Abstract

We present an analytical algorithm for deriving the shapes of the deformable mirrors to be used for multiconjugate adaptive correction on a large telescope. The algorithm is optimal in the limit where the overlap of the wave-front contributions from relevant atmospheric layers probed by the guide stars is close to the size of the pupil. The fundamental principle for correction is based on a minimization of the sum of the residual power spectra of the phase fluctuations seen by the guide stars after correction. On the basis of the expressions for the mirror shapes, so-called layer transfer functions describing the distribution of the correction of a single atmospheric layer among the deformable mirrors and the resulting correction of that layer have been derived. It is shown that for five guide stars distributed in a regular cross, two- and three-mirror correction will be possible only up to a maximum frequency defined by the largest separation of the conjugate altitudes of the mirrors and by the angular separation of the guide stars. The performance of the algorithm is investigated in the K band by using a standard seven-layer atmosphere. We present results obtained for two guide-star configurations: a continuous distribution within a given angular radius and a five-star cross pattern with a given angular arm length. The wave-front fluctuations are subjected to correction using one, two, and three deformable mirrors. The needed mirror dynamic range is derived as required root-mean-square stroke and actuator pitch. Finally the performance is estimated in terms of the Strehl ratio obtained by the correction as a function of field angle. No noise has been included in the present analysis, and the guide stars are assumed to be at infinity.

© 2002 Optical Society of America

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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] [PubMed]
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    [CrossRef]
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    [CrossRef]
  16. N. Hubin, M. le Louarn, M. Sarazin, A. Tokovinin, E. Viard, “New challenges for adaptive optics: the OWL 100 m telescope,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1100–1107 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2000 (2)

B. Ellerbrook, F. Rigaut, “Optics adapt to the whole sky,” Nature 403, 25–26 (2000).
[CrossRef]

R. Ragazzoni, E. Marchetti, G. Valente, “Adaptive-optics corrections available for the whole sky,” Nature 403, 54–56 (2000).
[CrossRef] [PubMed]

1999 (1)

1994 (2)

1976 (1)

Agabi, A.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “Gemini site characterization report,” (Gemini Observatory, Tucson, Ariz., 2000).

Andersen, T.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 72–82.

Ardeberg, A.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 72–82.

Avila, R.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “Gemini site characterization report,” (Gemini Observatory, Tucson, Ariz., 2000).

Azouit, M.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “Gemini site characterization report,” (Gemini Observatory, Tucson, Ariz., 2000).

Beckers, J.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 72–82.

Beckers, J. M.

J. M. Beckers, “Multi-conjugate adaptive optics: experiments in atmospheric tomography,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1056–1065 (2000).
[CrossRef]

J. M. Beckers, “Increasing the size of the isoplanatic patch with multiconjugate adaptive optics,” in Proceedings of the ESO Conference on Very Large Telescopes and Their Instrumentation (European Southern Observatory, Garching, Germany, 1988), p. 693.

J. M. Beckers, “Detailed compensation of atmospheric seeing using multiconjugate adaptive optics,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 215–217 (1989).
[CrossRef]

Chun, M.

F. Rigaut, R. Ragazzoni, M. Chun, M. Mountain, “Adaptive optics challenges for the ELT’s,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 168–174.

Conan, J. M.

T. Fusco, J. M. Conan, V. Michau, L. Mugnier, G. Rousset, “Efficient phase estimation for large-field-of-view adaptive optics,” Opt. Lett. 24, 1472–1474 (1999).
[CrossRef]

T. Fusco, J. M. Conan, V. Michau, L. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation through the Atmosphere III, M. C. Roggeman, L. R. Bisonnette, eds., Proc. SPIE3763, 125–133 (1999).
[CrossRef]

Conan, J.-M.

T. Fusco, J.-M. Conan, V. Micheau, G. Rousset, L. M. Mugnier, “Isoplanatic angle and optimal guide star separation for multiconjugate adaptive optics,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1044–1055 (2000).
[CrossRef]

Conan, R.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “Gemini site characterization report,” (Gemini Observatory, Tucson, Ariz., 2000).

Ellerbroek, B. L.

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

F. J. Rigaut, B. L. Ellerbroek, R. Flicker, “Principles, limitations and performance of multi-conjugate adaptive optics,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1022–1031 (2000).
[CrossRef]

R. Flicker, F. J. Rigaut, B. L. Ellerbroek, “Comparison of multiconjugate adaptive optics configurations and control algorithms for the Gemini-South 8-m Telescope,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1032–1043 (2000).
[CrossRef]

B. L. Ellerbroek, F. J. Rigaut, “Scaling multi-conjugate adaptive optics performance estimates to extremely large telescopes,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1088–1099 (2000).
[CrossRef]

Ellerbrook, B.

B. Ellerbrook, F. Rigaut, “Optics adapt to the whole sky,” Nature 403, 25–26 (2000).
[CrossRef]

Farinato, J.

R. Ragazzoni, J. Farinato, E. Marchetti, “Adaptive optics for 100 m class telescopes: new challenges require new solutions,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1076–1087 (2000).
[CrossRef]

Flicker, R.

F. J. Rigaut, B. L. Ellerbroek, R. Flicker, “Principles, limitations and performance of multi-conjugate adaptive optics,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1022–1031 (2000).
[CrossRef]

R. Flicker, F. J. Rigaut, B. L. Ellerbroek, “Comparison of multiconjugate adaptive optics configurations and control algorithms for the Gemini-South 8-m Telescope,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1032–1043 (2000).
[CrossRef]

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 72–82.

Fried, D. L.

D. L. Fried, “Fundamental limits in field widening for a multiple deformable mirror adaptive optics system,” (The Optical Sciences Company, Anaheim, Calif., 1992).

Fusco, T.

T. Fusco, J. M. Conan, V. Michau, L. Mugnier, G. Rousset, “Efficient phase estimation for large-field-of-view adaptive optics,” Opt. Lett. 24, 1472–1474 (1999).
[CrossRef]

T. Fusco, J.-M. Conan, V. Micheau, G. Rousset, L. M. Mugnier, “Isoplanatic angle and optimal guide star separation for multiconjugate adaptive optics,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1044–1055 (2000).
[CrossRef]

T. Fusco, J. M. Conan, V. Michau, L. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation through the Atmosphere III, M. C. Roggeman, L. R. Bisonnette, eds., Proc. SPIE3763, 125–133 (1999).
[CrossRef]

Gontcharov, A.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 72–82.

A. Gontcharov, M. Owner-Petersen, “Multiconjugate adaptive optics for the Swedish ELT,” in Telescope Structures, Enclosures, Controls, Assembly/Integration/Validation and Commissioning, T. A. Seebring, T. Andersen, eds., Proc. SPIE4004, 309–395 (2000).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, Oxford, UK, 1998).

Hubin, N.

N. Hubin, M. le Louarn, M. Sarazin, A. Tokovinin, E. Viard, “New challenges for adaptive optics: the OWL 100 m telescope,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1100–1107 (2000).
[CrossRef]

Jessen, N. C.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 72–82.

Johnston, D. C.

le Louarn, M.

N. Hubin, M. le Louarn, M. Sarazin, A. Tokovinin, E. Viard, “New challenges for adaptive optics: the OWL 100 m telescope,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1100–1107 (2000).
[CrossRef]

Mannery, E.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 72–82.

Marchetti, E.

R. Ragazzoni, E. Marchetti, G. Valente, “Adaptive-optics corrections available for the whole sky,” Nature 403, 54–56 (2000).
[CrossRef] [PubMed]

R. Ragazzoni, J. Farinato, E. Marchetti, “Adaptive optics for 100 m class telescopes: new challenges require new solutions,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1076–1087 (2000).
[CrossRef]

Martin, F.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “Gemini site characterization report,” (Gemini Observatory, Tucson, Ariz., 2000).

Masciadri, E.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “Gemini site characterization report,” (Gemini Observatory, Tucson, Ariz., 2000).

Michau, V.

T. Fusco, J. M. Conan, V. Michau, L. Mugnier, G. Rousset, “Efficient phase estimation for large-field-of-view adaptive optics,” Opt. Lett. 24, 1472–1474 (1999).
[CrossRef]

T. Fusco, J. M. Conan, V. Michau, L. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation through the Atmosphere III, M. C. Roggeman, L. R. Bisonnette, eds., Proc. SPIE3763, 125–133 (1999).
[CrossRef]

Micheau, V.

T. Fusco, J.-M. Conan, V. Micheau, G. Rousset, L. M. Mugnier, “Isoplanatic angle and optimal guide star separation for multiconjugate adaptive optics,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1044–1055 (2000).
[CrossRef]

Mountain, M.

F. Rigaut, R. Ragazzoni, M. Chun, M. Mountain, “Adaptive optics challenges for the ELT’s,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 168–174.

Mugnier, L.

T. Fusco, J. M. Conan, V. Michau, L. Mugnier, G. Rousset, “Efficient phase estimation for large-field-of-view adaptive optics,” Opt. Lett. 24, 1472–1474 (1999).
[CrossRef]

T. Fusco, J. M. Conan, V. Michau, L. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation through the Atmosphere III, M. C. Roggeman, L. R. Bisonnette, eds., Proc. SPIE3763, 125–133 (1999).
[CrossRef]

Mugnier, L. M.

T. Fusco, J.-M. Conan, V. Micheau, G. Rousset, L. M. Mugnier, “Isoplanatic angle and optimal guide star separation for multiconjugate adaptive optics,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1044–1055 (2000).
[CrossRef]

Noll, R. J.

Owner-Petersen, M.

A. Gontcharov, M. Owner-Petersen, “Multiconjugate adaptive optics for the Swedish ELT,” in Telescope Structures, Enclosures, Controls, Assembly/Integration/Validation and Commissioning, T. A. Seebring, T. Andersen, eds., Proc. SPIE4004, 309–395 (2000).

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 72–82.

Ragazzoni, R.

R. Ragazzoni, E. Marchetti, G. Valente, “Adaptive-optics corrections available for the whole sky,” Nature 403, 54–56 (2000).
[CrossRef] [PubMed]

R. Ragazzoni, “Adaptive optics for giant telescopes,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 175–180.

R. Ragazzoni, J. Farinato, E. Marchetti, “Adaptive optics for 100 m class telescopes: new challenges require new solutions,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1076–1087 (2000).
[CrossRef]

F. Rigaut, R. Ragazzoni, M. Chun, M. Mountain, “Adaptive optics challenges for the ELT’s,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 168–174.

Rigaut, F.

B. Ellerbrook, F. Rigaut, “Optics adapt to the whole sky,” Nature 403, 25–26 (2000).
[CrossRef]

F. Rigaut, R. Ragazzoni, M. Chun, M. Mountain, “Adaptive optics challenges for the ELT’s,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds. (Lund University, Lund, Sweden, 1999), pp. 168–174.

Rigaut, F. J.

F. J. Rigaut, B. L. Ellerbroek, R. Flicker, “Principles, limitations and performance of multi-conjugate adaptive optics,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1022–1031 (2000).
[CrossRef]

B. L. Ellerbroek, F. J. Rigaut, “Scaling multi-conjugate adaptive optics performance estimates to extremely large telescopes,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1088–1099 (2000).
[CrossRef]

R. Flicker, F. J. Rigaut, B. L. Ellerbroek, “Comparison of multiconjugate adaptive optics configurations and control algorithms for the Gemini-South 8-m Telescope,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1032–1043 (2000).
[CrossRef]

Rousset, G.

T. Fusco, J. M. Conan, V. Michau, L. Mugnier, G. Rousset, “Efficient phase estimation for large-field-of-view adaptive optics,” Opt. Lett. 24, 1472–1474 (1999).
[CrossRef]

T. Fusco, J. M. Conan, V. Michau, L. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation through the Atmosphere III, M. C. Roggeman, L. R. Bisonnette, eds., Proc. SPIE3763, 125–133 (1999).
[CrossRef]

T. Fusco, J.-M. Conan, V. Micheau, G. Rousset, L. M. Mugnier, “Isoplanatic angle and optimal guide star separation for multiconjugate adaptive optics,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1044–1055 (2000).
[CrossRef]

Sanchez, L.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “Gemini site characterization report,” (Gemini Observatory, Tucson, Ariz., 2000).

Sarazin, M.

N. Hubin, M. le Louarn, M. Sarazin, A. Tokovinin, E. Viard, “New challenges for adaptive optics: the OWL 100 m telescope,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1100–1107 (2000).
[CrossRef]

Tokovinin, A.

N. Hubin, M. le Louarn, M. Sarazin, A. Tokovinin, E. Viard, “New challenges for adaptive optics: the OWL 100 m telescope,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1100–1107 (2000).
[CrossRef]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, Boston, Mass., 1998).

Valente, G.

R. Ragazzoni, E. Marchetti, G. Valente, “Adaptive-optics corrections available for the whole sky,” Nature 403, 54–56 (2000).
[CrossRef] [PubMed]

Vernin, J.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “Gemini site characterization report,” (Gemini Observatory, Tucson, Ariz., 2000).

Viard, E.

N. Hubin, M. le Louarn, M. Sarazin, A. Tokovinin, E. Viard, “New challenges for adaptive optics: the OWL 100 m telescope,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1100–1107 (2000).
[CrossRef]

Welsh, B. M.

Ziad, A.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “Gemini site characterization report,” (Gemini Observatory, Tucson, Ariz., 2000).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nature (2)

B. Ellerbrook, F. Rigaut, “Optics adapt to the whole sky,” Nature 403, 25–26 (2000).
[CrossRef]

R. Ragazzoni, E. Marchetti, G. Valente, “Adaptive-optics corrections available for the whole sky,” Nature 403, 54–56 (2000).
[CrossRef] [PubMed]

Opt. Lett. (1)

Other (19)

T. Fusco, J. M. Conan, V. Michau, L. Mugnier, G. Rousset, “Phase estimation for large field of view: application to multiconjugate adaptive optics,” in Propagation through the Atmosphere III, M. C. Roggeman, L. R. Bisonnette, eds., Proc. SPIE3763, 125–133 (1999).
[CrossRef]

T. Fusco, J.-M. Conan, V. Micheau, G. Rousset, L. M. Mugnier, “Isoplanatic angle and optimal guide star separation for multiconjugate adaptive optics,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1044–1055 (2000).
[CrossRef]

J. M. Beckers, “Multi-conjugate adaptive optics: experiments in atmospheric tomography,” in Adaptive Systems Technology, P. Wizinowich, ed., Proc. SPIE4007, 1056–1065 (2000).
[CrossRef]

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[CrossRef]

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[CrossRef]

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A. Gontcharov, M. Owner-Petersen, “Multiconjugate adaptive optics for the Swedish ELT,” in Telescope Structures, Enclosures, Controls, Assembly/Integration/Validation and Commissioning, T. A. Seebring, T. Andersen, eds., Proc. SPIE4004, 309–395 (2000).

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Figures (16)

Fig. 1
Fig. 1

Layout of the optics for the Swedish ELT including a proposed scheme for dual conjugate adaptive control. The guide-star wave fronts subjected only to corrections performed on DM1 are measured by the LGS WFS (wave-front sensor). A test source is placed in the infinite conjugate focus of the Cassegrain with the purpose of probing the shape of DM2, which is located in the Offner relay system. This is done by the Test Source WFS. Since the shape of DM2 and the wave fronts measured by the LGS WFS are known, new wave fronts subjected to corrections performed on both DM1 and DM2 can be synthesized. This is done by the Virtual WFS. When these wave fronts are known, the analytical estimator described in this paper derives the corrections to be imposed on the two DMs. The converters transform the corrective mirror shapes into actuator commands. Note that both real wave-front sensors, the LGS WFS and the Test Source WFS, must operate at high bias levels, whereas the Virtual WFS operates in a global null-seeking mode. Note also that the diagram shows only the flow of information. Details regarding control loops have been omitted.

Fig. 2
Fig. 2

Two-DM correction. Layer transfer functions for DM1 at -575 m as a function of frequency in inverse meters for a homogeneous guide-star distribution with radius α0=31 arcsec. The layer altitudes are (0) km to (16) km in steps of 2 km. DM2 is at 8000 m. Note the negative low-frequency transfer for the outer layers.

Fig. 3
Fig. 3

Two-DM correction. Layer transfer functions for DM2 at 8000 m as a function of frequency in inverse meters for a homogeneous guide-star distribution with radius α0=31 arcsec. Layer altitudes as in Fig. 2. DM1 is at -575 m. Note the large positive low-frequency transfer for the outer layers.

Fig. 4
Fig. 4

Two-DM correction. Spectral amplification as a function of frequency in inverse meters for a homogeneous guide-star distribution with radius α0=31 arcsec. Layer altitudes as in Fig. 2. DM1 is at -575 m, and DM2 is at 8000 m. Note the good compensation of layers close to the mirrors.

Fig. 5
Fig. 5

Two-DM correction. Layer transfer functions for DM1 at -575 m as a function of frequency in inverse meters along the cross arm for a cross guide-star distribution with arm length α0=31 arcsec. Layer altitudes as in Fig. 2. DM2 is at 8000 m. Note the negative low-frequency transfer for the outer layers. Note also the numerical increase in the transfer functions when the critical frequency is approached.

Fig. 6
Fig. 6

Two-DM correction. Layer transfer functions for DM2 at 8000 m as a function of frequency in inverse meters along the cross arm for a cross guide-star distribution with arm length α0=31 arcsec. Layer altitudes as in Fig. 2. DM1 is at -575 m. Note the large positive low-frequency transfer for the outer layers. Note also the numerical increase in the transfer functions when the critical frequency is approached.

Fig. 7
Fig. 7

Two-DM correction. Spectral amplification as a function of frequency in inverse meters along the cross arms for a cross guide-star distribution with arm length α0=31 arcsec. Layer altitudes as in Fig. 2. DM1 is at -575 m and DM2 is at 8000 m. Note the good compensation of layers close to the mirrors.

Fig. 8
Fig. 8

Strehl ratio as a function of angle in arcseconds for conventional AO and a homogeneous guide-star distribution with radius α0=31 arcsec. For mirror conjugate altitudes see Subsection 3.F. Curve (1), conventional AO correction (one axial guide star), one-DM correction; curve (2), one-DM correction; curve (3), two-DM correction; curve (4), three-DM correction.

Fig. 9
Fig. 9

Strehl ratio as a function of angle in arcseconds along the cross arm for a cross guide-star distribution with arm length α0=31 arcsec. For mirror conjugate altitudes see Subsection 3.F. Curve (1), one-DM correction; curve (2), two-DM correction; curve (3), three-DM correction.

Fig. 10
Fig. 10

Strehl ratio as a function of angle in arcseconds along the diagonal for a cross guide-star distribution with arm length α0=31 arcsec. For mirror conjugate altitudes see Subsection 3.F. Curve definitions as in Fig. 9.

Fig. 11
Fig. 11

Two-DM correction. Strehl ratio as a function of angle in arcseconds for a homogeneous guide-star distribution. For mirror conjugate altitudes see Subsection 3.F. Curve (1), guide-star field radius α0=31 arcsec; curve (2), α0=62 arcsec; curve (3), α0=93 arcsec.

Fig. 12
Fig. 12

Two-DM correction. Strehl ratio as a function of angle in arcseconds along the cross arm for a cross guide-star distribution. For mirror conjugate altitudes see Subsection 3.F. Curve definitions as in Fig. 11.

Fig. 13
Fig. 13

Two-DM correction. Strehl ratio as a function of angle in arcseconds along the diagonal for a cross guide-star distribution. For mirror conjugate altitudes see Subsection 3.F. Curve definitions as in Fig. 11.

Fig. 14
Fig. 14

Three-DM correction. Strehl ratio as a function of angle in arcseconds for a homogeneous guide star distribution. For mirror conjugate altitudes see Subsection 3.F. Curve definitions as in Fig. 11.

Fig. 15
Fig. 15

Three-DM correction. Strehl ratio as a function of angle in arcseconds along the cross arm for a cross guide-star distribution. For mirror conjugate altitudes see Subsection 3.F. Curve definitions as in Fig. 11.

Fig. 16
Fig. 16

Three-DM correction. Strehl ratio as a function of angle in arcseconds along the diagonal for a cross guide-star distribution. For mirror conjugate altitudes see Subsection 3.F Curve definitions as in Fig. 11.

Tables (5)

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Table 1 Cerro–Pacon Atmosphere at 2.2 µm

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Table 2 One-Mirror Correctiona

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Table 3 Two-Mirror Correctiona

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Table 4 Two-Mirror Correctiona

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Table 5 Three-Mirror Correctiona

Equations (75)

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P(r)P(r+Lmαq).
1-O=2LmαqD1.
q(r)=ϕqa(r)+m=1Mϕm(r+Lmαq).
Eq(f)=Φqa(f)+m=1MΦm(f)exp(2πiLmf·αq).
S(f)=q=1Q|Eq(f)|2.
Am(f)+m=1MGm,m(f) Φm(f)=0,
 Am(f)=1Qq=1QΦqa(f) exp(-2πiLmf·αq),
Gm,m(f)=1Qq=1Q exp[2πi(Lm-Lm)f·αq].
Φqa(f)=Φnl(f)exp(2πilnf·αq),
Am(f)=Gm,nl(f)Φnl(f),
Gm,nl(f)=1Qq=1Q exp[2πi(ln-Lm)f·αq].
Φm(f)=-Tn,m(f)Φnl(f),
Gm,nl(f)-m=1MGm,m(f)Tn,m(f)=0.
Pn(f)=1Qq=1Q|Φnl(f)|2exp(2πilnf·αq)-m=1MTn,m(f)exp(2πiLmf·αq)2.
Rn(f)=1+m=1M|Tn,m(f)|2-2 Rem=1MTn,m*(f)Gm,nl(f)-m=1Mm>mMTn,m*(f)Tn,m(f)Gm,n(f).
RMSϕ2=--Pϕ(f)1-2J1(πfD)πfD2df,
RMSav2=n=1N--Pn,ϕ(f)Rn(f)×1-2J1(πfD)πfD2df,
Pn,ϕ(f)=0.023r0,n5/3(f2+f0,n2)-11/6.
RMSϕ2=n=1N2π0fPn,ϕ(f)1-2J1(πfD)πfD2df.
RMSm2=n=1N--Pn,ϕ(f)|Tn,m(f)|21-2J1(πfD)πfD2df,
RMS2(α)=n=1N--Pn,ϕ(f)Rn(α, f)×1-2J1(πfD)πfD2df,
Rn(α, f)=1+m=1M|Tn,m(f)|2-2 Rem=1MTn,m*(f)Gm,nl(α, f)-m=1Mm>mMTn,m*(f)Tn,m(f)Gm,m(α, f),
Gm,nl(α, f)=exp[2πi(ln-Lm)f·α],
Gm,m(α, f)=exp(2πi(Lm-Lm)f·α).
ST(α)=exp[-RMS2(α)].
Gm,m(f)=2J1[2π(Lm-Lm)fα0]2π(Lm-Lm)fα0,
Gm,nl(f)=2J1(2π(ln-Lm)fα0)2π(ln-Lm)fα0,
Gm,m(f)=15{1+2 cos[2π(Lm-Lm)fxα0]+2 cos[2π(Lm-Lm)fyα0]},
Gm,nl(f)=15{1+2 cos[2π(ln-Lm)fxα0]+2 cos[2π(ln-Lm)fyα0]}.
1r0,eff5/3=n=1N1r0,n5/3,
r0,eff=0.877m.
1αi5/3=6.88n=1N|ln-L1|r0,n5/3,
αi=13.4 arsec.
RMSϕ=1.019Dr0,eff5/6,
RMSϕ=29.6 rad.
G1,1(f)=G1,nl(f)=Tn,1(f)=1,
Rn(f)=0.
RMSav=0 corresponding to Strehlav=1
RMS1=29.6 corresponding to
RMS1,stroke=5.18 μm at λ=2.2 μm.
RMS2(α)=a=1N4π0fPn,ϕ(f)1-2J1(πfD)πfD2×{1-J0[2π(ln-L1)fα]}df,
fmax=512r0,eff=2.85 m-1,
Na=2Dfmax=5Dr0,eff=285.
G1,1(f)=1,
Tn,1(f)=G1,nl(f),
Rn(f)=1-|Tn,1(f)|2,
RMS2(α)=n=1N2π0fPn,ϕ(f)1-2J1(πfD)πfD2×{1+Tn,1(f)2-2Tn,1(f)×J0[2π(ln-L1)fα]}df.
G1,1(f)=G2,2(f)=1,
Tn,1(f)=[G1,nl(f)-G1,2(f)G2,nl(f)]/Det,
Tn,2(f)=[G2,nl(f)-G1,2*(f)G1,nl(f)]/Det,
Det=1-|G1,2(f)|2,
Rn(f)=1+|Tn,1|2+|Tn,2|2-2 Re(Tn,1*G1,nl+Tn,2*G2,nl-Tn,1*Tn,2G1,2),
f>f0=1(L2-L1)α0,
f012r0,eff,
fmax=0.95f0=0.95(L2-L1)α0,
Na=2fmaxD.
G1,1(f)=G2,2(f)=G3,3(f)=1,
Tn,1(f)=[G1,nl(1-|G2,3|2)-G2,nl(G1,2-G1,3G3,2)-G3,nl(G1,3-G1,2G2,3)]/Det,
Tn,2(f)=[G2,nl(1-|G3,1|2)-G3,nl(G2,3-G2,1G1,3)-G1,nl(G2,1-G2,3G3,1)]/Det,
Tn,3(f)=[G3,nl(1-|G1,2|2)-G1,nl(G3,1-G3,2G2,1)-G2,nl(G3,2-G3,1G1,2)]/Det,
Det=1-|G1,2(f)|2-|G2,3|2-|G3,1|2+2 Re(G1,2G2,3G3,1),
Rn(f)=1+|Tn,1|2+|Tn,2|2+|Tn,3|2-2 Re(Tn,1*G1,nl+Tn,2*G2,nl+Tn,3*G3,nl-Tn,1*Tn,2G1,2-Tn,2*Tn,3G2,3-Tn,3*Tn,1G3,1),
G1,3=15{3+2 cos[2π(L3-L1)fxα0]}15[3+2 cos(2v)],
G2,3=15{3+2 cos[2π(L3-L2)fxα0]}15[3+2 cos(v+u)],
G1,2=15{3+2 cos[2π(L2-L1)fxα0]}15[3+2 cos(v-u)],
fx=f0=1(L3-L1)α0,G1,3=1,
Det=-(G1,2-G2,3)2=0,
G1,2-G2,3=45sin(v)sin(u)=0.
One-DMcorrection:L1=-575 m,
Two-DMcorrection:L1=-575 m,L2=8000 m,
Three-DMcorrection:L1=-575 m,L2=4000 m,
L3=800 m.
α0=1.5×10-4(31 arcsec),
α0=3.0×10-4(62 arcsec),
α0=4.5×10-4(93 arcsec).

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