Abstract

Although the wave-front correction provided by an adaptive optics system should be as complete as possible, only a partial compensation is attainable in the visible. An estimate of the residual phase variance in the compensated wave front can be used to calibrate system performance, but it is not a simple task when errors affect the compensation process. We propose a simple method for estimation of the residual phase variance that requires only the measurement of the Strehl ratio value. It provides good results over the whole range of compensation degrees. The estimate of the effective residual phase variance is useful not only for system calibration but also for determining the light intensity statistics to be expected in the image as a function of the degree of compensation introduced.

© 2000 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,” Astron. 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. A. Glindemann, “Improved performance of adaptive optics in the visible,” J. Opt. Soc. Am. A 11, 1370–1375 (1994).
    [CrossRef]
  6. P. Nisenson, R. Barakat, “Partial atmospheric correction with adaptive optics,” J. Opt. Soc. Am. A 4, 2249–2253 (1987).
    [CrossRef]
  7. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  8. J. Beckers, “Adaptive optics for astronomy,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
    [CrossRef]
  9. M. P. Cagigal, V. F. Canales, “Generalized Fried parameter after adaptive optics partial wave-front compensation,” J. Opt. Soc. Am. A 17, 903–910 (2000).
    [CrossRef]
  10. G. Rousset, P.-Y. Madec, D. Rabaud, “Adaptive optics partial correction simulation for two telescope interferometry,” in Proceedings of the ESO Symposium on High Resolution Imaging by Interferometry, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104.
  11. M. P. Cagigal, V. F. Canales, “Speckle statistics in partially corrected wave fronts,” Opt. Lett. 23, 1072–1074 (1998).
    [CrossRef]
  12. V. F. Canales, M. P. Cagigal, “Rician distribution to describe speckle statistics in adaptive optics,” Appl. Opt. 38, 766–771 (1999).
    [CrossRef]
  13. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North- Holland, Amsterdam, 1981), Vol. XIX, pp. 281–376.
  14. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  15. J. Conan, “Etude de la correction partielle en optique adaptative,” Ph.D. dissertation, Pub. 1995-1 (Office National d’Etudes et de Recherches Aerospatiales, Paris, 1995).
  16. G. C. Valley, S. M. Wandzura, “Spatial correlation of phase-expansion coefficients for propagation through atmospheric turbulence,” J. Opt. Soc. Am. 69, 712–717 (1979).
    [CrossRef]
  17. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).
  18. J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).
  19. F. Roddier, C. Roddier, “National Optical Astronomy Observatories (NOAO) Infrared Adaptive Optics Program II: modeling atmospheric effects in adaptive optics systems for astronomical telescopes,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 298–304 (1986).
    [CrossRef]
  20. R. C. Smithson, M. L. Peri, R. S. Benson, “Quantitative simulation of image correction for astronomy with a segmented mirror,” Appl. Opt. 27, 1615–1620 (1988).
    [CrossRef] [PubMed]
  21. B. R. Frieden, Probability, Statistical Optics and Data Testing (Springer-Verlag, Berlin, 1983).
  22. V. F. Canales, M. P. Cagigal, “Photon statistics in partially compensated wave fronts,” J. Opt. Soc. Am. A 10, 2550–2555 (1999).
    [CrossRef]

2000 (1)

1999 (2)

V. F. Canales, M. P. Cagigal, “Rician distribution to describe speckle statistics in adaptive optics,” Appl. Opt. 38, 766–771 (1999).
[CrossRef]

V. F. Canales, M. P. Cagigal, “Photon statistics in partially compensated wave fronts,” J. Opt. Soc. Am. A 10, 2550–2555 (1999).
[CrossRef]

1998 (1)

1994 (1)

1993 (1)

J. Beckers, “Adaptive optics for astronomy,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[CrossRef]

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)

1979 (1)

1978 (1)

1976 (1)

Barakat, R.

Beckers, J.

J. Beckers, “Adaptive optics for astronomy,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[CrossRef]

Benson, R. S.

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, Pub. 1995-1 (Office National d’Etudes et de Recherches Aerospatiales, Paris, 1995).

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).

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics and Data Testing (Springer-Verlag, Berlin, 1983).

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).

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).

Madec, P.-Y.

G. Rousset, P.-Y. Madec, D. Rabaud, “Adaptive optics partial correction simulation for two telescope interferometry,” in Proceedings of the ESO Symposium on High Resolution Imaging by Interferometry, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104.

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.

Peri, M. L.

Rabaud, D.

G. Rousset, P.-Y. Madec, D. Rabaud, “Adaptive optics partial correction simulation for two telescope interferometry,” in Proceedings of the ESO Symposium on High Resolution Imaging by Interferometry, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104.

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

Roddier, C.

F. Roddier, C. Roddier, “National Optical Astronomy Observatories (NOAO) Infrared Adaptive Optics Program II: modeling atmospheric effects in adaptive optics systems for astronomical telescopes,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 298–304 (1986).
[CrossRef]

Roddier, F.

F. Roddier, C. Roddier, “National Optical Astronomy Observatories (NOAO) Infrared Adaptive Optics Program II: modeling atmospheric effects in adaptive optics systems for astronomical telescopes,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 298–304 (1986).
[CrossRef]

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

G. Rousset, P.-Y. Madec, D. Rabaud, “Adaptive optics partial correction simulation for two telescope interferometry,” in Proceedings of the ESO Symposium on High Resolution Imaging by Interferometry, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104.

Smithson, R. C.

Valley, G. C.

Wandzura, S. M.

Wang, J. Y.

Welsh, B.

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

Wolf, E.

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

Annu. Rev. Astron. Astrophys. (1)

J. Beckers, “Adaptive optics for astronomy,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[CrossRef]

Appl. Opt. (3)

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. (3)

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

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

Opt. Lett. (1)

Other (8)

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

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

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

F. Roddier, C. Roddier, “National Optical Astronomy Observatories (NOAO) Infrared Adaptive Optics Program II: modeling atmospheric effects in adaptive optics systems for astronomical telescopes,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 298–304 (1986).
[CrossRef]

G. Rousset, P.-Y. Madec, D. Rabaud, “Adaptive optics partial correction simulation for two telescope interferometry,” in Proceedings of the ESO Symposium on High Resolution Imaging by Interferometry, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104.

B. R. Frieden, Probability, Statistical Optics and Data Testing (Springer-Verlag, Berlin, 1983).

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

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

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Figures (7)

Fig. 1
Fig. 1

Phase variance at the telescope aperture as a function of the position: piston removed (solid curve) and the first ten Zernike polynomials corrected (dashed curve, rescaled for comparison).

Fig. 2
Fig. 2

Structure function of the wave-front phase for D/r0=38.4 and 1 (short-dashed curve), 6 (solid curve), 21 (circles), 41 (thin long-dashed curve), and 81 (triangles) corrected polynomials.

Fig. 3
Fig. 3

Values of ρ0 in the compensated wave front obtained from the fitting of the structure function as a function of the number of corrected polynomials for D/r0=38.4.

Fig. 4
Fig. 4

Correlation-length values obtained from the structure function series for r0=1/38.4 (circles), for r0=1 (crosses) and values of the fitting curve expressed by relation (16) (solid curve) as a function of the number of corrected polynomials. The correlation length decreases quickly at the beginning and tends to a constant later on when compensation increases.

Fig. 5
Fig. 5

(a) Values of the residual phase variance obtained by simulation (solid curve) compared with those obtained from Eq. (17) (circles) as a function of the number of actuators in the system. To simulate errors in the compensation process, the coefficients to be corrected are reduced to 2.5% of their original values, instead of setting them to a zero value. The Marechal values (dashed curve) given by Eq. (13) are also shown. (b) Values of the residual phase variance obtained by simulation with an error of 1% (solid curve) and 10% (dashed curve) compared with those obtained from Eq. (17) with the same errors (circles and triangles, respectively) as a function of the number of actuators.

Fig. 6
Fig. 6

Evolution of the distribution parameters M(1)2 [Fig. 6(a)] and 2σ2 [Fig. 6(b)] as a function of the number of corrected Zernike polynomials.

Fig. 7
Fig. 7

Light intensity probability distribution at the central core for (a) 11, (b) 21, (c) 41, and (d) 81 corrected polynomials. Solid curves, theoretical Rician values; dashed curves, simulated values. All the intensity values have been normalized to the central value of the Airy pattern with full correction. The error in the compensation process is the same as in Fig. 5(a).

Equations (28)

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ϕ(r, θ)=i=1aiZi(r, θ),
Δj=i=j+1|ai|2,
Dϕ(r-r)=[ϕ(r)-ϕ(r)]2.
Dϕ(r)=6.88rr05/3,
Dϕ(r)=2σϕ21-ϕ(r)ϕ(r+r)σϕ2=2σϕ2[1-γ(r)]
Dϕ(r)=6.88rρ05/3,
pupilIhalo(r)dr=π4 1.27λρ02Ihalo(0).
ρ0=DSR-exp(-σϕ2)1-exp(-σϕ2)1/2,
ρ0SR1/2D.
6.88(lc/ρ0)5/3=2σϕ2.
σϕ2=3.44lcSR-exp(-σϕ2)1-exp(-σϕ2) D2-1/25/3,
σϕ2=3.44[lc(SR D2)-1/2]5/3.
σϕ2=1-SR,
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,
i(x)=|FT{P(r)exp[iϕ(r)]}|2.
lc0.286j-0.362D,
σϕ2=3.440.286j-0.362SR-exp(-σϕ2)1-exp(-σϕ2)-1/25/3.
σϕ2=3.440.286j-0.362SR1/2.
P(ϕ)=1/(2πσϕ2)exp[-ϕ2/(2σϕ2)].
Mϕ(ω)=- exp(jωϕ)P(ϕ)dϕ=exp-σϕ2ω22.
P(Ar, Ai)=12πσr2 exp[-(Ar-Ar)2/(2σr2)]×12πσi2exp[-A(Ai)2/(2σi2)],
P(I, θ)=12 12πσr2×exp-(I cos θ-Ar)22σr2 12πσi2×exp-(I sin θ)22σi2,
P(I)=-ππP(I, θ)d θ.
P(I)=12σ2 exp-I+a22σ2I0aIσ2.
a2NIPMϕ(1)2=NIP exp(-σϕ2),
2σ2=I-a2I-NIPMϕ(1)2,
P(I)=1I¯-a2 exp-I+a2I¯-a2I02 aII¯-a2.
P(I)1I¯-a2 exp-I+a2I¯-a2 1+aII¯-a22.

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