Abstract

Current wavefront sensors for high resolution imaging have either a large dynamic range or a high sensitivity. A new kind of wavefront sensor is developed which can have both: the Generalised Optical Differentiation wavefront sensor. This new wavefront sensor is based on the principles of optical differentiation by amplitude filters. We have extended the theory behind linear optical differentiation and generalised it to nonlinear filters. We used numerical simulations and laboratory experiments to investigate the properties of the generalised wavefront sensor. With this we created a new filter that can decouple the dynamic range from the sensitivity. These properties make it suitable for adaptive optic systems where a large range of phase aberrations have to be measured with high precision.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]

2015 (1)

2014 (1)

2010 (3)

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

E. Gendron, M. Brangier, G. Chenegros, F. Vidal, Z. Hubert, G. Rousset, and F. Pouplard, “A new sensor for laser tomography on ELTs,” Adaptative Optics for Extremely Large Telescopes Conference, 05003 (2010).
[Crossref]

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

2008 (1)

D. Peter, M. Feldt, B. Dorner, T. Henning, S. Hippler, and J. Aceituno, “PYRAMIR: Calibration and Operation of a Pyramid Near-Infrared Wavefront Sensor,” Pacific 120, 872–886 (2008).

2007 (1)

M. Schwertner, M. J. Booth, and T. Wilson, “Specimen-induced distortions in light microscopy,” J. Microsc. 228, 97–102 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (3)

O. Guyon, “Limits of adaptive optics for high-contrast imaging,” Astrophys. J. 629, 592–614 (2005).
[Crossref]

F. Hénault, “Wavefront sensor based on varying transmission filters: theory and expected performance,” J. Mod. Opt. 52, 1917–1931 (2005).
[Crossref]

J.E. Oti, V. F. Canales, and M. P. Cagigal, “Improvements on the optical differentiation wavefront sensor,” Mon. Not. R. Astron. Soc. 360, 1448–1454 (2005).
[Crossref]

2004 (3)

C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Comm. 233 (2004).
[Crossref]

M. Schwertner, M. J. Booth, M. A. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[Crossref]

L. A. Poyneer and B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. A 21, 810–819 (2004).
[Crossref]

2003 (2)

N. O ’Mahony, “RoboDIMM - The ING’s new seeing monitor,” The newsletter of the Isaac Newton Group of telescopes 7, 22–24 (2003).

J. E. Oti, V. F. Canales, and M. P. Cagigal, “Analysis of the signal-to-noise ratio in the optical differentiation wavefront sensor,” Opt. Express 11, 2783–2790 (2003)
[Crossref] [PubMed]

2002 (1)

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[Crossref]

1999 (1)

R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. textbf350, L23–L26 (1999).

1997 (1)

B. Welsh, “A Fourier series based atmospheric phase screen generator for simulating nonisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327 (1997).
[Crossref]

1996 (3)

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

J. Davis and W. Tango, “Measurement of the atmospheric coherence time,” Publ. Astron. Soc. Pac. 108, 456 (1996).
[Crossref]

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

1994 (1)

B. A. Horwitz, “New pupil-plane wavefront gradient sensor,” Proc. SPIE 2201, 496 (May31, 1994).
[Crossref]

1984 (1)

1981 (1)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19281–376 (1981).
[Crossref]

1980 (1)

1972 (1)

1971 (1)

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

1948 (1)

T. von Karman, “Progress in the statistical theory of turbulence,” PNAS 34, 530–539 (1948).
[Crossref] [PubMed]

1938 (1)

G. I. Taylor, “The spectrum of turbulence,” (ISO 4) Proc. R. Soc. A 164, 476–490 (1938).
[Crossref]

’Mahony, N. O

N. O ’Mahony, “RoboDIMM - The ING’s new seeing monitor,” The newsletter of the Isaac Newton Group of telescopes 7, 22–24 (2003).

Aceituno, J.

D. Peter, M. Feldt, B. Dorner, T. Henning, S. Hippler, and J. Aceituno, “PYRAMIR: Calibration and Operation of a Pyramid Near-Infrared Wavefront Sensor,” Pacific 120, 872–886 (2008).

Agapito, G.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Arcidiacono, C.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Berdja, A.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Bergomi, M.

D. Greggio, D. Magrin, J. Farinato, R. Ragazzoni, M. Bergomi, M. Dima, L. Marafatto, and V. Viotto, “Avoiding to trade sensitivity for linearity in a real world WFS,” Proceedings of the Third AO4ELT conference, 36 (2013)

Booth, M. J.

M. Schwertner, M. J. Booth, and T. Wilson, “Specimen-induced distortions in light microscopy,” J. Microsc. 228, 97–102 (2007).
[Crossref] [PubMed]

M. Schwertner, M. J. Booth, M. A. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[Crossref]

Borgnino, J.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Bortz, J.C.

Brangier, M.

E. Gendron, M. Brangier, G. Chenegros, F. Vidal, Z. Hubert, G. Rousset, and F. Pouplard, “A new sensor for laser tomography on ELTs,” Adaptative Optics for Extremely Large Telescopes Conference, 05003 (2010).
[Crossref]

Briguglio, R.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Brusa-Zappellini, G.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Brynnel, J.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Burvall, A.

Busoni, L.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Bustos, E.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Cagigal, M. P.

J.E. Oti, V. F. Canales, and M. P. Cagigal, “Improvements on the optical differentiation wavefront sensor,” Mon. Not. R. Astron. Soc. 360, 1448–1454 (2005).
[Crossref]

J. E. Oti, V. F. Canales, and M. P. Cagigal, “Analysis of the signal-to-noise ratio in the optical differentiation wavefront sensor,” Opt. Express 11, 2783–2790 (2003)
[Crossref] [PubMed]

Canales, V. F.

J.E. Oti, V. F. Canales, and M. P. Cagigal, “Improvements on the optical differentiation wavefront sensor,” Mon. Not. R. Astron. Soc. 360, 1448–1454 (2005).
[Crossref]

J. E. Oti, V. F. Canales, and M. P. Cagigal, “Analysis of the signal-to-noise ratio in the optical differentiation wavefront sensor,” Opt. Express 11, 2783–2790 (2003)
[Crossref] [PubMed]

Cannon, R. C.

Chamot, S. R.

Chenegros, G.

E. Gendron, M. Brangier, G. Chenegros, F. Vidal, Z. Hubert, G. Rousset, and F. Pouplard, “A new sensor for laser tomography on ELTs,” Adaptative Optics for Extremely Large Telescopes Conference, 05003 (2010).
[Crossref]

Conan, J. M.

Dainty, C.

DaliAli, W.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Daly, E.

Davis, J.

J. Davis and W. Tango, “Measurement of the atmospheric coherence time,” Publ. Astron. Soc. Pac. 108, 456 (1996).
[Crossref]

Delgado, J. M.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Demers, R.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Dima, M.

D. Greggio, D. Magrin, J. Farinato, R. Ragazzoni, M. Bergomi, M. Dima, L. Marafatto, and V. Viotto, “Avoiding to trade sensitivity for linearity in a real world WFS,” Proceedings of the Third AO4ELT conference, 36 (2013)

Diolaiti, E.

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[Crossref]

Dorner, B.

D. Peter, M. Feldt, B. Dorner, T. Henning, S. Hippler, and J. Aceituno, “PYRAMIR: Calibration and Operation of a Pyramid Near-Infrared Wavefront Sensor,” Pacific 120, 872–886 (2008).

Escuti, M. J.

M. N. Miskiewicz and M. J. Escuti, “Direct-writing of complex liquid crystal patterns,” Opt. Express 22, 12691–12706 (2014).
[Crossref] [PubMed]

R.K. Komanduri, K. F. Lawler, and M. J. Escuti, “Multi-twist retarders: broadband retardation control using self-aligning reactive liquid crystal layers,” Opt. Express21, 404–420 (2013).

Esposito, S.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Farinato, J.

R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. textbf350, L23–L26 (1999).

D. Greggio, D. Magrin, J. Farinato, R. Ragazzoni, M. Bergomi, M. Dima, L. Marafatto, and V. Viotto, “Avoiding to trade sensitivity for linearity in a real world WFS,” Proceedings of the Third AO4ELT conference, 36 (2013)

Feldt, M.

D. Peter, M. Feldt, B. Dorner, T. Henning, S. Hippler, and J. Aceituno, “PYRAMIR: Calibration and Operation of a Pyramid Near-Infrared Wavefront Sensor,” Pacific 120, 872–886 (2008).

Fini, L.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Fuensalida, J. J.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Fusco, T.

Gendron, E.

E. Gendron, M. Brangier, G. Chenegros, F. Vidal, Z. Hubert, G. Rousset, and F. Pouplard, “A new sensor for laser tomography on ELTs,” Adaptative Optics for Extremely Large Telescopes Conference, 05003 (2010).
[Crossref]

Greggio, D.

D. Greggio, D. Magrin, J. Farinato, R. Ragazzoni, M. Bergomi, M. Dima, L. Marafatto, and V. Viotto, “Avoiding to trade sensitivity for linearity in a real world WFS,” Proceedings of the Third AO4ELT conference, 36 (2013)

Guerra, J. C.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Guyon, O.

O. Guyon, “Limits of adaptive optics for high-contrast imaging,” Astrophys. J. 629, 592–614 (2005).
[Crossref]

Hénault, F.

F. Hénault, “Wavefront sensor based on varying transmission filters: theory and expected performance,” J. Mod. Opt. 52, 1917–1931 (2005).
[Crossref]

Henning, T.

D. Peter, M. Feldt, B. Dorner, T. Henning, S. Hippler, and J. Aceituno, “PYRAMIR: Calibration and Operation of a Pyramid Near-Infrared Wavefront Sensor,” Pacific 120, 872–886 (2008).

Hippler, S.

D. Peter, M. Feldt, B. Dorner, T. Henning, S. Hippler, and J. Aceituno, “PYRAMIR: Calibration and Operation of a Pyramid Near-Infrared Wavefront Sensor,” Pacific 120, 872–886 (2008).

Horwitz, B. A.

B. A. Horwitz, “New pupil-plane wavefront gradient sensor,” Proc. SPIE 2201, 496 (May31, 1994).
[Crossref]

Hubert, Z.

E. Gendron, M. Brangier, G. Chenegros, F. Vidal, Z. Hubert, G. Rousset, and F. Pouplard, “A new sensor for laser tomography on ELTs,” Adaptative Optics for Extremely Large Telescopes Conference, 05003 (2010).
[Crossref]

Komanduri, R.K.

R.K. Komanduri, K. F. Lawler, and M. J. Escuti, “Multi-twist retarders: broadband retardation control using self-aligning reactive liquid crystal layers,” Opt. Express21, 404–420 (2013).

Lawler, K. F.

R.K. Komanduri, K. F. Lawler, and M. J. Escuti, “Multi-twist retarders: broadband retardation control using self-aligning reactive liquid crystal layers,” Opt. Express21, 404–420 (2013).

Lombardi, G.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Macintosh, B.

L. A. Poyneer and B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. A 21, 810–819 (2004).
[Crossref]

Magrin, D.

D. Greggio, D. Magrin, J. Farinato, R. Ragazzoni, M. Bergomi, M. Dima, L. Marafatto, and V. Viotto, “Avoiding to trade sensitivity for linearity in a real world WFS,” Proceedings of the Third AO4ELT conference, 36 (2013)

Maire, J.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Marafatto, L.

D. Greggio, D. Magrin, J. Farinato, R. Ragazzoni, M. Bergomi, M. Dima, L. Marafatto, and V. Viotto, “Avoiding to trade sensitivity for linearity in a real world WFS,” Proceedings of the Third AO4ELT conference, 36 (2013)

Meimon, S.

Miskiewicz, M. N.

Navarrete, J.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Neil, M. A. A.

M. Schwertner, M. J. Booth, M. A. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[Crossref]

Oti, J. E.

Oti, J.E.

J.E. Oti, V. F. Canales, and M. P. Cagigal, “Improvements on the optical differentiation wavefront sensor,” Mon. Not. R. Astron. Soc. 360, 1448–1454 (2005).
[Crossref]

Peter, D.

D. Peter, M. Feldt, B. Dorner, T. Henning, S. Hippler, and J. Aceituno, “PYRAMIR: Calibration and Operation of a Pyramid Near-Infrared Wavefront Sensor,” Pacific 120, 872–886 (2008).

Pieralli, F.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Pinna, E.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Plantet, C.

Platt, B. C.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Pouplard, F.

E. Gendron, M. Brangier, G. Chenegros, F. Vidal, Z. Hubert, G. Rousset, and F. Pouplard, “A new sensor for laser tomography on ELTs,” Adaptative Optics for Extremely Large Telescopes Conference, 05003 (2010).
[Crossref]

Poyneer, L. A.

L. A. Poyneer and B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. A 21, 810–819 (2004).
[Crossref]

Puglisi, A. T.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Quirós-Pacheco, F.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Ragazzoni, R.

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[Crossref]

R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. textbf350, L23–L26 (1999).

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

D. Greggio, D. Magrin, J. Farinato, R. Ragazzoni, M. Bergomi, M. Dima, L. Marafatto, and V. Viotto, “Avoiding to trade sensitivity for linearity in a real world WFS,” Proceedings of the Third AO4ELT conference, 36 (2013)

Reyes, M.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Riccardi, A.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19281–376 (1981).
[Crossref]

Rousset, G.

E. Gendron, M. Brangier, G. Chenegros, F. Vidal, Z. Hubert, G. Rousset, and F. Pouplard, “A new sensor for laser tomography on ELTs,” Adaptative Optics for Extremely Large Telescopes Conference, 05003 (2010).
[Crossref]

Salinari, P.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Sarazin, M.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Schwertner, M.

M. Schwertner, M. J. Booth, and T. Wilson, “Specimen-induced distortions in light microscopy,” J. Microsc. 228, 97–102 (2007).
[Crossref] [PubMed]

M. Schwertner, M. J. Booth, M. A. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[Crossref]

Shack, R. V.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Southwell, W. H.

Sprague, R. A.

Stefanini, P.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Tango, W.

J. Davis and W. Tango, “Measurement of the atmospheric coherence time,” Publ. Astron. Soc. Pac. 108, 456 (1996).
[Crossref]

Taylor, G. I.

G. I. Taylor, “The spectrum of turbulence,” (ISO 4) Proc. R. Soc. A 164, 476–490 (1938).
[Crossref]

Thompson, B. J.

Tokovinin, A.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Tozzi, A.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Vazquez Ramio, H.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

Vérinaud, C.

C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Comm. 233 (2004).
[Crossref]

Vernet, E.

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[Crossref]

Vidal, F.

E. Gendron, M. Brangier, G. Chenegros, F. Vidal, Z. Hubert, G. Rousset, and F. Pouplard, “A new sensor for laser tomography on ELTs,” Adaptative Optics for Extremely Large Telescopes Conference, 05003 (2010).
[Crossref]

Viotto, V.

D. Greggio, D. Magrin, J. Farinato, R. Ragazzoni, M. Bergomi, M. Dima, L. Marafatto, and V. Viotto, “Avoiding to trade sensitivity for linearity in a real world WFS,” Proceedings of the Third AO4ELT conference, 36 (2013)

von Karman, T.

T. von Karman, “Progress in the statistical theory of turbulence,” PNAS 34, 530–539 (1948).
[Crossref] [PubMed]

Welsh, B.

B. Welsh, “A Fourier series based atmospheric phase screen generator for simulating nonisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327 (1997).
[Crossref]

Wilson, T.

M. Schwertner, M. J. Booth, and T. Wilson, “Specimen-induced distortions in light microscopy,” J. Microsc. 228, 97–102 (2007).
[Crossref] [PubMed]

M. Schwertner, M. J. Booth, M. A. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[Crossref]

Xompero, M.

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

Ziad, A.

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

(ISO 4) Proc. R. Soc. A (1)

G. I. Taylor, “The spectrum of turbulence,” (ISO 4) Proc. R. Soc. A 164, 476–490 (1938).
[Crossref]

A new sensor for laser tomography on ELTs (1)

E. Gendron, M. Brangier, G. Chenegros, F. Vidal, Z. Hubert, G. Rousset, and F. Pouplard, “A new sensor for laser tomography on ELTs,” Adaptative Optics for Extremely Large Telescopes Conference, 05003 (2010).
[Crossref]

Appl. Opt. (1)

Astron. Astrophys. (2)

W. DaliAli, A. Ziad, A. Berdja, J. Maire, J. Borgnino, M. Sarazin, G. Lombardi, J. Navarrete, H. Vazquez Ramio, M. Reyes, J. M. Delgado, J. J. Fuensalida, A. Tokovinin, and E. Bustos, “Multi-instrument measurement campaign at Paranal in 2007. Characterization of the outer scale and the seeing of the surface layer,” Astron. Astrophys. 524, A73 (2010).
[Crossref]

R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. textbf350, L23–L26 (1999).

Astrophys. J. (1)

O. Guyon, “Limits of adaptive optics for high-contrast imaging,” Astrophys. J. 629, 592–614 (2005).
[Crossref]

J. Microsc. (2)

M. Schwertner, M. J. Booth, M. A. A. Neil, and T. Wilson, “Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry,” J. Microsc. 213, 11–19 (2004).
[Crossref]

M. Schwertner, M. J. Booth, and T. Wilson, “Specimen-induced distortions in light microscopy,” J. Microsc. 228, 97–102 (2007).
[Crossref] [PubMed]

J. Mod. Opt. (2)

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

F. Hénault, “Wavefront sensor based on varying transmission filters: theory and expected performance,” J. Mod. Opt. 52, 1917–1931 (2005).
[Crossref]

J. Opt. Soc. A (1)

L. A. Poyneer and B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. A 21, 810–819 (2004).
[Crossref]

J. Opt. Soc. Am. (2)

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
[Crossref]

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

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

Mon. Not. R. Astron. Soc. (1)

J.E. Oti, V. F. Canales, and M. P. Cagigal, “Improvements on the optical differentiation wavefront sensor,” Mon. Not. R. Astron. Soc. 360, 1448–1454 (2005).
[Crossref]

Opt. Comm. (1)

C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Comm. 233 (2004).
[Crossref]

Opt. Commun. (1)

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
[Crossref]

Opt. Express (4)

Pacific (1)

D. Peter, M. Feldt, B. Dorner, T. Henning, S. Hippler, and J. Aceituno, “PYRAMIR: Calibration and Operation of a Pyramid Near-Infrared Wavefront Sensor,” Pacific 120, 872–886 (2008).

PNAS (1)

T. von Karman, “Progress in the statistical theory of turbulence,” PNAS 34, 530–539 (1948).
[Crossref] [PubMed]

Proc. SPIE (2)

B. A. Horwitz, “New pupil-plane wavefront gradient sensor,” Proc. SPIE 2201, 496 (May31, 1994).
[Crossref]

B. Welsh, “A Fourier series based atmospheric phase screen generator for simulating nonisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327 (1997).
[Crossref]

Prog. Opt. (1)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19281–376 (1981).
[Crossref]

Publ. Astron. Soc. Pac. (1)

J. Davis and W. Tango, “Measurement of the atmospheric coherence time,” Publ. Astron. Soc. Pac. 108, 456 (1996).
[Crossref]

SPIE (1)

S. Esposito, A. Riccardi, L. Fini, A. T. Puglisi, E. Pinna, M. Xompero, R. Briguglio, F. Quirós-Pacheco, P. Stefanini, J. C. Guerra, L. Busoni, A. Tozzi, F. Pieralli, G. Agapito, G. Brusa-Zappellini, R. Demers, J. Brynnel, C. Arcidiacono, and P. Salinari, “First light AO (FLAO) system for LBT: final integration, acceptance test in Europe, and preliminary on-sky commissioning results,” SPIE 7736, 773609 (2010).

The newsletter of the Isaac Newton Group of telescopes (1)

N. O ’Mahony, “RoboDIMM - The ING’s new seeing monitor,” The newsletter of the Isaac Newton Group of telescopes 7, 22–24 (2003).

Other (2)

R.K. Komanduri, K. F. Lawler, and M. J. Escuti, “Multi-twist retarders: broadband retardation control using self-aligning reactive liquid crystal layers,” Opt. Express21, 404–420 (2013).

D. Greggio, D. Magrin, J. Farinato, R. Ragazzoni, M. Bergomi, M. Dima, L. Marafatto, and V. Viotto, “Avoiding to trade sensitivity for linearity in a real world WFS,” Proceedings of the Third AO4ELT conference, 36 (2013)

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Figures (13)

Fig. 1
Fig. 1 The three filters used in the simulations: linear, step and sigmoid. The figure on the right shows the derivative of the filters. The derivative of the step filter is not shown since that is an infinitely high delta function.
Fig. 2
Fig. 2 In this figure a single measurement loop is shown. The measurements start with a phase distortion caused by either the atmosphere or a non flat shape of the deformable mirror. This creates an aberrated psf, which is then split up in four copies. Each copy is filtered by a differently oriented amplitude filter. The amplitude filter in this example is the hybrid filter, with β = 1/3. Another propagation creates four pupil images. The pupil images are then combined to retrieve the wavefront gradients. From the wavefront gradients the input wavefront is reconstructed with the matrix-vector multiplication method.
Fig. 3
Fig. 3 These two figures show the dynamic range of the filters. The figure on the right is a zoomed version of the left figure. The input tilt coefficients on the x axis are plotted against the measured tilt coefficients.
Fig. 4
Fig. 4 In these figures the deviation from unity of the Strehl ratio is plotted against time. The left figure shows the change in strehl when the phase screen is created by taking a superposition of deformable mirror modes. The figure on the right shows the increase in Strehl when a static atmospheric phase screen is used. All wavefront sensors start with the same phase screen.
Fig. 5
Fig. 5 The singular value is plotted against mode number. The pyramid has the highest singular values. The sigmoid is in between the pyramid and the linear.
Fig. 6
Fig. 6 Two different dynamical closed loop simulations. In the left figure the turbulence strength is D/r0 = 84, and for the right figure it corresponds to median conditions in the visible at D/r0 = 21. All wavefront sensors are looking at the same sky so that they correct the same phase screen at every point in time. For both atmospheric conditions 10 simulations were done. The mean 1 − S of these simulations are plotted with solid lines. The standard deviation of each wavefront sensor is given by the coloured region. 1 − S is plotted for the different wavefront sensors and the natural effects of the atmosphere. In both cases the sigmoid filter out-performs both wavefront sensors.
Fig. 7
Fig. 7 Snapshot of a single dynamical simulation with D/r0 = 21. The top row shows the residual phase distortion due to the atmosphere. Red is a positive π phase difference and blue is a negative π phase difference. Up to four Airy rings can be clearly seen in the corrected PSF of the pyramid and the sigmoid. The linear filter cannot compensate enough to stabilize the first Airy ring. The PSFs are normalised with respect to the maximum of the diffraction limited PSF.
Fig. 8
Fig. 8 In this figure the schematic lab set-up is drawn. The focal lengths are not to scale. A laser at 636.3 nm is used as input. The laser is focused at a 10 µm pinhole to simulate a point source. The point source is collimated on a deformable mirror which is used to create and correct pupil aberrations. The new pupil is focused through a beam splitter from which one part goes to the new wavefront sensor and the other to a Shack-Hartmann wavefront sensor. The beam to the new wavefront sensor is magnified such that the spatial light modulator is Nyquist sampling the PSF. Amplitude filtering is done by placing the spatial light modulator between two orthogonal polarisers. The final pupil is then re-imaged onto the camera. The fold mirrors are excluded from this drawing.
Fig. 9
Fig. 9 In these figures the response due to a quadrofoil aberration is shown for different filters. The left side corresponds to measured responses from the lab setup. The responses on the right side are simulated responses for each filter. For each measurement the same exposure time is used. The first and third column are the x gradients and the second and fourth columns are the y gradients. Blue corresponds to a saturated negative measurement and red to a saturated positive response. White is a response of zero. The top figure is the measurement from a linear filter. The middle is from the G-ODWFS sigmoid filter. And the bottom figures are the responses from the pyramid.
Fig. 10
Fig. 10 The responses curves for the tilt mode are shown. On the left the whole range of input tilt is plotted against the response for the SHWFS and the ODWFS. The figure on the right shows the same thing but for the G-ODWFS and the PWFS.
Fig. 11
Fig. 11 The response curves for the different step size filters. On the x-axis are the input tilt coefficients and on the y-axis the responses. The figure on the left shows the response for small steps, and the right figure shows the response for large steps.
Fig. 12
Fig. 12 Here the Cramer-Rao lower bound is shown against the mode index of the deformable mirror. The CR bound is shown in units of λ. The pyramid has the lowest bound and the linear the highest. The sigmoid filter here has a β = 1/3.
Fig. 13
Fig. 13 The optical set-up is fed by an unpolarised focused beam. The first Wollaston prism splits the light in horizontal and vertical polarisation. Then the beams are filtered by the varying half wave plates in the focal plane. Another Wollaston prism will then split the two beams into four. The lens that transforms the focused beams to pupil images is not shown, but should be behind the second Wollaston prism.

Tables (1)

Tables Icon

Table 1 Overview of different wavefront sensors. The dynamic range is the maximal local tilt that can be measured and the sensitivity is the variance in the wavefront measurements due to photon noise. The variables stand for; D the entrance aperture diameter, λ the wavelength, N is the number of photons per phase measurement, Nsubap is the amount of sub-apertures of the SHWFS, and Na is the number of Airy rings that fit within the field of view of the WFS. For the ODWFS the field of view is the size of the filter, for the MPWFS it is the modulation radius and for the SHWFS it is the lenslet field of view. These relations were adapted from [6,7,14]

Equations (26)

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U in = A ( r ) exp ( i ϕ ( r ) )
U T = { T { U in } }
T ( k x , k y ) = n = 0 t n ( k x k m ) n ,
U T = { n = 0 t n ( k x k m ) n { U in } } .
U T = n = 0 t n { ( k x k m ) n { U in } } .
I T = n = 0 t n 2 | U n | 2 + 2 m n t n t m { U n U m * } .
T ( k x , k y ) = R ( k x , k y ) ,
t n = ( 1 ) n r n ,
t n 2 = r n 2 .
I R I T = m n , n + m = o d d 4 t n t m { U n U m * } .
{ k x n { U ( r ) } } = ( i ) n n U ( r ) x n .
I R I T = 4 t 0 t 1 k m ϕ x A 2 .
I R + I T = 2 n = 0 t n 2 | U n | 2 + 4 n m n + m = e v e n t n t m { U n U m * } .
I T + I R = 2 t 0 2 A 2 ( 1 + [ t 1 k m t 0 ] 2 { ( A x A ) 2 + ϕ x 2 } ) .
I R I T I R + I T 2 t 1 k m t 0 ϕ x .
I R I T I R + I T 2 t 1 k m t 0 i = 0 N i A i 2 ϕ x , i i = 0 N i A i 2 = 2 t 1 k m t 0 ϕ x I .
I R I T I R + I T 2 t 1 t 0 λ 0 Δ λ / 2 λ 0 + Δ λ / 2 A 2 ϕ x k m 1 d λ λ 0 Δ λ / 2 λ 0 + Δ λ / 2 A 2 d λ = 2 t 1 t 0 ϕ x k m 1 λ .
ϕ x k m 1 = k m 1 k W x .
Var [ ϕ x ] = ( k m t 0 2 t 1 ) 2 Var [ I R I T I R + I T ] .
Var [ ϕ x ] ( k m t 0 2 t 1 ) 2 ( 2 N + 8 σ read 2 N 2 ) .
T ( k x , k y ) = ( 1 β ) 2 ( 1 + k x k m ) + β 1 + exp ( k x σ k m ) .
T ( k x , k y ) = 1 β 2 ( 1 + k x k m ) + β 2 ,
I R I T I R + I T ( 1 β ) 2 t 1 k m t 0 ϕ + β D π ϕ .
g H g PWFS = β + 1 β N A .
s = A a ,
A i = s + s a + a ,

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