Abstract

Astronomical images obtained with adaptive optics systems can be enhanced by using image restoration techniques. However, this usually requires an accurate knowledge of the system point-spread function (PSF) which is variable in time. We present a method to estimate the PSF related to each image, using data from the adaptive optics control computer, namely, the wave-front sensor measurements and the commands to the deformable mirror, accumulated in synchronization with the acquisition. This method requires no extra observing time and has been successfully tested on PUEO, the Canada–France–Hawaii Telescope adaptive optics system. With this system, accurate PSF estimations could be achieved for guide stars of magnitude 13 or brighter.

© 1997 Optical Society of America

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    [CrossRef]
  4. M. C. Roggemann, B. L. Ellerbroek, T. A. Rhoadarmer, “Widening the effective field-of-view of adaptive-optics telescopesusing deconvolution from wave-front sensing: average and signal-to-noise ratiobehavior,” Appl. Opt. 34, 1432–1444 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
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  8. M. J. Northcott, “Observing with adaptive optics,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 115–117.
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  12. J.-M. Conan, G. Rousset, P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
    [CrossRef]
  13. F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations:time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10, 957–965 (1993).
    [CrossRef]
  14. D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmosphericoptical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991).
    [CrossRef]
  15. G. Rousset, “Wavefront sensing,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, ed., Vol. 423 of NATO Advanced Science Institutes Series (Kluwer Academic, Dordrecht, The Netherlands, 1994), pp. 115–138.
  16. B. L. Ellerbroek, M. J. Northcott, F. Rigaut, D. Simons, “Comparison of adaptive optics technologies for large astronomical telescopes,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 220–222.
  17. F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.
  18. E. Gendron, P. Léna, “Astronomical adaptive optics I: modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).
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    [CrossRef]
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    [CrossRef]
  21. W. H. Southwell, “What's wrong with cross coupling in modal wave-front estimation?,” in Adaptive Optics Systems and Technology, R. J. Becherer, B. A. Horwitz, eds., Proc. SPIE365, 97–104 (1982).
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    [CrossRef]
  23. J.-M. Conan, “Étude de la correction partielle en optique adaptative,” Ph.D. dissertation (Université Paris XI, Orsay, France, 1994).
  24. G. M. Dai, “Modal compensation of atmospheric turbulence with the use of Zernikepolynomials and Karhunen–Loève functions,” J. Opt. Soc. Am. A 12, 2182–2193 (1995).
    [CrossRef]
  25. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  26. R. Arsenault, D. A. Salmon, J. Kerr, F. Rigaut, “PUEO: the Canada-France-Hawaii adaptive optics bonnette I: system description,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 833–842 (1994).
    [CrossRef]
  27. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991).

1996 (2)

1995 (3)

1994 (1)

E. Gendron, P. Léna, “Astronomical adaptive optics I: modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).

1993 (2)

1991 (1)

1990 (1)

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

1981 (1)

1979 (1)

1978 (1)

1976 (1)

1941 (1)

A. N. Kolmogorov, “Local structure of turbulence in incompressible fluids with very highReynolds number,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).

Arsenault, R.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

R. Arsenault, D. A. Salmon, J. Kerr, F. Rigaut, “PUEO: the Canada-France-Hawaii adaptive optics bonnette I: system description,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 833–842 (1994).
[CrossRef]

Boyer, C.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Conan, J.-M.

J.-M. Conan, G. Rousset, P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
[CrossRef]

J.-M. Conan, “Étude de la correction partielle en optique adaptative,” Ph.D. dissertation (Université Paris XI, Orsay, France, 1994).

Couvée, P.

F. Rigaut, E. Gendron, P. Léna, P. Y. Madec, P. Couvée, G. Rousset, “Partial correction with the adaptive optics system Come-On,” in High Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, ed., Vol. 39 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1991), pp. 1105–1112.

Crampton, D.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Cubalchini, R.

Dai, G. M.

Ellerbroek, B. L.

M. C. Roggemann, B. L. Ellerbroek, T. A. Rhoadarmer, “Widening the effective field-of-view of adaptive-optics telescopesusing deconvolution from wave-front sensing: average and signal-to-noise ratiobehavior,” Appl. Opt. 34, 1432–1444 (1995).
[CrossRef] [PubMed]

B. L. Ellerbroek, M. J. Northcott, F. Rigaut, D. Simons, “Comparison of adaptive optics technologies for large astronomical telescopes,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 220–222.

Fletcher, M.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Gaffard, J.-P.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Gendron, E.

E. Gendron, P. Léna, “Astronomical adaptive optics I: modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).

F. Rigaut, E. Gendron, P. Léna, P. Y. Madec, P. Couvée, G. Rousset, “Partial correction with the adaptive optics system Come-On,” in High Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, ed., Vol. 39 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1991), pp. 1105–1112.

Gigan, P.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Graves, J. E.

Hatzinakos, D.

D. Kundur, D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13, 43–64 (1996).
[CrossRef]

Herrmann, J.

Jagourel, P.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Kerr, J.

R. Arsenault, D. A. Salmon, J. Kerr, F. Rigaut, “PUEO: the Canada-France-Hawaii adaptive optics bonnette I: system description,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 833–842 (1994).
[CrossRef]

Kolmogorov, A. N.

A. N. Kolmogorov, “Local structure of turbulence in incompressible fluids with very highReynolds number,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).

Kundur, D.

D. Kundur, D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13, 43–64 (1996).
[CrossRef]

Lai, O.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Leckie, B.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Léna, P.

E. Gendron, P. Léna, “Astronomical adaptive optics I: modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).

F. Rigaut, E. Gendron, P. Léna, P. Y. Madec, P. Couvée, G. Rousset, “Partial correction with the adaptive optics system Come-On,” in High Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, ed., Vol. 39 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1991), pp. 1105–1112.

Madec, P. Y.

F. Rigaut, E. Gendron, P. Léna, P. Y. Madec, P. Couvée, G. Rousset, “Partial correction with the adaptive optics system Come-On,” in High Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, ed., Vol. 39 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1991), pp. 1105–1112.

Madec, P.-Y.

Markey, J. K.

McArthur, S.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

McKenna, D. L.

Meinhardt, J. A.

Noll, R.-J.

Northcott, M. J.

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations:time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10, 957–965 (1993).
[CrossRef]

B. L. Ellerbroek, M. J. Northcott, F. Rigaut, D. Simons, “Comparison of adaptive optics technologies for large astronomical telescopes,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 220–222.

M. J. Northcott, “Observing with adaptive optics,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 115–117.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991).

Rhoadarmer, T. A.

Rigaut, F.

F. Rigaut, E. Gendron, P. Léna, P. Y. Madec, P. Couvée, G. Rousset, “Partial correction with the adaptive optics system Come-On,” in High Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, ed., Vol. 39 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1991), pp. 1105–1112.

B. L. Ellerbroek, M. J. Northcott, F. Rigaut, D. Simons, “Comparison of adaptive optics technologies for large astronomical telescopes,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 220–222.

R. Arsenault, D. A. Salmon, J. Kerr, F. Rigaut, “PUEO: the Canada-France-Hawaii adaptive optics bonnette I: system description,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 833–842 (1994).
[CrossRef]

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Roberts, S.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Roddier, D.

Roddier, F.

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations:time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10, 957–965 (1993).
[CrossRef]

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in E. Wolf, ed., Progress in Optics (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.

Rouan, D.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Rousset, G.

J.-M. Conan, G. Rousset, P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
[CrossRef]

G. Rousset, “Wavefront sensing,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, ed., Vol. 423 of NATO Advanced Science Institutes Series (Kluwer Academic, Dordrecht, The Netherlands, 1994), pp. 115–138.

F. Rigaut, E. Gendron, P. Léna, P. Y. Madec, P. Couvée, G. Rousset, “Partial correction with the adaptive optics system Come-On,” in High Resolution Imaging by Interferometry II, J. M. Beckers, F. Merkle, ed., Vol. 39 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1991), pp. 1105–1112.

Salmon, D.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Salmon, D. A.

R. Arsenault, D. A. Salmon, J. Kerr, F. Rigaut, “PUEO: the Canada-France-Hawaii adaptive optics bonnette I: system description,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 833–842 (1994).
[CrossRef]

Simons, D.

B. L. Ellerbroek, M. J. Northcott, F. Rigaut, D. Simons, “Comparison of adaptive optics technologies for large astronomical telescopes,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 220–222.

Southwell, W. H.

W. H. Southwell, “What's wrong with cross coupling in modal wave-front estimation?,” in Adaptive Optics Systems and Technology, R. J. Becherer, B. A. Horwitz, eds., Proc. SPIE365, 97–104 (1982).

Stilburn, J.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Tessier, E.

E. Tessier, “Natural guide star adaptive optics data,” in Adaptive Optics, Vol. 54 of ESO Conference and Workshop Proceedings, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1996), pp. 465–470.

Thomas, J.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Véran, J.-P.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Wang, J. Y.

Winker, D. M.

Woof, R.

F. Rigaut, D. Salmon, R. Arsenault, S. McArthur, J. Thomas, O. Lai, D. Rouan, P. Gigan, J.-P. Véran, D. Crampton, M. Fletcher, J. Stilburn, B. Leckie, S. Roberts, R. Woof, C. Boyer, P. Jagourel, J.-P. Gaffard, “First results of the CFHT adaptive optics bonnette on the telescope,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 97–99.

Appl. Opt. (1)

Astron. Astrophys. (1)

E. Gendron, P. Léna, “Astronomical adaptive optics I: modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).

Dokl. Akad. Nauk SSSR (1)

A. N. Kolmogorov, “Local structure of turbulence in incompressible fluids with very highReynolds number,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941).

IEEE Signal Process. Mag. (1)

D. Kundur, D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13, 43–64 (1996).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Opt. Eng. (1)

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

Other (12)

R. Arsenault, D. A. Salmon, J. Kerr, F. Rigaut, “PUEO: the Canada-France-Hawaii adaptive optics bonnette I: system description,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 833–842 (1994).
[CrossRef]

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991).

J.-M. Conan, “Étude de la correction partielle en optique adaptative,” Ph.D. dissertation (Université Paris XI, Orsay, France, 1994).

E. Tessier, “Natural guide star adaptive optics data,” in Adaptive Optics, Vol. 54 of ESO Conference and Workshop Proceedings, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1996), pp. 465–470.

M. J. Northcott, “Observing with adaptive optics,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 115–117.

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Figures (5)

Fig. 1
Fig. 1

Solid curve: variance of the mirror modes in a turbulent Kolmogorov phase. Dotted curve: variance of the remaining error. Both graphs are for D / r 0 = 1 .

Fig. 2
Fig. 2

Closed-loop transfer function H cl ( g i ,   f ) for different modal gains g i and corresponding - 3 - d B cutoff frequencies f cl ( g i ) .

Fig. 3
Fig. 3

Example 1: bright guide star (magnitude 10.4). Comparison of the estimated PSF and the real PSF in the case of a bright guide star. The left column (plots a, b, and c) shows the image, displayed with a square-root vertical scale. The right column (plots d, e, and f) shows the modulus of the OTF displayed with a logarithmic scale. In each column, the X cut (plots a and d), Y cut (plots b and e), and circular average (plots c and f) are presented. The real PSF is shown by the solid curve and the plus signs. The estimated PSF is shown by the diamond signs. The dotted curve is the absolute difference of the two curves, plotted with the same vertical scale (square root in the left column, logarithmic in the right column). In the left column, for a better comparison, both the real and the estimated PSF have been scaled so that the maximum of the real PSF is 1.

Fig. 4
Fig. 4

Example 2: fainter guide star (magnitude 12.7). Same legend as for example 1.

Fig. 5
Fig. 5

Example 3: very faint guide star (magnitude 15.4). Same legend as for example 1.    

Tables (2)

Tables Icon

Table 1 Numerical Atmospheric and AO Loop Values (Given at λ = 0.5   μ m)

Tables Icon

Table 2 Numerical Comparison of the PSF's

Equations (70)

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P ( x ) = 1 inside the aperture 0 outside the aperture .
( ϕ 1 | ϕ 2 ) E = 1 S   P ϕ 1 ( x ) ϕ 2 ( x ) d x
ϕ a ( x ,   t ) = i = 1 z i ( t ) Z i ( x ) .
z i z j = K ij D r 0 5 / 3 .
f c ( i ) = 0.3 ( n i + 1 ) V / D ,
ϕ ( x ,   t ) = ϕ a ( x ,   t ) - ϕ m ( x ,   t ) .
ϕ a ( x , t ) = ϕ a | | ( x , t ) M + ϕ ( x , t ) M .
ϕ ( x ,   t ) = ϕ ( x ,   t ) + ϕ ( x ,   t ) .
ϕ ( x ,   t ) = ϕ a ( x ,   t ) - ϕ m ( x ,   t ) .
ϕ a ( x ,   t ) = i = 1 m a i ( t ) M i ( x ) ,
ϕ m ( x ,   t ) = i = 1 m i ( t ) M i ( x ) ,
ϕ ( x ,   t ) = i = 1 m i ( t ) M i ( x ) .
a i a j = K ij D r 0 5 / 3
W [ ϕ ( x ,   t ) ] = W [ ϕ ( x ,   t ) ] + W [ ϕ ( x ,   t ) ] = D ( t ) + W [ ϕ ( x ,   t ) ] .
w ( t ) = D ( t ) + W [ ϕ ( x ,   t ) ] + n w ( t ) .
D + = ( D T D ) - 1 D T ,
ˆ ( t ) = D + w ( t ) .
ˆ ( t ) - ( t ) = n ( t ) + r ( t ) ,
n = D + n w ( t )
r ( t ) = D + W [ ϕ ( x ,   t ) ]
m ˜ i ( f ) = H cl ( g i ,   f ) a ˜ i ( f ) + H cl ( g i ,   f ) r ˜ i ( f ) + H n ( g i ,   f ) n ˜ i ( f ) .
˜ i ( f ) = H cor ( g i ,   f ) a ˜ i ( f ) - H cl ( g i ,   f ) r ˜ i ( f ) - H n ( g i ,   f ) n ˜ i ( f ) .
H cor ( g i ,   f ) = 1 - H cl ( g i ,   f ) .
B ( ρ / λ ,   t ) = 1 S   P P ( x ) P ( x + ρ ) exp [ i ϕ ( x ,   t ) ] × exp [ - i ϕ ( x + ρ ,   t ) ] d x .
B ( ρ / λ ) = B ( ρ / λ ,   t ) = 1 S   P P ( x ) P ( x + ρ ) exp - 1 2   D ϕ ( x ,   ρ ) d x ,
D ϕ ( x ,   ρ ) = | ϕ ( x ,   t ) - ϕ ( x + ρ ,   t ) | 2
PSF ( u ) = F [ B ( ρ / λ ) ] .
D ¯ ϕ ( ρ ) = P P ( x ) P ( x + ρ ) D ϕ ( x ,   ρ ) d x P P ( x ) P ( x + ρ ) d x
B ( ρ / λ ) = exp - 1 2   D ¯ ϕ ( ρ ) P P ( x ) P ( x + ρ ) d x .
D ¯ ϕ ( ρ ) = D ¯ ϕ ( ρ ) + D ¯ ϕ ( ρ ) + 2   P P ( x ) P ( x + ρ ) [ ϕ ( x ,   t ) - ϕ ( x + ρ ,   t ) ] [ ϕ ( x ,   t ) - ϕ ( x + ρ ,   t ) ] d x P P ( x ) P ( x + ρ ) d x
B ( ρ / λ ) = B ( ρ / λ ) B ( ρ / λ ) B tel ( ρ / λ ) .
B tel ( ρ / λ ) = 1 S   P P ( x ) P ( x + ρ ) d x .
B ( ρ / λ ) = exp - 1 2   D ¯ ϕ ( ρ ) ,
D ¯ ϕ ( ρ )
= P P ( x ) P ( x + ρ ) | ϕ ( x ,   t ) - ϕ ( x + ρ ,   t ) | 2 d x P P ( x ) P ( x + ρ ) d x .
 
B ( ρ / λ ) = exp - 1 2   D ¯ ϕ ( ρ ) .
D ¯ ϕ ( ρ ) = i , j n i j U ij ( ρ ) ,
U ij ( ρ ) = P P ( x ) P ( x + ρ ) [ M i ( x ) - M i ( x + ρ ) ] [ M j ( x ) - M j ( x + ρ ) ] d x P P ( x ) P ( x + ρ ) d x .
B ( ρ / λ ) = B ( ρ / λ ) B ( ρ / λ ) B s ( ρ / λ ) .
ˆ ( t ) = ( t ) + n ( t ) + r ( t )
C = C ˆ ˆ - C nn - C rr - 2 C r .
C ˆ ˆ = D + C ww ( D + ) T .
C nn = D + C n w n w ( D + ) T .
- ( C rr ) ij - 2 ( C r ) ij = ( C rr ) ij - 2 H cor * ( g i ,   f ) S r i r j ( f ) d f - 2 H cor * ( g i ,   f ) S a i r j ( f ) d f .
C C ˆ ˆ - C nn + C rr .
D ¯ ϕ ( x ,   ρ ) = ( D ¯ ϕ ( x ,   ρ ) ) D / r 0 = 1 ( D / r 0 ) 5 / 3 ,
C rr = ( C rr ) D / r 0 = 1 ( D / r 0 ) 5 / 3 .
( D / r 0 ) i = ( σ a i 2 / K ii ) 3 / 5 ,
σ m i 2 = | H cl ( g i ,   f ) | 2 S a i a i ( f ) d f + | H cl ( g i ,   f ) | 2 S r i r i ( f ) d f + σ n i 2 | H n ( g i ,   f ) | 2 d f .
σ a i 2 = σ m i 2 - σ r i 2 - σ n i 2 | H n ( g i ,   f ) | 2 d f .
W ( t ) = N 1 ( t ) - N 2 ( t ) N 1 ( t ) + N 2 ( t ) = D ( t ) S ( t ) ,
σ W 2 σ D 2 N 2 .
w ( t ) = n 1 ( t ) - n 2 ( t ) n 1 ( t ) + n 2 ( t ) = d ( t ) s ( t ) .
N = s ( t ) ,
σ S 2 = σ s 2 - N ,
σ D 2 = σ d 2 - N .
σ W 2 σ d 2 N 2 - 1 N .
σ n w 2 σ w 2 - σ W 2 .
σ w 2 σ d 2 N 2 .
σ n w 2 = 1 N ,
f M = 0.3 ( 7 + 1 ) 20 3.6 13   Hz .
σ ϕ 2 0.021 ( D / r 0 ) 5 / 3 ,
R xx ( τ ) = x ( t + τ ) x * ( t ) .
S xx ( f ) = R xx ( τ ) exp ( - 2 j π f τ ) d τ .
R xy ( τ ) = x ( t + τ ) y * ( t ) ,
S xy ( f ) = R xy ( τ ) exp ( - 2 j π f τ ) d τ .
R xy ( τ ) = S xy ( f ) exp ( 2 j π f τ ) d f .
x ( t ) y ( t ) = R xy ( 0 ) = S xy ( f ) d f .
S xz ( f ) = S xy ( f ) H * ( f ) .

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