Abstract

The Bethe–Salpeter equation for the second moment of a wave propagating in a random media is solved in the ladder approximation. The solution obtained applies to propagation problems in which the wavelength is much smaller than the scale of the inhomogeneities, and the product of the rms index fluctuation, the wavenumber, and a characteristic scale length of the inhomogeneities is also much smaller than unity. The field is assumed given on an initial plane and to have appreciable amplitude over a domain many wavelengths in extent in this plane. It is not required, however, that this domain be larger than the scale of the inhomogeneities. Quantitative results are given for a gaussian beam. We show that the ensemble-averaged distribution of irradiance remains gaussian and give the e-folding radius of this distribution. The two-point correlation function of the field in such a beam is shown to depend only on the distance between the points, not on their relative position within the beam. The coherence properties of the beam are expressed in terms of a coherence length. We also consider the ensemble-averaged interference pattern of the field from multiple apertures and verify that well-defined interference fringes occur only when the apertures are separated by a distance smaller than the coherence length of the individual beams. The solution of the Bethe–Salpeter equation obtained here satisfies an equation derived by Beran. We show that the local-independence assumption used by Beran is valid for sufficiently weak inhomogeneities.

© 1971 Optical Society of America

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References

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  1. A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, edited by R. A. Silverman(Prentice–Hall, Englewood Cliffs, N. J., 1962).
  2. V. I. Tatarski, Sov. Phys. JETP 19, 946 (1964).
  3. U. Frisch, in Probabilistic Methods in Applied Mathematics, edited by A. T. Bharucha-Reid (Academic, New York, 1968).
  4. W. P. Brown, IEEE Trans. AP-15, 81 (1967).
    [CrossRef]
  5. D. A. deWolf, Radio Sci. 2, 1379 (1967).
  6. W. P. Brown, in Modern Optics, edited by J. Fox (Polytechnic Institute Press, Brooklyn, 1967).
  7. M. Beran, J. Opt. Soc. Am. 60, 518 (1970).
    [CrossRef]

1970 (1)

1967 (2)

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

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

1964 (1)

V. I. Tatarski, Sov. Phys. JETP 19, 946 (1964).

Beran, M.

Brown, W. P.

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

W. P. Brown, in Modern Optics, edited by J. Fox (Polytechnic Institute Press, Brooklyn, 1967).

deWolf, D. A.

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

Frisch, U.

U. Frisch, in Probabilistic Methods in Applied Mathematics, edited by A. T. Bharucha-Reid (Academic, New York, 1968).

Tatarski, V. I.

V. I. Tatarski, Sov. Phys. JETP 19, 946 (1964).

Yaglom, A. M.

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, edited by R. A. Silverman(Prentice–Hall, Englewood Cliffs, N. J., 1962).

IEEE Trans. (1)

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

J. Opt. Soc. Am. (1)

Radio Sci. (1)

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

Sov. Phys. JETP (1)

V. I. Tatarski, Sov. Phys. JETP 19, 946 (1964).

Other (3)

U. Frisch, in Probabilistic Methods in Applied Mathematics, edited by A. T. Bharucha-Reid (Academic, New York, 1968).

W. P. Brown, in Modern Optics, edited by J. Fox (Polytechnic Institute Press, Brooklyn, 1967).

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, edited by R. A. Silverman(Prentice–Hall, Englewood Cliffs, N. J., 1962).

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Figures (2)

Fig. 1
Fig. 1

Coherence length normalized to the diameter of the ensemble-averaged irradiance distribution, βi(z/k) = 0.5, ξ = z/f.

Fig. 2
Fig. 2

Coherence length normalized to the diameter of the ensemble-averaged irradiance distribution, βi(z/k) = 1.0, ξ = z/f.

Equations (81)

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[ 2 + k 2 n 2 ( x ) ] u ( x ) = 0 ,             z > 0
lim r r [ ( u / r ) - i k n u ] = 0 ,             z > 0 ,
n 2 ( x ) = n 2 ( x ) + μ ( x ) ,
μ ( x ) = K d N μ ( K , z ) exp ( i K · x T ) ,             x = x T , z
d N μ ( K 1 , z 1 ) d N μ * ( K 2 , z 2 ) = d K 1 d K 2 δ ( K 1 - K 2 ) F μ ( K 1 , z 1 - z 2 ) .
d U ( K 1 , z ) d U * ( K 2 , z ) = d U ( K 1 , z ) d U * ( K 2 , z ) + ( π k 2 ) 2 × 0 z d z 1 0 z d z 2 G ( K 1 , z - z 1 ) G * ( K 2 , z - z 2 ) × K 11 K 21 d N μ ( K 1 - K 11 , z 1 ) d N μ * ( K 2 - K 21 , z 2 ) × d U ( K 11 , z 1 ) d U * ( K 21 , z 2 ) .
u ( x ) = K d U ( K , z ) exp ( i K · x T ) ,
G ( x i , x j ) = K d K G ( K , z i - z j ) exp [ i K · ( x T i - x T j ) ] .
k e = k + i α
k = k n 2 1 2
α = ( 2 k 2 π / 4 ) K d K Φ μ ( K , 0 ) 2 k 2 l / 4 ,
Φ μ ( K , K z ) = 1 2 π - d z exp ( i K z z ) F μ ( K , z ) .
d U ( K i , z ) ~ d K i S ( K i ) exp [ i ( k e 2 - K i 2 ) 1 2 z ] ,             i = 1 , 2
G ( K i , z - z i ) ~ i exp [ i ( k e 2 - K i 2 ) 1 2 z - z i ] 2 π ( k e 2 - K i 2 ) 1 2 ,             i = 1 , 2
d U ( K i , z i ) d U * ( K j , z j ) = d K i · d K j I ( K i , K j , z i , z j ) × exp [ i k ( z i - z j ) - α ( z i + z j ) - i ( K i 2 z i - K j 2 z j ) / 2 k ] ,
I ( K 1 , K 2 , z , z ) = I 0 ( K 1 , K 2 ) + ( k / 2 ) 2 0 z d z 1 0 z d z 2 K 11 K 21 × d N μ ( K 1 - K 11 , z 1 ) d N μ * ( K 2 - K 21 , z 2 ) × exp [ i ( K 1 2 - K 11 2 ) z 1 / 2 k - i ( K 2 2 - K 21 2 ) z 2 / 2 k ] × I ( K 11 , K 21 , z 1 , z 2 ) ,
I 0 ( K 1 , K 2 ) = S ( K 1 ) S * ( K 2 ) .
I ( K 1 , K 2 , z , z ) = I 0 ( K 1 , K 2 ) + ( k / 2 ) 2 0 z d z 1 0 z d z 2 K d K F μ ( K , z 1 - z 2 ) × exp [ i ( K 1 z 1 - K 2 z 2 ) · K / k - i K 2 ( z 1 - z 2 ) / 2 k ] × I ( K 1 - K , K 2 - K , z 1 , z 2 ) .
I ( K 1 , K 2 , z ) = I 0 ( K 1 , K 2 ) + ( k / 2 ) 2 0 z d z 1 0 z d z 2 K d K F μ ( K , z 1 - z 2 ) × exp [ i ( K 1 - K 2 ) · K z / k ] I ( K 1 - K , K 2 - K , z 1 ) ,
u ( x ) u * ( x 2 ) = K 1 d K 1 K 2 d K 2 exp ( i K 1 · x T 1 - i K 2 · x T 2 ) × exp [ - 2 α z - i ( K 1 2 - K 2 2 ) z / 2 k ] I ( K 1 , K 2 , z ) , = K a d K a K b d K b exp ( i K a · ϱ + + i K b · ϱ - ) × exp [ - 2 α z - i K a · K b z / k ] I ( K a , K b , z ) ,
ϱ + = ( x T 1 + x T 2 ) / 2 ,
ϱ - = x T 1 - x T 2 .
I ( K a , K b , z ) = I 0 ( K a , K b ) + ( k / 2 ) 2 0 z d z 1 0 z d z 2 K d K F μ ( K , z 1 - z 2 ) × exp ( i K a · K z 1 / k ) I ( K a , K b - K , z 1 ) .
I ( K a , ϱ - , z ) = I 0 ( K a , ϱ - ) + ( k / 2 ) 2 0 z d z 1 0 z d z 2 × C ( ϱ - + K a z / k , z 1 - z 2 ) I ( K a , ϱ - , z 1 ) ,
C ( x T , z ) = K d K exp ( i K · x T ) × - d K z exp ( i K z z ) Φ μ ( K , K z ) .
0 z d z 1 0 z d z 2 C ( ϱ - + K a z 1 / k , z 1 - z 2 ) I ( K a , ϱ - , z 1 ) = 0 z d v I ( K a , ϱ - , v ) - z + v v d u C ( ϱ - + K a v / k , u ) ~ 0 z d v I ( K a , ϱ - , v ) - d u C ( ϱ - + K a v / k , u ) .
I ( K a , ϱ - , z ) = I 0 ( K a , ϱ - ) + ( k / 2 ) 2 0 z d v I ( K a , ϱ - , v ) × - d u C ( ϱ - + K a v / k , u ) .
I ( K a , ϱ - , z ) = I 0 ( K a , ϱ - ) × exp [ ( k / 2 ) 2 0 z d v - d u C ( ϱ - + K a v / k , u ) ] .
u ( x 1 ) u * ( x 2 ) = K a d K a exp ( i K a · ϱ + ) × exp ( - 2 a z ) I ( K a , ϱ - - K a z / k , z ) ,
u ( x 1 ) u * ( x 2 ) = K a d K a exp ( i K a · ϱ + ) I 0 ( K a , ϱ - - K a z / k ) × exp { - ( k / 2 ) 2 0 z d v - d u × [ C ( 0 , u ) - C ( ρ - - K a v / k , u ) ] } ,
2 α = ( k / 2 ) 2 - d u C ( 0 , u ) .
u ( x 1 ) u * ( x 2 ) | plane wave = exp { - ( k / 2 ) 2 0 z d v - d u [ C ( 0 , u ) - C ( ϱ - , u ) ] } .
0 z d v - d u [ C ( 0 , u ) - C ( ϱ - - K a v / k , u ) ] = 2 π z K d K [ 1 - exp ( i K · ϱ - ) 1 - exp ( - i K · K a z / k ) i K · K a z / k ] × Φ μ ( K , 0 ) .
K d K [ 1 - exp ( i K · ϱ - ) 1 - exp ( - i K · K a z / k ) i K · K a z / k ] Φ μ ( K , 0 ) = 0 d K K Φ μ ( K , 0 ) 0 2 π d φ { 1 - exp [ i K ρ - cos ( θ - φ ) ] × 1 - exp ( - i K K a z cos φ / k ) i K K a z cos φ / k } ,
K a ( K a 0 2 π d φ { 1 - exp [ i K ρ - cos ( θ - φ ) ] × 1 - exp ( - i K K a z cos φ / k i K K a z cos φ / k } ) = 0 2 π d φ { 1 - exp [ i K ρ - cos ( θ - φ ) - i K K a z cos φ / k ] } = 2 π [ 1 - J 0 ( K ϱ - - K a z / k ) ] .
K d K [ 1 - exp ( i K · ϱ - ) 1 - exp ( - i K · K a z / k ) i K · K a z / k ] Φ μ ( K , 0 ) = 2 π 0 1 d ξ 0 d K K Φ μ ( K , 0 ) × [ 1 - J 0 ( K ρ - - K a z ξ / k ) ] .
( k / 2 ) 2 0 z d v - d u [ C ( 0 , u ) - C ( ϱ - - K a v / k , u ) ] = π 2 2 k 2 z [ P 2 AV 4 0 d K K 3 Φ μ ( K , 0 ) - P 4 AV 64 0 d K K 5 Φ μ ( K , 0 ) + + ( - 1 ) m + 1 × P 2 m AV 4 m ( m ! ) 2 0 d K K 2 m + 1 Φ μ ( K , 0 ) + ] ,
P 2 m AV = 0 1 d ξ ρ - - K a z ξ / k 2 m .
c 0 α z P 2 AV l 2 ( 1 + c 1 P 4 AV / l 4 P 2 AV / l 2 + c 2 P 6 AV / l 6 P 2 AV / l 2 + ) ,
( k / 2 ) 2 0 z d v - d u [ C ( 0 , u ) - C ( ϱ - - K a v / k , u ) ] ~ β 1 P 2 AV = β 1 [ ρ - 2 - K a ρ - z / k + 1 3 ( K a z / k ) 2 ] ,
β 1 = ( π k / 2 ) 2 z 0 d K K 3 Φ μ ( K , 0 ) .
( / 2 ) 2 Φ μ ( k , 0 ) = 0.033 K 0 - ( 11 / 3 ) C N 2 [ 1 + ( K / K 0 ) 2 ] 11 / 6 ,             K < K i
= 0 ,             K > K i
( k / 2 ) 2 0 z d v d u [ C ( 0 , u ) - C ( ρ - - K a v / k , u ) ] = E ( 0 , K 1 ) + E ( K 1 , ) ,
E ( 0 , K 1 ) = β 2 P 2 AV { 1 - K 1 2 P 4 AV 112 P 2 AV + K 1 4 P 6 AV 7488 P 2 AV - + ( - 1 ) m K 1 2 m P 2 m + 2 A V ( 6 m + 1 ) 4 m [ ( m + 1 ) ! ] 2 P 2 AV + } ,
β 2 = ( π k / 2 ) 2 z 0 K 1 d K K 3 Φ μ ( K , 0 ) ~ 0.1 π 2 C N 2 k 2 z K 1 1 3 .
E 1 ( K 1 ) = 0.132 π 2 C N 2 k 2 z K 0 - ( 11 / 3 ) × 0 1 d ξ K 1 K i d K K [ 1 + ( K / K 0 ) 2 ] - ( 11 / 6 ) × [ 1 - J 0 ( K ϱ - - K a z ξ / l k ) ] < 0.112 π 2 C N 2 k 2 z K 1 - ( 5 / 3 ) ,
[ 1 + ( K / K 0 ) 2 ] - ( 11 / 6 ) [ 1 - J 0 ( K ϱ - - K a z ξ / k ) ] < 1.41 ( K 0 / K ) 11 / 3 .
[ 1 + e 1 112 β 2 K 1 - 2 + e 2 7488 ( β 2 K 1 - 2 ) 2 - ] ,
K 1 = ( π 2 C N 2 k 2 z ) 3 / 5 .
( k / 2 ) 2 0 z d v - d u × [ C ( 0 , u ) - C ( ϱ - - K a v / k , u ) ] ~ β 2 P 2 AV = β 2 [ ρ - 2 - K a · ϱ - z / k + 1 3 ( K a z / k ) 2 ] .
β 2 = 0.1 ( π 2 C N 2 k 2 z ) 6 / 5 .
β 2 = ( π k / 2 ) 2 z 0 K 1 d K K 3 Φ μ ( K , 0 )
E ( K 1 , ) < 1.41 ( π k ) 2 z K 1 d K K Φ μ ( K , 0 ) = 0.1 .
u ( x T i , 0 ) = exp ( ρ i 2 2 ρ 0 2 - i k ρ i 2 2 f ) ,
u ( x T i - K a z / 2 k , 0 ) u * ( x T 2 + K a z / 2 k , 0 ) = exp [ - ρ + 2 ρ 0 2 - ( ϱ - - K a z / k ) 2 4 ρ 0 2 - i k ρ + · ( ϱ - - K a z / k ) f ]
I 0 ( K a , ϱ - - K a z / k ) = ( 1 / 2 π ) 2 ρ + d ϱ + exp ( - i K a · ϱ - ) × exp [ - ϱ + 2 ρ 0 2 - ( ϱ - - K a z / k ) 2 4 ρ 0 2 - i k ϱ + · ( ϱ - - K a z / k ) f ] = ρ 0 2 4 π exp [ - ϱ - 2 4 ρ 0 a 2 + γ 0 K a · ϱ - 2 - K a 2 ρ 0 b 2 4 ] ,
ρ 0 a 2 = ρ 0 2 [ 1 + ( k ρ 0 2 / f ) 2 ] - 1 ,
γ 0 = z / k ρ 0 2 - ( k ρ 0 2 / f ) [ 1 - ( z / f ) ] ,
ρ 0 b 2 = ρ 0 2 { [ 1 - ( z / f ) ] 2 + ( z / k ρ 0 2 ) 2 } .
( 1 / 2 π ) 2 ρ d ϱ exp ( - i K · ϱ ) exp ( - ρ 2 / a 2 ) = a 2 4 π exp [ - ( K a ) 2 / 4 ] .
u ( x 1 ) u * ( x 2 ) = ( ρ 0 2 / 4 π ) K a d K a exp ( i K a · ϱ + ) I 0 ( K a , ϱ - - K a z / k ) exp { - β i [ ρ - 2 - K a · ϱ - z / k + 1 3 ( K a z / k ) 2 ] } , = ( ρ 0 / ρ b ) 2 exp [ - ρ - 2 4 ρ a 2 - ρ + 2 ρ b 2 + i γ ϱ - · ϱ + p b 2 ] ,
ρ b 2 = ρ 0 2 [ ( 1 - z f ) 2 + ( z k ρ 0 2 ) 2 ( 1 + 4 3 β i ρ 0 2 ) ] ,
ρ a 2 = ρ b 2 1 + 4 β i ρ 0 2 [ 1 - 3 z f + 7 3 ( z f ) 2 + 7 3 ( z k ρ 0 2 ) 2 ] + 4 3 ( β i ρ 0 2 ) 2 ( z k ρ 0 2 ) 2 ,
γ = z k ρ 0 2 ( 1 - 2 β i ρ 0 2 ) - k ρ 0 2 f ( 1 - z f ) .
u ( x T , z ) u * ( x T , z ) = ( ρ 0 / ρ b ) 2 exp ( - ρ 2 / ρ b 2 ) .
C 12 ( x T 1 , x T 2 ) = u ( x 1 ) u * ( x 2 ) u ( x 1 ) u * ( x 1 ) 1 2 u ( x 2 ) u * ( x 2 ) 1 2 .
C 12 ( x T 1 , x T 2 ) = exp ( - ρ - 2 / ρ c 2 ) ,
ρ c = 2 [ 1 ρ a 2 - 1 ρ b 2 ] - 1 2 = 2 ρ 0 [ ( 1 - z f ) 2 + ( z k ρ 0 2 ) 2 ( 1 + 4 3 β i ρ 0 2 ) 4 β i ρ 0 2 [ 1 - 3 z f + 7 3 ( z f ) 2 + 7 3 ( z k ρ 0 2 ) 2 ] + 4 3 ( β i z k ) 2 ] 1 2 .
ρ c / 2 ρ b = [ ζ 4 β i z k [ ζ 2 ( 1 - 3 ξ + 7 3 ξ 2 ) + 1 3 β i z k ζ + 7 3 ] ] 1 2 ,
S 1 ( K 1 ) = exp ( - i K 1 · x 0 / 2 ) S ( K 1 ) ,
S 2 ( K 2 ) = exp ( i K 2 · x 0 / 2 ) S ( K 2 ) .
u 1 ( x ) + u 2 ( x ) 2 = u 1 ( x ) u 1 * ( x ) + u 2 ( x ) u 2 * ( x ) + 2 Re u 1 ( x ) u 2 * ( x ) ,
u i ( x ) u i * ( x ) = ( ρ 0 / ρ b ) 2 × exp [ - ( x T x 0 / 2 ) 2 / ρ b 2 ] ,             i = 1 , 2
u 1 ( x ) u 2 * ( x ) = ( ρ 0 / ρ b ) 2 × exp ( - ρ 2 / ρ b 2 - x 0 2 / 4 ρ a 2 - i γ x T · x 0 / ρ b 2 ) ,
u 1 ( x ) + u 2 ( x ) 2 = ( ρ 0 / ρ b ) 2 exp [ - ( ρ 2 + x 0 2 / 4 ) / ρ b 2 ] × [ exp ( x T · x 0 / ρ b 2 ) + exp ( - x T · x 0 / ρ b 2 ) + 2 exp ( - x 0 2 / ρ c 2 ) cos ( γ x T · x 0 / ρ b 2 ] .
z u ( x 1 ) u * ( x 2 ) = K 1 d K 1 K 2 d K 2 exp ( i K 1 · x T 1 - i K 2 · x T 2 ) × exp [ - 2 α z - i ( K 1 2 - K 2 2 ) z / 2 k ] × [ - 2 α - i ( K 1 2 - K 2 2 ) / 2 k + z ] I ( K 1 , K 2 , z ) = [ - 2 α + i ( T 1 2 - T 2 2 ) / 2 k ] u ( x 1 ) u * ( x 2 ) + K a d K a K b d K b exp ( i K a · ϱ + + i K b ϱ - ) × exp [ - 2 α z - i K a · K b z / k ] I ( K a , K b , z ) z ,
I ( K a , K b , z ) z = 1 ( 2 π ) 2 ρ d ϱ exp ( - i K b · ϱ ) ( k / 2 ) 2 × - d u C ( ρ + K a z / k , u ) I ( K a , ρ , z ) .
( k / 2 ) 2 K a d K a [ exp ( i K a · ϱ + ) ] × exp ( - 2 α z ) I ( K a , ϱ - - K a z / k ) - d u C ( ϱ - , u ) ,
[ ( k / 2 ) 2 - d u C ( ϱ - , u ) ] u ( x 1 ) u * ( x 2 ) .
z u ( x 1 ) u * ( x 2 ) = { i ( T 1 2 - T 2 2 ) / 2 k - ( k / 2 ) 2 × - d u [ C ( 0 , u ) - C ( ϱ - , u ) ] } u ( x 1 ) u * ( x 2 ) .