Abstract

Corrections to the long- and short-term (tilt removed) Strehl ratios of a laser beam propagating through atmospheric turbulence are given for turbulence with finite inner and outer scales. It is found that outer-scale corrections are important for long-term Strehl ratios, and inner-scale corrections are important for short-term Strehl ratios. The results are applicable to near-ground propagation where the outer scale may be of the same order as the beam diameter and to upper atmospheric propagation where the inner scale may become a significant fraction of the beam diameter. The results are also useful for determining length scales for wave optics propagation codes, since such codes introduce artificial outer and inner scales due to finite grid size and mesh-point separation.

© 1979 Optical Society of America

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References

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  1. W. P. Brown, J. Opt. Soc. Am. 61, 1051 (1971).
    [Crossref]
  2. R. F. Lutomirski, H. T. Yura, J. Opt. Soc. Am. 61, 482 (1971).
    [Crossref]
  3. H. T. Yura, Appl. Opt. 12, 107 (1973).
    [Crossref]
  4. R. L. Fante, J. Opt. Soc. Am. 64, 592 (1974).
    [Crossref]
  5. R. F. Lutomirski, W. L. Woodie, R. G. Buser, Appl. Opt. 16, 665 (1977).
    [Crossref] [PubMed]
  6. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [Crossref]
  7. W. P. Brown, “Computer Simulation of Adaptive Optical Systems,” Hughes Research Laboratories Report under contract N60921-74-C-0249 (1975).
  8. H. T. Yura, J. Opt. Soc. Am. 63, 567 (1973).
    [Crossref]
  9. R. L. Fante, Proc. IEEE 63, 1669 (1975).
    [Crossref]
  10. J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
    [Crossref]
  11. V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1967), p. 262.
  12. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 611.
  13. M. T. Tavis, H. T. Yura, Appl. Opt. 15, 2922 (1976).
    [Crossref] [PubMed]
  14. M. T. Tavis, H. T. Yura, “Short Term Irradiance Profile of an Optical Beam in a Turbulent Medium,” unpublished preprint (1976).

1977 (1)

1976 (1)

1975 (1)

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[Crossref]

1974 (1)

1973 (2)

1971 (2)

1968 (1)

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

1966 (1)

Brown, W. P.

W. P. Brown, J. Opt. Soc. Am. 61, 1051 (1971).
[Crossref]

W. P. Brown, “Computer Simulation of Adaptive Optical Systems,” Hughes Research Laboratories Report under contract N60921-74-C-0249 (1975).

Buser, R. G.

Fante, R. L.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 611.

Fried, D. L.

Lutomirski, R. F.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 611.

Strohbehn, J. W.

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1967), p. 262.

Tavis, M. T.

M. T. Tavis, H. T. Yura, Appl. Opt. 15, 2922 (1976).
[Crossref] [PubMed]

M. T. Tavis, H. T. Yura, “Short Term Irradiance Profile of an Optical Beam in a Turbulent Medium,” unpublished preprint (1976).

Woodie, W. L.

Yura, H. T.

M. T. Tavis, H. T. Yura, Appl. Opt. 15, 2922 (1976).
[Crossref] [PubMed]

H. T. Yura, Appl. Opt. 12, 107 (1973).
[Crossref]

H. T. Yura, J. Opt. Soc. Am. 63, 567 (1973).
[Crossref]

R. F. Lutomirski, H. T. Yura, J. Opt. Soc. Am. 61, 482 (1971).
[Crossref]

M. T. Tavis, H. T. Yura, “Short Term Irradiance Profile of an Optical Beam in a Turbulent Medium,” unpublished preprint (1976).

Appl. Opt. (3)

J. Opt. Soc. Am. (5)

Proc. IEEE (2)

R. L. Fante, Proc. IEEE 63, 1669 (1975).
[Crossref]

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

Other (4)

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1967), p. 262.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 611.

W. P. Brown, “Computer Simulation of Adaptive Optical Systems,” Hughes Research Laboratories Report under contract N60921-74-C-0249 (1975).

M. T. Tavis, H. T. Yura, “Short Term Irradiance Profile of an Optical Beam in a Turbulent Medium,” unpublished preprint (1976).

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Figures (6)

Fig. 1
Fig. 1

Strehl ratio as a function of D/ρ0. Long-term ratio includes beam wander. Short-term ratio means that beam wander is completely tracked out. The Fried approximation ignores the residual phase-tilt correlations. The 1.0 approximation indicates that the numerical factor in the Fried approximation is set equal to 1.0 (adapted from Tavis and Yura14).

Fig. 2
Fig. 2

Outer-scale dependence of the tilt-correction integral. Equation (12) as a function of D/L0.

Fig. 3
Fig. 3

Long-term Strehl ratio as a function of D/ρ0 for values of L0/D.

Fig. 4
Fig. 4

Short-term Strehl ratio as a function of D/ρ0 for values of L0/D.

Fig. 5
Fig. 5

Long-term Strehl ratio as a function of D/ρ0 for values of D/ρm.

Fig. 6
Fig. 6

Short-term Strehl ratio as a function of D/ρ0 for values of D/ρm.

Tables (1)

Tables Icon

Table I Finite Inner-Scale Coefficient of ρ 0 - 5 / 3 D - 1 / 3 ρ 2 Term in Tilt-Corrected MTF

Equations (14)

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S = 8 0 1 d ζ ζ M atm ( ζ ) M ap ( ζ ) ,
M atm ( ρ ) = exp [ - 1 2 D φ ( ρ ) ] = exp [ - ( ρ / ρ 0 ) 5 / 3 ] ,
M ap ( ρ ) = 2 π { arccos ( ρ / D ) - ( ρ / D ) [ 1 - ( ρ / D ) 2 ] 1 / 2 } 0 ρ D , = 0 D < ρ ,
ρ 0 = ( 1.45719 k 2 C n 2 L ) - 3 / 5
D φ ( ρ ) = 4 π 0 K d K [ 1 - J 0 ( K ρ ) ] Φ φ ( K ) ,
Φ φ ( K ) = 2 π k 2 Δ z Φ Δ n ( K ) ,
Φ Δ n ( K ) = 0.033 C n 2 L 0 11 / 3 ( 1 + K 2 L 0 2 ) 11 / 6 exp ( - K 2 ρ m 2 ) ,
D φ ( ρ ) = 2 8 / 3 Γ ( / 6 11 ) Γ ( ) ( ρ 0 / L 0 ) - 5 / 3 [ 1 - 5 ( ρ / L 0 ) 5 / 6 3 ( 2 5 / 6 ) Γ ( / 6 11 ) K 5 / 6 ( ρ / L 0 ) ] = 1.07302 ( ρ 0 / L 0 ) - 5 / 3 [ 1 - 0.994397 ( ρ / L 0 ) 5 / 6 K 5 / 6 ( ρ / L 0 ) ] ,
D φ ( ρ ) = 2 ( ρ / ρ 0 ) 5 / 3 Γ ( / 6 11 ) ( ρ / 2 ρ m ) - 5 / 3 × [ F 1 1 ( - ; 1 ; - ¼ ρ 2 / ρ m 2 ) - 1 ] .
F 1 1 ( a ; b ; - z ) = Γ ( b ) Γ ( b - a ) z - a
a x 2 = [ π 0 d r r 3 W ( r ) ] - 2 d 2 r φ ( r ) x W ( r ) d 2 r φ ( r ) x W ( r )
exp ( ½ a x 2 ρ 2 ) = exp ( 1.04332 ρ 0 - 5 / 3 D - 1 / 3 ρ 2 ) .
½ a x 2 = 1.04332 ρ o - 5 / 3 D - 1 / 3 × o d x x - 1 J 2 2 ( x ) / ( x 2 + D 2 / 4 L o 2 ) 11 / 6 0 d x x - 14 / 3 J 2 2 ( x ) .
½ a x 2 = 1.04332 ρ o - 5 / 3 × D - 1 / 3 o d x x - 14 / 3 J 2 2 ( x ) exp ( - 4 x 2 ρ m 2 / D 2 ) 0 d x x - 14 / 3 J 2 2 ( x ) .

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