Abstract

Equations are derived for the effect of amplifier saturation on the moments of intensity fluctuations in atmospheric laser propagation. The intensity is assumed to have the recently proposed log-normally modulated exponential probability density. The effect is also calculated for the K probability density.

© 1987 Optical Society of America

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References

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  1. J. H. Churnside, R. J. Hill, J. Opt. Soc. Am. A 4, 727 (1987).
    [Crossref]
  2. G. Parry, Opt. Acta 28, 715 (1981).
    [Crossref]
  3. A. Consortini, G. Conforti, J. Opt. Soc. Am. A 1, 1075 (1984).
    [Crossref]
  4. A. Consortini, E. Briccolani, G. Conforti, J. Opt. Soc. Am. A 3, 101 (1986).
    [Crossref]
  5. E. Jakeman, J. Phys. A 13, 31 (1980).
    [Crossref]
  6. E. Jakeman, P. N. Pusey, Phys. Rev. Lett. 40, 546 (1978).
    [Crossref]
  7. G. Parry, P. N. Pusey, J. Opt. Soc. Am. 69, 796 (1979).
    [Crossref]
  8. M. Abramowitz, A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1968).
  9. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966). Reference is made to Eq. (4), p. 133, Sec. 5.1, Chap. V, as well as to Eq. (5), p. 79, Sec. 3.71, Chap. III.

1987 (1)

1986 (1)

1984 (1)

1981 (1)

G. Parry, Opt. Acta 28, 715 (1981).
[Crossref]

1980 (1)

E. Jakeman, J. Phys. A 13, 31 (1980).
[Crossref]

1979 (1)

1978 (1)

E. Jakeman, P. N. Pusey, Phys. Rev. Lett. 40, 546 (1978).
[Crossref]

Briccolani, E.

Churnside, J. H.

Conforti, G.

Consortini, A.

Hill, R. J.

Jakeman, E.

E. Jakeman, J. Phys. A 13, 31 (1980).
[Crossref]

E. Jakeman, P. N. Pusey, Phys. Rev. Lett. 40, 546 (1978).
[Crossref]

Parry, G.

Pusey, P. N.

G. Parry, P. N. Pusey, J. Opt. Soc. Am. 69, 796 (1979).
[Crossref]

E. Jakeman, P. N. Pusey, Phys. Rev. Lett. 40, 546 (1978).
[Crossref]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966). Reference is made to Eq. (4), p. 133, Sec. 5.1, Chap. V, as well as to Eq. (5), p. 79, Sec. 3.71, Chap. III.

J. Opt. Soc. Am. (1)

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

J. Phys. A (1)

E. Jakeman, J. Phys. A 13, 31 (1980).
[Crossref]

Opt. Acta (1)

G. Parry, Opt. Acta 28, 715 (1981).
[Crossref]

Phys. Rev. Lett. (1)

E. Jakeman, P. N. Pusey, Phys. Rev. Lett. 40, 546 (1978).
[Crossref]

Other (2)

M. Abramowitz, A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1968).

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966). Reference is made to Eq. (4), p. 133, Sec. 5.1, Chap. V, as well as to Eq. (5), p. 79, Sec. 3.71, Chap. III.

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Figures (1)

Fig. 1
Fig. 1

The true and deteriorated moments of orders 2–5 are shown for several values of αDS for both the LnME and K PDF's. The true moments are given by the short-dashed and solid curves for the LnME and K PDF's, respectively. For the indicated values of αDS, the deteriorated moments are given by the medium-dashed and long-dashed curves for the LnME and K PDF's, respectively. The true moments of the exponential PDF are given by the triangles.

Equations (25)

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M n det = 0 α DS I n P ( I ) d I + α DS n α DS P ( I ) d I ,
M n det = M n + S n ;
S n = α DS I n P ( I ) d I + α DS n α DS P ( I ) d I .
P ( I ) = 1 2 π σ z 0 d z z 2 exp [ I z ( ln z + 1 / 2 σ z 2 ) 2 2 σ z 2 ] ,
σ z 2 = ln ( M 2 / 2 ) .
M n = 0 I n P ( I ) d I = n ! ( M 2 / 2 ) n ( n 1 ) / 2 .
S n = 1 2 π σ z j = 1 n n ! ( n j ) ! α DS n j × exp [ j y α DS e y ( y + 1 / 2 σ z 2 ) 2 2 σ z 2 ] d y .
S n = j = 1 n n ! ( n j ) ! α DS n j ( 1 + α DS σ z 2 ξ j ) 1 / 2 × exp [ j ln ξ j α DS ξ j σ z 2 2 ( j + α DS ξ j ) 2 ] ,
j + 1 2 α DS ξ j = 1 σ z 2 ln ξ j .
P ( I ) = 2 Γ ( y ) y ( y + 1 ) / 2 I ( y 1 ) / 2 K y 1 [ 2 ( I y ) 1 / 2 ] ,
y = 2 / ( M 2 2 ) ,
M n = n ! Γ ( n + y ) y n Γ ( y ) .
z x μ + 1 K ν ( x ) d x = + ( μ 2 ν 2 ) z x μ 1 K ν ( x ) d x + [ z μ + 1 K ν + 1 ( z ) + ( μ ν ) z μ K ν ( z ) ] .
z x ν + 1 K ν ( x ) d x = + z ν + 1 K ν + 1 ( z ) .
z x μ + 1 K ν ( x ) d x = z μ + 1 { [ 1 + W ( μ ν 2 , μ + ν 2 , z ) ] × K ν + 1 ( z ) + V ( μ ν 2 , μ + ν 2 , z ) K ν ( z ) } ,
W ( n , b , z ) = i = 1 n A i ( n ) A i ( b ) ( z / 2 ) 2 i ,
V ( n , b , z ) = i = 1 n A i ( n ) A i 1 ( b ) ( z / 2 ) 2 i + 1 .
A 0 ( b ) = 1 , A i ( b ) = A i 1 ( b ) ( b + 1 i ) ,
A i ( b ) = b ! ( b i ) ! .
S n = G n z DS x μ + 1 K ν ( x ) d x + G n z DS 2 n z DS x ν + 1 K ν ( x ) d x ,
G n y n 2 2 n y + 1 / Γ ( y ) ,
ν = y 1 , μ = 2 n + y 1 ,
z DS 2 ( α DS y ) 1 / 2 .
S n = G n z DS 2 n + y [ W ( n , n + y 1 , z DS ) K y ( z DS ) + V ( n , n + y 1 , z DS ) K y 1 ( z DS ) ]
= 2 y n Γ ( y ) i = 1 n Γ ( n + 1 ) Γ ( n + y ) Γ ( n + 1 i ) Γ ( n + y i ) × ( z DS 2 ) 2 n + y 2 i [ K y ( z DS ) + z DS 2 ( n + y i ) × K y 1 ( z DS ) ] .

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