Abstract

Zernike polynomials are widely used to describe common optical aberrations of a wavefront as they are well suited to the circular geometry of various optical apertures. Non-conventional optical systems, such as future large optical telescopes with highly segmented primary mirrors or advanced wavefront control devices using segmented mirror membrane facesheets, exhibit a hexagonal geometry, making the hexagonal orthogonal polynomials a valued basis. A cost-benefit trade-off study for deriving practical upper limits in, e.g., polishing, phasing, alignment, and stability of hexagons imposes analytical calculation to avoid time-consuming end-to-end simulations, for the sake of exactness. It is important to include global modes over the pupil for Zernike decomposition over a hexagonal segmented optical aperture into the error budget. However, numerically calculated Zernike decomposition is not optimal due to the discontinuities at the segment boundaries that result in imperfect hexagon sampling. In this paper, we present a novel approach for a rigorous Zernike and hexagonal mode decomposition adapted to hexagonal segmented pupils by means of analytical calculations. By contrast to numerical approaches that are dependent on the sampling of the segment, the decomposition expressed analytically only relies on the number and positions of segments comprising the pupil. Our analytical method allows extremely quick results minimizing computational and memory costs. Further, the proposed formulæ can be applied independently from the geometrical architecture of segmented optical apertures. Consequently, the method is universal and versatile per se. For instance, this work finds applications in optical metrology and active correction of phase aberrations. In modern astronomy with extremely large telescopes, it can contribute to sophisticated analytical specification of the contrast in the focal plane in presence of aberrations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Orthonormal polynomials in wavefront analysis: error analysis

Guang-ming Dai and Virendra N. Mahajan
Appl. Opt. 47(19) 3433-3445 (2008)

Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials

Xi Hou, Fan Wu, Li Yang, and Qiang Chen
Appl. Opt. 45(35) 8893-8901 (2006)

Zernike-like systems in polygons and polygonal facets

Chelo Ferreira, José L. López, Rafael Navarro, and Ester Pérez Sinusía
Appl. Opt. 54(21) 6575-6583 (2015)

References

  • View by:
  • |
  • |
  • |

  1. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999), 7th ed.
    [Crossref]
  2. V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis, vol. PM221 (SPIE Press, 2013).
  3. V. N. Mahajan, “Zernike polynomial and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (John Wiley, 2007), 3rd ed.
    [Crossref]
  4. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  5. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [Crossref]
  6. E. Pinna, F. Quirós-Pacheco, S. Esposito, A. Puglisi, and P. Stefanini, “Signal spatial filtering for co-phasing in seeing-limited conditions,” Opt. Lett. 32, 3465–3467 (2007).
    [Crossref] [PubMed]
  7. S. Manzanera, M. A. Helmbrecht, C. J. Kempf, and A. Roorda, “Mems segmented-based adaptive optics scanning laser ophthalmoscope,” Biomed. Opt. Express 2, 1204–1217 (2011).
    [Crossref] [PubMed]
  8. J. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
    [Crossref]
  9. B. M. Levine, E. A. Martinsen, A. Wirth, A. Jankevics, M. Toledo-Quinones, F. Landers, and T. L. Bruno, “Horizontal line-of-sight turbulence over near-ground paths and implications for adaptive optics corrections in laser communications,” Appl. Opt. 37, 4553–4560 (1998).
    [Crossref]
  10. R. Tamai, M. Cirasuolo, J. C. González, B. Koehler, and M. Tuti, “The E-ELT program status,” Proc. SPIE 9906, 99060W (2016).
    [Crossref]
  11. J. Nelson and G. H. Sanders, “TMT status report,” Proc. SPIE 6267, 626728 (2006).
    [Crossref]
  12. M. A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Segmented MEMS deformable-mirror technology for space applications,” Proc. SPIE 6223, 622305 (2006).
    [Crossref]
  13. M. A. Helmbrecht, M. He, P. Rhodes, and C. J. Kempf, “Preliminary results of large-actuator-count MEMS DM development,” Proc. SPIE 7595, 75950C (2010).
    [Crossref]
  14. M. A. Helmbrecht, M. He, C. J. Kempf, and M. Besse, “MEMS DM development at Iris AO, Inc.,” Proc. SPIE 7931, 793108 (2011).
    [Crossref]
  15. M. A. Helmbrecht, M. He, C. J. Kempf, and F. Marchis, “Long-term stability and temperature variability of Iris AO segmented MEMS deformable mirrors,” Proc. SPIE 9909, 990981 (2016).
    [Crossref]
  16. M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
    [Crossref]
  17. B. Pope, N. Cvetojevic, A. Cheetham, F. Martinache, B. Norris, and P. Tuthill, “A demonstration of wavefront sensing and mirror phasing from the image domain,” Mon. Not. R. Astron. Soc. 440, 125–133 (2014).
    [Crossref]
  18. P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).
  19. A. P. Norton, D. T. Gavel, M. Helmbrecht, C. Kempf, E. Gates, K. Chloros, D. Redel, R. Kupke, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” Proc. SPIE 9148, 91481C (2014).
  20. D. T. Miller, L. N. Thibos, and X. Hong, “Requirements for segmented correctors for diffraction-limited performance in the human eye,” Opt. Express 13, 275–289 (2005).
    [Crossref] [PubMed]
  21. D. Sinefeld, H. P. Paudel, D. G. Ouzounov, T. G. Bifano, and C. Xu, “Adaptive optics in multiphoton microscopy: comparison of two, three and four photon fluorescence,” Opt. Express 23, 31472–31483 (2015).
    [Crossref] [PubMed]
  22. B. Blochet, L. Bourdieu, and S. Gigan, “Focusing light through dynamical samples using fast continuous wavefront optimization,” Opt. Lett. 42, 4994–4997 (2017).
    [Crossref] [PubMed]
  23. K. L. Baker, E. A. Stappaerts, D. Gavel, S. C. Wilks, J. Tucker, D. A. Silva, J. Olsen, S. S. Olivier, P. E. Young, M. W. Kartz, L. M. Flath, P. Kruelevitch, J. Crawford, and O. Azucena, “High-speed horizontal-path atmospheric turbulence correction with a large-actuator-number microelectromechanical system spatial light modulator in an interferometric phase-conjugation engine,” Opt. Lett. 29, 1781–1783 (2004).
    [Crossref] [PubMed]
  24. V. N. Mahajan and G. ming Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
    [Crossref]
  25. V. N. Mahajan and G. ming Dai, “Orthonormal polynomials for hexagonal pupils,” Opt. Lett. 31, 2462–2464 (2006).
    [Crossref] [PubMed]
  26. L. Leboulleux, J.-F. Sauvage, L. Pueyo, T. Fusco, R. Soummer, J. Mazoyer, A. Sivaramakrishnan, M. N’Diaye, and O. Fauvarque, “Pair-based Analytical model for Segmented Telescopes Imaging from Space (PASTIS) for sensitivity analysis,” J. Astron. Telesc. Instruments, Syst. (2018, in press).
    [Crossref]
  27. F. Marchis and R. JL Faltick, Iris AO company, 2930 Shattuck Ave. #304, Berkeley, CA 94705, personal communication (2015).

2017 (1)

2016 (2)

R. Tamai, M. Cirasuolo, J. C. González, B. Koehler, and M. Tuti, “The E-ELT program status,” Proc. SPIE 9906, 99060W (2016).
[Crossref]

M. A. Helmbrecht, M. He, C. J. Kempf, and F. Marchis, “Long-term stability and temperature variability of Iris AO segmented MEMS deformable mirrors,” Proc. SPIE 9909, 990981 (2016).
[Crossref]

2015 (1)

2014 (3)

B. Pope, N. Cvetojevic, A. Cheetham, F. Martinache, B. Norris, and P. Tuthill, “A demonstration of wavefront sensing and mirror phasing from the image domain,” Mon. Not. R. Astron. Soc. 440, 125–133 (2014).
[Crossref]

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

A. P. Norton, D. T. Gavel, M. Helmbrecht, C. Kempf, E. Gates, K. Chloros, D. Redel, R. Kupke, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” Proc. SPIE 9148, 91481C (2014).

2013 (1)

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

2011 (2)

2010 (1)

M. A. Helmbrecht, M. He, P. Rhodes, and C. J. Kempf, “Preliminary results of large-actuator-count MEMS DM development,” Proc. SPIE 7595, 75950C (2010).
[Crossref]

2007 (2)

2006 (3)

V. N. Mahajan and G. ming Dai, “Orthonormal polynomials for hexagonal pupils,” Opt. Lett. 31, 2462–2464 (2006).
[Crossref] [PubMed]

J. Nelson and G. H. Sanders, “TMT status report,” Proc. SPIE 6267, 626728 (2006).
[Crossref]

M. A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Segmented MEMS deformable-mirror technology for space applications,” Proc. SPIE 6223, 622305 (2006).
[Crossref]

2005 (1)

2004 (1)

1998 (1)

1997 (1)

1990 (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[Crossref]

1976 (1)

Abe, L.

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Azucena, O.

Baker, K. L.

Beaulieu, M.

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Besse, M.

M. A. Helmbrecht, M. He, C. J. Kempf, and M. Besse, “MEMS DM development at Iris AO, Inc.,” Proc. SPIE 7931, 793108 (2011).
[Crossref]

Bifano, T. G.

Blochet, B.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999), 7th ed.
[Crossref]

Bourdieu, L.

Bruno, T. L.

Carlotti, A.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

Cheetham, A.

B. Pope, N. Cvetojevic, A. Cheetham, F. Martinache, B. Norris, and P. Tuthill, “A demonstration of wavefront sensing and mirror phasing from the image domain,” Mon. Not. R. Astron. Soc. 440, 125–133 (2014).
[Crossref]

Chloros, K.

A. P. Norton, D. T. Gavel, M. Helmbrecht, C. Kempf, E. Gates, K. Chloros, D. Redel, R. Kupke, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” Proc. SPIE 9148, 91481C (2014).

Choquet, E.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

Cirasuolo, M.

R. Tamai, M. Cirasuolo, J. C. González, B. Koehler, and M. Tuti, “The E-ELT program status,” Proc. SPIE 9906, 99060W (2016).
[Crossref]

Crawford, J.

Cvetojevic, N.

B. Pope, N. Cvetojevic, A. Cheetham, F. Martinache, B. Norris, and P. Tuthill, “A demonstration of wavefront sensing and mirror phasing from the image domain,” Mon. Not. R. Astron. Soc. 440, 125–133 (2014).
[Crossref]

Daban, J.-B.

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Dai, G. ming

Dejonghe, J.

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Dillon, D.

A. P. Norton, D. T. Gavel, M. Helmbrecht, C. Kempf, E. Gates, K. Chloros, D. Redel, R. Kupke, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” Proc. SPIE 9148, 91481C (2014).

Doble, N.

M. A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Segmented MEMS deformable-mirror technology for space applications,” Proc. SPIE 6223, 622305 (2006).
[Crossref]

Elliot, E.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

Esposito, S.

Faltick, R. JL

F. Marchis and R. JL Faltick, Iris AO company, 2930 Shattuck Ave. #304, Berkeley, CA 94705, personal communication (2015).

Fantei-Caujolle, Y.

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Fauvarque, O.

L. Leboulleux, J.-F. Sauvage, L. Pueyo, T. Fusco, R. Soummer, J. Mazoyer, A. Sivaramakrishnan, M. N’Diaye, and O. Fauvarque, “Pair-based Analytical model for Segmented Telescopes Imaging from Space (PASTIS) for sensitivity analysis,” J. Astron. Telesc. Instruments, Syst. (2018, in press).
[Crossref]

Flath, L. M.

Fusco, T.

L. Leboulleux, J.-F. Sauvage, L. Pueyo, T. Fusco, R. Soummer, J. Mazoyer, A. Sivaramakrishnan, M. N’Diaye, and O. Fauvarque, “Pair-based Analytical model for Segmented Telescopes Imaging from Space (PASTIS) for sensitivity analysis,” J. Astron. Telesc. Instruments, Syst. (2018, in press).
[Crossref]

Gates, E.

A. P. Norton, D. T. Gavel, M. Helmbrecht, C. Kempf, E. Gates, K. Chloros, D. Redel, R. Kupke, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” Proc. SPIE 9148, 91481C (2014).

Gavel, D.

Gavel, D. T.

A. P. Norton, D. T. Gavel, M. Helmbrecht, C. Kempf, E. Gates, K. Chloros, D. Redel, R. Kupke, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” Proc. SPIE 9148, 91481C (2014).

Gigan, S.

González, J. C.

R. Tamai, M. Cirasuolo, J. C. González, B. Koehler, and M. Tuti, “The E-ELT program status,” Proc. SPIE 9906, 99060W (2016).
[Crossref]

Gouvret, C.

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Groff, T.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

Hart, M.

M. A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Segmented MEMS deformable-mirror technology for space applications,” Proc. SPIE 6223, 622305 (2006).
[Crossref]

He, M.

M. A. Helmbrecht, M. He, C. J. Kempf, and F. Marchis, “Long-term stability and temperature variability of Iris AO segmented MEMS deformable mirrors,” Proc. SPIE 9909, 990981 (2016).
[Crossref]

M. A. Helmbrecht, M. He, C. J. Kempf, and M. Besse, “MEMS DM development at Iris AO, Inc.,” Proc. SPIE 7931, 793108 (2011).
[Crossref]

M. A. Helmbrecht, M. He, P. Rhodes, and C. J. Kempf, “Preliminary results of large-actuator-count MEMS DM development,” Proc. SPIE 7595, 75950C (2010).
[Crossref]

Helmbrecht, M.

A. P. Norton, D. T. Gavel, M. Helmbrecht, C. Kempf, E. Gates, K. Chloros, D. Redel, R. Kupke, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” Proc. SPIE 9148, 91481C (2014).

Helmbrecht, M. A.

M. A. Helmbrecht, M. He, C. J. Kempf, and F. Marchis, “Long-term stability and temperature variability of Iris AO segmented MEMS deformable mirrors,” Proc. SPIE 9909, 990981 (2016).
[Crossref]

M. A. Helmbrecht, M. He, C. J. Kempf, and M. Besse, “MEMS DM development at Iris AO, Inc.,” Proc. SPIE 7931, 793108 (2011).
[Crossref]

S. Manzanera, M. A. Helmbrecht, C. J. Kempf, and A. Roorda, “Mems segmented-based adaptive optics scanning laser ophthalmoscope,” Biomed. Opt. Express 2, 1204–1217 (2011).
[Crossref] [PubMed]

M. A. Helmbrecht, M. He, P. Rhodes, and C. J. Kempf, “Preliminary results of large-actuator-count MEMS DM development,” Proc. SPIE 7595, 75950C (2010).
[Crossref]

M. A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Segmented MEMS deformable-mirror technology for space applications,” Proc. SPIE 6223, 622305 (2006).
[Crossref]

Hong, X.

Janin-Potiron, P.

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Jankevics, A.

Juneau, T.

M. A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Segmented MEMS deformable-mirror technology for space applications,” Proc. SPIE 6223, 622305 (2006).
[Crossref]

Kartz, M. W.

Kasdin, N. J.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

Kempf, C.

A. P. Norton, D. T. Gavel, M. Helmbrecht, C. Kempf, E. Gates, K. Chloros, D. Redel, R. Kupke, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” Proc. SPIE 9148, 91481C (2014).

Kempf, C. J.

M. A. Helmbrecht, M. He, C. J. Kempf, and F. Marchis, “Long-term stability and temperature variability of Iris AO segmented MEMS deformable mirrors,” Proc. SPIE 9909, 990981 (2016).
[Crossref]

M. A. Helmbrecht, M. He, C. J. Kempf, and M. Besse, “MEMS DM development at Iris AO, Inc.,” Proc. SPIE 7931, 793108 (2011).
[Crossref]

S. Manzanera, M. A. Helmbrecht, C. J. Kempf, and A. Roorda, “Mems segmented-based adaptive optics scanning laser ophthalmoscope,” Biomed. Opt. Express 2, 1204–1217 (2011).
[Crossref] [PubMed]

M. A. Helmbrecht, M. He, P. Rhodes, and C. J. Kempf, “Preliminary results of large-actuator-count MEMS DM development,” Proc. SPIE 7595, 75950C (2010).
[Crossref]

Koehler, B.

R. Tamai, M. Cirasuolo, J. C. González, B. Koehler, and M. Tuti, “The E-ELT program status,” Proc. SPIE 9906, 99060W (2016).
[Crossref]

Kruelevitch, P.

Kupke, R.

A. P. Norton, D. T. Gavel, M. Helmbrecht, C. Kempf, E. Gates, K. Chloros, D. Redel, R. Kupke, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” Proc. SPIE 9148, 91481C (2014).

Landers, F.

Leboulleux, L.

L. Leboulleux, J.-F. Sauvage, L. Pueyo, T. Fusco, R. Soummer, J. Mazoyer, A. Sivaramakrishnan, M. N’Diaye, and O. Fauvarque, “Pair-based Analytical model for Segmented Telescopes Imaging from Space (PASTIS) for sensitivity analysis,” J. Astron. Telesc. Instruments, Syst. (2018, in press).
[Crossref]

Levine, B. M.

Liang, J.

Macintosh, B.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

Mahajan, V. N.

V. N. Mahajan and G. ming Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
[Crossref]

V. N. Mahajan and G. ming Dai, “Orthonormal polynomials for hexagonal pupils,” Opt. Lett. 31, 2462–2464 (2006).
[Crossref] [PubMed]

V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis, vol. PM221 (SPIE Press, 2013).

V. N. Mahajan, “Zernike polynomial and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (John Wiley, 2007), 3rd ed.
[Crossref]

Manzanera, S.

Marchis, F.

M. A. Helmbrecht, M. He, C. J. Kempf, and F. Marchis, “Long-term stability and temperature variability of Iris AO segmented MEMS deformable mirrors,” Proc. SPIE 9909, 990981 (2016).
[Crossref]

F. Marchis and R. JL Faltick, Iris AO company, 2930 Shattuck Ave. #304, Berkeley, CA 94705, personal communication (2015).

Martinache, F.

B. Pope, N. Cvetojevic, A. Cheetham, F. Martinache, B. Norris, and P. Tuthill, “A demonstration of wavefront sensing and mirror phasing from the image domain,” Mon. Not. R. Astron. Soc. 440, 125–133 (2014).
[Crossref]

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Martinez, P.

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Martinsen, E. A.

Mattei, D.

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Mawet, D.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

Mazoyer, J.

L. Leboulleux, J.-F. Sauvage, L. Pueyo, T. Fusco, R. Soummer, J. Mazoyer, A. Sivaramakrishnan, M. N’Diaye, and O. Fauvarque, “Pair-based Analytical model for Segmented Telescopes Imaging from Space (PASTIS) for sensitivity analysis,” J. Astron. Telesc. Instruments, Syst. (2018, in press).
[Crossref]

Miller, D. T.

N’Diaye, M.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

L. Leboulleux, J.-F. Sauvage, L. Pueyo, T. Fusco, R. Soummer, J. Mazoyer, A. Sivaramakrishnan, M. N’Diaye, and O. Fauvarque, “Pair-based Analytical model for Segmented Telescopes Imaging from Space (PASTIS) for sensitivity analysis,” J. Astron. Telesc. Instruments, Syst. (2018, in press).
[Crossref]

Nelson, J.

J. Nelson and G. H. Sanders, “TMT status report,” Proc. SPIE 6267, 626728 (2006).
[Crossref]

Noll, R. J.

Norris, B.

B. Pope, N. Cvetojevic, A. Cheetham, F. Martinache, B. Norris, and P. Tuthill, “A demonstration of wavefront sensing and mirror phasing from the image domain,” Mon. Not. R. Astron. Soc. 440, 125–133 (2014).
[Crossref]

Norton, A. P.

A. P. Norton, D. T. Gavel, M. Helmbrecht, C. Kempf, E. Gates, K. Chloros, D. Redel, R. Kupke, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” Proc. SPIE 9148, 91481C (2014).

Olivier, S. S.

Olsen, J.

Ottogalli, S.

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Ouzounov, D. G.

Paudel, H. P.

Perrin, M. D.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

Pinna, E.

Pope, B.

B. Pope, N. Cvetojevic, A. Cheetham, F. Martinache, B. Norris, and P. Tuthill, “A demonstration of wavefront sensing and mirror phasing from the image domain,” Mon. Not. R. Astron. Soc. 440, 125–133 (2014).
[Crossref]

Preis, O.

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Pueyo, L.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

L. Leboulleux, J.-F. Sauvage, L. Pueyo, T. Fusco, R. Soummer, J. Mazoyer, A. Sivaramakrishnan, M. N’Diaye, and O. Fauvarque, “Pair-based Analytical model for Segmented Telescopes Imaging from Space (PASTIS) for sensitivity analysis,” J. Astron. Telesc. Instruments, Syst. (2018, in press).
[Crossref]

Puglisi, A.

Quirós-Pacheco, F.

Redel, D.

A. P. Norton, D. T. Gavel, M. Helmbrecht, C. Kempf, E. Gates, K. Chloros, D. Redel, R. Kupke, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” Proc. SPIE 9148, 91481C (2014).

Rhodes, P.

M. A. Helmbrecht, M. He, P. Rhodes, and C. J. Kempf, “Preliminary results of large-actuator-count MEMS DM development,” Proc. SPIE 7595, 75950C (2010).
[Crossref]

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[Crossref]

Roorda, A.

Sanders, G. H.

J. Nelson and G. H. Sanders, “TMT status report,” Proc. SPIE 6267, 626728 (2006).
[Crossref]

Sauvage, J.-F.

L. Leboulleux, J.-F. Sauvage, L. Pueyo, T. Fusco, R. Soummer, J. Mazoyer, A. Sivaramakrishnan, M. N’Diaye, and O. Fauvarque, “Pair-based Analytical model for Segmented Telescopes Imaging from Space (PASTIS) for sensitivity analysis,” J. Astron. Telesc. Instruments, Syst. (2018, in press).
[Crossref]

Shaklan, S.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

Sheckells, M.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

Silva, D. A.

Sinefeld, D.

Sivaramakrishnan, A.

L. Leboulleux, J.-F. Sauvage, L. Pueyo, T. Fusco, R. Soummer, J. Mazoyer, A. Sivaramakrishnan, M. N’Diaye, and O. Fauvarque, “Pair-based Analytical model for Segmented Telescopes Imaging from Space (PASTIS) for sensitivity analysis,” J. Astron. Telesc. Instruments, Syst. (2018, in press).
[Crossref]

Soummer, R.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

L. Leboulleux, J.-F. Sauvage, L. Pueyo, T. Fusco, R. Soummer, J. Mazoyer, A. Sivaramakrishnan, M. N’Diaye, and O. Fauvarque, “Pair-based Analytical model for Segmented Telescopes Imaging from Space (PASTIS) for sensitivity analysis,” J. Astron. Telesc. Instruments, Syst. (2018, in press).
[Crossref]

Spang, A.

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

Stappaerts, E. A.

Stefanini, P.

Tamai, R.

R. Tamai, M. Cirasuolo, J. C. González, B. Koehler, and M. Tuti, “The E-ELT program status,” Proc. SPIE 9906, 99060W (2016).
[Crossref]

Thibos, L. N.

Toledo-Quinones, M.

Tucker, J.

Tuthill, P.

B. Pope, N. Cvetojevic, A. Cheetham, F. Martinache, B. Norris, and P. Tuthill, “A demonstration of wavefront sensing and mirror phasing from the image domain,” Mon. Not. R. Astron. Soc. 440, 125–133 (2014).
[Crossref]

Tuti, M.

R. Tamai, M. Cirasuolo, J. C. González, B. Koehler, and M. Tuti, “The E-ELT program status,” Proc. SPIE 9906, 99060W (2016).
[Crossref]

Wallace, J. K.

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

Wilks, S. C.

Williams, D. R.

Wirth, A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999), 7th ed.
[Crossref]

Xu, C.

Young, P. E.

Appl. Opt. (1)

Biomed. Opt. Express (1)

J. Opt. Soc. Am. (1)

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

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

B. Pope, N. Cvetojevic, A. Cheetham, F. Martinache, B. Norris, and P. Tuthill, “A demonstration of wavefront sensing and mirror phasing from the image domain,” Mon. Not. R. Astron. Soc. 440, 125–133 (2014).
[Crossref]

Opt. Eng. (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[Crossref]

Opt. Express (2)

Opt. Lett. (4)

Proc. SPIE (9)

P. Martinez, O. Preis, C. Gouvret, J. Dejonghe, J.-B. Daban, A. Spang, F. Martinache, M. Beaulieu, P. Janin-Potiron, L. Abe, Y. Fantei-Caujolle, D. Mattei, and S. Ottogalli, “SPEED: the segmented pupil experiment for exoplanet detection,” Proc. SPIE 9145, 91454E (2014).

A. P. Norton, D. T. Gavel, M. Helmbrecht, C. Kempf, E. Gates, K. Chloros, D. Redel, R. Kupke, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” Proc. SPIE 9148, 91481C (2014).

R. Tamai, M. Cirasuolo, J. C. González, B. Koehler, and M. Tuti, “The E-ELT program status,” Proc. SPIE 9906, 99060W (2016).
[Crossref]

J. Nelson and G. H. Sanders, “TMT status report,” Proc. SPIE 6267, 626728 (2006).
[Crossref]

M. A. Helmbrecht, T. Juneau, M. Hart, and N. Doble, “Segmented MEMS deformable-mirror technology for space applications,” Proc. SPIE 6223, 622305 (2006).
[Crossref]

M. A. Helmbrecht, M. He, P. Rhodes, and C. J. Kempf, “Preliminary results of large-actuator-count MEMS DM development,” Proc. SPIE 7595, 75950C (2010).
[Crossref]

M. A. Helmbrecht, M. He, C. J. Kempf, and M. Besse, “MEMS DM development at Iris AO, Inc.,” Proc. SPIE 7931, 793108 (2011).
[Crossref]

M. A. Helmbrecht, M. He, C. J. Kempf, and F. Marchis, “Long-term stability and temperature variability of Iris AO segmented MEMS deformable mirrors,” Proc. SPIE 9909, 990981 (2016).
[Crossref]

M. N’Diaye, E. Choquet, L. Pueyo, E. Elliot, M. D. Perrin, J. K. Wallace, T. Groff, A. Carlotti, D. Mawet, M. Sheckells, S. Shaklan, B. Macintosh, N. J. Kasdin, and R. Soummer, “High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design,” Proc. SPIE 8864, 88641K (2013).
[Crossref]

Other (5)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999), 7th ed.
[Crossref]

V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis, vol. PM221 (SPIE Press, 2013).

V. N. Mahajan, “Zernike polynomial and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (John Wiley, 2007), 3rd ed.
[Crossref]

L. Leboulleux, J.-F. Sauvage, L. Pueyo, T. Fusco, R. Soummer, J. Mazoyer, A. Sivaramakrishnan, M. N’Diaye, and O. Fauvarque, “Pair-based Analytical model for Segmented Telescopes Imaging from Space (PASTIS) for sensitivity analysis,” J. Astron. Telesc. Instruments, Syst. (2018, in press).
[Crossref]

F. Marchis and R. JL Faltick, Iris AO company, 2930 Shattuck Ave. #304, Berkeley, CA 94705, personal communication (2015).

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 (4)

Fig. 1
Fig. 1 Representation of a M = 2 (N = 19) segmented pupil. The dashed red and solid orange circles stand for the inscribed and circumscribed circle to the full mirror respectively.
Fig. 2
Fig. 2 Representation of the absolute errors | ε n ( Z j ) | obtained by means of simulations for the first ten Zernike modes. Three different pupils are considered (N = 19, 37 and 91).
Fig. 3
Fig. 3 Representation of the total fitting error for the first 10 Zernike polynomials (left) and hexagonal polynomials (right) for different numbers of segments comprising the mirror. The dots represent the values obtained by means of numerical simulations while the lines represent the theoretical values given in Table 3.
Fig. 4
Fig. 4 Representation of the total fitting error σ ( j ) (left) and its relative error with regard to Eq. (14) (right) for the first 10 hexagonal polynomials for different values of rsim used to numerically compute the piston, tip and tilt coefficients on each segment. The pupil used for this simulation is composed of N = 37 segments.

Tables (3)

Tables Icon

Table 1 Piston, tip, and tilt coefficients on the segment n for the decomposition of the 10 first Zernike polynomials Zj over a segmented aperture including an arbitrary rotation of the segment by θn.

Tables Icon

Table 2 Piston, tip, and tilt coefficients on the segment n for the decomposition of the 10 first hexagonal polynomials Hj over a segmented aperture.

Tables Icon

Table 3 Mean fitting error values over the segment n for the 10 first Zernike and hexagonal modes.

Equations (16)

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

H j ( x , y ) = i = 1 j α i Z i ( x , y ) ,
p n ( Z j ) = 1 A n S Z j ( x R , y R ) H 1 ( x X n r n , y Y n r n ) d x d y ,
t n ( Z j ) = 1 A n S Z j ( x R , y R ) H 2 ( x X n r n , y Y n r n ) d x d y ,
T n ( Z j ) = 1 A n S Z j ( x R , y R ) H 3 ( x X n r n , y Y n r n ) d x d y ,
ζ 1 ( x ) = { Y n 3 ( X n + x + r ) for r < x X n < r 2 Y n 3 r / 2 for r 2 < x X n < r 2 Y n + 3 ( X n + x r ) for r 2 < x X n < r
ζ 2 ( x ) = { Y n + 3 ( X n + x + r ) for r < x X n < r 2 Y n + 3 r / 2 for r 2 < x X n < r 2 Y n 3 ( X n + x r ) for r 2 < x X n < r
p n ( H j ) = 1 A n S H j ( x R , y R ) H 1 ( x X n r n , y Y n r n ) d x d y ,
t n ( H j ) = 1 A n S H j ( x R , y R ) H 2 ( x X n r n , y Y n r n ) d x d y ,
T n ( H j ) = 1 A n S H j ( x R , y R ) H 3 ( x X n r n , y Y n r n ) d x d y .
x R X n + ( x X n ) cos θ n ( y Y n ) sin θ n R , y R Y n + ( x X n ) sin θ n + ( y Y n ) cos θ n R .
ε n ( x , y , Z j ) = [ p n ( Z j ) × H 1 ( x X n r n , y Y n r n ) + t n ( Z j ) × H 2 ( x X n r n , y Y n r n ) + T n ( Z j ) × H 3 ( x X n r n , y Y n r n ) ] Z j ( X n + ( x X n ) cos θ n ( y Y n ) sin θ n R , Y n + ( x X n ) sin θ n + ( y Y n ) cos θ n R ) ,
ε n ( x , y , H j ) = [ p n ( H j ) × H 1 ( x X n r n , y Y n r n ) + t n ( H j ) × H 2 ( x X n r n , y Y n r n ) + T n ( H j ) × H 3 ( x X n r n , y Y n r n ) ] H j ( X n + ( x X n ) cos θ n ( y Y n ) sin θ n R , Y n + ( x X n ) sin θ n + ( y Y n ) cos θ n R ) .
ε n 2 ( Z j ) ¯ = 1 A n S ε n 2 ( x , y , Z j ) d x d y ,
ε n 2 ( H j ) ¯ = 1 A n S ε n 2 ( x , y , H j ) d x d y .
σ ( Z j ) = A π R 2 × n ε n 2 ( Z j ) ¯ = 3 3 r 2 2 π R 2 × n ε n 2 ( Z j ) ¯ ,
σ ( H j ) = 2 A 3 3 R 2 × n ε n 2 ( H j ) ¯ = r 2 R 2 × n ε n 2 ( H j ) ¯ ,