Abstract

Recent studies have used the superposition principle (extended Huygens-Fresnel principle) to characterize completely the statistics of a field that has propagated through a thick slab of turbulent air in terms of the statistics for spherical-wave sources. In this paper, we consider the normal-mode decomposition associated with this linear system propagation model. In particular, we use the statistics of the atmospheric impulse response (Green’s function) to show that the atmospheric mode decomposition exhibits far-field and near-field regimes very similar to those of free-space propagation. The significance of these results for optical communication through the atmosphere is briefly discussed.

© 1974 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  2. R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
    [CrossRef]
  3. E. Brookner, IEEE Trans. COM-18, 396 (1970).
    [CrossRef]
  4. J. W. Strohbehn, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1971), Vol 9.
    [CrossRef]
  5. R. F. Lutomirski, H. T. Yura, Appl. Opt. 10, 1652 (1971).
    [CrossRef] [PubMed]
  6. H. T. Yura, Appl. Opt. 11, 1399 (1972).
    [CrossRef] [PubMed]
  7. H. T. Yura, J. Opt. Soc. Am. 62, 889 (1972).
    [CrossRef]
  8. H. S. Lin, “Communication Model for the Turbulent Atmosphere,” Ph.D. thesis, Case Western Reserve U., 1973.
  9. D. Slepian, J. Opt. Soc. Am. 55, 1110 (1965).
    [CrossRef]
  10. G. V. Borgiotti, IEEE Trans. Antennas Prop. AP-14, 158 (1966).
    [CrossRef]
  11. C. K. Rushforth, R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).
    [CrossRef]
  12. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
    [CrossRef] [PubMed]
  13. J. H. Shapiro, IEEE Trans. COM-19, 410 (1971).
    [CrossRef]
  14. J. H. Shapiro, J. Opt. Soc. Am. 61, 492 (1971).
    [CrossRef]
  15. J. H. Shapiro, Appl. Opt. 13, 2709 (1974).
  16. D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).
  17. J. B. Thomas, An Introduction to Applied Probability and Random Processes (Wiley, New York, 1971), p. 107.
  18. In the ensuing discussion, we shall assume that the aperture diameters are not equal and that the smaller diameter is less than the phase-coherence length.
  19. E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
    [CrossRef]
  20. M. C. Teich, S. Rosenberg, Appl. Opt. 12, 2616 (1973).
    [CrossRef] [PubMed]
  21. S. Rosenberg, M. C. Teich, Appl. Opt. 12, 2625 (1973).
    [CrossRef] [PubMed]
  22. D. L. Fried, H. T. Yura, J. Opt. Soc. Am. 62, 600 (1972).
    [CrossRef]
  23. G. Q. McDowell, “Pre-Distortion of Local Oscillator Wavefront for Improved Optical Heterodyne Detection Through a Turbulent Atmosphere,” Sc.D. thesis, MIT, 1971.

1974 (1)

J. H. Shapiro, Appl. Opt. 13, 2709 (1974).

1973 (2)

1972 (3)

1971 (3)

1970 (3)

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[CrossRef]

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

E. Brookner, IEEE Trans. COM-18, 396 (1970).
[CrossRef]

1969 (1)

1968 (1)

1966 (1)

G. V. Borgiotti, IEEE Trans. Antennas Prop. AP-14, 158 (1966).
[CrossRef]

1965 (1)

1964 (1)

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).

Borgiotti, G. V.

G. V. Borgiotti, IEEE Trans. Antennas Prop. AP-14, 158 (1966).
[CrossRef]

Brookner, E.

E. Brookner, IEEE Trans. COM-18, 396 (1970).
[CrossRef]

Fried, D. L.

Halme, S. J.

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[CrossRef]

Harger, R. O.

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[CrossRef]

Harris, R. W.

Hoversten, E. V.

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[CrossRef]

Lawrence, R. S.

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

Lin, H. S.

H. S. Lin, “Communication Model for the Turbulent Atmosphere,” Ph.D. thesis, Case Western Reserve U., 1973.

Lutomirski, R. F.

McDowell, G. Q.

G. Q. McDowell, “Pre-Distortion of Local Oscillator Wavefront for Improved Optical Heterodyne Detection Through a Turbulent Atmosphere,” Sc.D. thesis, MIT, 1971.

Rosenberg, S.

Rushforth, C. K.

Shapiro, J. H.

J. H. Shapiro, Appl. Opt. 13, 2709 (1974).

J. H. Shapiro, IEEE Trans. COM-19, 410 (1971).
[CrossRef]

J. H. Shapiro, J. Opt. Soc. Am. 61, 492 (1971).
[CrossRef]

Slepian, D.

D. Slepian, J. Opt. Soc. Am. 55, 1110 (1965).
[CrossRef]

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).

Strohbehn, J. W.

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

J. W. Strohbehn, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1971), Vol 9.
[CrossRef]

Tatarski, V. I.

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

Teich, M. C.

Thomas, J. B.

J. B. Thomas, An Introduction to Applied Probability and Random Processes (Wiley, New York, 1971), p. 107.

Toraldo di Francia, G.

Yura, H. T.

Appl. Opt. (5)

Bell Syst. Tech. J. (1)

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).

IEEE Trans. (2)

J. H. Shapiro, IEEE Trans. COM-19, 410 (1971).
[CrossRef]

E. Brookner, IEEE Trans. COM-18, 396 (1970).
[CrossRef]

IEEE Trans. Antennas Prop. (1)

G. V. Borgiotti, IEEE Trans. Antennas Prop. AP-14, 158 (1966).
[CrossRef]

J. Opt. Soc. Am. (6)

Proc. IEEE (2)

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[CrossRef]

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

Other (6)

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

J. W. Strohbehn, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1971), Vol 9.
[CrossRef]

H. S. Lin, “Communication Model for the Turbulent Atmosphere,” Ph.D. thesis, Case Western Reserve U., 1973.

J. B. Thomas, An Introduction to Applied Probability and Random Processes (Wiley, New York, 1971), p. 107.

In the ensuing discussion, we shall assume that the aperture diameters are not equal and that the smaller diameter is less than the phase-coherence length.

G. Q. McDowell, “Pre-Distortion of Local Oscillator Wavefront for Improved Optical Heterodyne Detection Through a Turbulent Atmosphere,” Sc.D. thesis, MIT, 1971.

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Figures (6)

Fig. 1
Fig. 1

Propagation geometry.

Fig. 2
Fig. 2

η ˆ vs Dfo for free-space propagation (solid line) and worst case atmospheric propagation (broken line).

Fig. 3
Fig. 3

η ˆ vs Dfo for two cases of atmospheric propagation. The solid line corresponds to C = 10.2, the broken line corresponds to C = 10.2 Dfo5/12.

Fig. 4
Fig. 4

η ˆ (broken line) and γ (solid line) vs Dfo for worst case atmospheric propagation.

Fig. 5
Fig. 5

γ(f) vs f′ for free-space propagation (solid line) and worst case atmospheric propagation (broken line). Dfo = 100 for both curves.

Fig. 6
Fig. 6

γ(f) vs f′ for free-space propagation (solid line) and worst case atmospheric propagation (broken line). Dfo = 500 for both curves.

Equations (31)

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E 0 ( r ¯ ) = R 1 d r ¯ E i ( r ¯ ) h 21 ( r ¯ , r ¯ ) r ¯ R 2 ,
h 21 ( r ¯ , r ¯ ) = { exp [ i 2 π z ( 1 + | r ¯ r ¯ | 2 / 2 z 2 ) / λ ] / i λ z } × exp [ χ ( r ¯ , r ¯ ) + i ϕ ( r ¯ , r ¯ ) ] ,
exp { χ ( r ¯ + ρ ¯ , r ¯ + ρ ¯ ) + χ ( r ¯ , r ¯ ) + i [ ϕ ( r ¯ , r ¯ ) ϕ ( r ¯ + ρ ¯ , r ¯ + ρ ¯ ) ] } = exp ( D ( ρ ¯ , ρ ¯ ) / 2 ) ,
D ( ρ ¯ , ρ ¯ ) = [ χ ( r ¯ + ρ ¯ , r ¯ + ρ ¯ ) χ ( r ¯ , r ¯ ) ] 2 + [ ϕ ( r ¯ + ρ ¯ , r ¯ + ρ ¯ ) ϕ ( r ¯ , r ¯ ) ] 2 = 2.91 k 2 C n 2 z 0 1 d s | ρ ¯ s + ρ ¯ ( 1 s ) | 5 / 3
R 1 d r ¯ 2 K ( r ¯ 1 , r ¯ 2 ) Φ m ( r ¯ 2 ) = η m Φ m ( r ¯ 1 ) , r ¯ 1 R 1 ,
η m 1 / 2 ϕ m ( r ¯ ) = R 1 d r ¯ Φ m ( r ¯ ) h 21 ( r ¯ , r ¯ ) , r ¯ R 2 ,
K ( r ¯ 1 , r ¯ 2 ) = R 2 d r ¯ h 21 * ( r ¯ , r ¯ 1 ) h 21 ( r ¯ , r ¯ 2 ) .
D f = R 1 d r ¯ R 2 d r ¯ | h 21 ( r ¯ , r ¯ ) | 2 = D fo R 1 d r ¯ R 2 d r ¯ { exp [ 2 χ ( r ¯ , r ¯ ) ] } / ( π d 1 d 2 / 4 ) 2 ,
η ˆ = ( R 1 d r ¯ 1 R 1 d r ¯ 2 | K ( r ¯ 1 , r ¯ 2 ) | 2 ) / R 1 d r ¯ K ( r ¯ , r ¯ ) .
η ˆ = 0 1 d x ( 4 x / π ) exp [ D ( d 1 x ) ] [ cos 1 x x ( 1 x 2 ) 1 / 2 ] × [ J 1 ( 4 x D fo 1 / 2 ) / x ] 2 ,
D ( ρ ) = 1.09 k 2 C n 2 z ρ 5 / 3 .
D f = m = 1 η m ,
Φ ( r ¯ ) ( 2 / π 1 / 2 d 1 ) exp ( i π | r ¯ | 2 / λ z ) , r ¯ R 1 ,
ϕ ( r ¯ ) ( 2 / π 1 / 2 d 2 ) exp ( i π | r ¯ | 2 / λ z ) , r ¯ R 2 ,
Φ m ( r ¯ ) ( 2 / π 1 / 2 d 1 ) exp [ i 2 π ( | r ¯ | 2 / 2 λ z + f ¯ m · r ¯ ) ] , r ¯ R 1 , 1 m D fo ,
γ = R 2 d r ¯ | R 1 d r ¯ ( 2 / π 1 / 2 d 1 ) exp ( i π | r ¯ | 2 / λ z ) h 21 ( r ¯ , r ¯ ) | 2 ,
γ = 0 1 d x ( 8 / π ) exp [ D ( d 1 x ) / 2 ] J 1 ( 4 x D fo 1 / 2 ) × D fo 1 / 2 [ cos 1 x x ( 1 x 2 ) 1 / 2 ] ,
ϕ ( r ¯ ) R 1 d r ¯ h 21 ( r ¯ , r ¯ ) [ R 2 d ρ ¯ | R 1 d ρ ¯ h 21 ( ρ ¯ , ρ ¯ ) | 2 ] 1 / 2 , r ¯ R 2 .
ϕ ( r ¯ ) ( 2 / π 1 / 2 d 2 ) , r ¯ R 2 ;
Φ ( r ¯ ) R 2 d r ¯ h 21 * ( r ¯ , r ¯ ) [ R 1 d ρ ¯ | R 2 d ρ ¯ h 21 * ( ρ ¯ , ρ ¯ ) | 2 ] 1 / 2 , r ¯ R 1 .
γ ( f ) = R 2 d r ¯ | R 1 d r ¯ ( 2 / π 1 / 2 d 1 ) exp [ i 2 π ( | r ¯ | 2 / 2 λ z + f ¯ m · r ¯ ) ] h 21 ( r ¯ , r ¯ ) | 2 ,
γ ( f ) = 0 1 d x ( 8 / π ) exp [ D ( d 1 x ) / 2 ] J 1 ( 4 x D fo 1 / 2 ) × D fo 1 / 2 [ cos 1 x x ( 1 x 2 ) 1 / 2 ] J o ( 4 f D fo 1 / 2 x ) .
D f = R 1 d r ¯ R 2 d r ¯ | h 21 ( r ¯ , r ¯ ) | 2
0 η m 1 , for 1 m < ,
R 1 d r ¯ R 2 d r ¯ | h 21 ( r ¯ , r ¯ ) | 2 = m = 1 η m .
η ˆ = ( m = 1 η m 2 ) / D f .
0 η ˆ η 1 min ( 1 , D f ) .
m = 1 η m 2
m = 1 η m = R 1 d r ¯ R 2 d r ¯ | h 21 ( r ¯ , r ¯ ) | 2 = m = 1 η m ,
m = 1 η m 2 = R 1 d r ¯ 1 R 1 d r ¯ 2 | K ( r ¯ 1 , r ¯ 2 ) | 2 m = 1 η m 2 .
( m = 1 η m 2 ) / D fo ,

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