Abstract

This paper discusses the performance and cost of two computationally efficient Fourier-based tomographic wavefront reconstruction algorithms for wide-field laser guide star (LGS) adaptive optics (AO). The first algorithm is the iterative Fourier domain preconditioned conjugate gradient (FDPCG) algorithm developed by Yang et al. [Appl. Opt. 45, 5281 (2006)], combined with pseudo-open-loop control (POLC). FDPCG’s computational cost is proportional to N log(N), where N denotes the dimensionality of the tomography problem. The second algorithm is the distributed Kalman filter (DKF) developed by Massioni et al. [J. Opt. Soc. Am. A 28, 2298 (2011)], which is a noniterative spatially invariant controller. When implemented in the Fourier domain, DKF’s cost is also proportional to N log(N). Both algorithms are capable of estimating spatial frequency components of the residual phase beyond the wavefront sensor (WFS) cutoff frequency thanks to regularization, thereby reducing WFS spatial aliasing at the expense of more computations. We present performance and cost analyses for the LGS multiconjugate AO system under design for the Thirty Meter Telescope, as well as DKF’s sensitivity to uncertainties in wind profile prior information. We found that, provided the wind profile is known to better than 10% wind speed accuracy and 20 deg wind direction accuracy, DKF, despite its spatial invariance assumptions, delivers a significantly reduced wavefront error compared to the static FDPCG minimum variance estimator combined with POLC. Due to its nonsequential nature and high degree of parallelism, DKF is particularly well suited for real-time implementation on inexpensive off-the-shelf graphics processing units.

© 2013 Optical Society of America

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2012

R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
[CrossRef]

L. Wang and B. Ellerbroek, “Computer simulations and real-time control of ELT AO systems using graphical processing units,” Proc. SPIE 8447, 844723 (2012).
[CrossRef]

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48, 1939–1954 (2012).
[CrossRef]

2011

2010

2008

2007

2006

2004

2002

2001

Ammans, P.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Andersen, D.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Anderson, B. D. O.

B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods (Prentice-Hall, 1990).

Atwood, J.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Baruffolo, A.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Béchet, C.

M. Tallon, C. Béchet, I. Tallon-Bosc, M. Le Louarn, E. Thiébaut, R. Clare, and E. Marchetti, “Performance of MCAO on the E-ELT using the fractal iterative method for fast atmospheric tomography,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Bellazzini, M.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Bouchez, A. H.

R. Conan, B. Espeland, M. A. Van Dam, and A. H. Bouchez, “Design of the laser tomography adaptive optics system for the giant Magellan telescope,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Boyer, C.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Bregoli, G.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Butler, C. R.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Byrnes, P.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Caputa, K.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Ciliegi, P.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Clare, R.

M. Tallon, C. Béchet, I. Tallon-Bosc, M. Le Louarn, E. Thiébaut, R. Clare, and E. Marchetti, “Performance of MCAO on the E-ELT using the fractal iterative method for fast atmospheric tomography,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Clarke, F.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Clenet, Y.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Cohen, M.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Conan, J. M.

Conan, J.-M.

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48, 1939–1954 (2012).
[CrossRef]

Conan, R.

R. Conan, B. Espeland, M. A. Van Dam, and A. H. Bouchez, “Design of the laser tomography adaptive optics system for the giant Magellan telescope,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Correia, C.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Cosentino, G.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Davies, R. I.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

de Lesegno, P. V.

Delabre, B.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Diolaiti, E.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Doelman, N.

Dournaux, J. L.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Dunn, J.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Ellerbroek, B.

L. Wang and B. Ellerbroek, “Computer simulations and real-time control of ELT AO systems using graphical processing units,” Proc. SPIE 8447, 844723 (2012).
[CrossRef]

L. Wang, L. Gilles, and B. Ellerbroek, “Analysis of the improvement in sky coverage for multiconjugate adaptive optics systems obtained using minimum variance split tomography,” Appl. Opt. 50, 3000–3010 (2011).
[CrossRef]

L. Gilles and B. Ellerbroek, “Split atmospheric tomography using laser and natural guide stars,” J. Opt. Soc. Am. A 25, 2427–2435 (2008).
[CrossRef]

L. Gilles and B. Ellerbroek, “Constrained matched filtering for extended dynamic range and improved noise rejection for Shack–Hartmann wavefront sensing,” Opt. Lett. 33, 1159–1161 (2008).
[CrossRef]

L. Gilles, B. Ellerbroek, and J. P. Veran, “Laser guide star multi-conjugate adaptive optics performance of the thirty meter telescope with elongated beacons and matched filtering,” Proc. SPIE 6272, 627236 (2006).
[CrossRef]

Q. Yang, C. Vogel, and B. Ellerbroek, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. 45, 5281–5293 (2006).
[CrossRef]

B. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors using sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
[CrossRef]

B. Ellerbroek and F. Rigaut, “Methods for correcting tilt anisoplanatism in laser guide star based multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2539–2547 (2001).
[CrossRef]

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

L. Gilles and B. Ellerbroek, “LAOS: linear adaptive optics simulator,” Thirty Meter Telescope Observatory Corporation, Document-7988, TMT.AOS.TEC.07.003 (2007).

Ellerbroek, E.

L. Gilles, L. Wang, and E. Ellerbroek, “Minimum variance split tomography for laser guide star adaptive optics,” Eur. J. Control 17, 327–334 (2011).
[CrossRef]

Espeland, B.

R. Conan, B. Espeland, M. A. Van Dam, and A. H. Bouchez, “Design of the laser tomography adaptive optics system for the giant Magellan telescope,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Ferrari, M.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Fitzsimmons, J.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Foppiani, I.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Fraanje, R.

Fusco, T.

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

T. Fusco, J. M. Conan, G. Rousset, L. M. Mugnier, and V. Michau, “Optimal wave-front reconstruction strategies for multi-conjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2527–2538 (2001).
[CrossRef]

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Gilles, L.

L. Gilles, L. Wang, and E. Ellerbroek, “Minimum variance split tomography for laser guide star adaptive optics,” Eur. J. Control 17, 327–334 (2011).
[CrossRef]

L. Wang, L. Gilles, and B. Ellerbroek, “Analysis of the improvement in sky coverage for multiconjugate adaptive optics systems obtained using minimum variance split tomography,” Appl. Opt. 50, 3000–3010 (2011).
[CrossRef]

L. Gilles and B. Ellerbroek, “Split atmospheric tomography using laser and natural guide stars,” J. Opt. Soc. Am. A 25, 2427–2435 (2008).
[CrossRef]

L. Gilles and B. Ellerbroek, “Constrained matched filtering for extended dynamic range and improved noise rejection for Shack–Hartmann wavefront sensing,” Opt. Lett. 33, 1159–1161 (2008).
[CrossRef]

L. Gilles, B. Ellerbroek, and J. P. Veran, “Laser guide star multi-conjugate adaptive optics performance of the thirty meter telescope with elongated beacons and matched filtering,” Proc. SPIE 6272, 627236 (2006).
[CrossRef]

L. Gilles and B. Ellerbroek, “LAOS: linear adaptive optics simulator,” Thirty Meter Telescope Observatory Corporation, Document-7988, TMT.AOS.TEC.07.003 (2007).

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Gratadour, D.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Herriot, G.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Hickson, P.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

T. Pfrommer and P. Hickson, “Properties and dynamics of mesospheric sodium and the impact on sodium LGS AO systems,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Hill, A.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Hubin, N.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Jagourel, P.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Kulcsár, C.

Le Louarn, M.

M. Tallon, C. Béchet, I. Tallon-Bosc, M. Le Louarn, E. Thiébaut, R. Clare, and E. Marchetti, “Performance of MCAO on the E-ELT using the fractal iterative method for fast atmospheric tomography,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Le Roux, B.

Lombini, M.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Marchetti, E.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

M. Tallon, C. Béchet, I. Tallon-Bosc, M. Le Louarn, E. Thiébaut, R. Clare, and E. Marchetti, “Performance of MCAO on the E-ELT using the fractal iterative method for fast atmospheric tomography,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Massioni, P.

Meimon, S.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Michau, V.

T. Fusco, J. M. Conan, G. Rousset, L. M. Mugnier, and V. Michau, “Optimal wave-front reconstruction strategies for multi-conjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2527–2538 (2001).
[CrossRef]

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Moore, J. B.

B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods (Prentice-Hall, 1990).

Mugnier, L.

Mugnier, L. M.

Paufique, J.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Pazder, J.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Petit, C.

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48, 1939–1954 (2012).
[CrossRef]

C. Kulcsár, H. F. Raynaud, C. Petit, J. M. Conan, J. M. Conan, and P. V. de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14, 7464–7564(2006).
[CrossRef]

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Pfrommer, T.

T. Pfrommer and P. Hickson, “Properties and dynamics of mesospheric sodium and the impact on sodium LGS AO systems,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Piatrou, P.

Ramlau, R.

R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
[CrossRef]

Raynaud, H. F.

Raynaud, H.-F.

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48, 1939–1954 (2012).
[CrossRef]

Reshetov, V.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Rice, J.

Rigaut, F.

Robert, C.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Roggemann, M.

Rosensteiner, M.

R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
[CrossRef]

Rousset, G.

Schnetler, H.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Schreiber, L.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Smith, M.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Tallon, M.

E. Thiébaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescopes,” J. Opt. Soc. Am. A 27, 1046–1059 (2010).
[CrossRef]

M. Tallon, C. Béchet, I. Tallon-Bosc, M. Le Louarn, E. Thiébaut, R. Clare, and E. Marchetti, “Performance of MCAO on the E-ELT using the fractal iterative method for fast atmospheric tomography,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Tallon-Bosc, I.

M. Tallon, C. Béchet, I. Tallon-Bosc, M. Le Louarn, E. Thiébaut, R. Clare, and E. Marchetti, “Performance of MCAO on the E-ELT using the fractal iterative method for fast atmospheric tomography,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Tecza, M.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Thatte, N.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Thiébaut, E.

E. Thiébaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescopes,” J. Opt. Soc. Am. A 27, 1046–1059 (2010).
[CrossRef]

M. Tallon, C. Béchet, I. Tallon-Bosc, M. Le Louarn, E. Thiébaut, R. Clare, and E. Marchetti, “Performance of MCAO on the E-ELT using the fractal iterative method for fast atmospheric tomography,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Tokovinin, A.

Van Dam, M. A.

R. Conan, B. Espeland, M. A. Van Dam, and A. H. Bouchez, “Design of the laser tomography adaptive optics system for the giant Magellan telescope,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Veran, J. P.

L. Gilles, B. Ellerbroek, and J. P. Veran, “Laser guide star multi-conjugate adaptive optics performance of the thirty meter telescope with elongated beacons and matched filtering,” Proc. SPIE 6272, 627236 (2006).
[CrossRef]

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Verhaegen, M.

Viard, E.

Vogel, C.

Vogel, C. R.

C. R. Vogel, “Sparse matrix methods for wavefront reconstruction revisited,” Proc. SPIE 5490, 1327–1335 (2004).
[CrossRef]

Wang, L.

L. Wang and B. Ellerbroek, “Computer simulations and real-time control of ELT AO systems using graphical processing units,” Proc. SPIE 8447, 844723 (2012).
[CrossRef]

L. Wang, L. Gilles, and B. Ellerbroek, “Analysis of the improvement in sky coverage for multiconjugate adaptive optics systems obtained using minimum variance split tomography,” Appl. Opt. 50, 3000–3010 (2011).
[CrossRef]

L. Gilles, L. Wang, and E. Ellerbroek, “Minimum variance split tomography for laser guide star adaptive optics,” Eur. J. Control 17, 327–334 (2011).
[CrossRef]

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Welsh, B.

M. Roggemann and B. Welsh, Imaging Through Turbulence (CRC, 1996).

Wevers, I.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

Yang, Q.

Appl. Opt.

Automatica

C. Kulcsár, H.-F. Raynaud, C. Petit, and J.-M. Conan, “Minimum variance prediction and control for adaptive optics,” Automatica 48, 1939–1954 (2012).
[CrossRef]

Eur. J. Control

L. Gilles, L. Wang, and E. Ellerbroek, “Minimum variance split tomography for laser guide star adaptive optics,” Eur. J. Control 17, 327–334 (2011).
[CrossRef]

Inverse Probl.

R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
[CrossRef]

J. Opt. Soc. Am. A

R. Fraanje, J. Rice, M. Verhaegen, and N. Doelman, “Fast reconstruction and prediction of frozen flow turbulence on structured Kalman filtering,” J. Opt. Soc. Am. A 27, A235–A245 (2010).
[CrossRef]

P. Massioni, C. Kulcsár, H. F. Raynaud, and J. M. Conan, “Fast computation of an optimal controller for large-scale adaptive optics,” J. Opt. Soc. Am. A 28, 2298–2309 (2011).
[CrossRef]

T. Fusco, J. M. Conan, G. Rousset, L. M. Mugnier, and V. Michau, “Optimal wave-front reconstruction strategies for multi-conjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2527–2538 (2001).
[CrossRef]

L. Gilles and B. Ellerbroek, “Split atmospheric tomography using laser and natural guide stars,” J. Opt. Soc. Am. A 25, 2427–2435 (2008).
[CrossRef]

B. Ellerbroek and F. Rigaut, “Methods for correcting tilt anisoplanatism in laser guide star based multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2539–2547 (2001).
[CrossRef]

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

E. Thiébaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescopes,” J. Opt. Soc. Am. A 27, 1046–1059 (2010).
[CrossRef]

B. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors using sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
[CrossRef]

A. Tokovinin and E. Viard, “Limiting precision of tomographic phase estimation,” J. Opt. Soc. Am. A 18, 873–882 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

C. R. Vogel, “Sparse matrix methods for wavefront reconstruction revisited,” Proc. SPIE 5490, 1327–1335 (2004).
[CrossRef]

L. Gilles, B. Ellerbroek, and J. P. Veran, “Laser guide star multi-conjugate adaptive optics performance of the thirty meter telescope with elongated beacons and matched filtering,” Proc. SPIE 6272, 627236 (2006).
[CrossRef]

L. Wang and B. Ellerbroek, “Computer simulations and real-time control of ELT AO systems using graphical processing units,” Proc. SPIE 8447, 844723 (2012).
[CrossRef]

Other

“AO tomography workshop 2012: presentations and discussions,” available online http://www.dur.ac.uk/cfai/adaptiveoptics/aotw12/agenda/ .

T. Pfrommer and P. Hickson, “Properties and dynamics of mesospheric sodium and the impact on sodium LGS AO systems,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

R. Conan, B. Espeland, M. A. Van Dam, and A. H. Bouchez, “Design of the laser tomography adaptive optics system for the giant Magellan telescope,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

T. Fusco, S. Meimon, N. Thatte, H. Schnetler, Y. Clenet, M. Cohen, J. Paufique, P. Ammans, F. Clarke, J. L. Dournaux, M. Ferrari, D. Gratadour, N. Hubin, P. Jagourel, V. Michau, C. Petit, and M. Tecza, “Laser tomographic AO system for an integral field spectrograph on the E-ELT: the ATLAS project,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

G. Herriot, D. Andersen, J. Atwood, P. Byrnes, C. Boyer, K. Caputa, C. Correia, J. Dunn, B. Ellerbroek, J. Fitzsimmons, L. Gilles, P. Hickson, A. Hill, J. Pazder, V. Reshetov, M. Smith, J. P. Veran, L. Wang, and I. Wevers, “NFIRAOS: multi conjugate AO system for TMT,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

E. Diolaiti, I. Foppiani, J. M. Conan, C. R. Butler, R. I. Davies, A. Baruffolo, M. Bellazzini, G. Bregoli, P. Ciliegi, G. Cosentino, B. Delabre, T. Fusco, N. Hubin, M. Lombini, E. Marchetti, C. Petit, C. Robert, and L. Schreiber, “The E-ELT multi conjugate adaptive optics module,” presented at the Second International Conference on Adaptive Optics for Extremely Large Telescopes, British Columbia, Canada, 25–30 September 2011.

L. Gilles and B. Ellerbroek, “LAOS: linear adaptive optics simulator,” Thirty Meter Telescope Observatory Corporation, Document-7988, TMT.AOS.TEC.07.003 (2007).

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Figures (3)

Fig. 1.
Fig. 1.

Illustration of the key operations involved in the FD preconditioning step. The case Nps=3, Nos=2, nf=4 (toy problem of order 2×2) is illustrated. The origin has been marked with an “X” symbol. For each point j=1,,nf2 of the nf×nf WFS Fourier plane, nb=4Nos+(NpsNos) locations distributed across the Nps atmospheric Fourier planes are read out, multiplied by a precomputed complex-valued nb×nb FD matrix M^j, and written back to their original locations.

Fig. 2.
Fig. 2.

Sample 60×60 convolution kernels for DKF’s SD implementation for TMT (Nlgs=6 LGS WFSs) for the ground-level atmospheric layer (“ps1”) (left panel) and top layer at 16 km altitude (“ps6”) (right panel).

Fig. 3.
Fig. 3.

FDPCG and DKF performance estimates (cumulative LGS mode WFE over a 17 arcsec square field of view, capturing all fundamental AO error budget terms) for zenith observations with TMT and median Mauna Kea seeing. Left is for algorithms without prediction (i.e., a zero wind speed assumption) and for algorithms with perfect wind profile knowledge (indicated with the letter “P”). Right is for algorithms with 20 or 30 deg wind direction error, and either 0% or 10% wind speed error on each atmospheric layer, denoted, respectively, as P20P0, P20P10, P30P0, and P30P10. Note that the y axes in the left and right figures scale differently.

Tables (8)

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Table 1. FDPCG Tomography Algorithm Initialization Steps

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Table 2. FDPCG Tomography Algorithm Iterations and Output Approximate Solution

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Table 3. DM Fitting and Temporal Filtering

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Table 4. DKF Tomography Algorithm Description

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Table 5. Required Atmospheric and Aperture-Plane Grid Sizes for LGS MCAO Observations with TMT at 60 deg Zenith Angle and for a 2 arcmin Diameter Fitting Field of Viewa

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Table 6. Computation Requirements for Three FDPCG Iterations for the Case Nos=0, 2, Nps, Denoted, Respectively, as FD3, FD3o2, and FD3o6a

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Table 7. DKF Tomography Algorithm Computational Cost Analysis for the Case of the Kalman Gain Multiplication Implemented via FFTs for the Case Nos=0, Nps, Denoted, Respectively, as DKFP and DKFo6Pa

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Table 8. DKF Tomography Algorithm Computational Cost Analysis for the Case of the Kalman Gain Multiplication Implemented via SD Convolutions, for the Case Nos=0, Nps, Denoted, Respectively, as DKFP-SD and DKFo6P-SDa

Equations (36)

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sn=ΓH˜xn1Gaan1+ηn1,
snOL=sn+Gaan1.
Axn1=bn,
[z]k=Fk1[r^]k,r^B^r^,
nb=4Nos+(NpsNos).
B^=j=1nf2PjTM^jPj,
r^j=1nf2PjTM^j[rb]j,[rb]j=Pjr^.
M^j=Q^j1,Q^j=PjA^PjT.
xn=Hvxn1+νn1,
Cν=CxHvCxHvT.
[[u]k]p,q=i=1Nlgsp,q=nK/2nK/21([[τx]ki]pp,qq[[e¯x]i]p,q+[[τy]ki]pp,qq[[e¯y]i]p,q),
C^t+1=f(C^t),
f(C^t)=H^v(IK^tG^)C^tH^vT+C^ν,
K^t=C^tG^T(G^C^tG^T+σ2)1,
[K^x(y)]ki=diag(vec([τ^x(y)]ki)),
stencil(L¯k)=[0ω30ω4ω1ω20ω50],
ck2=Nk/xkTL¯k2xk.
xkTL¯k2xk=Nk(j=15ωj[xk]j)2.
(j=15ωj[xk]j)2=12j,j=15ωjωj([xk]j[xk]j)2=3.44r0,k5/3βk,
βk=j,j=15ωjωjpk,jpk,j5/3=(221/621/3)Δk5/3.
ck=0.24(r0,k/Δk)5/6.
Γx(y)=ΩΓ¯x(y),
Ω=diag(vec(ω)),[ω]jj={m0=nsa/(Nfft/2)2ifj,jare even,else0,
stencil(Γ¯x)=1(2Δ0)[1/401/41/201/21/401/4],stencil(Γ¯y)=stencil(Γ¯x)T.
[A]kl=i=1Nlgs[G]ikTσi2[G]ik+δk,lLk2,
[G]ik=[ΓxΓy][H˜]ik.
[A^]kl=Fk[A]klFl1=i=1Nlgs[G^]ikTσi2[G^]ik+δk,lL^k2,
[G^]ik=[Γ^xΓ^y][H˜^]ik,[H˜^]ik=F0[H˜]ikFk1,Γ^x(y)=F0Γx(y)F01,
[H˜^]ik={diag(vec(Λik))ifαk=1,elsediag(vec(Θ0))Tdiag(vec(Λik)),
Λik=exp(i¯πhk[fk,xθix+fk,yθiy]),
[DBDBCACADBDBCACA]=array(Tv),v=vec([ACBD]),
Θ0=12(1+cos(φ0,x)+cos(φ0,y)+cos(φ0,x)cos(φ0,y)),
L^k=diag(vec(ψ^k)),ψ^k=4ck(sin2(φk,x/2)+sin2(φk,y/2)),
Γ^x(y)=Ω^diag(vec(g^x(y))),
ϕjj=vec(ei¯(jφ0,x+jφ0,y))Tvec(ϕ^),
g^x=ei¯(φ0,x+φ0,y)8Δ0[e2i¯φ0,y+2ei¯(2φ0,y+φ0,x)+e2i¯(φ0,x+φ0,y)12ei¯φ0,xe2i¯φ0,x],g^y=g^xy.

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