Abstract

For closed-loop adaptive optics systems limited by time delay and measurement noise, we demonstrate that the ideal rejection transfer function is proportional to the frequency signal-to-noise ratio of the wave-front input. We describe a new modal linear predictive controller that approaches this ideal transfer function. Its parameters are optimized by minimization of the residual wave-front error with a modified recursive least-squares algorithm. The optimization can be performed with closed-loop data in the case of evolving turbulent conditions. We present numerical simulations to show the significant improvements brought by the predictor.

© 1998 Optical Society of America

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  1. G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).
  2. O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
    [CrossRef]
  3. E. Gendron, P. Léna, “Astronomical adaptive optics I. Modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).
  4. B. L. Ellerbroek, C. Van Loan, N. P. Pitsianis, R. J. Plemmons, “Optimizing closed-loop adaptive optics performance using multiple control bandwidths,” J. Opt. Soc. Am. A 11, 2871–2886 (1994).
    [CrossRef]
  5. C. Schwartz, G. Baum, E. N. Ribak, “Turbulence-degraded wave fronts as fractal surfaces,” J. Opt. Soc. Am. A 11, 444–451 (1994).
    [CrossRef]
  6. G. J. M. Aitken, D. McGaughey, “Predictability of atmospherically-distorted stellar wavefronts,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 89–94.
  7. V. Zuev, V. Lukin, “Dynamic characteristics of optical adaptive systems,” Appl. Opt. 26, 139–144 (1987).
    [CrossRef] [PubMed]
  8. M. B. Jorgenson, G. J. M. Aitken, “Prediction of atmospherically induced wave-front degradations,” Opt. Lett. 17, 466–468 (1992).
    [CrossRef] [PubMed]
  9. D. Bonaccini, D. Gallieni, R. Giampieretti, “Prediction of star wander path for adaptive optics,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 103–108.
  10. M. Lloyd-Hart, P. McGuire, “Spatio-temporal prediction for adaptive optics wavefront reconstructors,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 95–101.
  11. D. A. Montera, B. M. Welsh, M. C. Roggemann, D. W. Ruck, “Prediction of wave-front sensor slope measurements with artificial neural networks,” Appl. Opt. 36, 675–681 (1997).
    [CrossRef] [PubMed]
  12. R. N. Paschall, D. J. Anderson, “Linear quadratic Gaussian control of a deformable mirror adaptive optics system with time-delayed measurements,” Appl. Opt. 32, 6347–6358 (1993).
    [CrossRef] [PubMed]
  13. W. J. Wild, “Predictive optimal estimators for adaptive-optics systems,” Opt. Lett. 21, 1433–1435 (1996).
    [CrossRef] [PubMed]
  14. M. Demerlé, P.-Y. Madec, G. Rousset, “Servo-loop analysis for adaptive optics astronomy,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds., NATO ASI Series C423 (Kluwer, Dordrecht, The Netherlands, 1994), pp. 73–88.
    [CrossRef]
  15. G. F. Franklin, J. D. Powell, M. L. Workman, Digital Control of Dynamic Systems, 2nd ed. (Addison-Wesley, Reading, Mass., 1990).
  16. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1993), Chap. 9, pp. 460–468.
  17. F. Roddier, Institute for Astronomy, University of Hawaii, Honolulu, Haw. 96822 (personal communication, 1990).
  18. J. Makhoul, “Linear prediction,” Proc. IEEE 63, 561–581 (1975).
    [CrossRef]
  19. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1984), Chap. 11, pp. 337–350.
  20. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap. 10, pp. 305–352.
  21. G. Rousset, P.-Y. Madec, D. Rabaud, “Adaptive optics partial correction simulations for two telescope interferometry,” in Proceedings of the Conference on High-Resolution Imaging by Interferometry, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104.
  22. R. E. Hufnagel, “Propagation through atmospheric turbulence,” in The Infrared Handbook (Infrared Information Analysis Center, Washington, D.C., 1989), Chap. 6, pp. 6.1–6.56.
  23. M. G. Miller, P. L. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. SPIE75, 30–38 (1976).
    [CrossRef]
  24. 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]
  25. P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).
  26. C. Dessenne, P.-Y. Madec, G. Rousset, “Modal prediction for closed-loop adaptive optics,” Opt. Lett. 22, 1535–1537 (1997).
    [CrossRef]

1997 (3)

D. A. Montera, B. M. Welsh, M. C. Roggemann, D. W. Ruck, “Prediction of wave-front sensor slope measurements with artificial neural networks,” Appl. Opt. 36, 675–681 (1997).
[CrossRef] [PubMed]

P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).

C. Dessenne, P.-Y. Madec, G. Rousset, “Modal prediction for closed-loop adaptive optics,” Opt. Lett. 22, 1535–1537 (1997).
[CrossRef]

1996 (1)

1995 (1)

1994 (3)

1993 (1)

R. N. Paschall, D. J. Anderson, “Linear quadratic Gaussian control of a deformable mirror adaptive optics system with time-delayed measurements,” Appl. Opt. 32, 6347–6358 (1993).
[CrossRef] [PubMed]

1992 (1)

1990 (1)

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

1987 (1)

1975 (1)

J. Makhoul, “Linear prediction,” Proc. IEEE 63, 561–581 (1975).
[CrossRef]

Aitken, G. J. M.

M. B. Jorgenson, G. J. M. Aitken, “Prediction of atmospherically induced wave-front degradations,” Opt. Lett. 17, 466–468 (1992).
[CrossRef] [PubMed]

G. J. M. Aitken, D. McGaughey, “Predictability of atmospherically-distorted stellar wavefronts,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 89–94.

Anderson, D. J.

R. N. Paschall, D. J. Anderson, “Linear quadratic Gaussian control of a deformable mirror adaptive optics system with time-delayed measurements,” Appl. Opt. 32, 6347–6358 (1993).
[CrossRef] [PubMed]

Arsenault, R.

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Baum, G.

Bonaccini, D.

D. Bonaccini, D. Gallieni, R. Giampieretti, “Prediction of star wander path for adaptive optics,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 103–108.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1993), Chap. 9, pp. 460–468.

Boyer, C.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Conan, J.-M.

P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).

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]

Crampton, D.

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Demerlé, M.

M. Demerlé, P.-Y. Madec, G. Rousset, “Servo-loop analysis for adaptive optics astronomy,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds., NATO ASI Series C423 (Kluwer, Dordrecht, The Netherlands, 1994), pp. 73–88.
[CrossRef]

Dessenne, C.

Ellerbroek, B. L.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap. 10, pp. 305–352.

Fletcher, J.

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Fleury, B.

P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).

Fontanella, J.-C.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Franklin, G. F.

G. F. Franklin, J. D. Powell, M. L. Workman, Digital Control of Dynamic Systems, 2nd ed. (Addison-Wesley, Reading, Mass., 1990).

Gaffard, J. P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Gallieni, D.

D. Bonaccini, D. Gallieni, R. Giampieretti, “Prediction of star wander path for adaptive optics,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 103–108.

Gendron, E.

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

Giampieretti, R.

D. Bonaccini, D. Gallieni, R. Giampieretti, “Prediction of star wander path for adaptive optics,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 103–108.

Gigan, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Hanson, D.

M. G. Miller, P. L. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. SPIE75, 30–38 (1976).
[CrossRef]

Hufnagel, R. E.

R. E. Hufnagel, “Propagation through atmospheric turbulence,” in The Infrared Handbook (Infrared Information Analysis Center, Washington, D.C., 1989), Chap. 6, pp. 6.1–6.56.

Jagourel, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Jorgenson, M. B.

Kern, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Lacombe, F.

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Lai, O.

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Léna, P.

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

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Lloyd-Hart, M.

M. Lloyd-Hart, P. McGuire, “Spatio-temporal prediction for adaptive optics wavefront reconstructors,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 95–101.

Lukin, V.

Madec, P.-Y.

P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).

C. Dessenne, P.-Y. Madec, G. Rousset, “Modal prediction for closed-loop adaptive optics,” Opt. Lett. 22, 1535–1537 (1997).
[CrossRef]

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]

M. Demerlé, P.-Y. Madec, G. Rousset, “Servo-loop analysis for adaptive optics astronomy,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds., NATO ASI Series C423 (Kluwer, Dordrecht, The Netherlands, 1994), pp. 73–88.
[CrossRef]

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

Makhoul, J.

J. Makhoul, “Linear prediction,” Proc. IEEE 63, 561–581 (1975).
[CrossRef]

McGaughey, D.

G. J. M. Aitken, D. McGaughey, “Predictability of atmospherically-distorted stellar wavefronts,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 89–94.

McGuire, P.

M. Lloyd-Hart, P. McGuire, “Spatio-temporal prediction for adaptive optics wavefront reconstructors,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 95–101.

Mendez, F.

P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).

Merkle, F.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Michau, V.

P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).

Miller, M. G.

M. G. Miller, P. L. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. SPIE75, 30–38 (1976).
[CrossRef]

Montera, D. A.

Montri, J.

P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1984), Chap. 11, pp. 337–350.

Paschall, R. N.

R. N. Paschall, D. J. Anderson, “Linear quadratic Gaussian control of a deformable mirror adaptive optics system with time-delayed measurements,” Appl. Opt. 32, 6347–6358 (1993).
[CrossRef] [PubMed]

Pitsianis, N. P.

Plemmons, R. J.

Powell, J. D.

G. F. Franklin, J. D. Powell, M. L. Workman, Digital Control of Dynamic Systems, 2nd ed. (Addison-Wesley, Reading, Mass., 1990).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap. 10, pp. 305–352.

Rabaud, D.

P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).

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

Ribak, E. N.

Rigaut, F.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Roddier, F.

F. Roddier, Institute for Astronomy, University of Hawaii, Honolulu, Haw. 96822 (personal communication, 1990).

Roggemann, M. C.

Rouan, D.

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Rousset, G.

P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).

C. Dessenne, P.-Y. Madec, G. Rousset, “Modal prediction for closed-loop adaptive optics,” Opt. Lett. 22, 1535–1537 (1997).
[CrossRef]

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, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

M. Demerlé, P.-Y. Madec, G. Rousset, “Servo-loop analysis for adaptive optics astronomy,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds., NATO ASI Series C423 (Kluwer, Dordrecht, The Netherlands, 1994), pp. 73–88.
[CrossRef]

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

Rousset-Rouvière, L.

P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).

Ruck, D. W.

Salmon, D.

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Schwartz, C.

Séchaud, M.

P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).

Stilburn, J.

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap. 10, pp. 305–352.

Thomas, J.

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Van Loan, C.

Véran, J.-P.

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap. 10, pp. 305–352.

Welsh, B. M.

Wild, W. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1993), Chap. 9, pp. 460–468.

Workman, M. L.

G. F. Franklin, J. D. Powell, M. L. Workman, Digital Control of Dynamic Systems, 2nd ed. (Addison-Wesley, Reading, Mass., 1990).

Zieske, P. L.

M. G. Miller, P. L. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. SPIE75, 30–38 (1976).
[CrossRef]

Zuev, V.

Appl. Opt. (1)

R. N. Paschall, D. J. Anderson, “Linear quadratic Gaussian control of a deformable mirror adaptive optics system with time-delayed measurements,” Appl. Opt. 32, 6347–6358 (1993).
[CrossRef] [PubMed]

Appl. Opt. (2)

Astron. Astrophys. (2)

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

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

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

Lett. Observat. Haute Provence (1)

P.-Y. Madec, D. Rabaud, B. Fleury, J.-M. Conan, L. Rousset-Rouvière, F. Mendez, J. Montri, V. Michau, G. Rousset, M. Séchaud, “Essais du banc d’optique adaptative ONERA à l’Observatoire de Haute Provence,” Lett. Observat. Haute Provence 16, 2–8 (1997).

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[CrossRef]

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A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1984), Chap. 11, pp. 337–350.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), Chap. 10, pp. 305–352.

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

R. E. Hufnagel, “Propagation through atmospheric turbulence,” in The Infrared Handbook (Infrared Information Analysis Center, Washington, D.C., 1989), Chap. 6, pp. 6.1–6.56.

M. G. Miller, P. L. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. SPIE75, 30–38 (1976).
[CrossRef]

M. Demerlé, P.-Y. Madec, G. Rousset, “Servo-loop analysis for adaptive optics astronomy,” in Adaptive Optics for Astronomy, D. M. Alloin, J.-M. Mariotti, eds., NATO ASI Series C423 (Kluwer, Dordrecht, The Netherlands, 1994), pp. 73–88.
[CrossRef]

G. F. Franklin, J. D. Powell, M. L. Workman, Digital Control of Dynamic Systems, 2nd ed. (Addison-Wesley, Reading, Mass., 1990).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1993), Chap. 9, pp. 460–468.

F. Roddier, Institute for Astronomy, University of Hawaii, Honolulu, Haw. 96822 (personal communication, 1990).

D. Bonaccini, D. Gallieni, R. Giampieretti, “Prediction of star wander path for adaptive optics,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 103–108.

M. Lloyd-Hart, P. McGuire, “Spatio-temporal prediction for adaptive optics wavefront reconstructors,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 95–101.

G. J. M. Aitken, D. McGaughey, “Predictability of atmospherically-distorted stellar wavefronts,” in Proceedings of Topical Meeting on Adaptive Optics, M. Cullum, ed. (European Southern Observatory, Garching, Germany, 1995), Vol. 54, pp. 89–94.

O. Lai, J.-P. Véran, F. Rigaut, D. Rouan, P. Gigan, F. Lacombe, P. Léna, R. Arsenault, D. Salmon, J. Thomas, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “CFHT adaptive optics: first results at the telescope,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 859–870 (1996).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Block diagram of an AO system. D/A, digital-to-analog.

Fig. 2
Fig. 2

Discrete block diagram of an AO system for τ = T.

Fig. 3
Fig. 3

Theoretical spectral SNRω of a turbulent tilt mode perturbed by white noise. Conditions: SNR = 20, f s = 100 Hz, / D = 2 Hz.

Fig. 4
Fig. 4

Nyquist diagram for τ = T. ℋ and 1 + ℋ have been drawn in the complex plane to indicate that part of 1 + ℋ is always inside the unit circle if ℋ does not encircle the (-1, 0) point.

Fig. 5
Fig. 5

Convergence of the predictor parameters for the third Karhunen–Loeve mode. Conditions: τ = T, f s = 100 Hz, / D = 2 Hz, p = 4, q = 3.

Fig. 6
Fig. 6

Spectral SNRω for the third mode. ‖1 + ℋ‖2 for a predictor and an integral controller with τ = T and τ = 0 are superimposed. Conditions: SNR = 20, f s = 100 Hz, / D = 2 Hz, p = 4, q = 3.

Fig. 7
Fig. 7

Gain η versus SNR for two different sampling frequencies. Conditions: τ = T, / D = 2 Hz, p = 4, q = 3.

Fig. 8
Fig. 8

Strehl ratio versus SNR for two different sampling frequencies. Comparison between the predictor and the integral controller is shown. Conditions: τ = T, / D = 2 Hz, D/ r 0 = 10, p = 4, q = 3.

Fig. 9
Fig. 9

Modal gain ηmode versus Zernike mode number. A linear regression for modes other than the two tilts is superimposed. Conditions: SNR = 50, f s = 50 Hz, τ = T, / D = 2 Hz, p = 4, q = 3.

Fig. 10
Fig. 10

Spectral SNR w and optimal ‖1 + ℋ‖2 for a tilt mode. A vibration at 25 Hz has been introduced. Conditions: SNR = 20, f s = 100 Hz, τ = T, / D = 2 Hz, p = 5, q = 4.

Equations (25)

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s = S s s C z = e + Ts D s = 1 - e - Ts Ts 2 e - τ s C z = e + Ts
z = z - 2 C z .
z = z - l m 1 - z - 1 + z - 1 C z .
e ˜ meas = S e ˜ + ñ ,
˜ meas = S e ˜ - C D ˜ meas + ñ .
˜ meas = e ˜ meas 1 + .
C z = i = 0 , q - 1   a i z - i 1 + i = 1 , p - 1   b i z - i ,
J = 2 = ω   1 1 + 2 e ˜ j ω 2 d ω + ω   C D 1 + 2 ñ j ω 2 d ω .
meas 2 = 2 + n 2 .
J = meas 2 = ω   1 1 + 2 e ˜ meas j ω 2 d ω ,
ω ,   ˜ meas j ω 2 = n 2 / 2 π ,
ω ,   1 + z = e + j ω 2 = e ˜ meas j ω 2 n 2 / 2 π .
ω ,   C z = e + j ω 2 = e ˜ j ω 2 n 2 / 2 π ,
1 + z = 1 + z - l m 1 - z - 1 + z - 1 C z .
ω ,   1 + z = e + j ω 2 1 .
J = k   meas 2 k .
meas k = - i = 1 p - 1   b i meas k - i - i = 0 q - 1   a i meas k - i - 2 + e meas k + i = 1 p - 1   b i e meas k - i .
y ˜ = - i = 1 p - 1   b i z - i y ˜ - z - 2 i = 0 q - 1   a i z - i y ˜ - e ˜ meas ,
meas = e meas - y .
θ = b 1     b p - 1 ,   a 0     a q - 1 T , ϕ i = y i - 1     y i - p + 1 ,
y - e meas i - 2     y - e meas i - q - 1 ] ,
y i = - θ · ϕ i , meas i = e meas i + θ · ϕ i , J = i = 0 N s - 1   meas 2 i ,
θ i = b 1 i       b p - 1 i ,   a 0 i     a q - 1 i T , ϕ i = y alg i - 1     y alg i - p + 1 ,   y alg - e meas × i - 2     y alg - e meas i - q - 1 , y alg i = - θ i · ϕ i , alg i = e meas i + θ i · ϕ i , J alg i = k i   meas 2 k .
ϕ 2 = mode = 2 mode = 45   mode 2 + mode = 46 mode = 861   e mode 2 ,
η = 1 - ϕ 2 pred / ϕ 2 integ , η mode = 1 - mode 2 pred / mode 2 integ ,

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