Abstract

We introduce in this paper a general formalism for Fourier-based wave front sensing. To do so, we consider the filtering mask as a free parameter. Such an approach allows us to unify sensors like the pyramid wave front sensor (PWFS) and the Zernike wave front sensor (ZWFS). In particular, we take the opportunity to generalize these two sensors in terms of sensors’ class, where optical quantities such as the apex angle for the PWFS or the depth of the Zernike mask for the ZWFS become free parameters. In order to compare all the generated sensors of these two classes thanks to common performance criteria, we first define a general phase-linear quantity that we call meta-intensity. Analytical developments allow us to then split the perfectly phase-linear behavior of a WFS from the nonlinear contributions, making robust and analytic definitions of the sensitivity and the linearity range possible. Moreover, we define a new quantity called the SD factor, which characterizes the trade-off between these two antagonistic quantities. These developments are generalized for a modulation device and polychromatic light. A nonexhaustive study is finally conducted on the two classes, allowing us to retrieve the usual results and also make explicit the influence of the optical parameters introduced above.

© 2016 Optical Society of America

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  1. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
    [Crossref]
  2. V. Akondi, S. Castillo, and B. Vohnsen, “Multi-faceted digital pyramid wavefront sensor,” Opt. Commun. 323, 77–86 (2014).
    [Crossref]
  3. B. Vohnsen, S. Castillo, and D. Rativa, “Wavefront sensing with an axicon,” Opt. Lett. 36, 846–848 (2011).
    [Crossref]
  4. O. Fauvarque, B. Neichel, T. Fusco, and J.-F. Sauvage, “Variation around a pyramid theme: optical recombination and optimal use of photons,” Opt. Lett. 40, 3528–3531 (2015).
    [Crossref]
  5. F. Zernike, “Diffraction theory of the knife-edge test and its improved form, the phase-contrast method,” Mon. Not. R. Astron. Soc. 94, 377–384 (1934).
    [Crossref]
  6. K. Dohlen, “Phase masks in astronomy: from the Mach–Zehnder interferometer to coronagraphs,” EAS Pub. Ser. 12, 33–44 (2004).
    [Crossref]
  7. A. Toepler, Beobachtung nach einer neuen optischen Method (1864).
  8. R. N. Smartt and W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn J. Appl. Phys. 14, 351 (1975).
  9. 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]
  10. O. Fauvarque, B. Neichel, T. Fusco, and J.-F. Sauvage, “Optimization of the imaging processing in the context of Fourier based wave front sensing,” in preparation.
  11. F. Rigaut and E. Gendron, “Laser guide star in adaptive optics, the tilt determination problem,” Astron. Astrophys. 261, 677–684 (1992).
  12. A. Wang, J. Yao, D. Cai, and H. Ren, “Design and fabrication of a pyramid wavefront sensor,” Opt. Eng. 49, 073401 (2010).
    [Crossref]
  13. C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27–38 (2004).
    [Crossref]
  14. I. Shatokhina, A. Obereder, and R. Ramlau, “Fast algorithm for wavefront reconstruction in XAO/SCAO with pyramid wavefront sensor,” Proc. SPIE 9148, 91480P (2014).
    [Crossref]
  15. M. N’Diaye, K. Dohlen, A. Caillat, and A. Costille, “Design optimization and lab demonstration of ZELDA: a Zernike sensor for near-coronagraph quasi-static measurements,” Proc. SPIE 9148, 91485H (2014).
    [Crossref]
  16. S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
    [Crossref]
  17. M. N’Diaye, K. Dohlen, T. Fusco, and B. Paul, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor,” Astron. Astrophys. 555, A94 (2013).
    [Crossref]
  18. O. Fauvarque, B. Neichel, T. Fusco, J.-F. Sauvage, and O. Giraut, “A general formalism for Fourier based wave front sensing: application to the pyramid wave front sensors,” Proc. SPIE 9909, 990960 (2016).
    [Crossref]

2016 (1)

O. Fauvarque, B. Neichel, T. Fusco, J.-F. Sauvage, and O. Giraut, “A general formalism for Fourier based wave front sensing: application to the pyramid wave front sensors,” Proc. SPIE 9909, 990960 (2016).
[Crossref]

2015 (1)

2014 (3)

V. Akondi, S. Castillo, and B. Vohnsen, “Multi-faceted digital pyramid wavefront sensor,” Opt. Commun. 323, 77–86 (2014).
[Crossref]

I. Shatokhina, A. Obereder, and R. Ramlau, “Fast algorithm for wavefront reconstruction in XAO/SCAO with pyramid wavefront sensor,” Proc. SPIE 9148, 91480P (2014).
[Crossref]

M. N’Diaye, K. Dohlen, A. Caillat, and A. Costille, “Design optimization and lab demonstration of ZELDA: a Zernike sensor for near-coronagraph quasi-static measurements,” Proc. SPIE 9148, 91485H (2014).
[Crossref]

2013 (1)

M. N’Diaye, K. Dohlen, T. Fusco, and B. Paul, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor,” Astron. Astrophys. 555, A94 (2013).
[Crossref]

2011 (1)

2010 (1)

A. Wang, J. Yao, D. Cai, and H. Ren, “Design and fabrication of a pyramid wavefront sensor,” Opt. Eng. 49, 073401 (2010).
[Crossref]

2004 (2)

C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27–38 (2004).
[Crossref]

K. Dohlen, “Phase masks in astronomy: from the Mach–Zehnder interferometer to coronagraphs,” EAS Pub. Ser. 12, 33–44 (2004).
[Crossref]

2003 (2)

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]

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

1996 (1)

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

1992 (1)

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

1975 (1)

R. N. Smartt and W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn J. Appl. Phys. 14, 351 (1975).

1934 (1)

F. Zernike, “Diffraction theory of the knife-edge test and its improved form, the phase-contrast method,” Mon. Not. R. Astron. Soc. 94, 377–384 (1934).
[Crossref]

Accardo, M.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Akondi, V.

V. Akondi, S. Castillo, and B. Vohnsen, “Multi-faceted digital pyramid wavefront sensor,” Opt. Commun. 323, 77–86 (2014).
[Crossref]

Baffa, C.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Biasi, R.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Biliotti, V.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Brusa, G.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Cagigal, M. P.

Cai, D.

A. Wang, J. Yao, D. Cai, and H. Ren, “Design and fabrication of a pyramid wavefront sensor,” Opt. Eng. 49, 073401 (2010).
[Crossref]

Caillat, A.

M. N’Diaye, K. Dohlen, A. Caillat, and A. Costille, “Design optimization and lab demonstration of ZELDA: a Zernike sensor for near-coronagraph quasi-static measurements,” Proc. SPIE 9148, 91485H (2014).
[Crossref]

Canales, V. F.

Carbillet, M.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Castillo, S.

V. Akondi, S. Castillo, and B. Vohnsen, “Multi-faceted digital pyramid wavefront sensor,” Opt. Commun. 323, 77–86 (2014).
[Crossref]

B. Vohnsen, S. Castillo, and D. Rativa, “Wavefront sensing with an axicon,” Opt. Lett. 36, 846–848 (2011).
[Crossref]

Costille, A.

M. N’Diaye, K. Dohlen, A. Caillat, and A. Costille, “Design optimization and lab demonstration of ZELDA: a Zernike sensor for near-coronagraph quasi-static measurements,” Proc. SPIE 9148, 91485H (2014).
[Crossref]

Dohlen, K.

M. N’Diaye, K. Dohlen, A. Caillat, and A. Costille, “Design optimization and lab demonstration of ZELDA: a Zernike sensor for near-coronagraph quasi-static measurements,” Proc. SPIE 9148, 91485H (2014).
[Crossref]

M. N’Diaye, K. Dohlen, T. Fusco, and B. Paul, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor,” Astron. Astrophys. 555, A94 (2013).
[Crossref]

K. Dohlen, “Phase masks in astronomy: from the Mach–Zehnder interferometer to coronagraphs,” EAS Pub. Ser. 12, 33–44 (2004).
[Crossref]

Esposito, S.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Fauvarque, O.

O. Fauvarque, B. Neichel, T. Fusco, J.-F. Sauvage, and O. Giraut, “A general formalism for Fourier based wave front sensing: application to the pyramid wave front sensors,” Proc. SPIE 9909, 990960 (2016).
[Crossref]

O. Fauvarque, B. Neichel, T. Fusco, and J.-F. Sauvage, “Variation around a pyramid theme: optical recombination and optimal use of photons,” Opt. Lett. 40, 3528–3531 (2015).
[Crossref]

O. Fauvarque, B. Neichel, T. Fusco, and J.-F. Sauvage, “Optimization of the imaging processing in the context of Fourier based wave front sensing,” in preparation.

Ferruzzi, D.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Fini, L.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Foppiani, I.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Fusco, T.

O. Fauvarque, B. Neichel, T. Fusco, J.-F. Sauvage, and O. Giraut, “A general formalism for Fourier based wave front sensing: application to the pyramid wave front sensors,” Proc. SPIE 9909, 990960 (2016).
[Crossref]

O. Fauvarque, B. Neichel, T. Fusco, and J.-F. Sauvage, “Variation around a pyramid theme: optical recombination and optimal use of photons,” Opt. Lett. 40, 3528–3531 (2015).
[Crossref]

M. N’Diaye, K. Dohlen, T. Fusco, and B. Paul, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor,” Astron. Astrophys. 555, A94 (2013).
[Crossref]

O. Fauvarque, B. Neichel, T. Fusco, and J.-F. Sauvage, “Optimization of the imaging processing in the context of Fourier based wave front sensing,” in preparation.

Gallieni, D.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Gendron, E.

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

Giraut, O.

O. Fauvarque, B. Neichel, T. Fusco, J.-F. Sauvage, and O. Giraut, “A general formalism for Fourier based wave front sensing: application to the pyramid wave front sensors,” Proc. SPIE 9909, 990960 (2016).
[Crossref]

N’Diaye, M.

M. N’Diaye, K. Dohlen, A. Caillat, and A. Costille, “Design optimization and lab demonstration of ZELDA: a Zernike sensor for near-coronagraph quasi-static measurements,” Proc. SPIE 9148, 91485H (2014).
[Crossref]

M. N’Diaye, K. Dohlen, T. Fusco, and B. Paul, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor,” Astron. Astrophys. 555, A94 (2013).
[Crossref]

Neichel, B.

O. Fauvarque, B. Neichel, T. Fusco, J.-F. Sauvage, and O. Giraut, “A general formalism for Fourier based wave front sensing: application to the pyramid wave front sensors,” Proc. SPIE 9909, 990960 (2016).
[Crossref]

O. Fauvarque, B. Neichel, T. Fusco, and J.-F. Sauvage, “Variation around a pyramid theme: optical recombination and optimal use of photons,” Opt. Lett. 40, 3528–3531 (2015).
[Crossref]

O. Fauvarque, B. Neichel, T. Fusco, and J.-F. Sauvage, “Optimization of the imaging processing in the context of Fourier based wave front sensing,” in preparation.

Obereder, A.

I. Shatokhina, A. Obereder, and R. Ramlau, “Fast algorithm for wavefront reconstruction in XAO/SCAO with pyramid wavefront sensor,” Proc. SPIE 9148, 91480P (2014).
[Crossref]

Oti, J. E.

Paul, B.

M. N’Diaye, K. Dohlen, T. Fusco, and B. Paul, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor,” Astron. Astrophys. 555, A94 (2013).
[Crossref]

Puglisi, A.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Ragazzoni, R.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

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

Ramlau, R.

I. Shatokhina, A. Obereder, and R. Ramlau, “Fast algorithm for wavefront reconstruction in XAO/SCAO with pyramid wavefront sensor,” Proc. SPIE 9148, 91480P (2014).
[Crossref]

Ranfagni, P.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Rativa, D.

Ren, H.

A. Wang, J. Yao, D. Cai, and H. Ren, “Design and fabrication of a pyramid wavefront sensor,” Opt. Eng. 49, 073401 (2010).
[Crossref]

Riccardi, A.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Rigaut, F.

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

Salinari, P.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Sauvage, J.-F.

O. Fauvarque, B. Neichel, T. Fusco, J.-F. Sauvage, and O. Giraut, “A general formalism for Fourier based wave front sensing: application to the pyramid wave front sensors,” Proc. SPIE 9909, 990960 (2016).
[Crossref]

O. Fauvarque, B. Neichel, T. Fusco, and J.-F. Sauvage, “Variation around a pyramid theme: optical recombination and optimal use of photons,” Opt. Lett. 40, 3528–3531 (2015).
[Crossref]

O. Fauvarque, B. Neichel, T. Fusco, and J.-F. Sauvage, “Optimization of the imaging processing in the context of Fourier based wave front sensing,” in preparation.

Seifert, W.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Shatokhina, I.

I. Shatokhina, A. Obereder, and R. Ramlau, “Fast algorithm for wavefront reconstruction in XAO/SCAO with pyramid wavefront sensor,” Proc. SPIE 9148, 91480P (2014).
[Crossref]

Smartt, R. N.

R. N. Smartt and W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn J. Appl. Phys. 14, 351 (1975).

Steel, W. H.

R. N. Smartt and W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn J. Appl. Phys. 14, 351 (1975).

Stefanini, P.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Storm, J.

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Toepler, A.

A. Toepler, Beobachtung nach einer neuen optischen Method (1864).

Tozzi, A.

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[Crossref]

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[Crossref]

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[Crossref]

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

F. Zernike, “Diffraction theory of the knife-edge test and its improved form, the phase-contrast method,” Mon. Not. R. Astron. Soc. 94, 377–384 (1934).
[Crossref]

Opt. Commun. (2)

V. Akondi, S. Castillo, and B. Vohnsen, “Multi-faceted digital pyramid wavefront sensor,” Opt. Commun. 323, 77–86 (2014).
[Crossref]

C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27–38 (2004).
[Crossref]

Opt. Eng. (1)

A. Wang, J. Yao, D. Cai, and H. Ren, “Design and fabrication of a pyramid wavefront sensor,” Opt. Eng. 49, 073401 (2010).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (4)

O. Fauvarque, B. Neichel, T. Fusco, J.-F. Sauvage, and O. Giraut, “A general formalism for Fourier based wave front sensing: application to the pyramid wave front sensors,” Proc. SPIE 9909, 990960 (2016).
[Crossref]

I. Shatokhina, A. Obereder, and R. Ramlau, “Fast algorithm for wavefront reconstruction in XAO/SCAO with pyramid wavefront sensor,” Proc. SPIE 9148, 91480P (2014).
[Crossref]

M. N’Diaye, K. Dohlen, A. Caillat, and A. Costille, “Design optimization and lab demonstration of ZELDA: a Zernike sensor for near-coronagraph quasi-static measurements,” Proc. SPIE 9148, 91485H (2014).
[Crossref]

S. Esposito, A. Tozzi, D. Ferruzzi, M. Carbillet, A. Riccardi, L. Fini, C. Vérinaud, M. Accardo, G. Brusa, D. Gallieni, R. Biasi, C. Baffa, V. Biliotti, I. Foppiani, A. Puglisi, R. Ragazzoni, P. Ranfagni, P. Stefanini, P. Salinari, W. Seifert, and J. Storm, “First-light adaptive optics system for large binocular telescope,” Proc. SPIE 4839, 164–173 (2003).
[Crossref]

Other (2)

O. Fauvarque, B. Neichel, T. Fusco, and J.-F. Sauvage, “Optimization of the imaging processing in the context of Fourier based wave front sensing,” in preparation.

A. Toepler, Beobachtung nach einer neuen optischen Method (1864).

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Figures (10)

Fig. 1.
Fig. 1. Schematic view (in 1D) of a Fourier filtering optical system.

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Fig. 2.
Fig. 2. Two classical tessellations. Cartesian splitting (left insert) and polar splitting (right insert).

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Fig. 3.
Fig. 3. Typical graph ( * ) of the gap between the effective meta-intensity and the linear intensity depending on a , and the input phase amplitude [Eq. (20)]. The slope of the red line equals the 2-norm of the quadratic intensity. The input phase is the vertical coma, and the considered WFS is the Zernike WFS.

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Fig. 4.
Fig. 4. Modulus (left) and argument (right) of the 2DFT with an apex angle that equals 1.5 D 2 f , where D is the entrance pupil diameter.

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Fig. 5.
Fig. 5. Intensity on the detector for a circular pupil and a flat incoming phase. θ equals 0.1, 1, 1.5, and 3 D 2 f .

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Fig. 6.
Fig. 6. Sensitivity, linearity range, and SD factor of the pyramid WFS with respect to the spatial frequencies. The apex angle equals 0.05 ( * ), 0.1 ( + ), and 3 ( Δ ) 2 θ f D . The phase basis corresponds to the 24 first Zernike radial orders.

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Fig. 7.
Fig. 7. Sensitivity, linearity range, and SD factor of the modulated pyramid WFS with respect to the spatial frequencies. The modulation radii equal 0 ( Δ ), 1 ( + ). and 3 ( * ) λ / D . The apex angle equals 3 2 θ f D , i.e., the 4 pupil images are widely separated.

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Fig. 8.
Fig. 8. Distance between the effective and linear meta-intensities ( + , G e l m I ) and the effective and linear slope maps ( * , G e l S ) as functions of the input phase amplitude. The slope of the red line equals the 2-norm of the quadratic intensity associated with the meta-intensities. The input phase is the vertical coma, and the considered WFS is the PWFS with an infinite apex angle.

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Fig. 9.
Fig. 9. Sensitivity, linearity range, and SD factor of the Zernike WFS with respect to the spatial frequencies. Depth of the Zernike mask δ equals 1/8 ( + ), 1/4 ( Δ ), and 3/8 ( * ) λ 0 . The sensitivity is identical for δ equals 1/8 and 3/8 λ 0 .

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Fig. 10.
Fig. 10. Sensitivity, linearity range, and SD factor of the Zernike WFS with respect to the spatial frequencies. Size of the Zernike mask ρ equals 0.5 ( * ), 1 ( Δ ), and 1.5 ( + ) 1.06 λ 0 / D .

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Equations (85)

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ψ p ( x p , y p , λ ) = n ( λ ) I P ( x p , y p ) exp ( 2 ı π λ Δ ( x p , y p ) ) ,
m ( f x , f y , λ ) = a ( f x , f y , λ ) exp ( 2 ı π λ OS ( f x , f y , λ ) ) .
ψ d ( x d , y d , λ ) ∝ ∬ d x p d y p f 2 ψ p ( x p − x d , y p − y d , λ ) , ∬ d f x d f y λ 2 m ( f x , f y , λ ) exp ( − 2 ı π f λ [ x p f x + y p f y ] ) .
ψ p ( x d , y d ) = n I P ( x d , y d ) exp ( ı φ ( x d , y d ) ) , m ( f x , f y ) = a ( f x , f y ) exp ( 2 ı π λ 0 OS ( f x , f y ) ) ,
ψ d = ψ p ⋆ F [ m ] ,
I = | ψ p ⋆ F [ m ] | 2 .
∪ i Ω i = R 2 and Ω i ∩ Ω j = Ø if    i ≠ j .
m ( f x , f y ) = ∑ i I Ω i ( f x , f y ) exp ( 2 ı π λ 0 ( δ i + f x α i + f y β i ) ) ,
{ Ω i , δ i , α i , β i } i .
F [ m ] ( x d , y d ) = ∑ i   exp ( 2 ı π δ i λ 0 ) F [ I Ω i ] ( x d − f α i , y d − f β i ) ,
I Ω + + ( f x , f y ) = Θ ( f x ) Θ ( f y ) ,
F [ I Ω + + ] ( x d , y d ) = 1 4 ( δ ( x d ) δ ( y d ) − 1 π 2 x d y d ) − ı 4 ( δ ( x d ) π y d + δ ( y d ) π x d ) .
F [ I Ω − + ] ( x d , y d ) = 1 4 ( δ ( x d ) δ ( y d ) + 1 π 2 x d y d ) − ı 4 ( δ ( x d ) π y d − δ ( y d ) π x d ) ,
F [ I Ω − − ] ( x d , y d ) = 1 4 ( δ ( x d ) δ ( y d ) − 1 π 2 x d y d ) + ı 4 ( δ ( x d ) π y d + δ ( y d ) π x d ) ,
F [ I Ω + − ] ( x d , y d ) = 1 4 ( δ ( x d ) δ ( y d ) + 1 π 2 x d y d ) + ı 4 ( δ ( x d ) π y d − δ ( y d ) π x d ) .
F [ I Ω ρ ] ( r d , θ d ) = ρ r d J 1 ( 2 π ρ r d ) ,
φ = φ r + φ t ,
m I ( φ t + a Φ t ) = m I ( φ t ) + a m I ( Φ t ) ∀    φ t , Φ t ∈ Phase space    and ∀    a ∈ R .
I ( φ , n ) = I ( φ r + φ t , n ) = n ∑ q = 0 ∞ ( − ı ) q q ! ∑ k = 0 q ( − 1 ) k ( q k ) φ t k ⋆ φ t q − k ⋆ ¯ where    φ t k ⋆ ≙ ( I p e ı φ r φ t k ) ⋆ F [ m ] .
Constant term , q = 0 : I c ≙ | I P e ı φ r ⋆ F [ m ] | 2 .
Linear term , q = 1 : I l ( φ t ) ≙ 2 I [ ( I P e ı φ r ⋆ F [ m ] ) ( I P e ı φ r φ t ⋆ F [ m ] ¯ ) ] .
Quadratic term , q = 2 : I q ( φ t ) ≙ | I P e ı φ r φ t ⋆ F [ m ] | 2 − R [ ( I P e ı i φ r ⋆ F [ m ] ) ( I P e ı i φ r φ t 2 ⋆ F [ m ] ¯ ) ] .
m I ( φ t ) = I ( φ r + φ t , n ) − I ( φ r , n ) n .
I l ( φ t ) = 1 n d I ( φ r + a φ t , n ) d a | a = 0 I q ( φ t ) = 1 2 n d 2 I ( φ r + a φ t , n ) d a 2 | a = 0 .
s ( φ t ) = ‖ I l ( φ t ) ‖ 2 .
σ WFS 2 ( φ t ) = σ noise 2 s ( φ t ) − 2 .
G e l ( φ t , a ) = ‖ m I ( a φ t ) a − I l ( φ t ) ‖ 2 ,
m I ( a φ t ) a − I l ( φ t ) = ∑ q = 2 ∞ a q − 1 ( − ı ) q q ! ∑ k = 0 q ( − 1 ) k ( q k ) φ t k ⋆ φ t q − k ⋆ ¯ = a I q ( φ t ) + a 2 ( … ) .
G e l ( φ t , a ) = a ‖ I q ( φ t ) ‖ 2 + a 2 ( … ) .
d ( φ t ) = ( ‖ I q ( φ t ) ‖ 2 ) − 1 ,
s ( φ t ) d ( φ t ) = ( ‖ I l ( φ t ) ‖ 2 ) ( ‖ I q ( φ t ) ‖ 2 ) − 1 ,
s η d 1 / η = ( ‖ I l ( φ t ) ‖ 2 ) η · ( ‖ I q ( φ t ) ‖ 2 ) − 1 / η ,
φ m ( s ) = ∑ k = 0 ∞ m k ( s ) Φ k with ∀ k    m k ( 0 ) = m k ( 1 ) w ( s ) with w ( s ) ≥ 0 , w ( 0 ) = w ( 1 ) , ∫ 0 1 w ( s ) d s = 1 ,
I m ( φ , n ) = ∫ 0 1 I ( φ + φ m ( s ) , n · w ( s ) ) d s .
I m ( φ t , n ) n = ∑ q = 0 ∞ ( − ı ) q q ! ∑ k = 0 q ( − 1 ) k ( q k ) ∫ 0 1 w ( s ) d s φ t k ⋆ ( s ) φ t q − k ⋆ ( s ) ¯ where    φ t k ⋆ ( s ) = ( I p e ı φ m ( s ) φ t k ) ⋆ F [ m ] .
I m c = ∫ 0 1 | I P e ı φ m ( s ) ⋆ F [ m ] | 2 w ( s ) d s I m l ( φ t ) = ∫ 0 1 2 I [ ( I P e ı φ m ( s ) ⋆ F [ m ] ) ( I P e ı φ m ( s ) φ t ⋆ F [ m ] ¯ ) ] w ( s ) d s I m q ( φ t ) = ∫ 0 1 [ | I P e ı φ m ( s ) φ t ⋆ F [ m ] | 2 ] w ( s ) d s − ∫ 0 1 [ R [ ( I P e ı φ m ( s ) ⋆ F [ m ] ) ( I P e ı φ m ( s ) φ t 2 ⋆ F [ m ] ¯ ) ] ] w ( s ) d s .
m I m ( φ t ) = I m ( φ t , n ) − I m ( 0 , n ) n .
Ω − + : 0 , − θ , θ Ω + + : 0 , θ , θ Ω − − : 0 , − θ , − θ Ω + − : 0 , θ , − θ .
F [ m Δ ] ( x d , y d ) = F [ I Ω + + ] ( x d − f θ , y d − f θ ) + F [ I Ω − − ] ( x d + f θ , y d + f θ ) + F [ I Ω − + ] ( x d + f θ , y d − f θ ) + F [ I Ω + − ] ( x d − f θ , y d + f θ ) .
φ m ( s ) = r m [ cos ( 2 π s ) Z 1 − 1 + sin ( 2 π s ) Z 1 1 ] and w ( s ) = 1 .
I − + = | ψ p ⋆ F [ I Ω − + ] | 2 I + + = | ψ p ⋆ F [ I Ω + + ] | 2 I − − = | ψ p ⋆ F [ I Ω − − ] | 2 I + − = | ψ p ⋆ F [ I Ω + − ] | 2 .
I l + + ( φ t ) = 2 I [ ( I P ⋆ CIR [ I Ω + + ] ) ( I P φ t ⋆ CIR [ I Ω + + ] ¯ ) ] .
I l + + ( φ t ) = 1 8 [ ( I + H x y 2 ) [ I P ] ( H x + H y ) [ I P φ t ] ] − 1 8 [ ( I + H x y 2 ) [ I P φ t ] ( H x + H y ) [ I P ] ] ,
I l + − ( φ t ) = 1 8 [ ( I − H x y 2 ) [ I P ] ( H x − H y ) [ I P φ t ] ] − 1 8 [ ( I − H x y 2 ) [ I P φ t ] ( H x − H y ) [ I P ] ] I l − + ( φ t ) = 1 8 [ ( I − H x y 2 ) [ I P ] ( − H x + H y ) [ I P φ t ] ] − 1 8 [ ( I − H x y 2 ) [ I P φ t ] ( − H x + H y ) [ I P ] ] I l − − ( φ t ) = 1 8 [ ( I + H x y 2 ) [ I P ] ( − H x − H y ) [ I P φ t ] ] − 1 8 [ ( I + H x y 2 ) [ I P φ t ] ( − H x − H y ) [ I P ] ] .
s ( φ t ) = ‖ I l + − ( φ t ) ‖ 2 + ‖ I l − + ( φ t ) ‖ 2 + ‖ I l − − ( φ t ) ‖ 2 + ‖ I l + + ( φ t ) ‖ 2 .
d ( φ t ) = ( ‖ I q + − ( φ t ) ‖ 2 + ‖ I q − + ( φ t ) ‖ 2 + ‖ I q − − ( φ t ) ‖ 2 + ‖ I q + + ( φ t ) ‖ 2 ) − 1 .
S x = I + + + I + − − I − + − I − − n ,
S y = I + + − I − + + I + − − I − − n .
S x = m I + + + m I + − − m I − + − m I − − ,
S y = m I + + − m I − + + m I + − − m I − − .
S l x = I l + + + I l + − − I l − + − I l − − S l y = I l + + − I l − + + I l + − − I l − − S q x = I q + + + I q + − − I q − + − I q − − S q y = I q + + − I q − + + I q + − − I q − − .
S l x ( φ t ) = 1 2 ( I [ I P ] H x [ I P φ t ] + H x y 2 [ I P ] H y [ I P φ t ] ) − 1 2 ( I [ I P φ t ] H x [ I P ] − H x y 2 [ I P φ t ] H y [ I P ] ) S l y ( φ t ) = 1 2 ( I [ I P ] H y [ I P φ t ] + H x y 2 [ I P ] H x [ I P φ t ] ) − 1 2 ( I [ I P φ t ] H y [ I P ] − H x y 2 [ I P φ t ] H x [ I P ] ) .
s S ( φ t ) = ‖ S l x ( φ t ) ‖ 2 + ‖ S l y ( φ t ) ‖ 2 .
S q x ( φ t ) = 0 and S q y ( φ t ) = 0 .
G e l S ( φ t , a ) = ‖ S x ( a φ t ) a − S l x ( φ t ) ‖ 2 + ‖ S y ( a φ t ) a − S l y ( φ t ) ‖ 2 ,
G e l m I ( φ t , a ) = ∑ 4 pupils ‖ m I ± ± ( a φ t ) a − I l ± ± ( φ t ) ‖ 2 .
Ω ρ : δ , 0 , 0 Ω ¯ ρ : 0 , 0 , 0 .
F [ m Z ] ( r d ) = F [ I Ω ¯ ρ ] ( r d ) + exp ( 2 ı π λ 0 δ ) F [ I Ω ρ ] ( r d ) ,
= δ ( r d ) + ( exp ( 2 ı π λ 0 δ ) − 1 ) ρ r d J 1 ( 2 π ρ r d ) .
Z ρ [ f ] = f ⋆ ρ r d J 1 ( 2 π ρ r d ) .
I l ( φ t ) = 2   sin ( 2 π λ 0 δ ) ⏞ a I P ( φ t Z ρ [ I P ] − Z ρ [ I P φ t ] ) ⏞ b .
I q ( φ t ) = [ 1 − cos ( 2 π λ 0 δ ) ] ( 2 Z ρ 2 [ φ t ] − 2 φ t Z ρ [ φ t ] + Z ρ [ φ t 2 ] + φ t 2 Z ρ [ I P ] − 2 Z ρ [ I P ] Z ρ [ φ t 2 ] ) .
s ( φ t ) = 2 | sin ( 2 π λ 0 δ ) | ‖ φ t Z ρ [ I P ] − Z ρ [ I P φ t ] ‖ 2 .
d ( φ t ) = 1 1 − cos ( 2 π λ 0 δ ) ‖ 2 Z ρ 2 [ φ t ] − 2 φ t Z ρ [ φ t ] + Z ρ [ φ t 2 ] + φ t 2 Z ρ [ I P ] − 2 Z ρ [ I P ] Z ρ [ φ t 2 ] ‖ 2 − 1 .
ψ p ( x p , y p , λ ) = n ( λ ) I P ( x p , y p ) exp ( 2 ı π λ Δ ( x p , y p ) ) ,
φ = 2 π λ ( Δ t + Δ r ) = 2 π λ 0 λ 0 λ ( Δ t + Δ r ) ≙ λ 0 λ ( φ t + φ r ) ,
m ( f x , f y , λ ) = exp ( 2 ı π λ OS ( f x , f y , λ ) ) .
OS ( f x , f y , λ ) = n r ( λ ) GS ( f x , f y ) .
ψ d ( x d , y d ) ∝ ∬ d x p d y p f 2 ψ p ( x p − x d , y p − y d , λ ) ∬ d f x d f y λ 2 m ( f x , f y , λ ) exp ( − 2 ı π f λ [ x p f x + y p f y ] ) .
I λ ( φ t ) = | ψ p ( φ t , λ ) ⋆ F λ [ m ] | 2 .
I p ( φ t ) = ∫ d λ | ψ p ( φ t , λ ) ⋆ F λ [ m ] | 2 .
F λ [ m ] ( x p , y p ) = ∬ d f x d f y λ 2 m ( f x , f y , λ ) e − 2 ı π f λ [ x p f x + y p f y ] .
( u , v ) = ( f x λ , f y λ ) ,
I p ( φ t ) = ∫ n ( λ ) d λ | I P   exp ( ı λ 0 λ ( φ t + φ r ) ) ⋆ F [ m ] | 2 .
I p ( φ t ) = ∑ q = 0 ∞ ( − ı ) q q ! ∫ n ( λ ) ( λ 0 λ ) q ∑ k = 0 q ( − 1 ) k ( q k ) φ t k ⋆ φ t q − k ⋆ ¯ d λ where    φ t k ⋆ ≙ ( I p e 2 ı π λ 0 λ φ r φ t k ) ⋆ F [ m ] .
I p ( φ t ) = ∑ q = 0 ∞ ( − ı ) q q ! ∫ n ( λ ) ( λ 0 λ ) q d λ ∑ k = 0 q ( − 1 ) k ( q k ) φ t k ⋆ φ t q − k ⋆ ¯ where    φ t k ⋆ ≙ ( I p φ t k ) ⋆ F [ m ] .
I p c = | I P ⋆ F [ m ] | 2 I p l ( φ t ) = 1 n ( ∫ n ( λ ) λ 0 λ d λ ) 2 I [ ( I P ⋆ F [ m ] ) ( I P φ t ⋆ F [ m ] ¯ ) ] I p q ( φ t ) = 1 n ( ∫ n ( λ ) ( λ 0 λ ) 2 d λ ) [ | I P φ t ⋆ F [ m ] | 2 ] − R [ ( I P ⋆ F [ m ] ) ( I P φ t 2 ⋆ F [ m ] ¯ ) ] ] ,
n = ∫ n ( λ ) d λ .
m I p ( φ t ) = I p ( φ t ) − I p ( 0 ) n .
I p l ( φ t ) = 1 n ( ∫ n ( λ ) λ 0 λ d λ ) I l ( φ t ) I p q ( φ t ) = 1 n ( ∫ n ( λ ) ( λ 0 λ ) 2 d λ ) I q ( φ t ) .
s p ( φ t ) = 1 n ( ∫ n ( λ ) λ 0 λ d λ ) s ( φ t ) d p ( φ t ) = n ( ∫ n ( λ ) ( λ 0 λ ) 2 d λ ) − 1 d ( φ t ) .
m Δ ( f x , f y , λ ) = exp ( 2 ı π λ n r ( λ ) θ ( | f x | + | f y | ) ) .
n r ( λ ) = B + C λ 2 + … ,
m Δ ( u , v ) = exp ( 2 ı π n air θ ( | u | + | v | ) ) .
m Z ( f x , f y , λ ) = 1 + ( e ı π 2 λ 0 λ − 1 ) Θ ( 1.06 λ 0 D − f x 2 + f y 2 ) .

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