Abstract

In a previous paper, we considered a perturbation solution for propagation of the fourth-order coherence function in a random medium. In this paper, it is shown how under certain conditions this solution may be extended to treat the propagation problem when the field fluctuations need not be small. A differential equation governing the fourth-order coherence function is derived. A solution is obtained in the geometric-optics limit when the four points of the coherence function are distinct.

© 1969 Optical Society of America

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References

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  1. T. L. Ho and M. J. Beran, J. Opt. Soc. Am. 58, 1335 (1968).
    [Crossref]
  2. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964). See p. 23 for the derivation of a similar relation for the Fourier transform of the mutual coherence function.
  3. In Ref. 1 we used the notation Ľ|z instead of {Lˆ0}z.
  4. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Co., New York1961). See Eqs. (13.13), (7.54), and (7.56).
  5. M. J. Beran, J. Opt. Soc. Am. 56, 1475 (1966).
    [Crossref]
  6. Equation (20) can also be derived by a diagram technique, i.e., a partial summation of the infinite series representing {Lˆ1}. See, V. I. Shishov, IVUZ-Radiophysics (Russian) 11, 866 (1968). Shishov used a gaussian form for the correlation function of refractive-index fluctuations.

1968 (2)

T. L. Ho and M. J. Beran, J. Opt. Soc. Am. 58, 1335 (1968).
[Crossref]

Equation (20) can also be derived by a diagram technique, i.e., a partial summation of the infinite series representing {Lˆ1}. See, V. I. Shishov, IVUZ-Radiophysics (Russian) 11, 866 (1968). Shishov used a gaussian form for the correlation function of refractive-index fluctuations.

1966 (1)

Beran, M. J.

T. L. Ho and M. J. Beran, J. Opt. Soc. Am. 58, 1335 (1968).
[Crossref]

M. J. Beran, J. Opt. Soc. Am. 56, 1475 (1966).
[Crossref]

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964). See p. 23 for the derivation of a similar relation for the Fourier transform of the mutual coherence function.

Ho, T. L.

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964). See p. 23 for the derivation of a similar relation for the Fourier transform of the mutual coherence function.

Shishov, V. I.

Equation (20) can also be derived by a diagram technique, i.e., a partial summation of the infinite series representing {Lˆ1}. See, V. I. Shishov, IVUZ-Radiophysics (Russian) 11, 866 (1968). Shishov used a gaussian form for the correlation function of refractive-index fluctuations.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Co., New York1961). See Eqs. (13.13), (7.54), and (7.56).

IVUZ-Radiophysics (Russian) (1)

Equation (20) can also be derived by a diagram technique, i.e., a partial summation of the infinite series representing {Lˆ1}. See, V. I. Shishov, IVUZ-Radiophysics (Russian) 11, 866 (1968). Shishov used a gaussian form for the correlation function of refractive-index fluctuations.

J. Opt. Soc. Am. (2)

Other (3)

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964). See p. 23 for the derivation of a similar relation for the Fourier transform of the mutual coherence function.

In Ref. 1 we used the notation Ľ|z instead of {Lˆ0}z.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Co., New York1961). See Eqs. (13.13), (7.54), and (7.56).

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Equations (43)

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L 1 ( x 1 , x 2 , x 3 , x 4 , τ 2 , τ 3 , τ 4 ) = V ( x 1 , t ) V * ( x 2 , t + τ 2 ) × V * ( x 3 , t + τ 3 ) V ( x 4 , t + τ 4 ) ;
{ L 1 ( x 1 , x 2 , x 3 , x 4 , τ 2 , τ 3 , τ 4 ) } ,
{ L 1 ( x 1 , x 2 , x 3 , x 4 , 0 , 0 , 0 ) } z 1 = z 2 = z 3 = z 4 = 0 = F ( r 12 , r 13 , r 14 ) ,
{ L 1 ( x 1 , x 2 , x 3 , x 4 , τ 2 , τ 3 , τ 4 ) } z 1 = z 2 = z 3 = z 4 = L
{ L ˆ 1 ( x 1 , x 2 , x 3 x 4 , ν 2 , ν 3 , ν 4 ) } z 1 = z 2 = z 3 = z 4 = L = - { L 1 ( x 1 , x 2 , x 3 , x 4 , τ 2 , τ 3 , τ 4 ) } z 1 = z 2 = z 3 = z 4 = L × exp [ 2 π i ( - ν 2 τ 2 - ν 3 τ 3 + ν 4 τ 4 ) ] d τ 2 d τ 3 d τ 4 .
{ L ˆ ( x 1 , x 2 , x 3 , x 4 , ν 2 , ν 3 , ν 4 ) } = { lim T 1 2 T V ˆ ( x 1 , ν 1 ) V ˆ * ( x 2 , ν 2 ) V ˆ * ( x 3 , ν 3 ) V ˆ ( x 4 , ν 4 ) } ,
{ L ˆ 1 } z = { L ˆ 0 } z - 1 8 ( k 1 2 + k 2 2 + k 3 2 + k 4 2 ) z { L ˆ 0 } z σ ¯ ( 0 ) + { L ˆ 1100 } + { L ˆ 1010 } + { L ˆ 1001 } + { L ˆ 0110 } + { L ˆ 0101 } + { L ˆ 0011 } ,
{ L ˆ 1 } z = L ˆ 0 { 1 + ( k 2 / 4 ) z ( σ ¯ ( r 12 ) + σ ¯ ( r 13 ) + σ ¯ ( r 24 ) + σ ¯ ( r 34 ) - 2 σ ¯ ( 0 ) - 2 π 0 σ ˜ ¯ ( η ) J 0 ( η r 14 ) η k i η 2 z [ 1 - exp ( - i η 2 z k ¯ ) ] d η + 2 π 0 σ ˜ ¯ ( η ) J 0 ( η r 23 ) η k i η 2 z [ 1 - exp ( i η 2 z k ¯ ) ] d η ) } .
{ L ˆ 1 } z = L ˆ 0 { 1 + ( k ¯ 2 / 4 ) z [ σ ¯ ( r 12 ) + σ ¯ ( r 13 ) + σ ¯ ( r 24 ) + σ ¯ ( r 34 ) - σ ¯ ( r 14 ) - σ ¯ ( r 23 ) - 2 σ ¯ ( 0 ) ] + i π k ¯ 4 z 2 0 σ ˜ ¯ ( η ) η 3 [ J 0 ( η r 14 ) - J 0 ( η r 23 ) ] d η + 1 12 π z 3 0 σ ˜ ¯ ( η ) η 5 [ J 0 ( η r 14 ) + J 0 ( η r 23 ) ] d η } .
{ L ˆ 1 } z L ˆ 0 { 1 + ( k ¯ 2 / 4 ) z [ σ ¯ ( r 12 ) + σ ¯ ( r 13 ) + σ ¯ ( r 24 ) + σ ¯ ( r 34 ) - σ ¯ ( r 14 ) - σ ¯ ( r 23 ) - 2 σ ¯ ( 0 ) ] } .
{ L ˆ 1 } z = { I ˆ ( x 1 , ν ¯ ) I ˆ ( x 2 , ν ¯ ) } ,
{ L ˆ 1 } z L ˆ 0 { 1 + 1 6 π z 3 0 σ ˜ ¯ ( η ) η 5 [ J 0 ( η r 14 ) ] d η } .
{ L ˆ 0 ( x 1 , x 2 , x 3 , x 4 , ν ¯ ) } = { L ˜ 0 ( k T 1 , k T 2 , k T 3 , k T 4 , ν ¯ ) } × exp [ i k T 1 · x T 1 - i k T 2 · x T 2 - i k T 3 · x T 3 + i k T 4 · x T 4 ] × exp [ i k z 1 z 1 - i k z 2 z 2 - i k z 3 z 3 + i k z 4 z 4 ] d k T 1 d k T 4 .
{ L ˆ 0 } z = 0 = { L ˜ 0 } exp [ i k T 1 · x T 1 - + i k T 4 · x T 4 ] × d k T 1 d k T 4 .
{ L ˆ 0 } z = { L ˜ 0 } exp [ i k T 1 · x T 1 - + i k T 4 · x T 4 ] × exp [ i ( k z 1 - k z 2 - k z 3 + k z 4 ) z ] d k T 1 d k T 4 .
k z i = ( k ¯ 2 - k x i 2 - k y i 2 ) 1 2
k z i k ¯ ( k x i 2 + k y i 2 ) / 2 k ˜ + .
{ L ˆ 0 } z { L ˜ 0 } exp [ i k T 1 · x T 1 - + i k T 4 · x T 4 ] × exp [ - i 2 k ¯ ( k x 1 2 + k y 2 2 - + k x 4 2 + k y 4 2 ) z ] × d k T 1 d k T 4 .
{ L ˆ 0 } z = { L ˜ 0 } exp [ i k T 1 · x T 1 - + i k T 4 · x T 4 ] × [ 1 - i 2 k ˜ ( k x 1 2 + k y 1 2 - + k x 4 2 + k y 4 2 ) z ] × d k T 1 d k T 4 .
( 2 / x 1 2 ) exp [ i k T 1 · x T 1 - + i k T 4 · x T 4 ] = - k x 1 2 exp [ i k T 1 · x T 1 - + i k T 4 · x T 4 ] ,
{ L ˆ 0 } z = [ 1 + ( i z / 2 k ¯ ) ( x T 1 2 - x T 2 2 - x T 3 2 + x T 4 2 ) ] { L ˆ 0 } 0 .
θ c 2 k ¯ z 1.
{ L ˆ 1 } z = [ 1 + ( k ¯ 2 / 4 ) z { σ ¯ ( r 12 ) + σ ¯ ( r 13 ) + σ ¯ ( r 24 ) + σ ¯ ( r 34 ) - σ ¯ ( r 14 ) - σ ¯ ( r 23 ) - 2 σ ¯ ( 0 ) } + ( i z / 2 k ¯ ) ( x T 1 2 - x T 2 2 - x T 3 2 + x T 4 2 ) ] { L ˆ 0 } 0 .
{ L ˆ 1 } z 0 + Δ z = [ 1 + ( k ¯ 2 / 4 ) Δ z { σ ¯ ( r 12 ) + σ ¯ ( r 13 ) + σ ¯ ( r 24 ) + σ ¯ ( r 34 ) - σ ¯ ( r 14 ) - σ ¯ ( r 23 ) - 2 σ ¯ ( 0 ) } + ( i Δ z / 2 k ¯ ) ( x T 1 2 - x T 2 2 - x T 3 2 + x T 4 2 ) ] { L ˆ 1 } z 0 .
| Δ z ( k 2 4 σ ¯ ( r i j ) + i 2 k ¯ x T i 2 ) { L ˆ 1 } z 0 | { L ˆ 1 } z 0
{ L ˆ 1 } z 0 + Δ z - { L ˆ 1 } z 0 Δ z d { L ˆ 1 } z d z ,
{ L ˆ 1 } z = { k ¯ 2 4 [ σ ¯ ( r 12 ) + σ ¯ ( r 13 ) + σ ¯ ( r 14 ) + σ ¯ ( r 34 ) - σ ¯ ( r 14 ) - σ ¯ ( r 23 ) - 2 σ ¯ ( 0 ) ] + ( i + 2 k ¯ ) ( x T 1 2 - x T 2 2 - x T 3 2 + x T 4 2 ) } { L ˆ 1 } .
{ L ˆ 1 } z = { k ¯ 2 4 [ σ ¯ ( r 12 ) + σ ¯ ( r 13 ) + σ ¯ ( r 24 ) + σ ¯ ( r 34 ) - σ ¯ ( r 14 ) - σ ¯ ( r 23 ) - 2 σ ¯ ( 0 ) ] + i k [ 2 x 12 x 13 + 2 y 12 y 13 + 2 x 12 x 14 + 2 y 12 y 14 + 2 x 13 x 14 + 2 y 13 y 14 + r 14 2 ] } { L ˆ 1 } .
( 2 x 12 x 13 + 2 y 12 y 13 ) f ( r 23 ) = - r 23 2 f ( r 23 ) , [ r 14 2 + ( 2 x 12 x 14 + 2 y 12 y 14 ) ] f ( r 24 ) = 0 , [ r 14 2 + ( 2 x 13 x 14 + 2 y 13 y 14 ) ] f ( r 34 ) = 0.
F ( r i j ) k ¯ 2 4 [ σ ¯ ( r 12 ) + σ ¯ ( r 13 ) + σ ¯ ( r 24 ) + σ ¯ ( r 34 ) - σ ¯ ( r 14 ) - σ ¯ ( r 23 ) - 2 σ ¯ ( 0 ) ] , A ( r i j ) i k ¯ [ 2 x 12 x 13 + 2 y 12 y 13 + 2 x 12 x 14 + 2 y 12 y 14 + 2 x 13 x 14 + 2 y 13 y 14 + r 14 2 ] .
( L ˆ 1 ) / z = F ( r i j ) { L ˆ 1 } + A ( r i j ) { L ˆ 1 } .
{ L ˆ 1 } z = { L ˆ 1 } 0 exp [ F ( r i j ) z ] + exp [ F ( r i j ) z ] × 0 z exp [ - F ( r i j ) z ] A ( r i j ) { L ˆ 1 } z d z ,
{ L ˆ 1 } z A { L ˆ 1 } 0 exp [ F ( r i j ) z ] .
K ¯ ( z ) = 0 z exp [ - F ( r i j ) z ] A ( r i j ) exp [ F ( r i j ) z ] d z .
i k ¯ 2 x 1 i x 1 j exp [ F ( r l m ) z ] = i k ¯ 2 F ( r l m ) x 1 i x 1 j z exp [ F ( r l m ) z ] + i k ¯ [ F ( r l m ) x 1 i ] [ F ( r l m ) x 1 j ] z 2 exp [ F ( r l m ) z ] .
A = ( i / k ) ( 2 F ( r l m ) / x 1 i x 1 j ) ( z 2 / 2 )
B = i k [ F ( r l m ) x 1 i ] [ F ( r l m ) x 1 j ] z 3 3 .
0 ( A ) = [ k ¯ 2 σ ¯ ( 0 ) z ] ( z / k ¯ l m 2 ) , 0 ( B ) = [ k ¯ 2 σ ¯ ( 0 ) z ] 2 ( z / k ¯ l m 2 ) .
F ( r i j ) all r i j = 0 = 0.
F ( r i j ) z [ k ¯ 2 σ ¯ ( 0 ) z ] 2 ( z / k ¯ l m 2 ) .
σ ¯ ( r F ) - σ ¯ ( 0 ) σ ¯ ( 0 ) r F 2 / l m 2 .
[ k ¯ 2 σ ¯ ( 0 ) z ] ( r F 2 / l m 2 ) [ k ¯ 2 σ ¯ ( 0 ) z ] 2 ( z / k ¯ l m 2 ) ,
z k ¯ r F 2 / [ k ¯ 2 σ ¯ ( 0 ) z ] .