Abstract

Atmospheric turbulence imposes the resolution limit attainable by large ground-based telescopes. This limit is λ/r0, where r0 is the Fried parameter or seeing cell size. Working in the visible, adaptive optics systems can partially compensate for turbulence-induced distortions. By analogy with the Fried parameter, r0, we have introduced a generalized Fried parameter, ρ0, that plays the same role as r0 but in partial compensation. Using this parameter and the residual phase variance, we have described the phase structure function, estimated the point-spread function halo size, and derived an expression for the Strehl ratio as a function of the degree of compensation. Finally, it is shown that ρ0 represents the diameter of the coherent cells in the pupil domain.

© 2000 Optical Society of America

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References

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  18. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
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1999 (2)

1998 (1)

1997 (1)

1994 (1)

1991 (2)

M. C. Roggemann, “Limited degree-of-freedom adaptive optics and image reconstruction,” Appl. Opt. 30, 4227–4233 (1991).
[CrossRef] [PubMed]

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

1990 (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

1988 (1)

1987 (1)

1982 (1)

1979 (2)

1978 (1)

1976 (1)

1973 (1)

1965 (1)

Barakat, R.

Benson, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).

Cagigal, M. P.

Canales, V. F.

Conan, J.

J. Conan, “Etude de la correction partielle en optique adaptative,” Ph.D. dissertation, Office National d’Etudes et de Recherches Aerospatiales Pub. 1995-1 (Paris, 1995).

Ellerbroek, B. L.

Fontanella, J. C.

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Fried, D. L.

Gaffard, J. P.

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Glindemann, A.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, Oxford, UK, 1998).

Kolmogorov, A.

A. Kolmogorov, Classic Papers on Statistical Theory, S. Friedlander, L. Topper, eds. (Interscience, New York, 1961).

Lena, P.

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Markey, J. K.

Merkle, F.

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Nisenson, P.

Noll, R. J.

Northcott, M. J.

Peri, M.

Rigaut, F.

F. Rigaut, B. L. Ellerbroek, M. J. Northcott, “Comparison of curvature-based and Shack–Hartmann-based adaptive optics for the Gemini telescope,” Appl. Opt. 36, 2856–2868 (1997).
[CrossRef] [PubMed]

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed., (North-Holland, Amsterdam, 1981), pp. 281–376.

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Roggemann, M. C.

Rousset, G.

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Smithson, R.

Valley, G.

Wandzura, S.

Wang, J. Y.

Welsh, B.

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

Winocur, J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).

Yura, H. T.

Appl. Opt. (6)

Astron. Astrophys. (1)

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

J. Opt. Soc. Am. (5)

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

Opt. Eng. (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Opt. Lett. (1)

Other (8)

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, Oxford, UK, 1998).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed., (North-Holland, Amsterdam, 1981), pp. 281–376.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).

A. Kolmogorov, Classic Papers on Statistical Theory, S. Friedlander, L. Topper, eds. (Interscience, New York, 1961).

J. Conan, “Etude de la correction partielle en optique adaptative,” Ph.D. dissertation, Office National d’Etudes et de Recherches Aerospatiales Pub. 1995-1 (Paris, 1995).

M. P. Cagigal, V. F. Canales, “Analysis of the photon statistics in partially corrected wavefronts,” in Propagation and Imaging through the Atmosphere, L. R. Bissonnette, C. Dainty, eds., Proc. SPIE3125, 320–326 (1997).
[CrossRef]

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

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Figures (6)

Fig. 1
Fig. 1

(a) Structure function of the wave-front phase for D/r0=38.4 and 100 (solid curve) corrected polynomials, assuming perfect compensation. It is compared with the structure function with 100 compensated polynomials when each coefficient is affected by a 10% error (circles). Dashed curve, reference line with a 5/3 slope. (b) Structure function of the wave-front phase for D/r0=38.4 and 1 (short-dashed curve), 6 (solid curve), 21 (circles), 41 (long-dashed curve), and 81 (triangles) corrected polynomials.

Fig. 2
Fig. 2

Correlation length values obtained from the structure function series for r0=1/38.4 (circles), for r0=1 (crosses), and for values of the fitting curve expressed by Eq. (27) (solid curve) as a function of the number of corrected polynomials.

Fig. 3
Fig. 3

Value of ρ0 in the compensated wave front as a function of the number of corrected polynomials for D/r0=38.4. Solid curve, values obtained from Eq. (28); circles, values obtained from the fitting of the structure function.

Fig. 4
Fig. 4

Solid curves, cross sections of the simulated PSF for a fixed value of D/r0=38.4 and (a) 11, (b) 21, (c) 41, and (d) 81 corrected polynomials. Dashed curves, cross section of PSF simulated with no compensation and a value D/r0=D/ρ0, where ρ0 is obtained from Eq. (28) with r0=1/38.4 and j=(a)11, (b) 21, (c) 41, and (d) 81 corrected polynomials.

Fig. 5
Fig. 5

Strehl ratio from simulated images (circles) compared with the theoretical values obtained from Eq. (22) (solid curve) [where ρ0 is estimated from Eq. (28)] and with approximated values obtained from Eq. (23) (crosses) and Eq. (24) (triangles).

Fig. 6
Fig. 6

Relative error in the estimation of the Strehl ratio from the theoretical expression given by Eq. (22) (solid curve) [where ρ0 is estimated from Eq. (28)] and from the approximated expressions given by Eq. (23) (crosses), and Eq. (24) (triangles).

Equations (33)

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ϕ(r,θ)=i=1aiZi(r,θ),
Δj=i=j+1|ai|2=coef(j)Dr05/3,
Dϕ(r-r)=[ϕ(r)-ϕ(r)]2.
Dϕ(r)=6.88rr05/3,
Dϕ(r)=2Δj1-ϕ(r)ϕ(r+r)Δj=2Δj[1-γ(r)],
Dϕ(r)=6.88rρ05/3,
6.88lcorrρ05/3=2Δj.
P(ϕ)=12πΔj1/2 exp-ϕ22Δj.
Mϕ(ω)=- exp(jωϕ)P(ϕ)dϕ=exp-Δjω22.
Ar=1Nk=1N|αk|cos ϕk,
Ai=1Nk=1N|αk|sin ϕk,
Ar=αM¯ϕ(1)N,Ai=0,
σr2=α2¯2[1+Mϕ(2)]-(α¯)2Mϕ2(1),
σi2=α2¯2[1-Mϕ(2)].
E=Ar2+σr2+Ai2+σi2
=[1+(N-1)exp(-Δj)](α¯)2.
acore=πλf2D2
SR=[1+(N-1)exp(-Δj)]N.
imageIhalo(x)dx=π4ω02Ihalo(0).
OTF=OTFTEL exp(-Dϕ/2)=OTFTEL exp[-Δj(1-γ)]=OTFTEL{exp(-Δj)+exp(-Δj)[exp(γΔj)-1]}.
PSF=PSFTEL exp(-Δj)+FT{exp(-Δj)[exp(γΔj)-1]}.
imageIhalo(x)dx=ET[1-exp(-Δj)].
E(0)=I(0)πλf2D2,
EC(0)=IC(0)πλf2D2.
Et[1-exp(-Δj)]=π(λf/2ρ0)2π(λf/2D)2[E(0)-Ec(0)]
E(0)=ET(D/ρ0)2{1+[(D/ρ0)2-1]exp(-Δj)},
SR=1(D/ρ0)2{1+[(D/ρ0)2-1]exp(-Δj)}.
SR1N=ρ0D2r0D2.
SRexp(-Δj).
Dϕ(r)=[ϕ2(r)+ϕ2(r+r)]P(r)P(r+r)dr-2ϕ(r)ϕ(r+r)P(r)P(r+r)drP(r)P(r+r)dr,
PSF(x)=|FT{P(r)exp[iϕ(r)]}|2.
lcorr0.286j-0.362D,
ρ0=3.44Δj3/5lcorr=3.44coef(j)3/50.286j-0.362r0.

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