Abstract

An expression is derived for the mutual coherence factor for a plane electromagnetic wave propagating in a region of dielectric turbulence. It is assumed that the medium is locally homogeneous and isotropic and that the scale of turbulence l is much larger than the wavelength λ. The formula is valid within ranges L determined by assuming small scattering angles (Ll43) and [(2π/λ)2〈(Δn)2lL]2≪1 where 〈(Δn)2〉 is the mean-square fluctuation of the refractive index. It is shown that this same expression can be derived within the limits of geometric optics for small separation distances.

© 1967 Optical Society of America

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References

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  1. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [Crossref]
  2. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961).
  3. A. W. Wheelon, J. Res. Natl. Bur. Std. 63D, 205 (1959).
  4. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Company, New York, 1960).
  5. J. B. Keller, in Proc. Symposia in Applied Mathematics, XVI. (American Mathematical Society, Providence, Rhode Island, 1964), p. 145.
    [Crossref]
  6. D. L. Fried and J. D. Cloud, in Proc. Conf. on Atmospheric Limitations to Optical Propagation, Boulder, Col. (1965).
  7. M. J. Beran, J. Opt. Soc. Am. 56, 1475 (1966).
    [Crossref]
  8. D. M. Chase, J. Opt. Soc. Am. 55, 1559 (1965).
    [Crossref]
  9. L. S. Taylor, J. Math. Phys. 4, 824 (1963).
    [Crossref]
  10. W. P. Brown, J. Opt. Soc. Am. 56, 1045 (1966).
    [Crossref]
  11. L. S. Taylor, J. Res. Natl. Bur. Std. Radio Sci./USNC (April, 1967).
  12. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

1967 (1)

L. S. Taylor, J. Res. Natl. Bur. Std. Radio Sci./USNC (April, 1967).

1966 (2)

1965 (1)

1964 (1)

1963 (1)

L. S. Taylor, J. Math. Phys. 4, 824 (1963).
[Crossref]

1959 (1)

A. W. Wheelon, J. Res. Natl. Bur. Std. 63D, 205 (1959).

Beran, M. J.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

Brown, W. P.

Chase, D. M.

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Company, New York, 1960).

Cloud, J. D.

D. L. Fried and J. D. Cloud, in Proc. Conf. on Atmospheric Limitations to Optical Propagation, Boulder, Col. (1965).

Fried, D. L.

D. L. Fried and J. D. Cloud, in Proc. Conf. on Atmospheric Limitations to Optical Propagation, Boulder, Col. (1965).

Hufnagel, R. E.

Keller, J. B.

J. B. Keller, in Proc. Symposia in Applied Mathematics, XVI. (American Mathematical Society, Providence, Rhode Island, 1964), p. 145.
[Crossref]

Stanley, N. R.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961).

Taylor, L. S.

L. S. Taylor, J. Res. Natl. Bur. Std. Radio Sci./USNC (April, 1967).

L. S. Taylor, J. Math. Phys. 4, 824 (1963).
[Crossref]

Wheelon, A. W.

A. W. Wheelon, J. Res. Natl. Bur. Std. 63D, 205 (1959).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

J. Math. Phys. (1)

L. S. Taylor, J. Math. Phys. 4, 824 (1963).
[Crossref]

J. Opt. Soc. Am. (4)

J. Res. Natl. Bur. Std. (1)

A. W. Wheelon, J. Res. Natl. Bur. Std. 63D, 205 (1959).

J. Res. Natl. Bur. Std. Radio Sci./USNC (1)

L. S. Taylor, J. Res. Natl. Bur. Std. Radio Sci./USNC (April, 1967).

Other (5)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Company, New York, 1960).

J. B. Keller, in Proc. Symposia in Applied Mathematics, XVI. (American Mathematical Society, Providence, Rhode Island, 1964), p. 145.
[Crossref]

D. L. Fried and J. D. Cloud, in Proc. Conf. on Atmospheric Limitations to Optical Propagation, Boulder, Col. (1965).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961).

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Equations (32)

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2 E - c 2 2 E t 2 = - [ 1 E · ] .
2 u - ( n 2 / c 2 ) ( 2 u / t 2 ) = 0.
v c ( r ) = v ( r ) v i ( r ) = v ( r ) - v c ( r ) .
M ( r 1 , r 2 ) = u ( r 1 , t ) u * ( r 2 , t ) = v c ( r 1 ) v c * ( r 2 ) + v i ( r 1 ) v i * ( r 2 ) .
k = k r + j k i .
v c ( p 1 , L ) v c * ( p 2 , L ) = v c ( p 1 , 0 ) v c * ( p 2 , 0 ) exp j ( k · r 1 - k * · r 2 ) = v c ( p 1 , 0 ) v c * ( p 2 , 0 ) exp ( - 2 k i L ) = exp ( - 2 k i L ) ,
M ( p 1 - p 2 , L ) = exp ( - 2 k i L ) + ( v i ( p 1 , L ) v i * ( p 2 , L ) .
( 2 + k 0 2 n 2 ) v i = k 0 2 n 2 v i + k 0 2 [ n 2 - n 2 ] v c ,
( 2 + k 0 2 ) v i = 2 k 0 2 [ μ v i - μ v i + ( μ - μ ) v c ] + 0 ( 2 ) .
( 2 + k 0 2 ) v i = 2 k 0 2 ( μ - μ ) v c + 0 ( 2 ) .
v i ( r ) = 2 k 0 2 G ( r , r ) [ μ - μ ( r ) ] v c ( r ) d r + 0 ( 2 ) ,
v i ( r 1 ) v i * ( r 2 ) = 4 2 k 0 4 G ( r 1 , r ) G * ( r 2 , r ) [ μ ( r ) μ ( r ) - μ 2 ] × v c ( r ) v c * ( r ) d r d r + 0 ( 3 ) .
C ( r - r ) = μ ( r ) μ ( r ) / μ 2 ,
M ( p 1 - p 2 , L ) = exp ( - 2 k i L ) + 4 k 0 4 2 μ 2 G ( r 1 , r ) G * ( r 2 , r ) C ( r - r ) exp j ( k z - k * z ) d r d r .
M ( p , L ) = exp ( - 2 k i L ) + k 0 2 μ 2 4 π 2 exp j ( k 0 r 1 - r - k 0 r 2 - r + k z - k * z ) r 1 - r · r 2 - r C ( r - r ) d r d r .
r 1 - r = L - z + 1 2 [ ( x - p ) 2 + y 2 ] / ( L - z ) r 2 - r = L - z + 1 2 ( x 2 + y 2 ) / ( L - z ) .
M ( p , L ) = exp ( - 2 k i L ) + k 0 4 2 μ 2 4 π 2 d r d r C ( r ) ( L - z ) ( L - z ) × exp j { k 0 ( z - z ) + k z - k * z + 1 2 k 0 [ ( x - p ) 2 + y 2 L - z - x 2 + y 2 L - z ] } .
x = x - x , y = y - y , z = z - z , X = 1 2 ( x + x ) , Y = 1 2 ( y + y ) , Z = 1 2 ( z + z ) ,
M ( p , L ) = exp ( - 2 k i L ) + k 0 4 2 μ 2 4 π 2 - L L 0 L - - exp [ j ( k r - k 0 ) z - 2 k i Z ] C ( r ) ( L - Z - 1 2 z ) ( L - Z + 1 2 z ) d z d Z d x d y × - - exp 1 2 j k 0 [ ( X + x / 2 - p ) 2 + ( Y + y / 2 ) 2 L - Z - 1 2 z - ( X - x / 2 ) 2 + ( Y - y / 2 ) 2 L - Z + 1 2 z ] d X d Y .
M ( p , L ) = exp ( - 2 k i L ) + j k 0 3 2 μ 2 2 π - L L 0 L z - 1 exp [ j ( k τ - k 0 ) z - 2 k i Z ] d z d Z × - - exp [ - j k 0 ( x - p ) 2 + y 2 2 z ] C ( r ) d x d y .
M ( p , L ) = exp ( - 2 k i L ) + ( k 0 2 / k i ) 2 μ 2 [ 1 - exp ( - 2 k i L ) ] 0 L C ( p , 0 , z ) exp [ - j ( k r - k 0 ) z ] d z .
M ( p , L ) = 1 - 2 k i L + 2 k 0 2 2 μ 2 L × 0 L C ( p , 0 , z ) exp [ - j ( k r - k 0 ) ] d z .
k 2 = k 0 2 ( 1 + 2 μ 2 ) + ( 4 2 μ 2 k 0 4 / k ) × 0 exp ( j k 0 r ) sin k r C ( r ) d r .
k 2 = k 0 2 ( 1 + 2 μ 2 ) - ( 2 j 2 μ 2 k 0 4 / k ) × 0 { exp ( 2 j k 0 r ) · exp - [ k i - j ( k r - k 0 ) ] r - exp [ k i - j ( k r - k 0 ) ] r } C ( r ) d r .
k i = k 0 2 2 μ 2 0 C ( r ) d r .
M ( p , L ) = 1 - 2 k 0 2 2 μ 2 L × 0 [ C ( 0 , 0 , z ) - C ( p , 0 , z ) ] d z .
2 A + 2 j k 0 ( A / z ) + 2 k 0 2 μ A = 0.
A ( x , y , L ) = exp [ i k 0 0 L μ ( x , y , z ) d z ]
M ( p , L ) = exp { i k 0 0 L [ μ ( p , 0 , z ) - μ ( 0 , 0 , z ) ] d z } .
M ( p , L ) = exp { - 1 2 k 0 2 2 0 L d z 1 0 L d z 2 [ μ ( p , 0 , z 1 ) - μ ( 0 , 0 , z 1 ) ] [ μ ( p , 0 , z 2 ) - μ ( 0 , 0 , z 2 ) ] } = exp { k 0 2 2 μ 2 0 L d z 1 0 L d z 2 [ C ( 0 , 0 , z 1 - z 2 ) - C ( p , 0 , z 1 - z 2 ) ] } .
M ( p , L ) = exp { - 2 k 0 2 2 μ 2 L 0 [ C ( 0 , 0 , z ) - C ( p , 0 , z ) ] d z } .
γ 2 = [ 2 k 0 2 2 μ 2 L 0 [ C ( 0 , 0 , z ) - C ( p , 0 , z ) ] d z ] 2 1.