Abstract

Predictive Fourier control is a temporal power spectral density–based adaptive method for adaptive optics that predicts the atmosphere under the assumption of frozen flow. The predictive controller is based on Kalman filtering and a Fourier decomposition of atmospheric turbulence using the Fourier transform reconstructor. It provides a stable way to compensate for arbitrary numbers of atmospheric layers. For each Fourier mode, efficient and accurate algorithms estimate the necessary atmospheric parameters from closed-loop telemetry and determine the predictive filter, adjusting as conditions change. This prediction improves atmospheric rejection, leading to significant improvements in system performance. For a 48×48 actuator system operating at 2kHz, five-layer prediction for all modes is achievable in under 2×109 floating-point operations/s.

© 2007 Optical Society of America

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2006 (5)

L. A. Poyneer and B. A. Macintosh, "Optimal Fourier control performance and speckle behavior in high-contrast imaging with adaptive optics," Opt. Express 17, 7499-7514 (2006).
[CrossRef]

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

D. Wiberg, L. Johnson, and D. Gavel, "Adaptive optics control of wind blown turbulence via translation and prediction," Proc. SPIE 6272, 62722X (2006).
[CrossRef]

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, "First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO," Proc. SPIE 6272, 62721T (2006).
[CrossRef]

2005 (2)

M. Frigo and S. G. Johnson, "The design and implementation of FFTW3," Proc. IEEE 93, 216-231 (2005).
[CrossRef]

L. A. Poyneer and J.-P. Véran, "Optimal modal Fourier transform wave-front control," J. Opt. Soc. Am. A 22, 1515-1526 (2005).
[CrossRef]

2004 (2)

2003 (2)

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of high Strehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[CrossRef]

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

2002 (3)

L. A. Poyneer, D. T. Gavel, and J. M. Brase, "Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform," J. Opt. Soc. Am. A 19, 2100-2111 (2002).
[CrossRef]

D. T. Gavel and D. Wiberg, "Towards Strehl-optimizing adaptive optics controllers," Proc. SPIE 4839, 890-901 (2002).
[CrossRef]

A. Sivaramakrishnan, J. P. Lloyd, P. Hodge, and B. A. Macintosh, "Speckle decorrelation and dynamic range in speckle noise-limited imaging," Astrophys. J. 581, L59-62 (2002).
[CrossRef]

1998 (1)

1995 (1)

1994 (2)

E. M. Johansson and D. T. Gavel, "Simulation of stellar speckle imaging," Proc. SPIE 1237, pp. 372-383 (1994).
[CrossRef]

E. Gendron and P. Léna, "Astronomical adaptive optics I. Modal control optimization," Astron. Astrophys. 291, 337-347 (1994).

Arsenault, R.

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Baudoz, P.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

Bauman, B.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Beuzit, J.-L.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

Brase, J. M.

Buck, J. R.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, 1999).

Charton, J.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Conan, J.-M.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, "First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO," Proc. SPIE 6272, 62721T (2006).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, "Optimal control law for classical and multiconjugate adaptive optics," J. Opt. Soc. Am. A 21, 1261-1276 (2004).
[CrossRef]

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

Dessenne, C.

Dohlen, K.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

Downing, M.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

Doyon, R.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Erikson, D.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Evans, J.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Feautrier, P.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Fedrigo, E.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

Frigo, M.

M. Frigo and S. G. Johnson, "The design and implementation of FFTW3," Proc. IEEE 93, 216-231 (2005).
[CrossRef]

Fusco, T.

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, "First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO," Proc. SPIE 6272, 62721T (2006).
[CrossRef]

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, "Optimal control law for classical and multiconjugate adaptive optics," J. Opt. Soc. Am. A 21, 1261-1276 (2004).
[CrossRef]

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Gavel, D.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

D. Wiberg, L. Johnson, and D. Gavel, "Adaptive optics control of wind blown turbulence via translation and prediction," Proc. SPIE 6272, 62722X (2006).
[CrossRef]

Gavel, D. T.

D. T. Gavel and D. Wiberg, "Towards Strehl-optimizing adaptive optics controllers," Proc. SPIE 4839, 890-901 (2002).
[CrossRef]

L. A. Poyneer, D. T. Gavel, and J. M. Brase, "Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform," J. Opt. Soc. Am. A 19, 2100-2111 (2002).
[CrossRef]

E. M. Johansson and D. T. Gavel, "Simulation of stellar speckle imaging," Proc. SPIE 1237, pp. 372-383 (1994).
[CrossRef]

Gendron, E.

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

E. Gendron and P. Léna, "Astronomical adaptive optics I. Modal control optimization," Astron. Astrophys. 291, 337-347 (1994).

Gigan, P.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Graham, J.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Graham, J. R.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of high Strehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[CrossRef]

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1999).

Hassibi, B.

T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation (Prentice Hall, 2000).

Hodge, P.

A. Sivaramakrishnan, J. P. Lloyd, P. Hodge, and B. A. Macintosh, "Speckle decorrelation and dynamic range in speckle noise-limited imaging," Astrophys. J. 581, L59-62 (2002).
[CrossRef]

Hubin, N.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

Hubin, N. N.

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Johansson, E. M.

E. M. Johansson and D. T. Gavel, "Simulation of stellar speckle imaging," Proc. SPIE 1237, pp. 372-383 (1994).
[CrossRef]

Johnson, L.

D. Wiberg, L. Johnson, and D. Gavel, "Adaptive optics control of wind blown turbulence via translation and prediction," Proc. SPIE 6272, 62722X (2006).
[CrossRef]

Johnson, S. G.

M. Frigo and S. G. Johnson, "The design and implementation of FFTW3," Proc. IEEE 93, 216-231 (2005).
[CrossRef]

Kailath, T.

T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation (Prentice Hall, 2000).

Kasper, M.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

Kern, P. Y.

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Kulcsar, C.

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, "First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO," Proc. SPIE 6272, 62721T (2006).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, "Optimal control law for classical and multiconjugate adaptive optics," J. Opt. Soc. Am. A 21, 1261-1276 (2004).
[CrossRef]

Lacombe, F.

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Lagrange, A.

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Larkin, J.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Le Roux, B.

Léna, P.

E. Gendron and P. Léna, "Astronomical adaptive optics I. Modal control optimization," Astron. Astrophys. 291, 337-347 (1994).

Lloyd, J. P.

A. Sivaramakrishnan, J. P. Lloyd, P. Hodge, and B. A. Macintosh, "Speckle decorrelation and dynamic range in speckle noise-limited imaging," Astrophys. J. 581, L59-62 (2002).
[CrossRef]

Macintosh, B.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

L. A. Poyneer and B. Macintosh, "Spatially filtered wave-front sensor for high-order adaptive optics," J. Opt. Soc. Am. A 21, 810-819 (2004).
[CrossRef]

Macintosh, B. A.

L. A. Poyneer and B. A. Macintosh, "Optimal Fourier control performance and speckle behavior in high-contrast imaging with adaptive optics," Opt. Express 17, 7499-7514 (2006).
[CrossRef]

A. Sivaramakrishnan, J. P. Lloyd, P. Hodge, and B. A. Macintosh, "Speckle decorrelation and dynamic range in speckle noise-limited imaging," Astrophys. J. 581, L59-62 (2002).
[CrossRef]

Madec, P.

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Madec, P.-Y.

Makidon, R. B.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of high Strehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[CrossRef]

Montri, J.

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, "First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO," Proc. SPIE 6272, 62721T (2006).
[CrossRef]

Morzinski, K.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Mouillet, D.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Mugnier, L. M.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, 1999).

Oppenheimer, B.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Oppenheimer, B. R.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of high Strehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[CrossRef]

Palmer, D.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Perrin, M. D.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of high Strehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[CrossRef]

Petit, C.

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, "First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO," Proc. SPIE 6272, 62721T (2006).
[CrossRef]

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

Phillion, D.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Poyneer, L.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Poyneer, L. A.

Puget, P.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Rabaud, D.

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, "First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO," Proc. SPIE 6272, 62721T (2006).
[CrossRef]

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Rabou, P.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Raynaud, H.-F.

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, "First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO," Proc. SPIE 6272, 62721T (2006).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, "Optimal control law for classical and multiconjugate adaptive optics," J. Opt. Soc. Am. A 21, 1261-1276 (2004).
[CrossRef]

Rousset, G.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

C. Dessenne, P.-Y. Madec, and G. Rousset, "Optimization of a predictive controller for closed-loop adaptive optics," Appl. Opt. 37, 4623-4633 (1998).
[CrossRef]

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

Saddlemyer, L.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Sauvage, J.-F.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

Sayed, A. H.

T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation (Prentice Hall, 2000).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, 1999).

Sivaramakrishnan, A.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of high Strehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[CrossRef]

A. Sivaramakrishnan, J. P. Lloyd, P. Hodge, and B. A. Macintosh, "Speckle decorrelation and dynamic range in speckle noise-limited imaging," Astrophys. J. 581, L59-62 (2002).
[CrossRef]

Soummer, R.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Stadler, E.

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Thibault, S.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Véran, J.-P.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

L. A. Poyneer and J.-P. Véran, "Optimal modal Fourier transform wave-front control," J. Opt. Soc. Am. A 22, 1515-1526 (2005).
[CrossRef]

J.-P. Véran, "Altair's optimiser," Tech. Rep. (Herzberg Institute of Astrophysics, 1998).

Wallace, J. K.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Wiberg, D.

D. Wiberg, L. Johnson, and D. Gavel, "Adaptive optics control of wind blown turbulence via translation and prediction," Proc. SPIE 6272, 62722X (2006).
[CrossRef]

D. T. Gavel and D. Wiberg, "Towards Strehl-optimizing adaptive optics controllers," Proc. SPIE 4839, 890-901 (2002).
[CrossRef]

Wildi, F.

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

Zins, G.

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (1)

E. Gendron and P. Léna, "Astronomical adaptive optics I. Modal control optimization," Astron. Astrophys. 291, 337-347 (1994).

Astrophys. J. (2)

A. Sivaramakrishnan, J. P. Lloyd, P. Hodge, and B. A. Macintosh, "Speckle decorrelation and dynamic range in speckle noise-limited imaging," Astrophys. J. 581, L59-62 (2002).
[CrossRef]

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of high Strehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Express (1)

L. A. Poyneer and B. A. Macintosh, "Optimal Fourier control performance and speckle behavior in high-contrast imaging with adaptive optics," Opt. Express 17, 7499-7514 (2006).
[CrossRef]

Proc. IEEE (1)

M. Frigo and S. G. Johnson, "The design and implementation of FFTW3," Proc. IEEE 93, 216-231 (2005).
[CrossRef]

Proc. SPIE (7)

G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A. Lagrange, P. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
[CrossRef]

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, "First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO," Proc. SPIE 6272, 62721T (2006).
[CrossRef]

E. M. Johansson and D. T. Gavel, "Simulation of stellar speckle imaging," Proc. SPIE 1237, pp. 372-383 (1994).
[CrossRef]

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, "The Gemini Planet Imager," Proc. SPIE 6272, 62720L (2006).
[CrossRef]

T. Fusco, C. Petit, G. Rousset, J.-F. Sauvage, K. Dohlen, D. Mouillet, J. Charton, P. Baudoz, M. Kasper, E. Fedrigo, P. Rabou, P. Feautrier, M. Downing, P. Gigan, J.-M. Conan, J.-L. Beuzit, N. Hubin, F. Wildi, and P. Puget, "Design of the extreme AO system for SPHERE, the planet finder instrument of the VLT," Proc. SPIE 6272, 62720K (2006).
[CrossRef]

D. T. Gavel and D. Wiberg, "Towards Strehl-optimizing adaptive optics controllers," Proc. SPIE 4839, 890-901 (2002).
[CrossRef]

D. Wiberg, L. Johnson, and D. Gavel, "Adaptive optics control of wind blown turbulence via translation and prediction," Proc. SPIE 6272, 62722X (2006).
[CrossRef]

Other (5)

J.-P. Véran, "Altair's optimiser," Tech. Rep. (Herzberg Institute of Astrophysics, 1998).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1999).

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, 1999).

T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation (Prentice Hall, 2000).

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Figures (13)

Fig. 1
Fig. 1

Temporal PSD of the complex Fourier modal coefficient k = 12 , l = 12 , estimated from spatially filtered, tip–tilt removed open-loop telemetry. The five circles represent the characteristic frequencies due to layer motion. The other temporal-frequency content is due to the fact that the phase content changes slowly as the layer moves across the aperture.

Fig. 2
Fig. 2

For comparison with Fig. 1, temporal PSDs for an actuator and for Zernike mode n = 4 , m = 2 are shown. Simulation agrees with theory as to the power-law structure and cutoff frequencies. Unlike the Fourier modes, under frozen flow neither the actuators nor the Zernike coefficients have concentrated temporal PSDs. (Power normalized to display both curves on the same axis.)

Fig. 3
Fig. 3

Block diagram of the two-step delay AO control system. All variables are complex-valued time series of a specific Fourier mode. The input atmosphere ϕ is compensated in a closed loop. The AO system has access only to y, the noisy measurement of the residual error. The control law C ( z ) will be optimized.

Fig. 4
Fig. 4

Structure of implementation of predictive control law C ( z ) . The residual error is integrated separately for each layer, with compensation for each layer’s translation. The weighted sum is then high-pass filtered to produce the estimated phase one time step ahead. Use of this structure is necessary for stable implementation of the predictor in the AO control system (see Section 6).

Fig. 5
Fig. 5

Magnitude squared of error transfer functions for k = 12 , l = 12 for OFC and PFC with DC + 4 layers. Since the response is different for positive and negative f, the reversed response is also shown. Each identified layer that is predicted results in a notch. In contrast to OFC, PFC has much lower overshoot, but a lower bandwidth.

Fig. 6
Fig. 6

Estimated joint open-loop temporal PSDs from closed-loop data, along with the Kalman layer model fit to it, DC + 4 . Layer peak frequencies and power levels, as well as the noise level, are all correctly fit.

Fig. 7
Fig. 7

Magnitude–phase plot for two predictors at high and low SNRs. The phase and gain margins are indicated by the circles. Even at very high SNRs, the predictors exhibit robust phase margins greater than 45 deg .

Fig. 8
Fig. 8

Timeline of the PFC process. At every time step the phase is reconstructed and the control law is applied to each mode. Accumulation of modal coefficients into buffers begins immediately. After s time steps, the half-overlapped windowed periodograms begin. After ( a + 0.5 ) s steps, periodogram estimation is complete. The optimization process then occurs to produce the new predictive control law for each mode. This involves finding layer frequencies and strengths and solving the ARE.

Fig. 9
Fig. 9

Composite 12 s PSFs for I mag = 6 , 2000 Hz GPI case, five-layer atmosphere. White–black log color scale ranges from 1e-4 to 1e-6.

Fig. 10
Fig. 10

Slices along the y axis through the center of the three PSFs shown in Fig. 9. The PSF intensity from the uniform gain controller is 1.8 times that provided by OFC. This OFC intensity, in turn, is 3 times higher than that provided by PFC. Use of the predictor significantly reduces scattered light in the PSF.

Fig. 11
Fig. 11

Magnitude squared of noise transfer functions for k = 12 , l = 12 for OFC and PFC with DC + 4 layers. PFC provides better correction of the layer components (see Fig. 5) and propagates less WFS noise than OFC does.

Fig. 12
Fig. 12

Residual MSE in the controllable range of spatial frequencies for the three different control laws. Data points from simulations at magnitudes I mag = 0 to I mag = 9 .

Fig. 13
Fig. 13

Predictor improvement is particularly noticeable outside the inner working distance (IWD) of 5 λ D . (Compare with Fig. 12.).

Tables (2)

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Table 1 Layer Strengths and Velocity Vectors for a Five-Layer Test Case with a Total r 0 of 16 cm

Tables Icon

Table 2 Iterations for Solution of ARE a

Equations (37)

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F k , l [ x , y ] = exp ( j 2 π [ k x + l y ] N ) = cos ( j 2 π [ k x + l y ] N ) + j sin ( j 2 π [ k x + l y ] N ) .
ϕ ( x , y , t ) = cos ( 2 π N d [ k ( x v x t ) + l ( y v y t ) ] ) ,
Φ k , l ( t ) = 1 2 exp ( j 2 π t [ k v x + l v y ] N d ) .
a [ t ] = α a [ t 1 ] + w [ t ] ,
P a ( ω ) = σ a 2 1 α exp ( j ω ) 2 ,
a = ( a 0 , a 1 , , a L ) ,
R = Diag ( α 0 , α 1 , , α L ) .
P w = Diag ( σ a 0 2 , σ a 1 2 , , σ a L 2 ) .
x [ t ] = ( a [ t ] , ϕ [ t + 1 ] , ϕ [ t ] , ϕ [ t 1 ] , d [ t 1 ] , d [ t 2 ] ) T .
A = [ R 0 1 × ( L + 1 ) 0 1 × ( L + 1 ) 0 1 × ( L + 1 ) 0 1 × ( L + 1 ) 0 1 × ( L + 1 ) 1 ( L + 1 ) × 1 0 0 0 0 0 0 ( L + 1 ) × 1 1 0 0 0 0 0 ( L + 1 ) × 1 0 1 0 0 0 0 ( L + 1 ) × 1 0 0 0 0 0 0 ( L + 1 ) × 1 0 0 0 1 0 ] .
G = ( 0 ( L + 1 ) × 1 , 0 , 0 , 0 , 1 , 0 ) T
B = [ I ( L + 1 ) × ( L + 1 ) 0 ( L + 1 ) × 5 ] .
C = ( 0 ( L + 1 ) × 1 , 0 , 0 , 1 , 1 , 0 ) .
P s = A P s A H + B P w B H A P s C H ( C P s C H + P v ) 1 C P s A H .
K s = P s C H ( C P s C H + P v ) 1 .
x ̂ [ t t ] = ( I K s C ) A x ̂ [ t 1 t 1 ] + ( I K s C ) G d [ t 1 ] + K s y [ t ] .
A i z 2 = A i α i z + p L + 3 , i p L + 3 , L + 3 + σ v 2 ( E z 1 + M z 2 + Y ) .
M = z i = 0 L A i + p L + 3 , L + 1 p L + 3 , L + 3 + σ v 2 ( E z 1 + M z 2 + Y ) .
E = M z 1 + p L + 3 , L + 2 p L + 3 , L + 3 + σ v 2 ( E z 1 + M z 2 + Y ) .
p L + 3 , L + 1 = i = 0 L p L + 3 , i α i ,
p L + 3 , L + 2 = i = 0 L p L + 3 , i α i 2 ,
p L + 3 , L + 3 = i = 0 L p L + 3 , i α i 3 .
C ( z ) = Q 1 ( 1 + p L + 3 , L + 2 Q 1 z 1 ) i = 0 L α i 1 p L + 3 , i ( 1 α i z 1 ) ,
Q = p L + 3 , L + 3 + σ v 2 .
P cl ( ω ) = [ A ( ω ) + N ( ω ) ] 1 1 + z 2 C 0 ( z ) 2 .
P ̂ ol ( ω ) = P ̂ cl ( ω ) 1 + z 2 C 0 ( z ) 2 .
σ ̂ v 2 = Mean [ P ̂ ol ( ω hsf ) ] .
σ ̂ a 2 = P a ( ω ) , P ̂ ol ( ω ) P a ( ω ) , P a ( ω ) .
α = 1 ω 0 20 .
F 0 = A ,
P 0 = B P w B H ,
O 0 = C H P v 1 C .
T i = ( I + P i O i ) 1 ,
F i + 1 = F i T i F i ,
P i + 1 = P i + F i P i T i H F i H ,
O i + 1 = O i + F i H O i T i F i .
C ( z ) = i = 0 L n i z i 1 i = 1 L + 2 d i z i ,

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