Abstract

Exoplanet direct imaging with large ground based telescopes requires eXtreme Adaptive Optics that couples high-order adaptive optics and coronagraphy. A key element of such systems is the high-order wavefront sensor. We study here several high-order wavefront sensing approaches, and more precisely compare their sensitivity to noise. Three techniques are considered: the classical Shack-Hartmann sensor, the pyramid sensor and the recently proposed LIFTed Shack-Hartmann sensor. They are compared in a unified framework based on precise diffractive models and on the Fisher information matrix, which conveys the information present in the data whatever the estimation method. The diagonal elements of the inverse of the Fisher information matrix, which we use as a figure of merit, are similar to noise propagation coefficients. With these diagonal elements, so called “Fisher coefficients”, we show that the LIFTed Shack-Hartmann and pyramid sensors outperform the classical Shack-Hartmann sensor. In photon noise regime, the LIFTed Shack-Hartmann and modulated pyramid sensors obtain a similar overall noise propagation. The LIFTed Shack-Hartmann sensor however provides attractive noise properties on high orders.

© 2015 Optical Society of America

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2014 (4)

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

R. M. Wagner, M. L. Edwards, O. Kuhn, D. Thompson, and C. Veillet, “An overview and the current status of instrumentation at the Large Binocular Telescope Observatory,” Proc. SPIE 9147, 914705 (2014).
[Crossref]

S. Meimon, T. Fusco, V. Michau, and C. Plantet, “Sensing more modes with fewer sub-apertures: the LIFTed Shack–Hartmann wavefront sensor,” Opt. Lett. 39, 2835–2837 (2014).
[Crossref] [PubMed]

J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
[Crossref]

2013 (1)

2012 (1)

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
[Crossref]

2011 (1)

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
[Crossref]

2010 (2)

2009 (1)

2007 (2)

2006 (2)

2005 (2)

O. Guyon, “Limits of adaptive optics for high-contrast imaging,” Astrophys. J. 629, 592 (2005).
[Crossref]

C. Vérinaud, M. Le Louarn, V. Korkiakoski, and M. Carbillet, “Adaptive optics for high-contrast imaging: pyramid sensor versus spatially filtered Shack-Hartmann sensor,” Monthly Notices Royal Astron. Soc. 357, 26–30 (2005).
[Crossref]

2004 (4)

2000 (1)

D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The Four-Quadrant Phase-Mask Coronagraph. I. Principle,” Pub Astron. Soc. Pacific. 112, 1479–1486 (2000).
[Crossref]

1999 (2)

L. Meynadier, V. Michau, M.-T. Velluet, J.-M. Conan, L. M. Mugnier, and G. Rousset, “Noise propagation in wave-front sensing with phase diversity,” Appl. Opt. 38, 4967–4979 (1999).
[Crossref]

R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astronomy Astrophys. 350, L23–L26 (1999).

1996 (2)

R. C. Cannon, “Optimal bases for wave-front simulation and reconstruction on annular apertures,” J. Opt. Soc. Am. A 13, 862–867 (1996).
[Crossref]

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[Crossref]

1995 (2)

1992 (2)

F. Rigaut and E. Gendron, “Laser guide star in adaptive optics - the tilt determination problem,” Astronomy Astrophys. 261, 677–684 (1992).

R. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992).
[Crossref] [PubMed]

1939 (1)

B. Lyot, “The study of the solar corona and prominences without eclipses,” Monthly Notices Royal Astron. Soc. 99, 580 (1939).

Agapito, G.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
[Crossref]

Anthony, A.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
[Crossref]

Arcidiacono, C.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
[Crossref]

Argomedo, J.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
[Crossref]

Atwood, J.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
[Crossref]

Barrett, H. H.

Barriga, N.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
[Crossref]

Barrufolo, A.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

Baruffolo, A.

J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
[Crossref]

Barwick, S.

Baudoz, P.

J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
[Crossref]

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

T. Fusco, G. Rousset, J.-F. Sauvage, C. Petit, J.-L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle, M. Kasper, P. Baudoz, and P. Puget, “High-order adaptive optics requirements for direct detection of extrasolar planets: application to the SPHERE instrument,” Opt. Express 14, 7515–7534 (2006).
[Crossref] [PubMed]

Bauman, B.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
[Crossref]

Beuzit, J.-L.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
[Crossref]

T. Fusco, G. Rousset, J.-F. Sauvage, C. Petit, J.-L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle, M. Kasper, P. Baudoz, and P. Puget, “High-order adaptive optics requirements for direct detection of extrasolar planets: application to the SPHERE instrument,” Opt. Express 14, 7515–7534 (2006).
[Crossref] [PubMed]

Boccaletti, A.

D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The Four-Quadrant Phase-Mask Coronagraph. I. Principle,” Pub Astron. Soc. Pacific. 112, 1479–1486 (2000).
[Crossref]

Briguglio, R.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
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Brusa, G.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
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Busoni, L.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
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Cannon, R. C.

Caputa, K.

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Carbillet, M.

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Carrara, D. A.

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J. R. Fienup, B. J. Thelen, R. G. Paxman, and D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Astronomical Telescopes and Instrumentation, (ISOP, 1998), pp. 930–940.

Charton, J.

Chilcote, J.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Clénet, Y.

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Conan, J.-M.

Costille, A.

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T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

Dai, G.-M.

Dainty, C.

Deboulbé, A.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

Denny, J.

Dillon, D.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Dohlen, K.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
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T. Fusco, G. Rousset, J.-F. Sauvage, C. Petit, J.-L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle, M. Kasper, P. Baudoz, and P. Puget, “High-order adaptive optics requirements for direct detection of extrasolar planets: application to the SPHERE instrument,” Opt. Express 14, 7515–7534 (2006).
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Downing, M.

J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
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T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

Doyon, R.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Dunn, J.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Edwards, M. L.

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Esposito, S.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
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Feautrier, F.

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Feautrier, P.

J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
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Fedrigo, E.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

Fienup, J. R.

J. R. Fienup, B. J. Thelen, R. G. Paxman, and D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Astronomical Telescopes and Instrumentation, (ISOP, 1998), pp. 930–940.

Fini, L.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
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Fusco, T.

J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
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T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

S. Meimon, T. Fusco, V. Michau, and C. Plantet, “Sensing more modes with fewer sub-apertures: the LIFTed Shack–Hartmann wavefront sensor,” Opt. Lett. 39, 2835–2837 (2014).
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S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack–Hartmann sensor,” Monthly Notices Royal Astron. Soc. 371, 323–336 (2006).
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M. Nicolle, T. Fusco, G. Rousset, and V. Michau, “Improvement of Shack-Hartmann wave-front sensor measurement for extreme adaptive optics,” Opt. Lett. 29, 2743–2745 (2004).
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T. Fusco and C. Petit, Onera, the French Aerospace Lab 92322 Chatillon, France, (personal communication, 2015).

Galvez, R.

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Gavel, D. T.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Gendron, E.

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Gherardi, A.

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Girard, J.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

Goodsell, S. J.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Graham, J. R.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Guerra, J. C.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
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Guyon, O.

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Hartung, M.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

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Kasper, M.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

T. Fusco, G. Rousset, J.-F. Sauvage, C. Petit, J.-L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle, M. Kasper, P. Baudoz, and P. Puget, “High-order adaptive optics requirements for direct detection of extrasolar planets: application to the SPHERE instrument,” Opt. Express 14, 7515–7534 (2006).
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J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
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Korkiakoski, V.

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Kuhn, O.

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B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Maire, J.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
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T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

T. Fusco, G. Rousset, J.-F. Sauvage, C. Petit, J.-L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle, M. Kasper, P. Baudoz, and P. Puget, “High-order adaptive optics requirements for direct detection of extrasolar planets: application to the SPHERE instrument,” Opt. Express 14, 7515–7534 (2006).
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Mugnier, L.

J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
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Pennington, T. L.

Perret, D.

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Perrin, M.

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J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
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T. Fusco, G. Rousset, J.-F. Sauvage, C. Petit, J.-L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle, M. Kasper, P. Baudoz, and P. Puget, “High-order adaptive optics requirements for direct detection of extrasolar planets: application to the SPHERE instrument,” Opt. Express 14, 7515–7534 (2006).
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Plantet, C.

Poyneer, L. A.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Poyneer, L.A.

Puget, P.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

T. Fusco, G. Rousset, J.-F. Sauvage, C. Petit, J.-L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle, M. Kasper, P. Baudoz, and P. Puget, “High-order adaptive optics requirements for direct detection of extrasolar planets: application to the SPHERE instrument,” Opt. Express 14, 7515–7534 (2006).
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Puglisi, A.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
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Quirez, C.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Quirós-Pacheco, F.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
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Ragazzoni, R.

R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astronomy Astrophys. 350, L23–L26 (1999).

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Rantakyro, F.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Reshtov, V.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Riaud, P.

D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The Four-Quadrant Phase-Mask Coronagraph. I. Principle,” Pub Astron. Soc. Pacific. 112, 1479–1486 (2000).
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Riccardi, A.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
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Rigaut, F.

F. Rigaut and E. Gendron, “Laser guide star in adaptive optics - the tilt determination problem,” Astronomy Astrophys. 261, 677–684 (1992).

Rochat, S.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

Roggemann, M. C.

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, and T. L. Pennington, “Fundamental performance comparison of a Hartmann and a shearing interferometer wave-front sensor,” Appl. Opt. 34, 4186–4195 (1995).
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T. J. Schulz, W. Sun, and M. C. Roggemann, “Cramér-Rao bounds for estimation of turbulence-induced wavefront aberrations,” in SPIE’s International Symposium on Optical Science, Engineering, and Instrumentation, (International Society for Optics and Photonics, 1999), pp. 23–28.

D. J. Lee, B. M. Welsh, and M. C. Roggemann, “Cramér-Rao analysis of phase diversity imaging,” in Optical Science, Engineering and Instrumentation’97, (ISOP, 1997), pp. 161–172.

Rouan, D.

D. Rouan, P. Riaud, A. Boccaletti, Y. Clénet, and A. Labeyrie, “The Four-Quadrant Phase-Mask Coronagraph. I. Principle,” Pub Astron. Soc. Pacific. 112, 1479–1486 (2000).
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Rousset, G.

Sadakuni, N.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Saddlemyer, L.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Salasnich, B.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

Salinari, P.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
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Sauvage, J.-F.

J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
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T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

T. Fusco, G. Rousset, J.-F. Sauvage, C. Petit, J.-L. Beuzit, K. Dohlen, D. Mouillet, J. Charton, M. Nicolle, M. Kasper, P. Baudoz, and P. Puget, “High-order adaptive optics requirements for direct detection of extrasolar planets: application to the SPHERE instrument,” Opt. Express 14, 7515–7534 (2006).
[Crossref] [PubMed]

Savran-sky, D.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Schulz, T. J.

T. J. Schulz, W. Sun, and M. C. Roggemann, “Cramér-Rao bounds for estimation of turbulence-induced wavefront aberrations,” in SPIE’s International Symposium on Optical Science, Engineering, and Instrumentation, (International Society for Optics and Photonics, 1999), pp. 23–28.

Sevin, A.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

Sivaramakrishnan, A.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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J. P. Lloyd and A. Sivaramakrishnan, “Tip-tilt error in Lyot coronagraphs,” Astrophys. J. 621, 1153–1158 (2010).
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Smith, M.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Soenke, C.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

Soummer, R.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Stefanini, P.

S. Esposito, A. Riccardi, E. Pinna, A. Puglisi, F. Quirós-Pacheco, C. Arcidiacono, M. Xompero, R. Briguglio, G. Agapito, L. Busoni, L. Fini, J. Argomedo, A. Gherardi, G. Brusa, D. Miller, J. C. Guerra, P. Stefanini, and P. Salinari, “Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011).
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Suarez, M.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

SuarezValles, M.

J.-F. Sauvage, T. Fusco, C. Petit, D. Mouillet, K. Dohlen, A. Costille, J.-L. Beuzit, A. Baruffolo, M. E. Kasper, M. SuarezValles, M. Downing, P. Feautrier, L. Mugnier, and P. Baudoz, “Wave-front sensor strategies for SPHERE: first on-sky results and future improvements,” Proc. SPIE 9148, 914847 (2014).
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Sun, W.

T. J. Schulz, W. Sun, and M. C. Roggemann, “Cramér-Rao bounds for estimation of turbulence-induced wavefront aberrations,” in SPIE’s International Symposium on Optical Science, Engineering, and Instrumentation, (International Society for Optics and Photonics, 1999), pp. 23–28.

Tallon, M.

Thelen, B. J.

J. R. Fienup, B. J. Thelen, R. G. Paxman, and D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Astronomical Telescopes and Instrumentation, (ISOP, 1998), pp. 930–940.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” in Optical Science, Engineering and Instrumentation’97, (ISOP, 1997), pp. 307–320.

Thiebaut, E.

E. Thiebaut, “Introduction to image reconstruction and inverse problems,” in Optics in Astrophysics, (Springer, 2005), pp. 397–422.

Thomas, S.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack–Hartmann sensor,” Monthly Notices Royal Astron. Soc. 371, 323–336 (2006).
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Thompson, D.

R. M. Wagner, M. L. Edwards, O. Kuhn, D. Thompson, and C. Veillet, “An overview and the current status of instrumentation at the Large Binocular Telescope Observatory,” Proc. SPIE 9147, 914705 (2014).
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Tokovinin, A.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack–Hartmann sensor,” Monthly Notices Royal Astron. Soc. 371, 323–336 (2006).
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Veillet, C.

R. M. Wagner, M. L. Edwards, O. Kuhn, D. Thompson, and C. Veillet, “An overview and the current status of instrumentation at the Large Binocular Telescope Observatory,” Proc. SPIE 9147, 914705 (2014).
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Velluet, M.-T.

Vérinaud, C.

C. Vérinaud, M. Le Louarn, V. Korkiakoski, and M. Carbillet, “Adaptive optics for high-contrast imaging: pyramid sensor versus spatially filtered Shack-Hartmann sensor,” Monthly Notices Royal Astron. Soc. 357, 26–30 (2005).
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C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27–38 (2004).
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Vigan, A.

T. Fusco, J.-F. Sauvage, C. Petit, A. Costille, K. Dohlen, D. Mouillet, J.-L. Beuzit, M. Kasper, M. Suarez, C. Soenke, E. Fedrigo, M. Downing, P. Baudoz, A. Sevin, D. Perret, A. Barrufolo, B. Salasnich, P. Puget, F. Feautrier, S. Rochat, T. Moulin, A. Deboulbé, E. Hugot, A. Vigan, D. Mawet, J. Girard, and N. Hubin, “Final performance and lesson-learned of SAXO, the VLT-SPHERE extreme AO: from early design to on-sky results,” Proc. SPIE 9148, 91481U (2014).

Wagner, R. F.

Wagner, R. M.

R. M. Wagner, M. L. Edwards, O. Kuhn, D. Thompson, and C. Veillet, “An overview and the current status of instrumentation at the Large Binocular Telescope Observatory,” Proc. SPIE 9147, 914705 (2014).
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Wallace, J. K.

B. A. Macintosh, A. Anthony, J. Atwood, N. Barriga, B. Bauman, K. Caputa, J. Chilcote, D. Dillon, R. Doyon, J. Dunn, D. T. Gavel, R. Galvez, S. J. Goodsell, J. R. Graham, M. Hartung, J. Isaacs, D. Kerley, Q. Konopacky, K. Labrie, J. E. Larkin, J. Maire, C. Marois, M. Millar-Blanchaer, A. Nunez, B. R. Oppenheimer, D. W. Palmer, J. Pazder, M. Perrin, L. A. Poyneer, C. Quirez, F. Rantakyro, V. Reshtov, L. Saddlemyer, N. Sadakuni, D. Savran-sky, A. Sivaramakrishnan, M. Smith, R. Soummer, S. Thomas, J. K. Wallace, J. Weiss, and S. Wiktorowicz, “The Gemini Planet Imager: integration and status,” Proc. SPIE 8446, 84461U (2012).
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Figures (4)

Fig. 1
Fig. 1 Fisher coefficients of the Shack-Hartmann sensor 40×40 and the LIFTed Shack-Hartmann sensor 20×20 for the estimation of 1000 Karhunen-Loève modes. The LIFTed Shack-Hartmann sensor estimates 8 modes per subaperture. The dotted line indicates the j 1 trend.
Fig. 2
Fig. 2 Fisher coefficients of the pyramid sensor for the estimation of 1000 Karhunen-Loève modes. The curves correspond to modulations from 0 λ/D to 6 λ/D. A modulation at 0.5 λ/D has been added to show the transition between 0 λ/D and 2 λ/D. The dotted line indicates the j 1 trend.
Fig. 3
Fig. 3 Fisher coefficients of the LIFTed Shack-Hartmann, the classical Shack-Hartmann and the pyramid sensors for the estimation of 1000 Karhunen-Loève modes. The modulation radii for the pyramid sensor are 0 λ/D, 2 λ/D, 3 λ/D and 6 λ/D. We consider a classical Shack-Hartmann estimating slopes with a CoG in photon noise regime, and a WCoG in read-out noise regime.
Fig. 4
Fig. 4 Photon noise Fisher coefficients of the LIFTed Shack-Hartmann, the classical Shack-Hartmann and the pyramid sensors for the estimation of 1000 Karhunen-Loève modes. Zoom over the modes 100 to 1000.

Equations (24)

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σ p 2 = I ¯ p + σ e 2
y = DA + n
y t = [ x 1 , y 1 , x 2 , y 2 , , x N s u b , y N s u b ]
S x [ k ] = ( P 1 [ k ] + P 3 [ k ] ) ( P 2 [ k ] + P 4 [ k ] ) N S y [ k ] = ( P 1 [ k ] + P 2 [ k ] ) ( P 3 [ k ] + P 4 [ k ] ) N with N = 1 N p i x k = 1 N p i x P 1 [ k ] + P 2 [ k ] + P 3 [ k ] + P 4 [ k ]
y t = [ S x [ 1 ] , S y [ 1 ] , S x [ 2 ] , , S y [ 2 ] , , S x [ N p i x ] , S y [ N p i x ] ]
y t = [ a 1 , 1 , a 2 , 1 , , a m , 1 , a 1 , 2 , a 2 , 2 , , a m , 2 , , a 1 , N s u b , a 2 , N s u b , , a m , N s u b ]
var { a ^ a } [ 1 + bias / a ] 2 Fisher ( a )
F i , j ( A p ) = E { [ a i ln p ( y | A ) ] | A = A P [ a j ln p ( y | A ) ] | A = A p }
F i , j ( A p ) = k = 1 N 1 σ k 2 y ¯ k ( A ) a i | A = A p y ¯ k ( A ) a j | A = A p
t r a c e ( F 1 ) = ( i a i ) a 1 n p h + ( i β i ) β ( σ e n p h ) 2
F i , j ( A P ) = E { [ a i ln p ( y | A ) ] | A = A p [ a j ln p ( y | A ) ] | A = A p }
p ( y | A ) = p b ( y ) = k = 1 N 1 2 π σ k 2 exp { [ y k y ¯ k ( A ) ] 2 2 σ k 2 }
ln p ( y | A ) = k = 1 N 1 2 ln ( 2 π σ k 2 ) [ y k y ¯ k ( A ) ] 2 2 σ k 2
a i ln p ( y | A ) | A = A p = k = 1 N 1 σ k 2 y ¯ k ( A ) a i | A = A p × [ y k y ¯ k ( A ) ]
a i ln p ( y | A ) | A = A p × a j ln p ( y | A ) | A = A p = k = 1 N 1 σ k 4 y ¯ k ( A ) a j | A = A p y ¯ k ( A ) a j | A = A p × [ y k y ¯ k ( A ) ] 2 + l k 1 σ l 2 σ k 2 y ¯ l ( A ) a i | A = A p y ¯ k ( A ) a j | A = A p × [ y k y ¯ k ( A ) ] × [ y l y ¯ l ( A ) ]
F i , j ( A P ) = k = 1 N 1 σ k 2 y ¯ k ( A ) a i | A = A p y ¯ k ( A ) a j | A = A p + l k σ k l 2 σ l 2 σ k 2 y ¯ l ( A ) a j | A = A p y ¯ k ( A ) a j | A = A p
F i , j ( A p ) = k = 1 N 1 σ k 2 y ¯ k ( A ) a i | A = A p y ¯ k ( A ) a j | A = A p
A ^ = ( D t C n 1 D ) 1 D t C n 1 y
< E E t > M L = ( D t C n 1 D ) 1
t r a c e ( < E E t > ) = ( i C p h , i ) C p h 1 n p h + ( i C d e t , i ) C d e t ( σ e n p h ) 2
D = y A
D = ( y 1 a 1 y 1 a 2 y 1 a m y 2 a 1 y 2 a 1 y 2 a m y p a 1 y p a 2 y p a m )
[ D t C n 1 D ] i , j = k 1 σ k 2 y k a i y k a j + l k k l 2 σ l 2 σ k 2 y l a i y k a j = F i , j
t r a c e ( F 1 ) = ( i α i ) α 1 n p h + ( i β i ) β ( σ e n p h ) 2

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