Abstract

Phase-structure and aperture-averaged slope-correlated functions with a finite outer scale are derived based on the Taylor hypothesis and a generalized spectrum, such as the von Kármán modal. The effects of the finite outer scale on measuring and determining the character of atmospheric-turbulence statistics are shown especially for an approximately 4-m class telescope and subaperture. The phase structure function and atmospheric coherent length based on the Kolmogorov model are approximations of the formalism we have derived. The analysis shows that it cannot be determined whether the deviation from the power-law parameter of Kolmogorov turbulence is caused by real variations of the spectrum or by the effect of the finite outer scale.

© 2002 Optical Society of America

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References

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  1. D. Dyton, B. Pierson, B. Spiebusch, J. Gonglewski, “Atmospheric structure function measurements with a Shack–Hartmann wave-front sensor,” Opt. Lett. 17, 1737–1739 (1992).
    [CrossRef]
  2. T. W. Nicholls, G. D. Boreman, J. C. Dainty, “Use of a Shack–Hartmann wave-front sensor to measure deviations from a Kolmogorov phase spectrum,” Opt. Lett. 20, 2460–2462 (1995).
    [CrossRef]
  3. E. E. Silbaugh, B. M. Welsh, M. C. Roggemann, “Characterization of atmospheric turbulence phase statistics using wave-front slope measurements,” J. Opt. Soc. Am. A 13, 2453–2460 (1996).
    [CrossRef]
  4. R. W.-H. Szeto, R. R. Butts, “Atmospheric characterization in the presence of strong additive measurement noise,” J. Opt. Soc. Am. A 15, 1698–1707 (1998).
    [CrossRef]
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    [CrossRef]
  6. M. Sarazzin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).
  7. D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
    [CrossRef]
  8. B. L. Ellerbroek, D. J. Lee, J. D. Barchers, “Analysis of atmospheric turbulence measurements obtained with the Starfire Optical Range 3.5-m telescope adaptive optics system,” In Adaptive Optics Systems and Technology, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 212–224 (1999).
    [CrossRef]
  9. B. L. Ellerbroek, “Including outer scale effects in zonal adaptive optics calculations,” Appl. Opt. 36, 9456–9467 (1997).
    [CrossRef]
  10. N. Takato, I. Yamaguchi, “Spatial correlation of Zernike phase-expansion coefficients for atmospheric turbulence with finite outer scale,” J. Opt. Soc. Am. A. 12, 958–963 (1995).
    [CrossRef]
  11. D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A. 8, 1568–1573 (1991).
    [CrossRef]
  12. 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]
  13. 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. SPIE, 2471, 181–196 (1995).
    [CrossRef]
  14. G. A. Tyler, “Bandwidth consideration for tracking through turbulence,” J. Opt. Soc. Am. A 11, 358–367 (1994).
    [CrossRef]
  15. E. P. Wallner, “Optimal wave front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]

1998 (1)

1997 (1)

1996 (1)

1995 (2)

T. W. Nicholls, G. D. Boreman, J. C. Dainty, “Use of a Shack–Hartmann wave-front sensor to measure deviations from a Kolmogorov phase spectrum,” Opt. Lett. 20, 2460–2462 (1995).
[CrossRef]

N. Takato, I. Yamaguchi, “Spatial correlation of Zernike phase-expansion coefficients for atmospheric turbulence with finite outer scale,” J. Opt. Soc. Am. A. 12, 958–963 (1995).
[CrossRef]

1994 (1)

1992 (1)

1991 (1)

D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A. 8, 1568–1573 (1991).
[CrossRef]

1990 (1)

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

1983 (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)

Barchers, J. D.

B. L. Ellerbroek, D. J. Lee, J. D. Barchers, “Analysis of atmospheric turbulence measurements obtained with the Starfire Optical Range 3.5-m telescope adaptive optics system,” In Adaptive Optics Systems and Technology, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 212–224 (1999).
[CrossRef]

Boreman, G. D.

Butts, R. R.

C. Dainty, J.

Dyton, D.

Ellerbroek, B. L.

B. L. Ellerbroek, “Including outer scale effects in zonal adaptive optics calculations,” Appl. Opt. 36, 9456–9467 (1997).
[CrossRef]

B. L. Ellerbroek, D. J. Lee, J. D. Barchers, “Analysis of atmospheric turbulence measurements obtained with the Starfire Optical Range 3.5-m telescope adaptive optics system,” In Adaptive Optics Systems and Technology, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 212–224 (1999).
[CrossRef]

Favier, D. L.

Fried, D. L.

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

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]

Gonglewski, J.

Lee, D. J.

B. L. Ellerbroek, D. J. Lee, J. D. Barchers, “Analysis of atmospheric turbulence measurements obtained with the Starfire Optical Range 3.5-m telescope adaptive optics system,” In Adaptive Optics Systems and Technology, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 212–224 (1999).
[CrossRef]

Nicholls, T. W.

Pierson, B.

R. Hines, J.

Roddier, F.

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

Roggemann, M. C.

E. E. Silbaugh, B. M. Welsh, M. C. Roggemann, “Characterization of atmospheric turbulence phase statistics using wave-front slope measurements,” J. Opt. Soc. Am. A 13, 2453–2460 (1996).
[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. SPIE, 2471, 181–196 (1995).
[CrossRef]

Sarazzin, M.

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

Silbaugh, E. E.

Spiebusch, 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. SPIE, 2471, 181–196 (1995).
[CrossRef]

Szeto, R. W.-H.

Takato, N.

N. Takato, I. Yamaguchi, “Spatial correlation of Zernike phase-expansion coefficients for atmospheric turbulence with finite outer scale,” J. Opt. Soc. Am. A. 12, 958–963 (1995).
[CrossRef]

Tyler, G. A.

Wallner, E. P.

Walters, D. L.

Welsh, B. M.

E. E. Silbaugh, B. M. Welsh, M. C. Roggemann, “Characterization of atmospheric turbulence phase statistics using wave-front slope measurements,” J. Opt. Soc. Am. A 13, 2453–2460 (1996).
[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. SPIE, 2471, 181–196 (1995).
[CrossRef]

Winker, D. M.

D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A. 8, 1568–1573 (1991).
[CrossRef]

Yamaguchi, I.

N. Takato, I. Yamaguchi, “Spatial correlation of Zernike phase-expansion coefficients for atmospheric turbulence with finite outer scale,” J. Opt. Soc. Am. A. 12, 958–963 (1995).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (1)

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. A. (2)

N. Takato, I. Yamaguchi, “Spatial correlation of Zernike phase-expansion coefficients for atmospheric turbulence with finite outer scale,” J. Opt. Soc. Am. A. 12, 958–963 (1995).
[CrossRef]

D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A. 8, 1568–1573 (1991).
[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 (2)

B. L. Ellerbroek, D. J. Lee, J. D. Barchers, “Analysis of atmospheric turbulence measurements obtained with the Starfire Optical Range 3.5-m telescope adaptive optics system,” In Adaptive Optics Systems and Technology, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3762, 212–224 (1999).
[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. SPIE, 2471, 181–196 (1995).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Effects of finite outer scale on atmospheric-turbulence phase-structure function. (a) The first term in the series of Eq. (32) (i = 1) is the most important in the deviations. (b) When the first term (i = 1) is considered, the deviations increase with the normalized value ρ/2L 0 for different values of p.

Fig. 2
Fig. 2

Theoretical related error of correlated function of sub-aperture-averaged slope measurement.

Equations (44)

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Dϕρ=|ϕr1-ϕr2|2,=6.88ρr05/3,
Φnκ, z=0.033 Cn2zκ11/3,
Φnκ, z=upCn2zκ2+κ02p+1,
Dϕρ=8π2k20Hdz  dκ 1-J0ρ1-z/HκΦnκ, z,
Dϕρ=8π2k2up0H Cn2zdz×κ0-2p2p1-2p2pΓp+1ρ 1-zHκ0×Kpρ 1-zHκ0,
2p2pΓp+1 ZpKpZ=1-π2 sin pπ ×2p12pΓp+12 Z2p+ 2pΓp+1×i=11i!1Γ-p+i+1-Z/22pΓp+i+1Z22i,
Dϕρ=Dϕ0ρ+i=1 Dϕiρ,κ0,
Dϕ0ρ=4π2 up k2Γ1-pp22p Γ1+p×ρ2p0HdzCn2z1-z/H2p.
up=22p-4 p2p-1π-3/2ΓpΓ1.5-p
Dϕ0ρ=w0ρρ02p.
w0=24p Γ1pp.
ρ0-2p=upΓ1-pp 22p Γ1+p w0 4π2k2×0HdzCn2z1-zH2p,
Dϕiρ, κ0=Dϕ1iρ, κ0-Dϕ2iρ,κ0,
Dϕ1iρ, κ0=4π2k2upΓ1-pκ02i-pΓ-p+i+1pi!22i×ρ2i0HdzCn2z1-zH2i,
Dϕ2iρ, κ0=4π2k2upΓ1-pκ02iΓp+i+1pi!22i+p×ρ2i+p0HdzCn2z1-zH2i+p.
Dϕiρ, κ0=w1ρρ1i2i,
Dϕ2iρ, κ0=w2ρρ2i2i+p.
RRmax=16πDρ0201cos-1u-u1-u21/2×exp- 12 DϕDρ0 uudu,
w1=24i Γ1ii,
w2=24i+p Γ1i+pi+p.
ρ01i-2i=4π2k2upw1Γ1-pκ02i-pΓ-p+i+1pi!22i×0HdzCn2z1-zH2i,
ρ02i-2i+p=4π2k2upw2Γ1-p κ02iΓp+i+1 pi!22i+p×0HdzCn2z1-zH2i+p.
εiρ, p, κ0=Dϕ1iρ, κ0-Dϕ2iρ, κ0Dϕ0ρ =10HCn2z1-zH2pdzΓ1+pi!Γ-p+i+1 t2i-p0HCn2z1-zH2idz-Γ1+pi!Γp+i+1 t2i0HCn2z1-zH2i+pdz.
sxn, t=drWr-rnϕr, t· i, syn, t=drWr-rnϕr, t · j,
Wr-rn=1d2rectx-xndrecty-ynd,
rectx=1|x|1/20otherwise.
sxn, t=drWr-rn · i ϕr, t.
Cxxn, n, τ=drdrWr-rn·i×Wr-rn·iϕr, Ttϕr, t,
ϕr=ψrdrWArψr=drWArψr-ψr.
ϕrϕr=-12 Dϕr, r+gr+gr-a,
gr=12dr2WAr2Dϕr, r2, a=12dr2dr3WAr2WAr3Dϕr2, r3.
Wr-rn=1d2iδx+d2+xn-δx-d2+xnrecty+ynd+jδy+d2+yn-δy-d2+ynrectx+xnd.
Cxxn, n, τ=-12d4drdrDϕr, r, τ×δx-d2+xn-δx+d2+xn×recty+yndδx-d2+xn-δx+d2+xnrecty+ynd,
u=y+ynd, v=y+ynd,
Cxxn, n, τ=-12d2dudv recturectv×Dϕd2-xn, y, d2-xn, y, τ+Dϕ-d2-xn, y, -d2-xn, y, τ-Dϕd2-xn, y, -d2-xn, y, τ-Dϕ-d2-xn, y, d2-xn, y, τ.
Cxxn, n, τ=Cxx0n, n, τ+i=1Cxx1in, n, τ-Cxx2in, n, τ,
Cxx0n, n, τ=dρ02pw0d2dudv recturectv×1+u-v+υτ2p-u-v+υτ2p,
Cxx1in, n, τ=dρ1i2iw1d2dudv recturectv×1+u-v+υτ2-u-v+υτ2,
Cxx2in, n, τ=dρ2i2i+pw2d2dudv recturectv×1+u-v+υτ2i+p-u-v+υτ2i+p,
Φsf=Cxxτexp-2πifτdτ=Φs0f+i=1Φs1if-Φs2if.
Φs0f=w0d2dρ02p sin c2fυ1υFTt,f/υ1+t2p-t2p=Dϕ0dd2sin c2fυ1υFTt,f/υ1+t2p-t2p,
εiρ, ρ, κ0=Γ1+p2p+1i!×t2i-pΓ-p+i+12i+1-t2iΓp+i+12i+2p+1,
t=κ0ρ2.
εc=Cxx 11τ-Cxx 21τCxx0τ =Γ1+p2p+1Γ-p+22p+3κ0d22dudv recturectv1+u-v+υτ21+p-u-v+υτ2i+pdudv recturectv1+u-v+υτ2p-u-v+υτ2p.

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