Abstract

Experiments indicate that the variance of irradiance fluctuations of a wave propagating through a turbulent atmosphere saturates with increasing path length or turbulence strength. In addition, many experimental data indicate that the saturation value depends on the path length. In this paper we calculate the variance of the irradiance fluctuations by a method originally proposed by Keller, but we view it in a conceptually different manner. The reason that this method is reconsidered is that it leads to solutions whose validity is essentially independent of distance for the coherent field and average energy flux. Our calculations yield a saturation curve that agrees with the experimental data, with a saturation value that depends on both the wavelength and the length of the propagation path. It also leads to a saturation curve with a pronounced maximum for the case of constant turbulence conditions and variable path length. Finally, the amplitude fluctuations are found to be Rayleigh distributed in the limit of infinite path length.

© 1970 Optical Society of America

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References

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  1. J. B. Keller, Proc. Symp. Appl. Math 16, 145 (1964).
    [CrossRef]
  2. W. P. Brown, IEEE Trans. Ant. Prop. AP 15, 81 (1967).
    [CrossRef]
  3. D. A. deWolf, Radio Sci. 2, 1379 (1967).
  4. U. Frisch, in Probability Methods in Applied Mathematics, Vol. 1, Bharucha-Reid, Ed. (Academic Press Inc., New York, 1968).
  5. A. D. Varvatsis and M. I. Sancer, Northrop Corp. Labs., Hawthorne, Calif., Rept. NCL 69-47R (1969).
  6. D. A. deWolf, thesis, Eindhoven, The Netherlands (1968).
  7. M. I. Sancer and A. D. Varvatsis, Northrop Corp. Labs., Hawthorne, Calif., Rept. NCL 69-28R (1969).
  8. D. A. deWolf, J. Opt. Soc. Am. 58, 461 (1968).
    [CrossRef]
  9. D. A. deWolf, J. Opt. Soc. Am. 59, 1455 (1969).
    [CrossRef]
  10. J. R. Kerr, J. Opt. Soc. Am. 59, 1513A (1969).
  11. P. Bassanini, C. Cercignani, F. Sernagiotto, and G. Tironi, Radio Sci. 2, 1 (1967).
  12. M. E. Gracheva, Izv. Vuz. Radiofiz. 10, 775 (1967).
  13. P. H. Deitz and N. J. Wright, J. Opt. Soc. Am. 59, 527 (1969).
    [CrossRef]
  14. J. W. Strohbehn, Proc. IEEE,  56, 1301 (1968).
    [CrossRef]
  15. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw–Hill Book Co., New York, 1965).
  16. M. E. Gracheva and A. S. Gurvich, Izv. Vuz. Radiofiz. 8, 717 (1965).

1969 (3)

1968 (2)

J. W. Strohbehn, Proc. IEEE,  56, 1301 (1968).
[CrossRef]

D. A. deWolf, J. Opt. Soc. Am. 58, 461 (1968).
[CrossRef]

1967 (4)

W. P. Brown, IEEE Trans. Ant. Prop. AP 15, 81 (1967).
[CrossRef]

D. A. deWolf, Radio Sci. 2, 1379 (1967).

P. Bassanini, C. Cercignani, F. Sernagiotto, and G. Tironi, Radio Sci. 2, 1 (1967).

M. E. Gracheva, Izv. Vuz. Radiofiz. 10, 775 (1967).

1965 (1)

M. E. Gracheva and A. S. Gurvich, Izv. Vuz. Radiofiz. 8, 717 (1965).

1964 (1)

J. B. Keller, Proc. Symp. Appl. Math 16, 145 (1964).
[CrossRef]

Bassanini, P.

P. Bassanini, C. Cercignani, F. Sernagiotto, and G. Tironi, Radio Sci. 2, 1 (1967).

Brown, W. P.

W. P. Brown, IEEE Trans. Ant. Prop. AP 15, 81 (1967).
[CrossRef]

Cercignani, C.

P. Bassanini, C. Cercignani, F. Sernagiotto, and G. Tironi, Radio Sci. 2, 1 (1967).

Deitz, P. H.

deWolf, D. A.

D. A. deWolf, J. Opt. Soc. Am. 59, 1455 (1969).
[CrossRef]

D. A. deWolf, J. Opt. Soc. Am. 58, 461 (1968).
[CrossRef]

D. A. deWolf, Radio Sci. 2, 1379 (1967).

D. A. deWolf, thesis, Eindhoven, The Netherlands (1968).

Frisch, U.

U. Frisch, in Probability Methods in Applied Mathematics, Vol. 1, Bharucha-Reid, Ed. (Academic Press Inc., New York, 1968).

Gracheva, M. E.

M. E. Gracheva, Izv. Vuz. Radiofiz. 10, 775 (1967).

M. E. Gracheva and A. S. Gurvich, Izv. Vuz. Radiofiz. 8, 717 (1965).

Gurvich, A. S.

M. E. Gracheva and A. S. Gurvich, Izv. Vuz. Radiofiz. 8, 717 (1965).

Keller, J. B.

J. B. Keller, Proc. Symp. Appl. Math 16, 145 (1964).
[CrossRef]

Kerr, J. R.

J. R. Kerr, J. Opt. Soc. Am. 59, 1513A (1969).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw–Hill Book Co., New York, 1965).

Sancer, M. I.

A. D. Varvatsis and M. I. Sancer, Northrop Corp. Labs., Hawthorne, Calif., Rept. NCL 69-47R (1969).

M. I. Sancer and A. D. Varvatsis, Northrop Corp. Labs., Hawthorne, Calif., Rept. NCL 69-28R (1969).

Sernagiotto, F.

P. Bassanini, C. Cercignani, F. Sernagiotto, and G. Tironi, Radio Sci. 2, 1 (1967).

Strohbehn, J. W.

J. W. Strohbehn, Proc. IEEE,  56, 1301 (1968).
[CrossRef]

Tironi, G.

P. Bassanini, C. Cercignani, F. Sernagiotto, and G. Tironi, Radio Sci. 2, 1 (1967).

Varvatsis, A. D.

M. I. Sancer and A. D. Varvatsis, Northrop Corp. Labs., Hawthorne, Calif., Rept. NCL 69-28R (1969).

A. D. Varvatsis and M. I. Sancer, Northrop Corp. Labs., Hawthorne, Calif., Rept. NCL 69-47R (1969).

Wright, N. J.

IEEE Trans. Ant. Prop. (1)

W. P. Brown, IEEE Trans. Ant. Prop. AP 15, 81 (1967).
[CrossRef]

Izv. Vuz. Radiofiz. (2)

M. E. Gracheva, Izv. Vuz. Radiofiz. 10, 775 (1967).

M. E. Gracheva and A. S. Gurvich, Izv. Vuz. Radiofiz. 8, 717 (1965).

J. Opt. Soc. Am. (4)

Proc. IEEE (1)

J. W. Strohbehn, Proc. IEEE,  56, 1301 (1968).
[CrossRef]

Proc. Symp. Appl. Math (1)

J. B. Keller, Proc. Symp. Appl. Math 16, 145 (1964).
[CrossRef]

Radio Sci. (2)

P. Bassanini, C. Cercignani, F. Sernagiotto, and G. Tironi, Radio Sci. 2, 1 (1967).

D. A. deWolf, Radio Sci. 2, 1379 (1967).

Other (5)

U. Frisch, in Probability Methods in Applied Mathematics, Vol. 1, Bharucha-Reid, Ed. (Academic Press Inc., New York, 1968).

A. D. Varvatsis and M. I. Sancer, Northrop Corp. Labs., Hawthorne, Calif., Rept. NCL 69-47R (1969).

D. A. deWolf, thesis, Eindhoven, The Netherlands (1968).

M. I. Sancer and A. D. Varvatsis, Northrop Corp. Labs., Hawthorne, Calif., Rept. NCL 69-28R (1969).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw–Hill Book Co., New York, 1965).

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Figures (2)

Fig. 1
Fig. 1

σ vs σ1 under constant turbulence conditions and variable path length for s = 20.

Fig. 2
Fig. 2

σ vs σ1: The data are taken from Gracheva12 and dots correspond to 1750 m while crosses to 500 m; the values of our parameter m are 0.7 and 0.2, respectively.

Equations (52)

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[ 2 + k 0 2 ( 1 + κ 1 ) ] U = f ,
L U c = - ξ U i + f
L U i = - ξ U c - ξ U i + ξ U i ,
L U i = - ξ U c
L U c = 2 ξ L - 1 ξ U c + f ,
L - 1 h ( r ) = - exp [ i k 0 r - r ] 4 π r - r h ( r ) d r .
Im [ U e U e * ] = Im U e f * ,
Im U i e U i e * = Im f * U c e - Im ( U c e U c e * ) .
Im [ U U * ] = 2 i     Im [ U C * ( ξ U i - ξ U i ) + U i * ξ U i + ξ U c 2 + f * U ] .
Im U i U i * = Im f * U c - Im ( U c U c * ) .
Im U i e U i e * Im U i U i * .
Im U i e U i e * Im U i U i *
Im U i U i * - k 0 e ˆ x U i 2
Im U i e U i e * - k 0 e ˆ x U i e 2 .
σ 2 = ln [ 1 + ( I - I ) 2 / I 2 ] = ln ( I 2 / I 2 ) ,
U c = exp [ i k 0 x - β x ] ,             x l ,
β = 2 k 0 2 4 0 B ( 0 , u ) d u ,             k 0 l 1
I = U c 2 + U i 2 ,
U i 2 = 2 k 0 4 G 0 ( r , r ) G 0 * ( r , r ) B ( r - r ) × U c ( r ) U c * ( r ) d r d r .
H ˆ ( k , x ) = ( 2 π ) - 2 H ( ϱ , x ) exp [ - i k ϱ ] d ϱ ,
U i 2 = ( 2 π ) 4 2 k 0 4 0 x 0 x d x d x d k G ˆ 0 ( k , x - x ) G ˆ 0 * ( k , x - x ) F n ( k , x - x ) × exp [ - β ( x + x ) ] exp [ i k 0 ( x - x ) ] ,
G ˆ 0 ( k , x ) = [ - i / 8 π 2 ( k 0 2 - k 2 ) 1 2 ]     exp [ i ( k 0 2 - k 2 ) 1 2 x ] .
( k 0 2 - k 2 ) 1 2 = k 0 [ 1 - 1 2 ( k / k 0 ) 2 - 3 8 ( k / k 0 ) 4 - ]
G ˆ 0 ( k , x ) = - i 8 π 2 k 0 exp [ i k 0 x - i 2 k 0 x k 2 ] ,
U i 2 = 2 k 0 2 0 x 0 x d x d x d k     exp [ i k 2 2 k 0 ( x - x ) ] × F n ( k x - x )     exp [ - β ( x + x ) ] .
U i 2 = 2 k 0 2 4 0 x 0 x d x d x B ( 0 , x - x ) × exp [ - β ( x + x ) ] .
U i 2 = 2 k 0 2 2 0 x d t     exp [ - 2 β t ] 0 B ( 0 , u ) d u .
U i 2 = 1 - exp [ - 2 β x ] ,             x l .
I 2 = U i 4 + U c 4 + 4 U i 2 U c 2 + 2     Re U c 2 * U i 2 .
U i 4 = 2 U i 2 2 + U i 2 U i 2 * .
U i 2 = k 0 4 B ( r , r ) G 0 ( r , r ) G 0 ( r , r ) × U c ( r ) U c ( r ) d r d r .
U i 2 = - k 0 2 4 exp [ 2 i k 0 x ] 0 x 0 x d x d x d k × exp [ - i x k 0 k 2 + i x + x 2 k 0 k 2 ] × F n ( k , x - x )     exp [ - β ( x + x ) ] .
U i 2 = - k 0 2 π 2 exp [ 2 i k 0 x ] 0 x d t     exp [ - 2 β t ] × 0 exp [ i t - x k 0 k 2 ] ϕ n ( k ) k d k ,
0 F n ( k , u ) d u = π ϕ n ( k )
B ( r ) = κ 1 2     exp [ - r 2 / 2 l 2 ] ,             ϕ n ( k ) = κ 1 2 l 3 ( 2 π ) 3 2     exp [ - k 2 l 2 / 2 ] ,
U i 2 = - k 0 2 π 2 κ 1 2 l 3 ( 2 π ) 3 2     exp [ 2 i k 0 x - 2 β ( x ) ] × 0 x     exp [ 2 β u ] d u 0     exp [ - i u k 2 k 0 - k 2 l 2 / 2 ] k d k ,
U i 2 = - k 0 2 π 2 κ 1 2 l 3 2 ( 2 π ) 3 2     ×     exp [ - 2 β x + 2 i k 0 x ] 0 x exp [ 2 β u ] l 2 / 2 + i u / k 0 d u .
E 1 ( z ) = z exp [ - t ] t d t ,             argz < π ,
U i 2 = i s exp [ 2 i k 0 x - 2 β x + i s ] × [ E 1 ( i s ) - E 1 ( i s - 2 β x ) ] ,
s = ( k 0 l ) ( β l ) ,             β = k 0 2 κ 1 2 l π / 4 2 .
σ 1 2 = 4 β x ,
σ 2 = ln ( 2 - exp [ - σ 1 2 ] + 2 exp [ - σ 1 2 ] × Re { i s     exp ( i s ) [ E 1 ( i s ) - E 1 ( i s - σ 1 2 2 ) ] } + exp [ - σ 1 2 ] s 2 | E 1 ( i s ) - E 1 ( i s - σ 1 2 2 ) | 2 ) .
σ 2 = ln ( 2 - exp ( - σ 1 2 ) + 2 exp ( - σ 1 2 ) ×     Re { i m σ 1 2     exp ( i m σ 1 2 ) [ E 1 ( i m σ 1 2 ) - E 1 ( i m σ 1 2 - σ 1 2 / 2 ) ] } + exp ( - σ 1 2 ) m 2 σ 1 2 E 1 ( i m σ 1 2 ) - E 1 ( i m σ 1 2 - σ 1 2 / 2 ) 2 ) ,
m = k 0 l 2 / 4 x .
lim σ 1 σ 2 = ln [ 2 + 4 m 2 / ( 4 m 2 + 1 ) ] .
U i 2 = ( Re U i ) 2 - ( Im U i ) 2 + 2 i Re U i     Im U i
U i 2 = ( Re U i ) 2 + ( Im U i ) 2 .
( Re U i ) 2 = ( Im U i ) 2 , ( Re U i ) ( Im U i ) = 0             ( x ) .
( Re U i ) 2 + ( Im U i ) 2 = 1             ( x ) .
U = [ ( Re U i ) 2 + ( Im U i ) 2 ] 1 2             ( x ) ,
p ( U ) = 2 U exp [ - U 2 ] ,
I n = U 2 n = n ! .