Abstract

By “atmospheric tomography” we mean the estimation of a layered atmospheric turbulence profile from measurements of the pupil–plane phase (or phase gradients) corresponding to several different guide star directions. We introduce what we believe to be a new Fourier domain preconditioned conjugate gradient (FD-PCG) algorithm for atmospheric tomography, and we compare its performance against an existing multigrid preconditioned conjugate gradient (MG-PCG) approach. Numerical results indicate that on conventional serial computers, FD-PCG is as accurate and robust as MG-PCG, but it is from one to two orders of magnitude faster for atmospheric tomography on 30 m class telescopes. Simulations are carried out for both natural guide stars and for a combination of finite-altitude laser guide stars and natural guide stars to resolve tip–tilt uncertainty.

© 2006 Optical Society of America

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  1. R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advancements in Adaptive Optics, D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).
    [CrossRef]
  2. T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
    [CrossRef]
  3. M. Johns, J. R. P. Angel, S. Shectman, R. Bernstein, D. G. Fabricant, P. McCarthy, and M. Phillips, "Status of the Giant Magellan Telescope (GMT) Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 441-453 (2004).
    [CrossRef]
  4. P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. L. Gilles, B. L. Ellerbroek, and C. R. Vogel, "Preconditioned conjugate gradient wavefront reconstructors for multiconjugate adaptive optics," Appl. Opt. 42, 5233-5250 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  19. D. Gavel, "Tomography for multiconjugate adaptive optics systems using laser guide stars," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 1356-1373 (2004).
    [CrossRef]
  20. B. L. Ellerbroek, "Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques," J. Opt. Soc. Am. A 19, 1803-1816 (2002).
    [CrossRef]
  21. C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
    [CrossRef]
  22. Y. Saad, Iterative Methods for Sparse Linear System, 2nd ed. (SIAM, 2003), pp. 261-282.
    [CrossRef]
  23. G. H. Golub and C. F. VanLoan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, 1989).
  24. M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC 1996).

2005 (1)

2004 (6)

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advancements in Adaptive Optics, D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).
[CrossRef]

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

M. Johns, J. R. P. Angel, S. Shectman, R. Bernstein, D. G. Fabricant, P. McCarthy, and M. Phillips, "Status of the Giant Magellan Telescope (GMT) Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 441-453 (2004).
[CrossRef]

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

D. Gavel, "Tomography for multiconjugate adaptive optics systems using laser guide stars," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 1356-1373 (2004).
[CrossRef]

H. Ren and R. Dekany, "Fast wave-front reconstruction by solving the Sylvester equation with the alternating direction implicit method," Opt. Express 12, 3279-3296 (2004).
[CrossRef] [PubMed]

2003 (2)

2002 (3)

2001 (2)

2000 (1)

1994 (2)

1989 (1)

1983 (1)

Andersen, T. E.

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

Angel, J. R. P.

M. Johns, J. R. P. Angel, S. Shectman, R. Bernstein, D. G. Fabricant, P. McCarthy, and M. Phillips, "Status of the Giant Magellan Telescope (GMT) Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 441-453 (2004).
[CrossRef]

Ardeberg, A.

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

Beckers, J. M.

J. M. Beckers, "Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics," in Proceedings of European Southern Observatory Conference and Workshop on Very Large Telescopes and Their Instrumentation, M. H. Ulrich, ed. (European Southern Observatory, 1988), Vol. 30, pp. 693-703.

Bernstein, R.

M. Johns, J. R. P. Angel, S. Shectman, R. Bernstein, D. G. Fabricant, P. McCarthy, and M. Phillips, "Status of the Giant Magellan Telescope (GMT) Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 441-453 (2004).
[CrossRef]

Brase, J. M.

Britton, M. C.

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advancements in Adaptive Optics, D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).
[CrossRef]

Brunetto, E. T.

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

Comeron, F.

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

Conan, J. M.

Dekany, R.

Dekany, R. G.

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advancements in Adaptive Optics, D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).
[CrossRef]

Dierickx, P.

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

Ellerbroek, B. L.

Enmark, A.

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

Fabricant, D. G.

M. Johns, J. R. P. Angel, S. Shectman, R. Bernstein, D. G. Fabricant, P. McCarthy, and M. Phillips, "Status of the Giant Magellan Telescope (GMT) Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 441-453 (2004).
[CrossRef]

Fan, C.

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

Fusco, T.

Gardner, C. S.

Gavel, D.

D. Gavel, "Tomography for multiconjugate adaptive optics systems using laser guide stars," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 1356-1373 (2004).
[CrossRef]

Gavel, D. T.

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advancements in Adaptive Optics, D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).
[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]

Gilles, L.

Gilmozzi, R.

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. VanLoan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, 1989).

Gonte, F. Y.

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

Herriot, G.

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advancements in Adaptive Optics, D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).
[CrossRef]

Johns, M.

M. Johns, J. R. P. Angel, S. Shectman, R. Bernstein, D. G. Fabricant, P. McCarthy, and M. Phillips, "Status of the Giant Magellan Telescope (GMT) Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 441-453 (2004).
[CrossRef]

Johnston, D. C.

Koch, F.

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

Lastiwka, M.

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

le Louarn, M.

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

A. Tokovinin and M. Le Louarn, "Isoplanatism in a multiconjugate adaptive optics system," J. Opt. Soc. Am. A 17, 1819-1827 (2000).
[CrossRef]

Max, C. E.

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advancements in Adaptive Optics, D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).
[CrossRef]

McCarthy, P.

M. Johns, J. R. P. Angel, S. Shectman, R. Bernstein, D. G. Fabricant, P. McCarthy, and M. Phillips, "Status of the Giant Magellan Telescope (GMT) Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 441-453 (2004).
[CrossRef]

McNamara, K.

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

Michau, V.

Monnet, G. J.

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

Moraru, D.

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

Mugnier, L. M.

Owner-Petersen, M.

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

Phillips, M.

M. Johns, J. R. P. Angel, S. Shectman, R. Bernstein, D. G. Fabricant, P. McCarthy, and M. Phillips, "Status of the Giant Magellan Telescope (GMT) Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 441-453 (2004).
[CrossRef]

Poyneer, L. A.

Quinlan, N.

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

Ren, H.

Riewaldt, H.

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

Roggemann, M. C.

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

Rousset, G.

Saad, Y.

Y. Saad, Iterative Methods for Sparse Linear System, 2nd ed. (SIAM, 2003), pp. 261-282.
[CrossRef]

Shearer, A.

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

Shectman, S.

M. Johns, J. R. P. Angel, S. Shectman, R. Bernstein, D. G. Fabricant, P. McCarthy, and M. Phillips, "Status of the Giant Magellan Telescope (GMT) Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 441-453 (2004).
[CrossRef]

Spyromilio, J.

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

Surdej, I.

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

Tokovinin, A.

VanLoan, C. F.

G. H. Golub and C. F. VanLoan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, 1989).

Veran, J.-P.

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advancements in Adaptive Optics, D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).
[CrossRef]

Verinaud, C.

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

Viard, E.

Vogel, C. R.

Wallner, E. P.

Wang, X.

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

Welsh, B.

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

Welsh, B. M.

Yaitskova, N.

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (10)

A. Tokovinin and M. Le Louarn, "Isoplanatism in a multiconjugate adaptive optics system," J. Opt. Soc. Am. A 17, 1819-1827 (2000).
[CrossRef]

A. Tokovinin and E. Viard, "Limiting precision of tomographic phase estimation," J. Opt. Soc. Am. A 18, 873-8827 (2001).
[CrossRef]

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

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

L. Gilles, C. R. Vogel, and B. L. Ellerbroek, "Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction," J. Opt. Soc. Am. A 19, 1817-1822 (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]

D. C. Johnston and B. M. Welsh, "Analysis of multiconjugate adaptive optics," J. Opt. Soc. Am. A 11, 394-408 (1994).
[CrossRef]

B. L. Ellerbroek, "First-order performance evaluation of adaptive-optics systems for atmospheric turbulence compensation in extended field-of-view astronomical telescopes," J. Opt. Soc. Am. A 11, 783-805 (1994).
[CrossRef]

B. M. Welsh and C. S. Gardner, "Performance analysis of adaptive-optics systems using laser guide stars and slope sensors," J. Opt. Soc. Am. A 6, 1913-1923 (1989).
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B. L. Ellerbroek, "Linear systems modeling of adaptive optics in the spatial-frequency domain," J. Opt. Soc. Am. A 22, 310-322 (2005).
[CrossRef]

Opt. Express (1)

Proc. SPIE (5)

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advancements in Adaptive Optics, D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).
[CrossRef]

T. E. Andersen, A. Ardeberg, H. Riewaldt, M. Lastiwka, N. Quinlan, K. McNamara, X. Wang, A. Enmark, M. Owner-Petersen, A. Shearer, C. Fan, and D. Moraru, "Status of the Euro50 Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 407-416 (2004).
[CrossRef]

M. Johns, J. R. P. Angel, S. Shectman, R. Bernstein, D. G. Fabricant, P. McCarthy, and M. Phillips, "Status of the Giant Magellan Telescope (GMT) Project," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 441-453 (2004).
[CrossRef]

P. Dierickx, E. T. Brunetto, F. Comeron, R. Gilmozzi, F. Y. Gonte, F. Koch, M. le Louarn, G. J. Monnet, J. Spyromilio, I. Surdej, C. Verinaud, and N. Yaitskova, "OWL phase A status report," in Ground-Based Telescopes, J. M. Oschmann, Jr., ed., Proc. SPIE 5489, 391-406 (2004).
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D. Gavel, "Tomography for multiconjugate adaptive optics systems using laser guide stars," in Advancements in Adaptive Optics, D. Bonaccini Calia, B. Ellerbroek, and R. Ragazzoni, eds., Proc. SPIE 5490, 1356-1373 (2004).
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Other (5)

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C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
[CrossRef]

Y. Saad, Iterative Methods for Sparse Linear System, 2nd ed. (SIAM, 2003), pp. 261-282.
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M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC 1996).

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Figures (9)

Fig. 1
Fig. 1

(a) A circular pupil mask and (b) its Fourier transform.

Fig. 2
Fig. 2

Truncations of the transformed pupil mask and the sparsity patterns of the corresponding BCCB matrix approximations to . (a) Shows the truncation points; subplot (b) shows the sparsity pattern of the BCCB matrix corresponding to 1-point truncation; (c) corresponds to 5-point truncation; and (d) corresponds to 9-point truncation.

Fig. 3
Fig. 3

Sparsity pattern for the representer C ^ of the Fourier domain preconditioner with 1-point truncation. (a) Shows the sparsity pattern without reordering, and (b) is with reordering.

Fig. 4
Fig. 4

Guide star constellations. (a) NGS and (b) LGS with NGSs added in to measure global tip–tilt. Phase errors will be evaluated along the dashed lines.

Fig. 5
Fig. 5

Normalized rms FOV-integrated phase errors for MG-PCG versus FD-PCG, with natural guide stars. In (a) we vary the FD-PCG truncation level with SNR = 5; in (b) we vary the truncation level with SNR = 10; and in (c) we vary the truncation level with SNR = 20. In (d) 1-point truncation is used for FD-PCG, and the noise level is varied. The grid size is 64 × 64 and horizontal axes give iteration count k.

Fig. 6
Fig. 6

Normalized rms phase errors when the science direction θ is sampled along an arc across the field of view. (a) 1-point FD-PCG, (b) 5-point FD-PCG, (c) 9-point FD-PCG, and (d) MG-PCG.

Fig. 7
Fig. 7

Grid resolution effects on PCG convergence, with natural guide stars. Normalized FOV-averaged phase errors are compared for MG-PCG and FD-PCG with 1-point truncation. Grid sizes are 64 × 64 and 128 × 128 and SNR = 10.

Fig. 8
Fig. 8

Normalized rms FOV-integrated phase errors for MG-PCG versus FD-PCG, with five laser guide stars and four natural guide stars. (a) SNR = 5, (b) SNR = 10, (c) SNR = 20, and (d) compares MG-PCG and 1-point FD-PCG under three different noise levels. 64 × 64 grid size.

Fig. 9
Fig. 9

Normalized rms FOV-integrated phase errors for LGS + NGS versus NGS. Systems Ax = b were solved using FD-PCG with 1-point truncation. The SNR = 10 and the grid size is 64 × 64.

Tables (2)

Tables Icon

Table 1 Atmospheric Turbulence Profile Used for Simulations

Tables Icon

Table 2 Computation Time per Iteration for MG-PCG versus FD-PCG for Tomography with Five Natural Guide Stars a

Equations (46)

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s = G ψ + η ,
ψ MV = E MV s ,
J ( E ) = E s ψ 2 = Trace ( E s ψ ) ( E s ψ ) T .
E MV = ( G T C η - 1 G + C ψ - 1 ) 1 G T C η - 1 ,
ψ MV = ( G T C η - 1 G + C ψ - 1 ) 1 G T C η - 1 s .
A = G T C η - 1 G + C ψ - 1 ,
b = G T C η - 1 s ,
C 1 A x = C 1 b ,
C = 1 C ^ .
C ψ = Block   Diag { C ψ } = 1 L .
C ψ = C n 2 ( z ) 1 | κ | 11 / 3 , l = 1 , , L .
C ψ - 1 = 1 B l o c k D i ag { C n - 2 ( z ) | κ | 11 / 3 } = 1 L .
ϕ j ( x ) = [ P ( θ j ) ψ ] ( x ) = def = 1 L ψ ( x + z θ j )
= = 1 L 1 exp ( i 2 π z κ θ j ) ψ .
s j = [ M Γ x ϕ j M Γ y ϕ j ] = [ M Γ x P ( θ j ) ψ M Γ y P ( θ j ) ψ ] , j = 1 , , N GS .
M s ( x ) = { s ( x ) , x Ω , 0 , otherwise,
Γ x ϕ = 1 2 ( S x 1 ) ϕ + 1 2 ( S x 1 ) S y ϕ = 1 2 ( S x 1 ) ( S y + 1 ) ϕ ,
Γ y ϕ = 1 2 ( S y 1 ) ( S x + 1 ) ϕ .
Γ x = 1 Γ ̂ x , Γ y = 1 Γ ̂ y ,
C η = σ 2 I ,
[ G T C η - 1 G ] m = σ 2 P T ( θ ) ( Γ x T M T M Γ x + Γ y T M T M Γ y ) × P ( θ m )
= σ 2 j = 1 N NGS P j       T ( Γ x T M T M Γ x + Γ y T M T M Γ y ) × P j m
= 1 { σ 2 j = 1 N GS P ̂ j * ( Γ ̂ x * M ̂ * M ̂ Γ ̂ x + Γ ̂ y * M ̂ * M ̂ Γ ̂ y ) P ̂ j m } .
[ C ^ ] m = σ 2 j = 1 N GS P ^ j * ( Γ ̂ x * M ˜ * M ˜ Γ ̂ x + Γ ̂ y * M ˜ * M ˜ Γ ̂ y ) P ^ j m + δ m C n - 2 ( z ) | κ | 11 / 3 ,
ϕ j ( x ) = [ P LGS ( θ j ) ] ( x ) = def = 1 L ψ ( H H z x + z θ j ) ,
ϕ ^ j ( κ ) = = 1 L 1 c 2 exp [ 2 π i ( κ / c ) ( z θ j ) ] ψ ^ ( κ / c ) ,
c = H H z .
s = [ s h s t ] = [ G h G t ] ψ + [ η h η t ] .
C η = [ N h + σ t     2 T T T 0 0 N t ] ,
C η - 1 = [ N h - 1 / 2 ( I Q ) N h - 1 / 2 0 0 N t     1 ] ,
Q = T ˜ ( T ˜ T T ˜ ) 1 T ˜ T , with   T ˜ = N h - 1 / 2 T .
G ˜ h = N h - 1 / 2 G h , G ˜ t = N t - 1 / 2 G t
A = G ˜ h T ( I Q ) G ˜ h + G ˜ t T G ˜ t + C ψ - 1 = A h ( G ˜ h T Q G ˜ h G ˜ t T G ˜ t ) ,
A h = G h T N h - 1 G h + C ψ - 1 .
U 2 = G ˜ t T = G t T N t     1 / 2 ,
U 1 = G ˜ h T T ˜ R = G h T N h - 1 T R .
A 1 = A h - 1 + L ,
L = ( A h - 1 U ) ( I V T A h - 1 U ) 1 ( A h - 1 V ) T = [ A h - 1 U 1 A h - 1 G t T ] [ I 1 U 1 T A h - 1 U 1 U 1 T A h - 1 G t T G t A h - 1 U 1 N t + G t A h - 1 G t T ] 1 [ ( A h - 1 U 1 ) T ( A h - 1 G t T ) T ] .
A 1 b = A h - 1 b + L b .
PE k ( θ ) = Ω { [ P ( θ ) ψ k ] ( x ) ( P ( θ ) ψ atmos ) ( x ) } 2 d x .
FOVPE ( k ) = FOV PE k ( θ ) d θ .
PE k ( θ ) Ω [ P ( θ ) ψ atmos ] 2 ( x ) d x     ,
FOVPE ( k ) FOV Ω [ P ( θ ) ψ atmos ] 2 ( x ) d x ,
γ = trace ( L 1 C 1 / 2 ) / trace ( L 2 ) .
SNR = trace ( G C ψ G T ) 2 σ 2 N gs N sensor ,
SNR = N ¯ N + σ 2 ,

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