Abstract

Optimal modal Fourier-transform wavefront control combines the speed of Fourier-transform reconstruction (FTR) with real-time optimization of modal gains to form a fast, adaptive wavefront control scheme. Our modal basis is the real Fourier basis, which allows direct control of specific regions of the point-spread function. We formulate FTR as modal control and show how to measure custom filters. Because the Fourier basis is a tight frame, we can use it on a circular aperture for modal control even though it is not an orthonormal basis. The modal coefficients are available during reconstruction, greatly reducing computational overhead for gain optimization. Simulation results show significant improvements in performance in low-signal-to-noise-ratio situations compared with nonadaptive control. This scheme is computationally efficient enough to be implemented with off-the-shelf technology for a 2.5kHz, 64×64 adaptive optics system.

© 2005 Optical Society of America

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2004 (1)

2003 (4)

2002 (4)

1998 (2)

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

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

1995 (1)

E. Gendron, P. Léna, “Astronomical adaptive optics: II. Experimental results of an optimized modal control,” Astron. Astrophys., Suppl. Ser. 111, 153–167 (1995).

1994 (2)

1986 (1)

1983 (1)

Arsenault, R.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Boyer, C.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Brack, G. L.

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse-matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 1035–1044 (2002).

Brase, J. M.

Briggs, W. L.

W. L. Briggs, V. E. Henson, The DFT: An Owner’s Manual for the Discrete Fourier Transform (SIAM, 1995).

Buck, J. R.

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

Burruss, R. S.

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse-matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 1035–1044 (2002).

Charton, J.

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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Crampton, D.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Dekany, R. G.

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse-matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 1035–1044 (2002).

Dessenne, C.

Ellerbroek, B.

Ellerbroek, B. L.

Feautrier, 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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Fletcher, J.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Freischlad, K.

K. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
[CrossRef]

K. Freischlad, C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, pp. 74–80 (1985).

Fusco, T.

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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Gavel, D.

Gavel, D. T.

L. A. Poyneer, D. T. Gavel, 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, D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 890–901 (2002).

E. M. Johansson, D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE1237, pp. 372–383 (1994).
[CrossRef]

Gendron, E.

E. Gendron, P. Léna, “Astronomical adaptive optics: II. Experimental results of an optimized modal control,” Astron. Astrophys., Suppl. Ser. 111, 153–167 (1995).

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

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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Gigan, P.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Gilles, L.

Graham, J. R.

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

Henson, V. E.

W. L. Briggs, V. E. Henson, The DFT: An Owner’s Manual for the Discrete Fourier Transform (SIAM, 1995).

Hodge, P.

A. Sivaramakrishnan, J. P. Lloyd, P. Hodge, B. A. Macintosh, “Speckle decorrelation and dynamic range in speckle noise-limited imaging,” Astrophys. J. Lett. 581, L59–L62 (2002).
[CrossRef]

Howell, S. B.

S. B. Howell, Handbook of CCD Astronomy (Cambridge U. Press, 2000).

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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Jagourel, P.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

Johansson, E. M.

E. M. Johansson, D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE1237, pp. 372–383 (1994).
[CrossRef]

Jolissaint, L.

L. Jolissaint, Herzberg Institute of Astrophysics, Victoria, British Columbia, Canada (personal communication, 2004).

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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Koliopoulos, C. L.

K. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
[CrossRef]

K. Freischlad, C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, pp. 74–80 (1985).

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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Lai, O.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
[CrossRef]

F. J. Rigaut, J.-P. Véran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini and R. K. Tyson, eds., Proc. SPIE3353, pp. 1038–1048 (1998).
[CrossRef]

Léna, P.

E. Gendron, P. Léna, “Astronomical adaptive optics: II. Experimental results of an optimized modal control,” Astron. Astrophys., Suppl. Ser. 111, 153–167 (1995).

E. Gendron, 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, B. A. Macintosh, “Speckle decorrelation and dynamic range in speckle noise-limited imaging,” Astrophys. J. Lett. 581, L59–L62 (2002).
[CrossRef]

Loan, C. V.

Macintosh, B.

Macintosh, B. A.

A. Sivaramakrishnan, J. P. Lloyd, P. Hodge, B. A. Macintosh, “Speckle decorrelation and dynamic range in speckle noise-limited imaging,” Astrophys. J. Lett. 581, L59–L62 (2002).
[CrossRef]

MacMartin, D. G.

D. G. MacMartin, “Local, hierarchic, and iterative reconstructors for adaptive optics,” J. Opt. Soc. Am. A 20, 1084–1093 (2003).
[CrossRef]

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse-matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 1035–1044 (2002).

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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Madec, P.-Y.

Makidon, R. B.

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

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).

Mouillet, D.

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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Oppenheim, A. V.

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

Oppenheimer, B. R.

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

Perrin, M. D.

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

Pitsianis, N.

Plemmons, R.

Poyneer, L. A.

Puget, 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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Rabaud, D.

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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Rabou, 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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Rigaut, F.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
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F. J. Rigaut, J.-P. Véran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini and R. K. Tyson, eds., Proc. SPIE3353, pp. 1038–1048 (1998).
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F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
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C. Dessenne, P.-Y. Madec, G. Rousset, “Optimization of a predictive controller for closed-loop adaptive optics,” Appl. Opt. 37, 4623–4633 (1998).
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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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

Salmon, D.

F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
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A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, 1999).

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F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse-matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 1035–1044 (2002).

Sivaramakrishnan, A.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. Lett. 596, 702–712 (2003).
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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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

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F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
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F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
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F. Rigaut, D. Salmon, R. Arsenault, J. Thomas, O. Lai, D. Rouan, J.-P. Véran, P. Gigan, D. Crampton, J. Fletcher, J. Stilburn, C. Boyer, P. Jagourel, “Performance of the Canada–France–Hawaii Telescope adaptive optics bonnette,” Publ. Astron. Soc. Pac. 110, 152–164 (1998).
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J.-P. Véran, “Altair’s optimiser,” Tech. Rep. (Herzberg Institute of Astrophysics, Victoria, British Columbia, Canada, 1998).

M. J. Smith, J.-P. Véran, “Implementation of the Altair optimization processes,” in Adaptive Optics Systems and Technology II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 964–971 (2002).

F. J. Rigaut, J.-P. Véran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini and R. K. Tyson, eds., Proc. SPIE3353, pp. 1038–1048 (1998).
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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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
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A. Sivaramakrishnan, J. P. Lloyd, P. Hodge, B. A. Macintosh, “Speckle decorrelation and dynamic range in speckle noise-limited imaging,” Astrophys. J. Lett. 581, L59–L62 (2002).
[CrossRef]

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[CrossRef]

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J.-P. Véran, “Altair’s optimiser,” Tech. Rep. (Herzberg Institute of Astrophysics, Victoria, British Columbia, Canada, 1998).

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, G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 140–149 (2003).
[CrossRef]

D. T. Gavel, D. Wiberg, “Toward Strehl-optimizing adaptive optics controllers,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 890–901 (2002).

K. Freischlad, C. L. Koliopoulos, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, pp. 74–80 (1985).

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse-matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 1035–1044 (2002).

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

L. A. Poyneer, “Advanced techniques for Fourier transform wavefront reconstruction,” in Adaptive Optical System Technologies II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 1023–1033 (2002).

E. M. Johansson, D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE1237, pp. 372–383 (1994).
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W. L. Briggs, V. E. Henson, The DFT: An Owner’s Manual for the Discrete Fourier Transform (SIAM, 1995).

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

F. J. Rigaut, J.-P. Véran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini and R. K. Tyson, eds., Proc. SPIE3353, pp. 1038–1048 (1998).
[CrossRef]

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).

M. J. Smith, J.-P. Véran, “Implementation of the Altair optimization processes,” in Adaptive Optics Systems and Technology II, P. L. Wizinowich and D. Bonaccini, eds., Proc. SPIE4839, pp. 964–971 (2002).

S. B. Howell, Handbook of CCD Astronomy (Cambridge U. Press, 2000).

L. Jolissaint, Herzberg Institute of Astrophysics, Victoria, British Columbia, Canada (personal communication, 2004).

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Figures (6)

Fig. 1
Fig. 1

Frequency grid showing numbering and location of cosine and sine modes for N = 8 . The four dark squares at [0, 0], [ 0 , N 2 ] , [ N 2 , 0 ] , and [ N 2 , N 2 ] are only cosines. The light gray squares have both a sine mode and a cosine mode. Modes in white are matched by Hermitian symmetry with a numbered mode.

Fig. 2
Fig. 2

Square-aperture noise-propagation PSDs for each of the three filters. The mod-Hud and ideal filters are very similar, while the custom filter has higher gains at high spatial frequencies to compensate for the DM response. The highest frequency N 2 is located in the center of each filter.

Fig. 3
Fig. 3

Block diagram of the control system model. The input mode coefficient m is compensated via feedback on the error signal ϵ, which is measured in the presence of noise n. The WFS is approximated as a pure delay, and the response H ( z ) is the control law we wish to optimize.

Fig. 4
Fig. 4

Measurement and temporal bandwidth MSEs as a function of WFS SNR for the N = 48 case. Constant gains lead to uniform bandwidth error and inverse-square measurement error. Gain optimization allows for balancing out these two components to lower overall MSE.

Fig. 5
Fig. 5

Fourier modes (and hence locations in the PSF) can be controlled independently even on a circular aperture. Slices along the x-frequency axis of long-exposure PSDs of the residual phase error from the end-to-end simulation, with the effective k value of the gain filter used as the index. “Constant gain” is for uniform gain of 0.6 on all modes. “Optimal gain” is for optimal gains on all modes. “Mixed gain” has optimal gains for 4 k 12 and a constant 0.6 gain elsewhere. The mixed curve follows the constant PSD up to k = 3 , where it transitions to the optimal PSD and then again at k = 13 begins to increase. This demonstrates that specific modes can be controlled effectively despite their not being orthogonal.

Fig. 6
Fig. 6

Gain optimization compensates for lack of knowledge about the DM in the reconstruction filter. “Empirical” is determined by putting modes through the simulation. “Ratio of optimal gains” is the ratio of the steady-state optimal gains for mod-Hud and custom filters on the same frozen-flow input.

Equations (42)

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X [ k , l ] = 1 N m = 0 N 1 n = 0 N 1 x [ m , n ] exp [ j 2 π ( k m + l n ) N ] ,
C k , l [ m , n ] = 1 N cos [ 2 π N ( k m + l n ) ] .
C k , l [ m , n ] = 2 N cos [ 2 π N ( k m + l n ) ] ,
S k , l [ m , n ] = 2 N sin [ 2 π N ( k m + l n ) ] .
X [ k , l ] = D k , l m = 0 N 1 n = 0 N 1 x [ m , n ] ( C k , l [ m , n ] j S k , l [ m , n ] ) ,
x [ m , n ] , C k , l [ m , n ] = 1 D k , l Re { X [ k , l ] }
x [ m , n ] , S k , l [ m , n ] = 1 D k , l Im { X [ k , l ] } ,
[ A B B A C D D C ] .
1 A 2 + B 2 + C 2 + D 2 [ A B C D B A D C ] .
X [ k , l ] = D k , l ( x [ m , n ] , C k , l [ m , n ] j x [ m , n ] , S k , l [ m , n ] ) ,
P ̂ [ k , l ] = ( A + j B ) X [ k , l ] + ( C + j D ) Y [ k , l ] ( A + j B ) 2 + ( C + j D ) 2 .
X [ k , l ] = M x [ k , l ] P [ k , l ] ,
Y [ k , l ] = M y [ k , l ] P [ k , l ] ,
P ̂ [ k , l ] = M x * [ k , l ] X [ k , l ] + M y * [ k , l ] Y [ k , l ] M x [ k , l ] 2 + M y [ k , l ] 2 ,
M x [ k , l ] = exp ( j π l N ) [ exp ( j 2 π k N ) 1 ]
M y [ k , l ] = exp ( j π k N ) [ exp ( j 2 π l N ) 1 ]
P ̂ [ k , l ] = exp ( j π l N ) [ exp ( j 2 π k N ) 1 ] X [ k , l ] + exp ( j π k N ) [ exp ( j 2 π l N ) 1 ] Y [ k , l ] 4 [ sin 2 ( π k N ) + sin 2 ( π l N ) ] .
A = { 0 k = 0 cos ( 2 π k N ) 1 k 0 , l = 0 ( 2 π l N ) 1 { [ cos ( 2 π l N ) 1 ] sin ( 2 π k N ) + [ cos ( 2 π k N ) 1 ] sin ( 2 π l N ) } k 0 , l 0 } ,
B = { 0 k = 0 sin ( 2 π k N ) k 0 , l = 0 ( 2 π l N ) 1 { [ cos ( 2 π k N ) 1 ] [ cos ( 2 π l N ) 1 ] sin ( 2 π k N ) sin ( 2 π l N ) } k 0 , l 0 } .
C = { 0 l = 0 cos ( 2 π l N ) 1 l 0 , k = 0 ( 2 π k N ) 1 { [ cos ( 2 π l N ) 1 ] sin ( 2 π k N ) + [ cos ( 2 π k N ) 1 ] sin ( 2 π l N ) } l 0 , k 0 } ,
D = { 0 l = 0 sin ( 2 π l N ) l 0 , k = 0 ( 2 π k N ) 1 { [ cos ( 2 π k N ) 1 ] [ cos ( 2 π l N ) 1 ] sin ( 2 π k N ) sin ( 2 π l N ) } l 0 , k 0 } .
P ̂ [ k , l ] = 0 .
P ̂ [ k , 0 ] = [ exp ( j 2 π k N ) 1 ] X [ k , l ] 4 sin 2 ( π k N ) ,
P ̂ [ 0 , l ] = [ exp ( j 2 π l N ) 1 ] Y [ k , l ] 4 sin 2 ( π l N ) .
P ̂ [ k , l ] = j 8 π [ exp ( j 2 π k N ) 1 ] [ exp ( j 2 π l N ) 1 ] sin 2 ( π k N ) sin 2 ( π l N ) ( 1 k 2 + l 2 ) ( X [ k , l ] l + Y [ k , l ] k ) .
A , B > 0 such that A x 2 k P x , m k 2 B x 2 ,
m ̃ k ( A T A ) 1 m k .
x = k P x , m k m ̃ k .
S cl ( z ) M ( z ) = z 1 1 + z 1 H ( z ) ,
S cl ( z ) N ( z ) = 1 1 + z 1 H ( z ) .
S ol ( z ) M ( z ) = z 1 ,
S ol ( z ) N ( z ) = 1 .
S ol ( z ) S cl ( z ) = 1 + z 1 H ( z ) .
J = S ( ω ) d ω .
J = 1 1 + exp ( j ω ) H ( ω ) 2 [ M ( ω ) + N ( ω ) ] d ω .
M ̂ ( ω ) + N ̂ ( ω ) = 1 + exp ( j ω ) H 0 ( ω ) 2 S ̂ ( ω ) ,
argmin H ( ω ) { 1 1 + exp ( j ω ) H ( ω ) 2 }
{ 1 2 × 1 + exp ( j ω ) H 0 ( ω ) 2 S ̂ ( ω ) d ω } .
H ( z ) = g 1 c z 1 ,
argmin H ( z ) { 1 1 + exp ( j ω ) H ( ω ) 2 }
{ 1 2 × 1 + exp ( j ω ) H 0 ( ω ) 2 [ S ̂ S ( ω ) + S ̂ C ( ω ) ] d ω } .
SNR ( F , I ) = 0.146202 ( 10 9 0.4 I ) F ( 0.146202 ( 10 9 0.4 I ) F + 256 ) 1 2 ,

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