Abstract

A new perturbation expansion for optical propagation in turbulence is presented. The method consists of expanding the power of the (Markov) refractive-index structure function about the trivial value 2. By providing a method of obtaining systematic corrections to results obtained by using a quadratic approximation to the structure function, the expansion permits quantitative estimation of the errors in these results. It is shown that this expansion necessarily introduces an arbitrary length, even in zeroth order (quadratic approximation). This length may be adjusted to improve the convergence of the expansion, thereby giving insight into the most-important-length scale of the problem. Essential differences are noted between optical processes in which the effects of overall wave-front tilts cancel and processes in which they do not. In the latter, the δ expansion gives (in low order) excellent approximations for all turbulence strengths, and the most important length decreases in increasing turbulence. In a tilt-canceling process, however, only moderate improvement over a weak-turbulence expansion is obtained, and the most important length appears to increase with the strength of the turbulence.

© 1981 Optical Society of America

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References

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  1. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Dept. of Commerce, National Technical Information Service, Springfield, Va., 1971).
  2. J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 45–106.
    [Crossref]
  3. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. N.Y. 20, 894–922 (1979).
    [Crossref]
  4. S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 9–43.
    [Crossref]
  5. S. F. Clifford and S. M. Wandzura, “The effect of the turbulent atmosphere on heterodyne lidar performance,” Appl. Opt. (to be published).
  6. S. F. Clifford and et al., “Study of a pulse coherent lidar for crosswind sensing,” (U.S. Department of Commerce, National Technical Information Service, Springfield, Va., 1980).
  7. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [Crossref] [PubMed]
  8. D. L. Fried, “Statistics of a geometrical representation of wave-front distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [Crossref]
  9. S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70, 745–747 (1980).
    [Crossref]
  10. G. C. Valley and S. M. Wandzura, “Spatial correlation of phase expansion coefficients for propagation through atmospheric turbulence,” J. Opt. Soc. Am. 69, 712–717 (1979).
    [Crossref]
  11. J. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175–185 (1978).
    [Crossref]
  12. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979).
    [Crossref]
  13. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [Crossref]

1980 (1)

1979 (3)

1978 (1)

1971 (1)

1966 (1)

1965 (1)

Clifford, S. F.

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 9–43.
[Crossref]

S. F. Clifford and S. M. Wandzura, “The effect of the turbulent atmosphere on heterodyne lidar performance,” Appl. Opt. (to be published).

S. F. Clifford and et al., “Study of a pulse coherent lidar for crosswind sensing,” (U.S. Department of Commerce, National Technical Information Service, Springfield, Va., 1980).

Dashen, R.

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. N.Y. 20, 894–922 (1979).
[Crossref]

Fried, D. L.

Leader, J. C.

Lutomirski, R. F.

Plonus, M. A.

Strohbehn, J. W.

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 45–106.
[Crossref]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Dept. of Commerce, National Technical Information Service, Springfield, Va., 1971).

Valley, G. C.

Wandzura, S. M.

Wang, S. C. H.

Yura, H. T.

Appl. Opt. (1)

J. Math. Phys. N.Y. (1)

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. N.Y. 20, 894–922 (1979).
[Crossref]

J. Opt. Soc. Am. (6)

Other (5)

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 9–43.
[Crossref]

S. F. Clifford and S. M. Wandzura, “The effect of the turbulent atmosphere on heterodyne lidar performance,” Appl. Opt. (to be published).

S. F. Clifford and et al., “Study of a pulse coherent lidar for crosswind sensing,” (U.S. Department of Commerce, National Technical Information Service, Springfield, Va., 1980).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Dept. of Commerce, National Technical Information Service, Springfield, Va., 1971).

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 45–106.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1

Long-term Strehl ratio; numerical integration of Eq. (14).

Fig. 2
Fig. 2

Comparison of first two orders of simple perturbation theory with S.

Fig. 3
Fig. 3

Fractional error of first two orders of δ expansion for S; ρ = R.

Fig. 4
Fig. 4

Fractional error of first two orders of δ expansion for S; ρ = ρ0.

Fig. 5
Fig. 5

α(1,2) (α0) for long-term Strehl ratio.

Fig. 6
Fig. 6

Fractional error in δ expansion for S; variable ρ.

Fig. 7
Fig. 7

α(2) (α0) for short-term Strehl ratio.

Fig. 8
Fig. 8

Comparison of various approximations to S ˜ with numerical integration of Eq. (27).

Equations (32)

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B ɛ ( r ) = ɛ 1 ( r ) ɛ 1 ( 0 )
B ɛ ( Markov ) ( r t r , z ) = δ ( z ) A ɛ ( r t r ) ,
r 2 = x 2 + y 2 + z 2 ,
r t r 2 = x 2 + y 2 ,
A ɛ ( r t r ) = - d z B ɛ ( r t r 2 + z 2 ) ,
x 5 / 3 = x 2 - δ = x 2 ( 1 - δ ln x + 1 / 2 δ 2 ln 2 x - ) ;             δ = 1 / 3.
D s ( x ) = 2 ( x ρ 0 ) 5 / 3 .
( x ρ 0 ) 5 / 3 = ( ρ ρ 0 ) 5 / 3 ( x ρ ) 2 - δ
( ρ ρ 0 ) 5 / 3 ( x ρ ) 2 - δ = ( ρ ρ 0 ) 5 / 3 ( x ρ ) 2 ( 1 - δ ln x ρ + ) ,
S ( R ρ 0 ) = d 2 x u ( x ) W 1 / 2 ( x ) 2 [ d 2 x u ( x ) W 1 / 2 ( x ) ] 2 .
u * ( x ) u ( x ) = exp [ - 1 / 2 D s ( x - x ) ] ,
D s ( x ) = 2 ( x ρ 0 ) 5 / 3 .
W 1 / 2 ( x ) = 1 π R 2 exp ( - x 2 / R 2 ) ,
S ( α 0 ) = 1 R 2 0 x d x exp [ - x 2 2 R 2 - ( α 0 x R ) 5 / 3 ] ,
α 0 R ρ 0 .
S = 1 - 1.676 α 0 5 / 3 + 2.388 α 0 10 / 3 .
D s ( x ) 2 ( ρ ρ 0 ) 5 / 3 ( x ρ ) 2 ( 1 - δ ln x ρ + δ 2 ln 2 x ρ ) ,
S ( α 0 , α , δ ) = 1 A + δ B A α 0 5 / 3 α 1 / 3 - δ 2 α 0 5 / 3 α 1 / 3 A 2 × { B 2 + π 2 6 - 1 4 - α 0 5 / 3 α 1 / 3 A [ ( B + 1 2 ) 2 + π 2 6 - 5 4 ] } + O ( δ 3 ) ,
A 1 + 2 α 0 5 / 3 α 1 / 3 ,
B 1 - γ + ln 2 α 2 A ,
α R ρ ,
( δ ) n S ( α 0 , α , δ ) δ = 0 = 0
u ˜ ( x ) = exp i [ ϕ ( x ) - b · x ] exp i ϕ ˜ ( x ) .
D ˜ s ( x - x ) = [ ϕ ˜ ( x ) - ϕ ˜ ( x ) ] 2 = D s ( x - x ) - 1 2 b 2 ( x - x ) 2 .
W 1 / 2 ( x ) = { 1 π R 2 , x R 0 , x > R
b = 4 π d 2 xx ϕ ( x ) W ( x ) ,
b 2 = 16 sin π δ 2 π ( 2 - δ 2 ) 2 ( ρ ρ 0 ) 5 / 3 1 R δ ρ 2 - δ Γ ( 5 - δ ) Γ ( δ 2 ) Γ ( 5 - δ 2 ) = 8 π R 1 / 3 ρ 0 5 / 3 ( 6 11 ) 2 Γ ( 14 3 ) Γ ( 1 6 ) Γ ( 29 6 ) ;             δ = .
S ˜ = 4 π R 2 0 2 R x d x exp [ - 1 2 D ˜ ( x ) ] × { cos - 1 x 2 R - x 2 R [ 1 - ( x 2 R ) 2 ] 1 / 2 } .
S ˜ ( α 0 ) = 1 - 0.1242 α 0 5 / 3 + 0.0089 α 0 10 / 3 - .
b 2 = 4 ( ρ ρ 0 ) 5 / 3 1 ρ 2 [ 1 - ( 13 24 ln R ρ ) δ + ( 13 24 ln R ρ + 1 2 ln 2 R ρ + π 2 24 - 187 576 ) δ 2 + O ( δ 3 ) ] .
S ˜ ( α , α 0 , δ ) = 1 - 3 δ 8 α 0 5 / 3 α 1 / 3 + δ 2 8 α 0 5 / 3 α 1 / 3 [ 1 8 + 3 2 ln α 2 + ( 5 π 2 9 - 2075 432 ) α 0 5 / 3 α 1 / 3 ] + O ( δ 3 ) .
ln α ( 2 ) + 1 24 + ( 5 π 2 27 - 2075 1296 ) α 0 5 / 3 ( α ( 2 ) ) 1 / 3 = 0.