effects on the statistics of intensity fluctuations of an atmospherically degraded laser beam,” Nuovo Cimento B 48, 63–73 (1978).
[CrossRef]

de Wolf, D. A.

Furutsu, K.

S. Ito, K. Furutsu, “Theoretical analysis of the high-order irradiance moments of light waves observed in turbulent air,”J. Opt. Soc. Am. 72, 760–764 (1982).
[CrossRef]

K. Furutsu, “Theory of irradiance distribution function in turbulent media—cluster approximation,”J. Math. Phys. 17, 1252–1263 (1976).
[CrossRef]

Ito, S.

Majumdar, A. K.

Pandolfini, P.

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Evidence of log-normal distributed intensity fluctuations of an atmospherically degraded laser beam,” Opt. Acta 26, 253–259 (1979).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Effect of recording process in evaluating higher-order propagation statistics,”J. Opt. Soc. Am. 68, 1350–1352 (1978).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Detection effects on the statistics of intensity fluctuations of an atmospherically degraded laser beam,” Nuovo Cimento B 48, 63–73 (1978).
[CrossRef]

Parry, G.

Pasqualetti, F.

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Evidence of log-normal distributed intensity fluctuations of an atmospherically degraded laser beam,” Opt. Acta 26, 253–259 (1979).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Effect of recording process in evaluating higher-order propagation statistics,”J. Opt. Soc. Am. 68, 1350–1352 (1978).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Detection effects on the statistics of intensity fluctuations of an atmospherically degraded laser beam,” Nuovo Cimento B 48, 63–73 (1978).
[CrossRef]

Phillips, R. L.

Pusey, P. N.

Ronchi, L.

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Evidence of log-normal distributed intensity fluctuations of an atmospherically degraded laser beam,” Opt. Acta 26, 253–259 (1979).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Effect of recording process in evaluating higher-order propagation statistics,”J. Opt. Soc. Am. 68, 1350–1352 (1978).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Detection effects on the statistics of intensity fluctuations of an atmospherically degraded laser beam,” Nuovo Cimento B 48, 63–73 (1978).
[CrossRef]

Vanni, R.

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Evidence of log-normal distributed intensity fluctuations of an atmospherically degraded laser beam,” Opt. Acta 26, 253–259 (1979).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Effect of recording process in evaluating higher-order propagation statistics,”J. Opt. Soc. Am. 68, 1350–1352 (1978).
[CrossRef]

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Detection effects on the statistics of intensity fluctuations of an atmospherically degraded laser beam,” Nuovo Cimento B 48, 63–73 (1978).
[CrossRef]

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1958).

J. Math. Phys. (1)

K. Furutsu, “Theory of irradiance distribution function in turbulent media—cluster approximation,”J. Math. Phys. 17, 1252–1263 (1976).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (3)

Nuovo Cimento B (1)

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Detection effects on the statistics of intensity fluctuations of an atmospherically degraded laser beam,” Nuovo Cimento B 48, 63–73 (1978).
[CrossRef]

Opt. Acta (1)

P. Ciolli, A. Consortini, P. Pandolfini, F. Pasqualetti, L. Ronchi, R. Vanni, “Evidence of log-normal distributed intensity fluctuations of an atmospherically degraded laser beam,” Opt. Acta 26, 253–259 (1979).
[CrossRef]

Other (1)

G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1958).

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Figures (4)

Fig. 1
Fig. 1

The quantity αc plotted versus M2 for a number of values of Δ′.

Fig. 2
Fig. 2

Furutsu’s distribution: Residual probability plotted versus M2 for three values of Δ′ in the case αDS = 40. For comparison, the residual probability corresponding to a log-normal distribution is also represented.

Fig. 3
Fig. 3

Solid lines, normalized deteriorated moments of order 2–5 plotted versus M2 for three values of Δ′ in the case αDS = 40. Dashed lines represent the corresponding moments of the distribution.

Fig. 4
Fig. 4

Normalized deteriorated moments of order 3, 4, and 5 plotted versus the normalized deteriorated second moment for three values of Δ′ in the case αDS = 40. Dashed lines refer to log-normal distribution. Stars and dots, measured normalized moments.

Equations (21)

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I n = 0 I DS I n p ( I ) d I + I DS n P res ,
P res = I DS p ( I ) d I = 1 - 0 I DS p ( I ) d I
E = ln I I ,
P ( E ) = e - a 2 / 2 δ ( E - E c ) + { ν 0 a b I 1 ( a b ) exp ( - a 2 + b 2 2 ) , E < E c 0 , E > E c ,
a 2 = ( 1 + ν 0 ) ln M 2 Δ ,             b 2 = 2 ν 0 ( E c - E ) , E c = ln M 2 2 Δ / ( 1 - Δ ) ,             ν 0 = 1 Δ - 2 , Δ = Δ 1 - Δ             M 2 = I 2 I 2 ,
M n = I n I n = M 2 n 2 ( n - 1 ) / [ 1 + ( n - 2 ) Δ ] ,
I D S I M 2 ( 1 - Δ ) / ( 2 Δ )
α c = I c I
α DS = I DS I
p ( I ) d I = d I 2 π i - i - + i - exp [ 1 2 ν ( ν - 1 ) ln M 2 1 + ( ν - 2 ) Δ - ν E ] d ν ,
I n ¯ I n = T n + α DS n P res ,
T n = 1 2 π i - i - i - ( n - ν ) - 1 × exp { 1 2 ν ( ν - 1 ) ln M 2 [ 1 + ( ν - 2 ) Δ ] + ( n - ν ) E DS } d ν ,
E DS = ln α DS ,
P res = 1 - T 0 .
t = b DS a ( 1 + ν ν 0 ) ,
b DS 2 = 2 ν 0 ( E c - ln α DS ) ,             E c > E DS ,
T n = 1 2 π i exp ( - a 2 + b DS 2 2 + n E DS ) × - i + i + exp [ a b DS 2 ( t + 1 t ) ] b DS ( n + ν 0 ν 0 a ) - t d t .
T n = 1 2 π i exp ( - a 2 + b DS 2 2 + n E DS ) × k = 0 ( a / b DS 1 + n / ν 0 ) k + 1 t k exp [ a b DS 2 ( t + 1 t ) ] d t .
T n = exp ( - a 2 + b DS 2 2 + n E DS ) × k = 1 [ a b DS ( 1 + n / ν 0 ) ] k I k ( a b DS ) .
P res = 1 - exp ( - a 2 + b DS 2 2 ) k = 1 ( a b DS ) k I k ( a b DS ) .
I n ¯ I n = α DS n { 1 + exp ( - a 2 + b DS 2 2 ) × k = 1 ( a b DS ) k [ 1 ( 1 + n / ν 0 ) k - 1 ] I k ( a b DS ) } .

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