Abstract

A simple derivation is given for the expression of one-dimensional spectra of turbulence-induced Zernike aberration terms in the case of a stellar source at infinity. Numerical results are presented for several terms. Expressions are given for the maximum acceptable delay and the maximum isoplanatic angle in a partially compensating adaptive optics system. Temporal-frequency spectra of the Zernike terms, recorded at the Canada–France–Hawaii telescope, are well reproduced when one convolves all the corresponding theoretical spectra with the same turbulence distribution. Only a few atmospheric layers are found to contribute to image degradation.

© 1993 Optical Society of America

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References

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  1. F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
    [Crossref]
  2. F. Roddier, J. E. Graves, D. McKenna, M. Northcott, “The University of Hawaii adaptive optics system. I. General approach,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 248–253 (1991).
    [Crossref]
  3. M. J. Northcott, “The University of Hawaii adaptive optics system. II. Computer simulation,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 254–261 (1991).
    [Crossref]
  4. J. E. Graves, D. L. McKenna, “A simple low-order adaptive optics system. III. The wave-front curvature sensor,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 262–272 (1991).
    [Crossref]
  5. J. E. Graves, F. Roddier, D. McKenna, M. Northcott, “Latest results from the University of Hawaii prototype adaptive optics system,” in Proceedings of the Workshop on Laser Guide Star Adaptive Optics, R. L. Fugate, ed. (Starfire Optical Range, Phillips Laboratory/LITE, Kirtland AFB, Albuquerque, N.M., 1992), Vol. 2, 511–521.
  6. S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).
  7. R. J. Noll, “Zernike polynomials and atmospheric turbulence,”J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  8. P. Y. Madec, J. M. Conan, G. Rousset, “Temporal characterization of atmospheric wave-front for adaptive optics,” in Proceedings of ESO Conference Number 42 on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1993), pp. 471–474.
  9. C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,”IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
    [Crossref]
  10. F. Chassat, “Calcul du domaine d’isoplanétisme d’un système d’optique adaptative fonctionnant à travers la turbulence atmosphérique,”J. Opt. (Paris) 20, 13–23 (1989).
    [Crossref]
  11. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
    [Crossref]
  12. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  13. D. L. Fried, “Time-delay-induced mean-square error in adaptive optics,” J. Opt. Soc. Am. A 7, 1224–1225 (1990).
    [Crossref]
  14. 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]
  15. D. L. Fried, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52–61 (1982).
    [Crossref]
  16. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [Crossref] [PubMed]
  17. J. Högbom, “Aperture synthesis with a non-regular distribution of interferometer base lines,” Astron. Astrophys. Suppl. Ser. 15, 417–426 (1974).
  18. F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
    [Crossref]
  19. R. D. McClure, J. Arnaud, J. M. Fletcher, J. L. Nieto, R. Racine, “A measurement of isoplanatism with HRCam at the CFHT,” Publ. Astron. Soc. Pac. 103, 570–576 (1991).
    [Crossref]

1991 (2)

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[Crossref]

R. D. McClure, J. Arnaud, J. M. Fletcher, J. L. Nieto, R. Racine, “A measurement of isoplanatism with HRCam at the CFHT,” Publ. Astron. Soc. Pac. 103, 570–576 (1991).
[Crossref]

1990 (1)

1989 (1)

F. Chassat, “Calcul du domaine d’isoplanétisme d’un système d’optique adaptative fonctionnant à travers la turbulence atmosphérique,”J. Opt. (Paris) 20, 13–23 (1989).
[Crossref]

1988 (1)

1982 (1)

1981 (1)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[Crossref]

1979 (1)

1976 (2)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,”J. Opt. Soc. Am. 66, 207–211 (1976).
[Crossref]

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,”IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[Crossref]

1974 (2)

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

J. Högbom, “Aperture synthesis with a non-regular distribution of interferometer base lines,” Astron. Astrophys. Suppl. Ser. 15, 417–426 (1974).

Arnaud, J.

R. D. McClure, J. Arnaud, J. M. Fletcher, J. L. Nieto, R. Racine, “A measurement of isoplanatism with HRCam at the CFHT,” Publ. Astron. Soc. Pac. 103, 570–576 (1991).
[Crossref]

Azouit, M.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

Beland, S.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

Bezdid’ko, S. N.

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Butts, R. R.

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,”IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[Crossref]

Caccia, J. L.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

Chassat, F.

F. Chassat, “Calcul du domaine d’isoplanétisme d’un système d’optique adaptative fonctionnant à travers la turbulence atmosphérique,”J. Opt. (Paris) 20, 13–23 (1989).
[Crossref]

Conan, J. M.

P. Y. Madec, J. M. Conan, G. Rousset, “Temporal characterization of atmospheric wave-front for adaptive optics,” in Proceedings of ESO Conference Number 42 on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1993), pp. 471–474.

Cowie, L.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

Cowley, D.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

Fletcher, J. M.

R. D. McClure, J. Arnaud, J. M. Fletcher, J. L. Nieto, R. Racine, “A measurement of isoplanatism with HRCam at the CFHT,” Publ. Astron. Soc. Pac. 103, 570–576 (1991).
[Crossref]

Fried, D. L.

Graves, J. E.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[Crossref]

F. Roddier, J. E. Graves, D. McKenna, M. Northcott, “The University of Hawaii adaptive optics system. I. General approach,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 248–253 (1991).
[Crossref]

J. E. Graves, D. L. McKenna, “A simple low-order adaptive optics system. III. The wave-front curvature sensor,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 262–272 (1991).
[Crossref]

J. E. Graves, F. Roddier, D. McKenna, M. Northcott, “Latest results from the University of Hawaii prototype adaptive optics system,” in Proceedings of the Workshop on Laser Guide Star Adaptive Optics, R. L. Fugate, ed. (Starfire Optical Range, Phillips Laboratory/LITE, Kirtland AFB, Albuquerque, N.M., 1992), Vol. 2, 511–521.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

Hill, S.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

Högbom, J.

J. Högbom, “Aperture synthesis with a non-regular distribution of interferometer base lines,” Astron. Astrophys. Suppl. Ser. 15, 417–426 (1974).

Hogge, C. B.

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,”IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[Crossref]

Limburg, E.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

Madec, P. Y.

P. Y. Madec, J. M. Conan, G. Rousset, “Temporal characterization of atmospheric wave-front for adaptive optics,” in Proceedings of ESO Conference Number 42 on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1993), pp. 471–474.

McClure, R. D.

R. D. McClure, J. Arnaud, J. M. Fletcher, J. L. Nieto, R. Racine, “A measurement of isoplanatism with HRCam at the CFHT,” Publ. Astron. Soc. Pac. 103, 570–576 (1991).
[Crossref]

McKenna, D.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

J. E. Graves, F. Roddier, D. McKenna, M. Northcott, “Latest results from the University of Hawaii prototype adaptive optics system,” in Proceedings of the Workshop on Laser Guide Star Adaptive Optics, R. L. Fugate, ed. (Starfire Optical Range, Phillips Laboratory/LITE, Kirtland AFB, Albuquerque, N.M., 1992), Vol. 2, 511–521.

F. Roddier, J. E. Graves, D. McKenna, M. Northcott, “The University of Hawaii adaptive optics system. I. General approach,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 248–253 (1991).
[Crossref]

McKenna, D. L.

J. E. Graves, D. L. McKenna, “A simple low-order adaptive optics system. III. The wave-front curvature sensor,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 262–272 (1991).
[Crossref]

Nieto, J. L.

R. D. McClure, J. Arnaud, J. M. Fletcher, J. L. Nieto, R. Racine, “A measurement of isoplanatism with HRCam at the CFHT,” Publ. Astron. Soc. Pac. 103, 570–576 (1991).
[Crossref]

Noll, R. J.

Northcott, M.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[Crossref]

F. Roddier, J. E. Graves, D. McKenna, M. Northcott, “The University of Hawaii adaptive optics system. I. General approach,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 248–253 (1991).
[Crossref]

J. E. Graves, F. Roddier, D. McKenna, M. Northcott, “Latest results from the University of Hawaii prototype adaptive optics system,” in Proceedings of the Workshop on Laser Guide Star Adaptive Optics, R. L. Fugate, ed. (Starfire Optical Range, Phillips Laboratory/LITE, Kirtland AFB, Albuquerque, N.M., 1992), Vol. 2, 511–521.

Northcott, M. J.

M. J. Northcott, “The University of Hawaii adaptive optics system. II. Computer simulation,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 254–261 (1991).
[Crossref]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Racine, R.

R. D. McClure, J. Arnaud, J. M. Fletcher, J. L. Nieto, R. Racine, “A measurement of isoplanatism with HRCam at the CFHT,” Publ. Astron. Soc. Pac. 103, 570–576 (1991).
[Crossref]

Roddier, C.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

Roddier, F.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[Crossref]

F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
[Crossref] [PubMed]

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[Crossref]

F. Roddier, J. E. Graves, D. McKenna, M. Northcott, “The University of Hawaii adaptive optics system. I. General approach,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 248–253 (1991).
[Crossref]

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

J. E. Graves, F. Roddier, D. McKenna, M. Northcott, “Latest results from the University of Hawaii prototype adaptive optics system,” in Proceedings of the Workshop on Laser Guide Star Adaptive Optics, R. L. Fugate, ed. (Starfire Optical Range, Phillips Laboratory/LITE, Kirtland AFB, Albuquerque, N.M., 1992), Vol. 2, 511–521.

Rousset, G.

P. Y. Madec, J. M. Conan, G. Rousset, “Temporal characterization of atmospheric wave-front for adaptive optics,” in Proceedings of ESO Conference Number 42 on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1993), pp. 471–474.

Salmon, D.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

Songaila, A.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

Valley, G. C.

Vernin, J.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

Wandzura, S. M.

Appl. Opt. (1)

Astron. Astrophys. Suppl. Ser. (1)

J. Högbom, “Aperture synthesis with a non-regular distribution of interferometer base lines,” Astron. Astrophys. Suppl. Ser. 15, 417–426 (1974).

IEEE Trans. Antennas Propag. (1)

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,”IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[Crossref]

J. Opt. (Paris) (1)

F. Chassat, “Calcul du domaine d’isoplanétisme d’un système d’optique adaptative fonctionnant à travers la turbulence atmosphérique,”J. Opt. (Paris) 20, 13–23 (1989).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Prog. Opt. (1)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[Crossref]

Publ. Astron. Soc. Pac. (2)

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[Crossref]

R. D. McClure, J. Arnaud, J. M. Fletcher, J. L. Nieto, R. Racine, “A measurement of isoplanatism with HRCam at the CFHT,” Publ. Astron. Soc. Pac. 103, 570–576 (1991).
[Crossref]

Sov. J. Opt. Technol. (1)

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Other (7)

P. Y. Madec, J. M. Conan, G. Rousset, “Temporal characterization of atmospheric wave-front for adaptive optics,” in Proceedings of ESO Conference Number 42 on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching bei Munchen, Germany, 1993), pp. 471–474.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH–UN–NOAO–CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[Crossref]

F. Roddier, J. E. Graves, D. McKenna, M. Northcott, “The University of Hawaii adaptive optics system. I. General approach,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 248–253 (1991).
[Crossref]

M. J. Northcott, “The University of Hawaii adaptive optics system. II. Computer simulation,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 254–261 (1991).
[Crossref]

J. E. Graves, D. L. McKenna, “A simple low-order adaptive optics system. III. The wave-front curvature sensor,” in Active and Adaptive Optical Systems, M. A Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 262–272 (1991).
[Crossref]

J. E. Graves, F. Roddier, D. McKenna, M. Northcott, “Latest results from the University of Hawaii prototype adaptive optics system,” in Proceedings of the Workshop on Laser Guide Star Adaptive Optics, R. L. Fugate, ed. (Starfire Optical Range, Phillips Laboratory/LITE, Kirtland AFB, Albuquerque, N.M., 1992), Vol. 2, 511–521.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

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Figures (14)

Fig. 1
Fig. 1

Theoretical power times frequency normalized spectra for wave-front tip and tilt as a function of log(ν). The frequency ν is expressed in terms of v/R.

Fig. 2
Fig. 2

Theoretical power times frequency normalized spectra for defocus (j = 4) and astigmatism (j = 5 and 6) as a function of log(ν). The frequency ν is expressed in terms of v/R.

Fig. 3
Fig. 3

Theoretical power times frequency normalized spectra for the two coma components as a function of log(ν). The frequency ν is expressed in terms of v/R.

Fig. 4
Fig. 4

Theoretical power times frequency normalized spectra for spherical aberration as a function of log(ν). The frequency ν is expressed in terms of v/R.

Fig. 5
Fig. 5

Theoretical power times frequency normalized spectra for j = 4, 5, 6, 11 in a log–log scale. All the power times frequency spectra decrease as ν−14/3 for large ν values.

Fig. 6
Fig. 6

Mean-square compensation error as a function of the delay τ in terms of R/v, or angle α in terms of R/h, for total tip and tilt, defocus, total coma, and spherical aberration. The error is expressed as a percentage of the mode variance.

Fig. 7
Fig. 7

Same as Fig. 6, in a log–log scale showing that, for small values of the delay τ or angle α, the error increases at τ2 or α2.

Fig. 8
Fig. 8

Mean-square compensation error as a function of the delay τ in terms of R/v, or angle α in terms of R/h, for a given degree n of compensation. The error is now expressed as a percentage of the total wave-front error variance.

Fig. 9
Fig. 9

Schematic layout of the wave-front sensor. A star image is focused on a vibrating membrane mirror. When the membrane is not activated, an image of the telescope entrance pupil is formed on the detector array. When the membrane vibrates, the pupil image is sequentially defocused in opposite directions at a rate of several kilohertz.

Fig. 10
Fig. 10

Observed power times frequency normalized spectrum for defocus (points) and model fit (solid curve). ν is expressed in hertz.

Fig. 11
Fig. 11

Observed power times frequency normalized spectrum for spherical aberration (points) and model fit (solid curve). ν is expressed in hertz.

Fig. 12
Fig. 12

Sum of the two observed power times frequency normalized spectra for coma (points) and model fit (solid curve). ν is expressed in hertz.

Fig. 13
Fig. 13

Difference between the two observed power times frequency normalized spectra for coma (points) and model fit (solid curve). ν is expressed in hertz.

Fig. 14
Fig. 14

Set of impulses obtained by application of the clean deconvolution algorithm, showing possible various contributions to image degradation.

Tables (3)

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Table 1 Characteristic Time τ0 in (R/v) Units or Characteristic Angle α0 in (R/h) Units, as a Function of n, the Degree of Compensation

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Table 2 Relative Wave-front Fitting Error f2, and Maximum Acceptable Delay τmax in (R/v) Units, or Maximum Isoplanatic Angle αmax in (R/h) Units

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Table 3 Characteristics of the Observed Main Turbulent Layers

Equations (37)

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a j = z j ( r ) ϕ ( r ) d r .
s ( t ) = a ( v t ) = z ( r ) ϕ ( r - v t ) d r ,
a ( r ) = ϕ ( r ) * z ( r ) .
W a ( f ) = W ϕ ( f ) · W z ( f ) ,
W z ( f ) | J n + 1 ( 2 π f ) f | 2 { cos 2 m θ sin 2 m θ 1             ( m = 0 ) ,
W ϕ ( f ) f - 11 / 3 .
W a ( f ) f - 17 / 3 J n + 1 ( 2 π f ) 2 { cos 2 m θ sin 2 m θ 1             ( m = 0 ) .
C a ( ρ ) = a ( r ) a ( r + ρ ) .
C a ( ρ ) = W a ( f ) exp ( 2 i π ρ . f ) d f .
C ( τ ) = s ( t ) s ( t + τ ) = a ( v t ) a ( v t + v τ ) = C a ( v τ ) .
C a ( ρ ) = C a ( ξ , η ) .
C a ( ξ , η ) = W a ( f x , f y ) exp [ 2 i π ( ξ f x + η f y ) ] d f x d f y ,
C ( τ ) = C a ( v τ , 0 ) = W a ( f x , f y ) exp ( 2 i π v f x τ ) d f x d f y = d f x exp ( 2 i π v f x τ ) d f y W a ( f x , f y ) ,
C ( τ ) = 1 ν d ν exp ( 2 i π ν τ ) d f y W a ( ν / v , f y ) .
Φ ( ν ) = 1 v W a ( ν / v , f y ) d f y .
Φ ( ν ) 1 v d f y ( ν 2 v 2 + f y 2 ) - 17 / 6 J n + 1 [ 2 π ( ν 2 / v 2 + f y 2 ) 1 / 2 ] 2 × { cos 2 m θ sin 2 m θ 1             ( m = 0 ) .
ν Φ j ( ν ) Φ j ( ν ) d ν ,
e 2 ( τ ) = s ( t ) - s ( t + τ ) 2 = s ( t ) 2 + s ( t + τ ) 2 - 2 s ( t ) s ( t + τ ) ,
e 2 ( τ ) = 2 [ C ( 0 ) - C ( τ ) ] .
j 2 ( τ ) = 100 e j 2 ( τ ) σ j 2 = 200 [ C j ( 0 ) - C j ( τ ) ] C j ( 0 ) ,
n 2 ( τ ) = j = 1 ( n + 1 ) ( n + 2 ) / 2 ( σ j 2 / σ 2 ) j 2 ( τ ) ,
σ 2 = j = 1 σ j 2 .
2 ( Δ t ) = 6.68 ( v Δ t / D ) 5 / 3 ,
2 ( τ ) = 2.10 τ 5 / 3 .
e 2 ( τ ) = ( τ / τ 0 ) 2 σ 2 .
e 2 ( Δ t ) k e k 2 ( Δ t ) = k [ Δ t Δ t 0 ( k ) ] 2 σ k 2 ,
Δ t 0 ( k ) = τ 0 R / v k .
e 2 ( Δ t ) = ( Δ t / τ 0 R ) 2 k σ k 2 v k 2 = ( Δ t / τ 0 R ) 2 σ 2 v ¯ 2 ,
v ¯ = ( k σ k 2 v k 2 k σ k 2 ) 1 / 2
C ( α ) = C a ( α h ) ,
h ¯ = ( k σ k 2 h k 2 k σ k 2 ) 1 / 2 .
Φ c ( ν ) Φ s ( ν ) Φ 0 ( ν ) } 1 v d f y ( ν 2 / v 2 + f y 2 ) 17 / 6 J n + 1 [ 2 π ( ν 2 / v 2 + f y 2 ) 1 / 2 ] 2 × { cos 2 m ( θ - θ 0 ) sin 2 m ( θ - θ 0 ) 1 - ( m = 0 ) .
Φ c ( ν ) = Φ s ( ν ) = Φ 0 ( ν ) ,
Φ c ( ν ) - Φ s ( ν ) 1 v d f y ( ν 2 / v 2 + f y 2 ) - 17 / 6 × J n + 1 [ 2 π ( ν 2 / v 2 + f y 2 ) 1 / 2 ] 2 × cos [ 2 m ( θ - θ 0 ) ] .
cos [ 2 m ( θ - θ 0 ) ] = cos ( 2 m θ ) cos ( 2 m θ 0 ) + sin ( 2 m θ ) sin ( 2 m θ 0 ) .
Φ c ( ν ) - Φ s ( ν ) cos ( 2 m θ 0 ) 1 v d f y ( ν 2 / v 2 + f y 2 ) - 17 / 6 × J n + 1 [ 2 π ( ν 2 / v 2 + f y 2 ) 1 / 2 ] 2 × [ cos 2 ( m θ ) - sin 2 ( m θ ) ] .
Φ c ( ν ) - Φ s ( ν ) = cos ( 2 m θ 0 ) [ Φ x ( ν ) - Φ y ( ν ) ] ,

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