Abstract

We present a fast method for the wavefront reconstruction from pyramid wavefront sensor (P-WFS) measurements. The method is based on an analytical relation between pyramid and Shack–Hartmann sensor (SH-WFS) data. The algorithm consists of two steps—a transformation of the P-WFS data to SH data, followed by the application of cumulative reconstructor with domain decomposition, a wavefront reconstructor from SH-WFS measurements. The closed loop simulations confirm that our method provides the same quality as the standard matrix vector multiplication method. A complexity analysis as well as speed tests confirm that the method is very fast. Thus, the method can be used on extremely large telescopes, e.g., for eXtreme adaptive optics systems.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. Korkiakoski and C. Verinaud, “Extreme adaptive optics simulations for EPICS,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 03007.
  2. F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
  3. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
    [CrossRef]
  4. R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208, 51–60 (2002).
    [CrossRef]
  5. R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. 350, L23–L26 (1999).
  6. S. Esposito and A. Riccardi, “Pyramid wavefront sensor behavior in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
    [CrossRef]
  7. S. Esposito, A. Riccardi, and O. Feeney, “Closed-loop performance of pyramid wavefront sensor,” Proc. SPIE 4034, 184–189 (2000).
    [CrossRef]
  8. M. Feldt, D. Peter, S. Hippler, T. Henning, J. Aceituno, and M. Goto, “PYRAMIR: first on-sky results from an infrared pyramid wavefront sensor,” Proc. SPIE 6272, 627218 (2006).
    [CrossRef]
  9. D. Peter, M. Feldt, T. Henning, S. Hippler, J. Aceituno, L. Montoya, J. Costa, and B. Dorner, “PYRAMIR: exploring the on-sky performance of the world’s first near-infrared pyramid wavefront sensor,” Publ. Astron. Soc. Pac. 122, 63–70 (2010).
    [CrossRef]
  10. V. Korkiakoski, C. Vérinaud, M. Le Louarn, and R. Conan, “Comparison between a model-based and a conventional pyramid sensor reconstructor,” Appl. Opt. 46, 6176–6184 (2007).
    [CrossRef]
  11. C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233, 27–38 (2004).
    [CrossRef]
  12. A. Burvall, E. Daly, S. R. Chamot, and C. Dainty, “Linearity of the pyramid wavefront sensor,” Opt. Express 14, 11925–11934 (2006).
    [CrossRef]
  13. D. W. Phillion and K. Baker, “Two-sided pyramid wavefront sensor in the direct phase mode,” Proc. SPIE 6274, 627228 (2006).
    [CrossRef]
  14. F. Quirós-Pacheco, C. Correia, and S. Esposito, “Fourier transform-wavefront reconstruction for the pyramid wavefront sensor,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 07005.
  15. M. Rosensteiner, “Cumulative reconstructor: fast wavefront reconstruction algorithm for extremely large telescopes,” J. Opt. Soc. Am. A 28, 2132–2138 (2011).
    [CrossRef]
  16. M. Rosensteiner, “Wavefront reconstruction for extremely large telescopes via CuRe with domain decomposition,” J. Opt. Soc. Am. A 29, 2328–2336 (2012).
    [CrossRef]
  17. M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” IPI 5, 893–913 (2011).
  18. J. LeDue, L. Jolissant, J.-P. Véran, and C. Bradley, “Calibration and testing with real turbulence of a pyramid sensor employing static modulation,” Opt. Express 17, 7186–7195 (2009).
    [CrossRef]
  19. M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European extremely large telescope,” Proc. SPIE 6272, 627234 (2006).
    [CrossRef]
  20. A. Garcia-Rissmann and M. Le Louarn, “SCAO simulation results with a pyramid sensor on an ELT-like telescope,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 03011.

2012 (1)

2011 (2)

M. Rosensteiner, “Cumulative reconstructor: fast wavefront reconstruction algorithm for extremely large telescopes,” J. Opt. Soc. Am. A 28, 2132–2138 (2011).
[CrossRef]

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” IPI 5, 893–913 (2011).

2010 (1)

D. Peter, M. Feldt, T. Henning, S. Hippler, J. Aceituno, L. Montoya, J. Costa, and B. Dorner, “PYRAMIR: exploring the on-sky performance of the world’s first near-infrared pyramid wavefront sensor,” Publ. Astron. Soc. Pac. 122, 63–70 (2010).
[CrossRef]

2009 (1)

2007 (1)

2006 (4)

D. W. Phillion and K. Baker, “Two-sided pyramid wavefront sensor in the direct phase mode,” Proc. SPIE 6274, 627228 (2006).
[CrossRef]

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European extremely large telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

A. Burvall, E. Daly, S. R. Chamot, and C. Dainty, “Linearity of the pyramid wavefront sensor,” Opt. Express 14, 11925–11934 (2006).
[CrossRef]

M. Feldt, D. Peter, S. Hippler, T. Henning, J. Aceituno, and M. Goto, “PYRAMIR: first on-sky results from an infrared pyramid wavefront sensor,” Proc. SPIE 6272, 627218 (2006).
[CrossRef]

2004 (1)

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

2002 (1)

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

2001 (1)

S. Esposito and A. Riccardi, “Pyramid wavefront sensor behavior in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

2000 (1)

S. Esposito, A. Riccardi, and O. Feeney, “Closed-loop performance of pyramid wavefront sensor,” Proc. SPIE 4034, 184–189 (2000).
[CrossRef]

1999 (1)

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

1996 (1)

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

Aceituno, J.

D. Peter, M. Feldt, T. Henning, S. Hippler, J. Aceituno, L. Montoya, J. Costa, and B. Dorner, “PYRAMIR: exploring the on-sky performance of the world’s first near-infrared pyramid wavefront sensor,” Publ. Astron. Soc. Pac. 122, 63–70 (2010).
[CrossRef]

M. Feldt, D. Peter, S. Hippler, T. Henning, J. Aceituno, and M. Goto, “PYRAMIR: first on-sky results from an infrared pyramid wavefront sensor,” Proc. SPIE 6272, 627218 (2006).
[CrossRef]

Baker, K.

D. W. Phillion and K. Baker, “Two-sided pyramid wavefront sensor in the direct phase mode,” Proc. SPIE 6274, 627228 (2006).
[CrossRef]

Bradley, C.

Burvall, A.

Chamot, S. R.

Conan, R.

Correia, C.

F. Quirós-Pacheco, C. Correia, and S. Esposito, “Fourier transform-wavefront reconstruction for the pyramid wavefront sensor,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 07005.

Costa, J.

D. Peter, M. Feldt, T. Henning, S. Hippler, J. Aceituno, L. Montoya, J. Costa, and B. Dorner, “PYRAMIR: exploring the on-sky performance of the world’s first near-infrared pyramid wavefront sensor,” Publ. Astron. Soc. Pac. 122, 63–70 (2010).
[CrossRef]

Dainty, C.

Daly, E.

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, T. Henning, S. Hippler, J. Aceituno, L. Montoya, J. Costa, and B. Dorner, “PYRAMIR: exploring the on-sky performance of the world’s first near-infrared pyramid wavefront sensor,” Publ. Astron. Soc. Pac. 122, 63–70 (2010).
[CrossRef]

Esposito, S.

S. Esposito and A. Riccardi, “Pyramid wavefront sensor behavior in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

S. Esposito, A. Riccardi, and O. Feeney, “Closed-loop performance of pyramid wavefront sensor,” Proc. SPIE 4034, 184–189 (2000).
[CrossRef]

F. Quirós-Pacheco, C. Correia, and S. Esposito, “Fourier transform-wavefront reconstruction for the pyramid wavefront sensor,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 07005.

Farinato, J.

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

Feeney, O.

S. Esposito, A. Riccardi, and O. Feeney, “Closed-loop performance of pyramid wavefront sensor,” Proc. SPIE 4034, 184–189 (2000).
[CrossRef]

Feldt, M.

D. Peter, M. Feldt, T. Henning, S. Hippler, J. Aceituno, L. Montoya, J. Costa, and B. Dorner, “PYRAMIR: exploring the on-sky performance of the world’s first near-infrared pyramid wavefront sensor,” Publ. Astron. Soc. Pac. 122, 63–70 (2010).
[CrossRef]

M. Feldt, D. Peter, S. Hippler, T. Henning, J. Aceituno, and M. Goto, “PYRAMIR: first on-sky results from an infrared pyramid wavefront sensor,” Proc. SPIE 6272, 627218 (2006).
[CrossRef]

Garcia-Rissmann, A.

A. Garcia-Rissmann and M. Le Louarn, “SCAO simulation results with a pyramid sensor on an ELT-like telescope,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 03011.

Goto, M.

M. Feldt, D. Peter, S. Hippler, T. Henning, J. Aceituno, and M. Goto, “PYRAMIR: first on-sky results from an infrared pyramid wavefront sensor,” Proc. SPIE 6272, 627218 (2006).
[CrossRef]

Henning, T.

D. Peter, M. Feldt, T. Henning, S. Hippler, J. Aceituno, L. Montoya, J. Costa, and B. Dorner, “PYRAMIR: exploring the on-sky performance of the world’s first near-infrared pyramid wavefront sensor,” Publ. Astron. Soc. Pac. 122, 63–70 (2010).
[CrossRef]

M. Feldt, D. Peter, S. Hippler, T. Henning, J. Aceituno, and M. Goto, “PYRAMIR: first on-sky results from an infrared pyramid wavefront sensor,” Proc. SPIE 6272, 627218 (2006).
[CrossRef]

Hippler, S.

D. Peter, M. Feldt, T. Henning, S. Hippler, J. Aceituno, L. Montoya, J. Costa, and B. Dorner, “PYRAMIR: exploring the on-sky performance of the world’s first near-infrared pyramid wavefront sensor,” Publ. Astron. Soc. Pac. 122, 63–70 (2010).
[CrossRef]

M. Feldt, D. Peter, S. Hippler, T. Henning, J. Aceituno, and M. Goto, “PYRAMIR: first on-sky results from an infrared pyramid wavefront sensor,” Proc. SPIE 6272, 627218 (2006).
[CrossRef]

Hubin, N.

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European extremely large telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

Jolissant, L.

Korkiakoski, V.

V. Korkiakoski, C. Vérinaud, M. Le Louarn, and R. Conan, “Comparison between a model-based and a conventional pyramid sensor reconstructor,” Appl. Opt. 46, 6176–6184 (2007).
[CrossRef]

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European extremely large telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

V. Korkiakoski and C. Verinaud, “Extreme adaptive optics simulations for EPICS,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 03007.

Le Louarn, M.

V. Korkiakoski, C. Vérinaud, M. Le Louarn, and R. Conan, “Comparison between a model-based and a conventional pyramid sensor reconstructor,” Appl. Opt. 46, 6176–6184 (2007).
[CrossRef]

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European extremely large telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

A. Garcia-Rissmann and M. Le Louarn, “SCAO simulation results with a pyramid sensor on an ELT-like telescope,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 03011.

LeDue, J.

Marchetti, E.

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European extremely large telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

Montoya, L.

D. Peter, M. Feldt, T. Henning, S. Hippler, J. Aceituno, L. Montoya, J. Costa, and B. Dorner, “PYRAMIR: exploring the on-sky performance of the world’s first near-infrared pyramid wavefront sensor,” Publ. Astron. Soc. Pac. 122, 63–70 (2010).
[CrossRef]

Neubauer, A.

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” IPI 5, 893–913 (2011).

Peter, D.

D. Peter, M. Feldt, T. Henning, S. Hippler, J. Aceituno, L. Montoya, J. Costa, and B. Dorner, “PYRAMIR: exploring the on-sky performance of the world’s first near-infrared pyramid wavefront sensor,” Publ. Astron. Soc. Pac. 122, 63–70 (2010).
[CrossRef]

M. Feldt, D. Peter, S. Hippler, T. Henning, J. Aceituno, and M. Goto, “PYRAMIR: first on-sky results from an infrared pyramid wavefront sensor,” Proc. SPIE 6272, 627218 (2006).
[CrossRef]

Phillion, D. W.

D. W. Phillion and K. Baker, “Two-sided pyramid wavefront sensor in the direct phase mode,” Proc. SPIE 6274, 627228 (2006).
[CrossRef]

Quirós-Pacheco, F.

F. Quirós-Pacheco, C. Correia, and S. Esposito, “Fourier transform-wavefront reconstruction for the pyramid wavefront sensor,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 07005.

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. 350, L23–L26 (1999).

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

Ramlau, R.

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” IPI 5, 893–913 (2011).

Riccardi, A.

S. Esposito and A. Riccardi, “Pyramid wavefront sensor behavior in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

S. Esposito, A. Riccardi, and O. Feeney, “Closed-loop performance of pyramid wavefront sensor,” Proc. SPIE 4034, 184–189 (2000).
[CrossRef]

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

Rosensteiner, M.

Véran, J.-P.

Verinaud, C.

V. Korkiakoski and C. Verinaud, “Extreme adaptive optics simulations for EPICS,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 03007.

Vérinaud, C.

V. Korkiakoski, C. Vérinaud, M. Le Louarn, and R. Conan, “Comparison between a model-based and a conventional pyramid sensor reconstructor,” Appl. Opt. 46, 6176–6184 (2007).
[CrossRef]

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European extremely large telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

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

Zhariy, M.

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” IPI 5, 893–913 (2011).

Appl. Opt. (1)

Astron. Astrophys. (2)

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

S. Esposito and A. Riccardi, “Pyramid wavefront sensor behavior in partial correction adaptive optic systems,” Astron. Astrophys. 369, L9–L12 (2001).
[CrossRef]

IPI (1)

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” IPI 5, 893–913 (2011).

J. Mod. Opt. (1)

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

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

Opt. Commun. (2)

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

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

Opt. Express (2)

Proc. SPIE (4)

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European extremely large telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

D. W. Phillion and K. Baker, “Two-sided pyramid wavefront sensor in the direct phase mode,” Proc. SPIE 6274, 627228 (2006).
[CrossRef]

S. Esposito, A. Riccardi, and O. Feeney, “Closed-loop performance of pyramid wavefront sensor,” Proc. SPIE 4034, 184–189 (2000).
[CrossRef]

M. Feldt, D. Peter, S. Hippler, T. Henning, J. Aceituno, and M. Goto, “PYRAMIR: first on-sky results from an infrared pyramid wavefront sensor,” Proc. SPIE 6272, 627218 (2006).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

D. Peter, M. Feldt, T. Henning, S. Hippler, J. Aceituno, L. Montoya, J. Costa, and B. Dorner, “PYRAMIR: exploring the on-sky performance of the world’s first near-infrared pyramid wavefront sensor,” Publ. Astron. Soc. Pac. 122, 63–70 (2010).
[CrossRef]

Other (4)

V. Korkiakoski and C. Verinaud, “Extreme adaptive optics simulations for EPICS,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 03007.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

F. Quirós-Pacheco, C. Correia, and S. Esposito, “Fourier transform-wavefront reconstruction for the pyramid wavefront sensor,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 07005.

A. Garcia-Rissmann and M. Le Louarn, “SCAO simulation results with a pyramid sensor on an ELT-like telescope,” in First AO4ELT Conference—Adaptive Optics for Extremely Large Telescopes, T. F. Y. Clénet, J.-M. Conan, and G. Rousset, eds. (EDP Sciences, 2010), p. 03011.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1.
Fig. 1.

Scheme of an optical setup of a pyramid WFS. The circular modulation path is shown in the dashed line.

Fig. 2.
Fig. 2.

SH to pyramid data transmission filters g sh / pyr n , l , m in the Fourier domain corresponding to the following AO system parameters: telescope diameter D = 42 m , number of WFS subapertures n = 200 , modulation amplitude (where applicable) 4 λ / D . The difference between the three filters is only in the low frequency domain.

Fig. 3.
Fig. 3.

Analytical space domain kernel p sh / pyr l corresponding to the following AO system parameters: telescope diameter D = 42 m , number of WFS subapertures n s = 200 × 200 , linear modulation with amplitude 4 λ / D .

Fig. 4.
Fig. 4.

Discretized space domain kernel p sh / pyr l computed in three different ways: using the FFT routine (large dashed line); with the analytical formula evaluated at a discretized grid (small dashed line); and by averaging the analytical formula over subapertures (solid line). The kernels correspond to the following AO system parameters: telescope diameter D = 42 m , number of WFS subapertures n s = 200 × 200 , linear modulation with amplitude 4 λ / D .

Fig. 5.
Fig. 5.

Simulated LE Strehl in K band obtained with the P-CuReD and with the MVM method using the parameters for the test case A and median atmosphere versus the detected natural guide star (NGS) photon flux.

Fig. 6.
Fig. 6.

Simulated LE PSF in K band obtained with the P-CuReD and with the MVM method using parameters for the test case A, median atmosphere and high photon flux level (1e4 photons/subaperture/frame). The two PSFs are equal up to approximately 800 mas.

Fig. 7.
Fig. 7.

Simulated SE Strehl in K band obtained with the P-CuReD and with the MVM method using the parameters for the test case A, median atmosphere and high photon flux level (1e4 photons/subaperture/frame).

Fig. 8.
Fig. 8.

Simulated LE Strehl in K band obtained with the P-CuReD method using parameters for the test case B versus the detected NGS photon flux.

Fig. 9.
Fig. 9.

Noise propagation of the P-CuReD method for annular aperture of a fixed diameter D = 42 m and different number of sensor subapertures n s (and correspondingly the subaperture sizes d ). Modulation amplitude α = 4 λ / D is assumed. The results are shown for three different levels of subdivision in the CuReD algorithm.

Fig. 10.
Fig. 10.

Noise propagation of the P-CuReD method for annular apertures of different diameters and a fixed subaperture size d = 0.21 m . The number n s of sensor subapertures changes correspondingly. Modulation amplitude α = 4 λ / D is assumed. The results are shown for two different levels of subdivision in the CuReD algorithm.

Fig. 11.
Fig. 11.

Noise propagation of the P-CuReD method for annular apertures of the fixed diameter D = 42 m , subaperture size d = 0.21 m , and number of sensor subapertures n s = 200 × 200 and varying modulation amplitudes α . The results are shown for two different levels of subdivision in the CuReD algorithm.

Fig. 12.
Fig. 12.

Reconstruction times obtained with the P-CuReD ( k ) algorithm for an XAO system with a 200 × 200 pyramid WFS. Here k indicates the number of subdivisions in the CuReD algorithm [16].

Fig. 13.
Fig. 13.

Computation time for applying the data preprocessing step to the 200 × 200 pyramid WFS measurements.

Tables (3)

Tables Icon

Table 1. Simulation Parameters

Tables Icon

Table 2. Test Case Parameters

Tables Icon

Table 3. Comparison of the LE Strehl Ratios Obtained with the MVM Method and with the P-CuReD for Different Atmospheres and Photon Fluxes Using the Parameters Defined in Test Case A

Equations (55)

Equations on this page are rendered with MathJax. Learn more.

S x ( x , y ) = [ I 1 ( x , y ) + I 2 ( x , y ) ] [ I 3 ( x , y ) + I 4 ( x , y ) ] I 0 ,
S y ( x , y ) = [ I 1 ( x , y ) + I 4 ( x , y ) ] [ I 2 ( x , y ) + I 3 ( x , y ) ] I 0 ,
S x n ( x , y ) = 1 π B ( y ) + B ( y ) sin [ ϕ ( x , y ) ϕ ( x , y ) ] x x d x ,
S x l ( x , y ) = 1 π B ( y ) + B ( y ) sin [ ϕ ( x , y ) ϕ ( x , y ) ] x x sinc ( α λ ( x x ) ) d x ,
S x c ( x , y ) = 1 π B ( y ) + B ( y ) sin [ ϕ ( x , y ) ϕ ( x , y ) ] x x J 0 ( α λ ( x x ) ) d x ,
S x n ( x , y ) = ϕ ( x , y ) * 1 π x = H ϕ .
S x l ( x , y ) = ϕ ( x , y ) * sinc ( α λ x ) π x .
S x c ( x , y ) = ϕ ( x , y ) * J 0 ( α λ x ) π x .
( F S pyr l ) ( u ) = ( F ϕ ) ( u ) · g pyr l ( u ) · sinc ( d u )
g pyr l ( u ) = { i sgn ( u ) , | u | > u mod , i u / u mod , | u | u mod ,
( F S sh ) ( u ) = ( F ϕ ) ( u ) · g sh ( u ) · sinc ( d u )
g sh ( u ) = 2 i π d u .
g pyr n ( u ) = i sgn ( u ) , u [ u cut , u cut ] .
g pyr c ( u ) = { i sgn ( u ) , | u | > u mod , 2 i π arcsin ( u / u mod ) , | u | u mod .
( F S sh ) ( u ) = ( F S pyr ) ( u ) · g sh / pyr ( u ) ,
g sh / pyr ( u ) := ( F S sh ) ( u ) ( F S pyr ) ( u ) = g sh ( u ) g pyr ( u ) .
S sh ( x ) = 1 2 π S pyr ( x ) * p sh / pyr ( x ) ,
p sh / pyr ( x ) := ( F 1 g sh / pyr ) ( x ) .
g sh / pyr n ( u ) = 2 π d u sgn ( u ) , u [ u cut , u cut ]
lim u 0 g sh / pyr n ( u ) = lim u 0 g sh ( u ) g pyr n ( u ) = lim u 0 2 i π d u i sgn ( u ) = lim u 0 2 i π d | u | sgn ( u ) i sgn ( u ) = lim u 0 2 i π d | u | = 0 .
g sh / pyr l ( u ) = { 2 π d u sgn ( u ) , | u | > u mod , 2 π d u mod , | u | u mod ,
lim u 0 g sh / pyr l ( u ) = lim u 0 g sh ( u ) g pyr l ( u ) = lim u 0 2 i π d u u mod i u = 2 π d u mod .
g sh / pyr c ( u ) = { 2 π d u sgn ( u ) , | u | > u mod , π 2 d u arcsin ( u / u mod ) , | u | u mod ,
lim u 0 g sh / pyr c ( u ) = lim u 0 g sh ( u ) g pyr c ( u ) = lim u 0 2 i π 2 d u 2 i arcsin ( u / u mod ) = lim u 0 ( π 2 d u ) u ( arcsin ( u / u mod ) ) u = lim u 0 π 2 d u mod 2 u 2 = π 2 d u mod .
p sh / pyr n ( x ) = 4 π d u cut 2 sinc ( 2 x u cut ) 2 π d u cut 2 sinc 2 ( x u cut ) .
p sh / pyr l ( x ) = 4 π d u cut 2 sinc ( 2 x u cut ) + 2 π d u mod 2 sinc 2 ( x u mod ) 2 π d u cut 2 sinc 2 ( x u cut ) .
mse = tr ( MM T ) n ,
S x c ( x , y ) = 1 π ϕ ( x , y ) * J 0 ( α λ x ) x ,
π g pyr c ( u ) = ( F ( J 0 ( α λ x ) x ) ) ( u ) = ( F ( J 0 ( α λ x ) ) ) ( u ) * ( F ( 1 x ) ) ( u ) ,
( F ( 1 x ) ) ( u ) = i π sgn ( u ) ,
( F ( J 0 ( α λ x ) ) ) ( u ) = 1 π u mod 1 u 2 u mod 2 Π ( u 2 u mod )
Π ( u ) := { 1 , | u | 1 2 , 0 , | u | > 1 2 .
π g pyr c ( u ) = i u mod + sgn ( u u ) 1 1 ( u ) 2 u mod 2 Π ( u 2 u mod ) d u = i u mod u mod u mod sgn ( u u ) 1 ( u ) 2 u mod 2 d u .
π g pyr c ( u ) = i u mod u u mod u + u mod sgn ( v ) 1 ( u v ) 2 u mod 2 d v .
π g pyr c ( u ) = i u mod u u mod u + u mod 1 1 ( u v ) 2 u mod 2 d v = i u mod u mod 1 1 1 1 k 2 d k = 2 i arcsin ( 1 ) = i π .
π g pyr c ( u ) = i u mod u u mod u + u mod 1 1 ( u v ) 2 u mod 2 d v = i π .
π g pyr c ( u ) = i u mod u u mod 0 1 1 ( u v ) 2 u mod 2 d v + i u mod 0 u + u mod 1 1 ( u v ) 2 u mod 2 d v = i u mod u mod u / u mod 1 1 1 k 2 d k + i u mod u mod 1 u / u mod 1 1 k 2 d k = 2 i arcsin ( u / u mod ) .
g pyr c ( u ) = { i sgn ( u ) , | u | > u mod , 2 i π arcsin ( u / u mod ) , | u | u mod .
g sh / pyr n ( u ) = 2 π d | u | , | u | u cut = { 2 π d u , u cut u < 0 , 2 π d u , 0 < u u cut ,
p sh / pyr n ( x ) = ( F 1 g sh / pyr n ) ( x ) = u cut u cut g sh / pyr n ( u ) exp ( 2 π i x u ) d u = 2 π d u cut 0 u exp ( 2 π i x u ) d u p 1 n ( x ) + 2 π d 0 u cut 2 π d u exp ( 2 π i x u ) d u p 2 n ( x ) ,
p 1 n ( x ) = u cut 2 π i x exp ( 2 π i x u cut ) + 1 4 π 2 x 2 1 4 π 2 x 2 exp ( 2 π i x u cut ) ,
p 2 n ( x ) = u cut 2 π i x exp ( 2 π i x u cut ) + 1 4 π 2 x 2 exp ( 2 π i x u cut ) 1 4 π 2 x 2 .
p sh / pyr n ( x ) = 2 π d p 1 ( x ) + 2 π d p 2 ( x ) = d u cut i x exp ( 2 π i x u cut ) + d 2 π x 2 exp ( 2 π i x u cut ) + d u cut i x exp ( 2 π i x u cut ) + d 2 π x 2 exp ( 2 π i x u cut ) d π x 2 .
exp ( i x ) exp ( i x ) 2 = i sin ( x ) ,
exp ( i x ) + exp ( i x ) 2 = cos ( x ) ,
p sh / pyr n ( x ) = 2 d u cut x sin ( 2 π x u cut ) + d π x 2 cos ( 2 π x u cut ) d π x 2 .
cos ( x ) = 1 2 sin 2 ( x 2 ) ,
p sh / pyr n ( x ) = 2 d u cut x sin ( 2 π x u cut ) 2 d π x 2 sin 2 ( π x u cut ) = 4 π d u cut 2 sinc ( 2 π x u cut ) 2 π d u cut 2 sinc 2 ( π x u cut ) .
g sh / pyr l ( u ) = { 2 π d u , u < u mod , 2 π d u mod , u mod u u mod , 2 π d u , u > u mod .
p sh / pyr l ( x ) = ( F 1 g sh / pyr ) ( x ) = u cut u cut g sh / pyr l ( u ) exp ( 2 π i x u ) d u = p 1 l ( x ) + p 2 l ( x ) + p 3 l ( x ) ,
p 2 l ( x ) = 2 π d u mod u mod u mod exp ( 2 π i x u ) d u = d u mod i x exp ( 2 π i x u mod ) d u mod i x exp ( 2 π i x u mod ) ,
p 1 ( x ) = 2 π d u cut u mod u exp ( 2 π i x u ) d u = d u mod i x exp ( 2 π i x u mod ) + d u cut i x exp ( 2 π i x u cut ) + d 2 π x 2 exp ( 2 π i x u mod ) d 2 π x 2 exp ( 2 π i x u cut ) ,
p 3 ( x ) = 2 π d u mod u cut u exp ( 2 π i x u ) d u = d u cut i x exp ( 2 π i x u cut ) d u mod i x exp ( 2 π i x u mod ) + d 2 π x 2 exp ( 2 π i x u cut ) d 2 π x 2 exp ( 2 π i x u mod ) .
p sh / pyr l ( x ) = 2 d u cut x sin ( 2 π x u cut ) d π x 2 cos ( 2 π x u mod ) + d π x 2 cos ( 2 π x u cut ) .
p sh / pyr l ( x ) = 4 π d u cut 2 sinc ( 2 π x u cut ) + 2 π d u mod 2 sinc 2 ( π x u mod ) 2 π d u cut 2 sinc 2 ( π x u cut ) .

Metrics