Abstract

Adaptive optics systems allow us to retrieve high-spatial-frequency information that is preserved in the wave fronts distorted by the atmosphere. Although wave-front correction should be as complete as possible, only partial compensation is attainable in the visible. We provide a procedure that uses the Rician distribution to predict the intensity statistics of the light at the image center as a function of the number of corrected Zernike polynomials.

© 1999 Optical Society of America

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References

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  1. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
    [CrossRef]
  2. 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,” Astronom. Astrophys. 250, 280–290 (1991).
  3. M. C. Roggemann, “Limited degree-of-freedom adaptive optics and image reconstruction,” Appl. Opt. 30, 4227–4233 (1991).
    [CrossRef] [PubMed]
  4. M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  5. J. W. Goodman, Statistical Optics (Wiley Interscience, New York, 1985).
  6. M. P. Cagigal, V. F. Canales, “Speckle statistics in partially corrected wave fronts,” Opt. Lett. 23, 1072–1074 (1998).
    [CrossRef]
  7. A. Labeyrie, “Images of exo-planets obtainable from dark speckles in adaptive telescopes,” Astronom. Astrophys. 298, 544–548 (1995).
  8. J. R. P. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368, 203–207 (1994).
    [CrossRef]
  9. S. M. Stahl, D. G. Sandler, “Optimization and performance of adaptive optics for imaging extrasolar planets,” Astrophys. J. 454, L153–L156 (1995).
    [CrossRef]
  10. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  11. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]

1998 (1)

1995 (2)

A. Labeyrie, “Images of exo-planets obtainable from dark speckles in adaptive telescopes,” Astronom. Astrophys. 298, 544–548 (1995).

S. M. Stahl, D. G. Sandler, “Optimization and performance of adaptive optics for imaging extrasolar planets,” Astrophys. J. 454, L153–L156 (1995).
[CrossRef]

1994 (1)

J. R. P. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368, 203–207 (1994).
[CrossRef]

1991 (2)

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,” Astronom. Astrophys. 250, 280–290 (1991).

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

1990 (1)

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

1978 (1)

1976 (1)

Angel, J. R. P.

J. R. P. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368, 203–207 (1994).
[CrossRef]

Cagigal, M. P.

Canales, V. F.

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,” Astronom. Astrophys. 250, 280–290 (1991).

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,” Astronom. Astrophys. 250, 280–290 (1991).

Goodman, J. W.

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

Labeyrie, A.

A. Labeyrie, “Images of exo-planets obtainable from dark speckles in adaptive telescopes,” Astronom. Astrophys. 298, 544–548 (1995).

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,” Astronom. 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,” Astronom. Astrophys. 250, 280–290 (1991).

Noll, R. J.

Rigaut, 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,” Astronom. Astrophys. 250, 280–290 (1991).

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,” Astronom. Astrophys. 250, 280–290 (1991).

Sandler, D. G.

S. M. Stahl, D. G. Sandler, “Optimization and performance of adaptive optics for imaging extrasolar planets,” Astrophys. J. 454, L153–L156 (1995).
[CrossRef]

Stahl, S. M.

S. M. Stahl, D. G. Sandler, “Optimization and performance of adaptive optics for imaging extrasolar planets,” Astrophys. J. 454, L153–L156 (1995).
[CrossRef]

Wang, J. Y.

Welsh, B.

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

Appl. Opt. (1)

Astronom. Astrophys. (2)

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,” Astronom. Astrophys. 250, 280–290 (1991).

A. Labeyrie, “Images of exo-planets obtainable from dark speckles in adaptive telescopes,” Astronom. Astrophys. 298, 544–548 (1995).

Astrophys. J. (1)

S. M. Stahl, D. G. Sandler, “Optimization and performance of adaptive optics for imaging extrasolar planets,” Astrophys. J. 454, L153–L156 (1995).
[CrossRef]

J. Opt. Soc. Am. (2)

Nature (1)

J. R. P. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature 368, 203–207 (1994).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (1)

Other (2)

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

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

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Figures (2)

Fig. 1
Fig. 1

Light-intensity probability distribution for (a) 21, (b) 41, (c) 71, and (d) 101 corrected polynomials. Solid curve, theoretical Rician values; points, theoretical exact values; dashed curve, simulated values. All the intensity values were normalized to the central value of the Airy pattern with full correction.

Fig. 2
Fig. 2

Light-intensity probability distribution for (a) 21, (b) 41, (c) 71, and (d) 101 corrected polynomials. Solid curve, semiempirical Rician values; long-dashed curve, theoretical Rician values; short-dashed curve, simulated values. All the intensity values were normalized to the central value of the Airy pattern with full correction. Theoretical values were obtained with a residual variance of the corrected wave front that was 10% less than that used in the simulation.

Equations (15)

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Φr, θ=i=1 aiZir, θ,
Δj=i=j+1 |ai|2=coefjDr05/3,
Dρ0=coefj0.1343/5Dr0.
Pϕ=12πΔj1/2 exp-ϕ2/2Δj.
Mϕω=-expjωϕPϕdϕ=exp-Δjω22.
Ar=1Nk=1N |αk|cos ϕk,Ai=1Nk=1N |αk|sin ϕk,
PAr, Ai=12πσr21/2exp-Ar-Ar2/2σr2×12πσi21/2exp-Ai2/2σi2,
Ar=N α¯Mϕ1,σr2=α2¯21+Mϕ2-α¯2Mϕ21,σi2=α2¯21-Mϕ2.
PI, θ=1212πσr21/2exp-Icos θ-Ar22σr2×12πσi21/2exp-Isin θ22σi2,
PI=-ππ PI, θdθ.
I=Ar2+Ai2,σI2=I2-I2=Ar4+Ai4+2Ar2Ai2-Ar2+Ai22.
I=σr2+σi2+Ar2,σI2=4σr2Ar2+2σr4+2σi4,
PI=12σ2exp-I+a22σ2I0aIσ2.
I=2σ2+a2,σI2=2σ2+4σ2a2.
2σ2=σr2+σi2+Ar2-Ar4+2Ar2σi2-σr2-σi2-σr221/2,a2=Ar4+2Ar2σi2-σr2-σi2-σr221/2.

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