Abstract

Turbulence has long been recognized as one of the most significant factors limiting the performance of optical systems operating in the presence of atmosphere. Atmospheric turbulence over vertical paths has been well characterized, both theoretically and experimentally. Much less is known about turbulence over long, horizontal paths. Perturbations of the wave-front phase can be measured with a Hartmann wave-front sensor (H-WFS). One can use these measurements to characterize atmospheric turbulence directly. Theoretical expressions for the slope structure function of the H-WFS measurements are derived and evaluated with the use of numerical quadrature. By concentrating on the slope structure function, we avoid the phase reconstruction step and use the slope measurements in a more direct fashion. The theoretical slope structure function is compared with estimated slope structure functions computed from H-WFS measurements collected in a series of experiments conducted by researchers at the U.S. Air Force’s Phillips Laboratory. These experiments involved H-WFS measurements over high-altitude (airborne) horizontal paths 20–200 km in length.

© 1996 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. B. M. Welsh, C. S. Gardner, “Effects of turbulence induced anisoplanatism on the imaging performance of adaptive astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. E. E. Silbaugh, “Characterization of Atmospheric Turbulence over Long Horizontal Paths Using Optical Slope Measurements,” Master’s thesis (School of Engineering, U.S. Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio 45433, 1995).
  16. B. E. Stribling, B. M. Welsh, M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 181–196 (1995).
    [CrossRef]
  17. M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).
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    [CrossRef]
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    [CrossRef]
  22. D. Washburn, “ABLE ACE: ABL Extended Atmospheric Characterization Experiment,” Tech. Note (U.S. Air Force Phillips Laboratory, Kirtland, Air Force Base, N.M. 87117-5776, 1994).

1995 (1)

1994 (2)

1992 (2)

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Elec. Eng. 18, 451–466 (1992).
[CrossRef]

D. Dayton, B. Pierson, B. Spielbusch, J. Gonglewski, “Atmospheric structure function measurements with a Shack–Hartmann wave-front sensor,” Opt. Lett. 17, 1737–1739 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

1989 (1)

1986 (1)

1985 (1)

1983 (1)

E. P. Wallner, “Optimal wave front correction using slope measurements,” J. Opt.Soc. Am. 73, 1771–1776 (1983).
[CrossRef]

1982 (1)

1979 (1)

1975 (1)

D. L. Fried, “Differential angle of arrival: theory, evaluation and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
[CrossRef]

1966 (1)

Boreman, G. D.

Butts, R. R.

R. R. Butts, L. D. Weaver, “ABLEX high-altitude laser propagation experiment,” in Laser Beam Propagation and Control, H. Weichel, L. F. DeSandre, eds., Proc. SPIE2120, 30–42 (1994).
[CrossRef]

Cho, K. H.

Dainty, J. C.

Dayton, D.

Eaton, F. D.

F. D. Eaton, W. A. Peterson, J. R. Hines, “Phase structure function measurements with multiple apertures,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. SPIE1115, 218–223 (1989).
[CrossRef]

Ellerbroek, B. L.

B. L. Ellerbroek, “First-order performance evaluation of adaptive optics systems for atmospheric turbulence compensation in extended field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

Favier, D. L.

Fried, D. L.

Gardner, C. S.

Gonglewski, J.

Hines, J. R.

D. L. Walters, D. L. Favier, J. R. Hines, “Vertical path atmospheric MTF Measurements,” J. Opt. Soc. Am. 69, 828–837 (1979).
[CrossRef]

F. D. Eaton, W. A. Peterson, J. R. Hines, “Phase structure function measurements with multiple apertures,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. SPIE1115, 218–223 (1989).
[CrossRef]

Nicholls, T. W.

Parenti, R. R.

Pennington, T. L.

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

Petersen, D. P.

Peterson, W. A.

F. D. Eaton, W. A. Peterson, J. R. Hines, “Phase structure function measurements with multiple apertures,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. SPIE1115, 218–223 (1989).
[CrossRef]

Pierson, B.

Roddier, F.

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

Roggemann, M. C.

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Elec. Eng. 18, 451–466 (1992).
[CrossRef]

B. E. Stribling, B. M. Welsh, M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 181–196 (1995).
[CrossRef]

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

Sarazin, M.

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

Sasiella, R. J.

Silbaugh, E. E.

E. E. Silbaugh, “Characterization of Atmospheric Turbulence over Long Horizontal Paths Using Optical Slope Measurements,” Master’s thesis (School of Engineering, U.S. Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio 45433, 1995).

Spielbusch, B.

Stribling, B. E.

B. E. Stribling, B. M. Welsh, M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 181–196 (1995).
[CrossRef]

Tavis, M. T.

Tyler, G. A.

Wallner, E. P.

E. P. Wallner, “Optimal wave front correction using slope measurements,” J. Opt.Soc. Am. 73, 1771–1776 (1983).
[CrossRef]

Walters, D. L.

Washburn, D.

D. Washburn, “ABLE ACE: ABL Extended Atmospheric Characterization Experiment,” Tech. Note (U.S. Air Force Phillips Laboratory, Kirtland, Air Force Base, N.M. 87117-5776, 1994).

Weaver, L. D.

R. R. Butts, L. D. Weaver, “ABLEX high-altitude laser propagation experiment,” in Laser Beam Propagation and Control, H. Weichel, L. F. DeSandre, eds., Proc. SPIE2120, 30–42 (1994).
[CrossRef]

Welsh, B. M.

B. M. Welsh, C. S. Gardner, “Effects of turbulence induced anisoplanatism on the imaging performance of adaptive astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[CrossRef]

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).

B. E. Stribling, B. M. Welsh, M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 181–196 (1995).
[CrossRef]

Yura, H. T.

Astron. Astrophys. (1)

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

Comput. Elec. Eng. (1)

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstruction,” Comput. Elec. Eng. 18, 451–466 (1992).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (6)

J. Opt.Soc. Am. (1)

E. P. Wallner, “Optimal wave front correction using slope measurements,” J. Opt.Soc. Am. 73, 1771–1776 (1983).
[CrossRef]

Opt. Lett. (2)

Radio Sci. (1)

D. L. Fried, “Differential angle of arrival: theory, evaluation and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
[CrossRef]

Other (7)

E. E. Silbaugh, “Characterization of Atmospheric Turbulence over Long Horizontal Paths Using Optical Slope Measurements,” Master’s thesis (School of Engineering, U.S. Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio 45433, 1995).

B. E. Stribling, B. M. Welsh, M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 181–196 (1995).
[CrossRef]

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

R. R. Butts, L. D. Weaver, “ABLEX high-altitude laser propagation experiment,” in Laser Beam Propagation and Control, H. Weichel, L. F. DeSandre, eds., Proc. SPIE2120, 30–42 (1994).
[CrossRef]

D. Washburn, “ABLE ACE: ABL Extended Atmospheric Characterization Experiment,” Tech. Note (U.S. Air Force Phillips Laboratory, Kirtland, Air Force Base, N.M. 87117-5776, 1994).

F. D. Eaton, W. A. Peterson, J. R. Hines, “Phase structure function measurements with multiple apertures,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. SPIE1115, 218–223 (1989).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Normalized x-slope structure function, D ̂ s x ̂, as a function of subaperture separation for x, y, and 45° (diagonal) separations. The power-law parameter β = 11/3.

Fig. 2
Fig. 2

Normalized x -slope structure function, D ̂ s x ̂, as a function of subaperture separation for x separations. The power-law parameter β ranges from 3 to 3.8.

Fig. 3
Fig. 3

Experimental and theoretical slope structure functions, D ̂ s x ̂, for (a) x-, (b)y-, (c) +45°-, and (d) −45°-directed separations. In all plots Δt = 0.

Equations (23)

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s â ( x , t ) = d r W ( r x ) [ ϕ ( r , t ) â ] ,
s â ( x , t ) = d r [ W ( r x ) â ] ϕ ( r , t ) .
W ( r ) = 1 d 2 rect ( r d ) = { d 2 | r x | d / 2 and | r y | d / 2 , 0 elsewhere
W ( r ) = x ̂ r x W ( r x , r y ) + ŷ r y W ( r x , r y ) = x ̂ d 2 [ δ ( r x + d 2 ) δ ( r x d 2 ) ] rect ( r y d ) + ŷ d 2 [ δ ( r y + d 2 ) δ ( r y d 2 ) ] rect ( r x d ) ,
D s â ( x , x , t , t ) = [ s â ( x , t ) s â ( x , t ) ] 2 = [ d r [ W ( r x ) â ] ϕ ( r , t ) d r [ W ( r x ) â ] ϕ ( r , t ) ] 2 ,
D s x ̂ ( x , x , t , t ) = d r 1 d r 2 [ W ( r 1 x ) x ̂ ] × [ W ( r 2 x ) x ̂ ] D ϕ ( r 1 , r 2 , t , t ) d r 1 d r 1 [ W ( r 1 ) x ̂ ] × [ W ( r 1 ) x ̂ ] D ϕ ( r 1 , r 1 , t , t ) ,
D ϕ ( x , x , t , t ) = [ ϕ ( x , t ) ϕ ( x , t ) ] 2 .
D x ̂ s ( x , x , t , t ) = 1 d 3 d u [ 2 D ϕ ( | x x , u | , t , t ) D ϕ ( | x x , d , u | , t , t ) D ϕ ( | x x , + d , u | , t , t ) ] tri [ u ( y y ) d ] 1 d 3 d u [ 2 D ϕ ( | 0 , u | , t , t ) D ϕ ( | d , u | , t , t ) D ϕ ( | d , u | , t , t ) ] tri ( u d ) ,
tri ( u ) = { 1 | u | | μ | 1 0 elsewhere
D ϕ ( x , x , t , t ) = γ β 0 d z [ | x x + v ( z ) ( t t ) | ρ 0 ] β 2 w ( z ) ,
0 d z w ( z ) = 1 .
ρ 0 = r 0 = 0.185 [ 4 π 2 k 2 0 d z C n 2 ( z ) ] 3 / 5 ,
D s x ̂ ( x , x , t , t ) = γ β d 3 ( d ρ 0 ) β 2 d u 0 d z w ( z ) tri ( u d ) { [ 2 | x x + υ x ( z ) ( t t ) d , u + y y + υ y ( z ) ( t t ) d | β 2 | x x d + υ x ( z ) ( t t ) d , u + y y + υ y ( z ) ( t t ) d | β 2 | x x + d + υ x ( z ) ( t t ) d , u + y y + υ y ( z ) ( t t ) d | β 2 ] ( 2 | 0 , u d | β 2 | 1 , u d | β 2 | 1 , u d | β 2 ) } ,
D s x ̂ ( Δ x s , Δ y s , Δ x t , Δ y t ) = γ β d 2 ( d ρ 0 ) β 2 d u 0 d z w ( z ) tri ( u ) × { [ 2 | Δ x s + Δ x t ( z ) , u + Δ y s + Δ y t ( z ) | β 2 | Δ x s + Δ x t ( z ) 1 , u + Δ y s + Δ y t ( z ) | β 2 | Δ x s + Δ x t ( z ) + 1 , u + Δ y s + Δ y t ( z ) | β 2 ] 2 ( | 0 , u | β 2 | 1 , u | β 2 ) } .
Δ x s = ( x x ) / d ,
Δ y s = ( y y ) / d ,
Δ x t ( z ) = υ x ( z ) ( t t ) / d ,
Δ y t ( z ) = υ y ( z ) ( t t ) / d .
D s x ̂ ( Δ x s , Δ y s , 0 , 0 ) = γ β d 2 ( d ρ 0 ) β 2 d u tri ( u ) [ ( 2 | Δ x s , u + Δ y s | β 2 | Δ x s 1 , u + Δ y s | β 2 | Δ x s + 1 , u + Δ y s | β 2 ) 2 ( | 0 , u | β 2 | 1 , u | β 2 ) ] .
D s x ̂ ( 0 , 0 , Δ x t , Δ y t ) = γ β d 2 ( d ρ 0 ) β 2 × d u 0 d z w ( z ) tri ( u ) × { [ 2 | Δ x t ( z ) , u + Δ y t ( z ) | β 2 | Δ x t ( z ) 1 , u + Δ y t ( z ) | β 2 | Δ x t ( z ) + 1 , u + Δ y t ( z ) | β 2 ] ( | 0 , u | β 2 | 1 , u | β 2 ) } .
D ̂ s â ( x n , x k , i Δ t ) = 1 M m = 1 M [ s â ( x n , t m ) s â ( x k , t m i Δ t ) ] 2 ,
D ̂ s â ( Δ x , i Δ t ) = 1 M N n m = 1 M [ s â ( x n , t m ) s â ( x n Δ x , t m i Δ t ) ] 2 ,
D ̂ s â ( x n , x k , i Δ t ) = 1 M N n m = 1 M [ s â ( x n , t m ) s â ( x n Δ x , t m i Δ t ) ] 2 2 σ n 2 M N .

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