Abstract

Study of autocorrelations of the fluctuations in a gaussian beam propagating through a random medium of gaussian inhomogeneity is extended in this paper to considerations of the spatially transverse autocorrelations of the log-amplitude or phase fluctuations at two off-axis positions over the cross section of the collimated and focused beams. Two-dimensional autocorrelation distributions with respect to one fixed off-axis point are obtained over the beam cross section at and near the focal position of a focused beam. The cross correlations of log-amplitude and phase fluctuations for the collimated and focused gaussian beams propagating through a random medium have also been evaluated as functions of the wave parameter, the normalized focal length, and the source-aperture size. A comparison is made among the cross-correlation coefficients for plane, spherical, and gaussian beams.

© 1969 Optical Society of America

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References

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  1. L. A. Chernov, Wave Propagation in a Random Medium, transl. by R. A. Silverman (McGraw–Hill Book Co., New York, 1960).
  2. V. I. Tatarski, Wave Propagation in a Turbulent Medium, transl. by R. A. Silverman (McGraw–Hill Book Co., New York, 1961).
  3. R. A. Schmeltzer, Quart. Appl. Math. 46, 339 (1966).
  4. D. L. Fried and J. B. Seidman, J. Opt. Soc. Am. 57, 181 (1967).
    [CrossRef]
  5. D. L. Fried, J. Opt. Soc. Am. 57, 980 (1967).
    [CrossRef]
  6. Y. Kinoshita, M. Suzuki, and T. Matsumoto, Radio Science 3, 287 (1968).
  7. Y. Kinoshita, T. Asakura, and M. Suzuki, J. Opt. Soc. Am. 58, 798 (1968).
    [CrossRef]
  8. T. Asakura and Y. Kinoshita, Japan. J. Appl. Phys. 8, 260 (1969).
    [CrossRef]
  9. Y. Kinoshita, T. Asakura, and M. Suzuki, J. Opt. Soc. Am. 58, 1040 (1968).
    [CrossRef]
  10. G. C. Knollman, J. Acoust. Soc. Am. 36, 2306 (1964).
    [CrossRef]
  11. I. Goldstein, P. A. Miles, and A. Chabot, Proc. IEEE 53, 1172 (1965).
    [CrossRef]
  12. D. M. Chase, J. Opt. Soc. Am. 56, 33 (1966).
    [CrossRef]
  13. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  14. W. P. Brown, J. Opt. Soc. Am. 56, 1045 (1966); J. Opt. Soc. Am. 57, 1539 (1967).
    [CrossRef]
  15. L. S. Taylor, Radio Science 2, 437 (1967).
  16. D. L. Fried, J. Opt. Soc. Am. 57, 268 (1967).
    [CrossRef]
  17. D. A. deWolf, J. Opt. Soc. Am. 55, 812 (1965); J. Opt. Soc. Am. 57, 1057 (1967); Radio Science 2, 1379 (1967).
    [CrossRef]
  18. V. I. Tatarski, Soviet Phys.—JETP 22, 1083 (1967).

1969 (1)

T. Asakura and Y. Kinoshita, Japan. J. Appl. Phys. 8, 260 (1969).
[CrossRef]

1968 (3)

1967 (6)

D. L. Fried, J. Opt. Soc. Am. 57, 980 (1967).
[CrossRef]

D. L. Fried and J. B. Seidman, J. Opt. Soc. Am. 57, 181 (1967).
[CrossRef]

D. L. Fried, J. Opt. Soc. Am. 57, 268 (1967).
[CrossRef]

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

L. S. Taylor, Radio Science 2, 437 (1967).

V. I. Tatarski, Soviet Phys.—JETP 22, 1083 (1967).

1966 (3)

1965 (2)

1964 (1)

G. C. Knollman, J. Acoust. Soc. Am. 36, 2306 (1964).
[CrossRef]

Asakura, T.

Brown, W. P.

Chabot, A.

I. Goldstein, P. A. Miles, and A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

Chase, D. M.

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium, transl. by R. A. Silverman (McGraw–Hill Book Co., New York, 1960).

deWolf, D. A.

Fried, D. L.

Goldstein, I.

I. Goldstein, P. A. Miles, and A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

Kinoshita, Y.

T. Asakura and Y. Kinoshita, Japan. J. Appl. Phys. 8, 260 (1969).
[CrossRef]

Y. Kinoshita, M. Suzuki, and T. Matsumoto, Radio Science 3, 287 (1968).

Y. Kinoshita, T. Asakura, and M. Suzuki, J. Opt. Soc. Am. 58, 798 (1968).
[CrossRef]

Y. Kinoshita, T. Asakura, and M. Suzuki, J. Opt. Soc. Am. 58, 1040 (1968).
[CrossRef]

Knollman, G. C.

G. C. Knollman, J. Acoust. Soc. Am. 36, 2306 (1964).
[CrossRef]

Matsumoto, T.

Y. Kinoshita, M. Suzuki, and T. Matsumoto, Radio Science 3, 287 (1968).

Miles, P. A.

I. Goldstein, P. A. Miles, and A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

Schmeltzer, R. A.

R. A. Schmeltzer, Quart. Appl. Math. 46, 339 (1966).

Seidman, J. B.

Suzuki, M.

Tatarski, V. I.

V. I. Tatarski, Soviet Phys.—JETP 22, 1083 (1967).

V. I. Tatarski, Wave Propagation in a Turbulent Medium, transl. by R. A. Silverman (McGraw–Hill Book Co., New York, 1961).

Taylor, L. S.

L. S. Taylor, Radio Science 2, 437 (1967).

J. Acoust. Soc. Am. (1)

G. C. Knollman, J. Acoust. Soc. Am. 36, 2306 (1964).
[CrossRef]

J. Opt. Soc. Am. (8)

Japan. J. Appl. Phys. (1)

T. Asakura and Y. Kinoshita, Japan. J. Appl. Phys. 8, 260 (1969).
[CrossRef]

Proc. IEEE (2)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

I. Goldstein, P. A. Miles, and A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

Quart. Appl. Math. (1)

R. A. Schmeltzer, Quart. Appl. Math. 46, 339 (1966).

Radio Science (2)

Y. Kinoshita, M. Suzuki, and T. Matsumoto, Radio Science 3, 287 (1968).

L. S. Taylor, Radio Science 2, 437 (1967).

Soviet Phys.—JETP (1)

V. I. Tatarski, Soviet Phys.—JETP 22, 1083 (1967).

Other (2)

L. A. Chernov, Wave Propagation in a Random Medium, transl. by R. A. Silverman (McGraw–Hill Book Co., New York, 1960).

V. I. Tatarski, Wave Propagation in a Turbulent Medium, transl. by R. A. Silverman (McGraw–Hill Book Co., New York, 1961).

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Figures (15)

Fig. 1
Fig. 1

Spatially transverse-autocorrelation coefficients Ca(ϱ, −ϱ) = Ca(2ρ) (indicated by solid curves) and Cp(ϱ, −ϱ) = Cp(2ρ) (by dashed curves) of log amplitude and phase fluctuations vs wave parameter Z ¯ for a collimated gaussian beam (l → ∞) whose source-aperture size is α/a = 1.0. The wave parameter Z ¯ is the normalized propagation distance by ka2/4, the length of the Fresnel region for the correlation length a of the random medium.

Fig. 2
Fig. 2

Transverse-autocorrelation coefficients Ca(2ρ) (solid curves) and Cp(2ρ) (dashed curves) of log amplitude and phase fluctuations at the two off-axis positions for a collimated beam having the source-aperture size α/a = 0.25.

Fig. 3
Fig. 3

Transverse-autocorrelation coefficients Ca(ϱ, −ϱ) = Ca(2ρ) of log-amplitude fluctuations at the two off-axis positions vs wave parameter Z ¯ for a focused gaussian beam of the source-aperture size α/a = 0.25 with the radius of wavefront curvature 4l/ka2 = 0.5.

Fig. 4
Fig. 4

Transverse autocorrelation coefficients Ca(ϱ, −ϱ) = Ca(2ρ) of log-amplitude fluctuations at the two off-axis positions vs wave parameter Z ¯ for a focused beam, where the source-aperture size is α/a = 1.0 and the radius of wavefront curvature is 4l/ka2 = 0.5.

Fig. 5
Fig. 5

Transverse autocorrelation coefficients Cp(ϱ, −ϱ) = Cp(2ρ) of phase fluctuations at the two off-axis positions vs wave parameter Z ¯ for a focused beam of the source-aperture size α/a = 0.25 with the radius of wavefront curvature 4l/ka2 = 0.5.

Fig. 6
Fig. 6

Transverse autocorrelation coefficients Cp(ϱ, −ϱ) = Cp(2ρ) of phase fluctuations at the two off-axis positions vs wave parameter Z ¯ for a focused beam where the source-aperture size is α/a = 1.0 and the radius of wavefront curvature is 4l/ka2 = 0.5.

Fig. 7
Fig. 7

Two-dimensional autocorrelation distribution Ca(ϱ1,ϱ2) of log-amplitude fluctuations over the cross section of a focused gaussian beam at the focal position Z ¯ = 0.5. The autocorrelations are taken between the fixed point ϱ1(0,α|B|) and the other varying points ϱ2 over the beam where the source-aperture size is α/a = 1.0 and the radius of wavefront curvature is 4l/ka2 = 0.5. In the figure, the contour lines of equal autocorrelation in log-amplitude fluctuations are shown.

Fig. 8
Fig. 8

Two-dimensional autocorrelation distribution Ca(ϱ1,ϱ2) of log-amplitude fluctuations over the cross section of the same focused beam of Fig. 7 just before the focus Z ¯ = 0.47.

Fig. 9
Fig. 9

Two-dimensional autocorrelation distribution Ca(ϱ1,ϱ2) of log-amplitude fluctuations over the cross section of the focused beam of Fig. 7 just behind the focus Z ¯ = 0.52.

Fig. 10
Fig. 10

Two-dimensional autocorrelation distribution Cp(ϱ1,ϱ2) of phase fluctuations over the cross section of the focused gaussian beam of Fig. 7 at the focal position Z ¯ = 0.5. The autocorrelations are taken between the fixed point ϱ1(0,α|B|) and the other varying points ϱ2 over the beam. The contour lines of equal autocorrelation in phase fluctuations are shown in the figure.

Fig. 11
Fig. 11

Two-dimensional autocorrelation distribution Cp(ϱ1,ϱ2) of phase fluctuations over the cross section of the focused beam of Fig. 7 just before the focus Z ¯ = 0.47.

Fig. 12
Fig. 12

Two-dimensional autocorrelation distribution Cp(ϱ1,ϱ2) of phase fluctuations over the cross section of the focused beam of Fig. 7 just behind the focus Z ¯ = 0.52.

Fig. 13
Fig. 13

Cross-correlation coefficients Dap(z) between log amplitude and phase fluctuations vs wave parameter Z ¯ for a collimated gaussian beam (l → ∞) as a function of the source-aperture size α/a. The cross-correlation coefficients for the plane (α → ∞) and spherical (α → 0) waves are added for comparison purposes.

Fig. 14
Fig. 14

Cross-correlation coefficients Dap(z) between log amplitude and phase fluctuations vs wave parameter Z ¯ for a focused gaussian beam 4l/ka2 = 0.5 as a function of the source-aperture size α/a.

Fig. 15
Fig. 15

Cross-correlation coefficients Dap(z) between log amplitude and phase fluctuations vs wave parameter Z ¯ for a focused gaussian beam 4l/ka2 = 2.0 as a function of the source-aperture size α/a.

Equations (10)

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δ n ( r 1 ) δ n ( r 2 ) = δ n 2 exp ( - | r 1 - r 2 a | 2 ) ,
ϕ 0 ( ϱ ) = 1 α exp { - ρ 2 α 2 + j k ρ 2 l }
χ ( r ) = Re δ S ( r ) = Im } k 2 2 π 0 z d z 1 - + d ϱ 1 δ n ( r 1 ) U 01 × exp { - j k U 01 ρ 1 2 2 + j k ϱ · ϱ 1 z - z 1 + j k U 10 ρ 2 2 } ,
U 01 = U ( z , z 1 ) = B ( z ) / ( z - z 1 ) B ( z 1 ) ,             U 10 = U ( z 1 , z ) .
χ ( z + ϱ 1 ) χ ( z + ϱ 2 ) δ S ( z + ϱ 1 ) δ S ( z + ϱ 2 ) } = π ( k a 2 ) 3 δ n 2 Re 0 Z ¯ d Z ¯ + ( I 1 ± I 2 ) ,
I 1 = { 1 + ( Z ¯ - Z ¯ + ) Im ( B + B ) } - 1 × exp { - [ ( B + B ) 2 ϱ 1 - ϱ 2 2 a 2 + j 2 Im ( B + B ) { ( B + B ) ρ 1 2 a 2 - ( B + B ) * ρ 2 2 a 2 } ] / [ 1 + ( Z ¯ - Z ¯ + ) Im ( B + / B ) ] } ; I 2 = { 1 - j ( Z ¯ - Z ¯ + ) ( B + B ) } - 1 exp { ( B + / B ) 2 [ ϱ 1 - ϱ 2 2 / a 2 ] 1 - j ( Z ¯ - Z ¯ + ) ( B + / B ) } ; B + = B ( z + ) ,             Z ¯ = 4 z / k a 2 ,             Z ¯ + = 4 z + / k a 2 .
χ ( r ) δ S ( r ) = π ( k a 2 ) 3 δ n 2 Re 1 W ln 2 - j Z ¯ ( 1 - W ) 2 - j Z ¯ ( 1 + W ) ,
W = { 1 + 1 2 ( 1 α 2 - j k l ) a 2 / B ( z ) } 1 2 ,             Z ¯ = 4 z k a 2 .
C a ( ϱ 1 , ϱ 2 ) = χ ( z + ϱ 1 ) χ ( z + ϱ 2 ) / { χ 2 ( z + ϱ 1 ) χ 2 ( z + ϱ 2 ) } 1 2 C p ( ϱ 1 , ϱ 2 ) = δ S ( z + ϱ 1 ) δ S ( z + ϱ 2 ) / { δ S 2 ( z + ϱ 1 ) δ S 2 ( z + ϱ 2 ) } 1 2 ,
D a p ( r ) = χ ( r ) δ S ( r ) / { χ 2 ( r ) δ S 2 ( r ) } 1 2