Abstract

A study is presented of a Mach–Zehnder interferometer for the measurement of phasing errors of the type found in segmented telescopes. We show that with a pinhole much larger than the Airy disk and an optical path difference between the arms equal to a quarter of the wavelength, the interferometric signal is related to the second derivative of the wave front. In this condition the signal is produced mostly by the segmentation errors and is marginally sensitive to other aberrations including atmospheric turbulence. The signal has distinguishable symmetric and antisymmetric properties that are related to segment aberrations. We suggest using the antisymmetric component of the signal to retrieve piston, tip, and tilt. The symmetric component of the signal serves as an estimate of the measurement error. In this way we proceed with a study of the errors associated with the misalignment of the interferometer, the segment edge imperfections, and the nonaveraged atmospheric perturbations. The entire study is performed on a theoretical basis, and numerical simulations are used to cross check the analytical results.

© 2005 Optical Society of America

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  1. P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).
  2. T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Own-Petersen, H. Riewaldt, R. Snel, D. Walker, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2002).
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    [CrossRef]
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    [CrossRef]
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  14. N. Yaitskova, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 62–71 (2003).
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  23. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  24. K. Dohlen, “Phase masks in astronomy: from the Mach–Zehnder interferometer to coronagraphs,” in Astronomy with High Contrast Imaging II, C. Aime and R. Soummer, eds., EAS Publ. Ser. 12 (European Astronomical Society, 2004), pp. 33–44; http://www2.iap.fr/eas.

2003 (1)

N. Yaitskova, K. Dohlen, P. Dierickx, “Analytical study of diffraction effects in extremely large segmented telescopes,” J. Opt. Soc. Am. A 20, 1563–1575 (2003).
[CrossRef]

1999 (1)

1998 (1)

G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck telescope: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

1994 (1)

R. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature (London) 368, 203 (1994).
[CrossRef]

1988 (1)

J. N. Aubrun, K. R. Lorell, T. W. Havas, W. C. Henninger, “Performance analysis of the segmented alignment control system for the ten-meter telescope,” Automatica 24, 437–454 (1988).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Andersen, T.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Own-Petersen, H. Riewaldt, R. Snel, D. Walker, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2002).

Angel, J. R.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Own-Petersen, H. Riewaldt, R. Snel, D. Walker, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2002).

J. Nelson, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 47–59 (2002).

S. Strom, L. Stepp, M. Mountain, B. Gregory, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2002).

L. Montoya, N. Yaitskova, P. Dierickx, K. Dohlen, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 564–573 (2002).

Angel, R.

R. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature (London) 368, 203 (1994).
[CrossRef]

M. Langlois, R. Angel, M. Lloyd-Hard, F. Widi, G. Love, A. Naumov, “High order, reconstructor-free adaptive optics for a 6–8 meter class telescope,” in Beyond Conventional Adaptive Optics, E. Vernet, R. Ragazzoni, S. Esposito, and N. Hubin, eds., ESO Conference Workshop Proceedings No. 58 (European Southern Observatory, Garching bei München, Germany, 2002), pp. 113–120 (2002).

Ardeberg, A. L.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Own-Petersen, H. Riewaldt, R. Snel, D. Walker, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2002).

Aubrun, J. N.

J. N. Aubrun, K. R. Lorell, T. W. Havas, W. C. Henninger, “Performance analysis of the segmented alignment control system for the ten-meter telescope,” Automatica 24, 437–454 (1988).
[CrossRef]

Beckers, J.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Own-Petersen, H. Riewaldt, R. Snel, D. Walker, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2002).

Beckers, J. L.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Brunetto, E.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Castro, F. J.

F. J. Castro, N. Devaney, L. Jochum, B. Ronquillo, L. Cavaller, P. Dierickx, ed., Proc. SPIE4003, 24–33 (2000).

Cavaller, L.

F. J. Castro, N. Devaney, L. Jochum, B. Ronquillo, L. Cavaller, P. Dierickx, ed., Proc. SPIE4003, 24–33 (2000).

Chanan, G.

G. Chanan, M. Troy, “Strehl ratio and modulation transfer function for segmented mirror telescope as function of segment phase error,” Appl. Opt. 38, 6642–6647 (1999).
[CrossRef]

G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck telescope: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

Conan, R.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Cuevas, S.

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).

Dekens, F.

G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck telescope: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

Devaney, N.

S. Esposito, E. Pinna, A. Tozzi, P. Stefenini, N. Devaney, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 214–225 (2003).

F. J. Castro, N. Devaney, L. Jochum, B. Ronquillo, L. Cavaller, P. Dierickx, ed., Proc. SPIE4003, 24–33 (2000).

Dierickx, P.

N. Yaitskova, K. Dohlen, P. Dierickx, “Analytical study of diffraction effects in extremely large segmented telescopes,” J. Opt. Soc. Am. A 20, 1563–1575 (2003).
[CrossRef]

F. J. Castro, N. Devaney, L. Jochum, B. Ronquillo, L. Cavaller, P. Dierickx, ed., Proc. SPIE4003, 24–33 (2000).

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

N. Yaitskova, L. Montoya, K. Dohlen, P. Dierickx, J. Oschmann, M. Tarenghi, eds., Proc. SPIE5489, 1139–1151 (2004).

L. Montoya, N. Yaitskova, P. Dierickx, K. Dohlen, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 564–573 (2002).

Dohlen, K.

N. Yaitskova, K. Dohlen, P. Dierickx, “Analytical study of diffraction effects in extremely large segmented telescopes,” J. Opt. Soc. Am. A 20, 1563–1575 (2003).
[CrossRef]

L. Montoya, N. Yaitskova, P. Dierickx, K. Dohlen, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 564–573 (2002).

N. Yaitskova, L. Montoya, K. Dohlen, P. Dierickx, J. Oschmann, M. Tarenghi, eds., Proc. SPIE5489, 1139–1151 (2004).

K. Dohlen, “Phase masks in astronomy: from the Mach–Zehnder interferometer to coronagraphs,” in Astronomy with High Contrast Imaging II, C. Aime and R. Soummer, eds., EAS Publ. Ser. 12 (European Astronomical Society, 2004), pp. 33–44; http://www2.iap.fr/eas.

Esposito, S.

S. Esposito, E. Pinna, A. Tozzi, P. Stefenini, N. Devaney, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 214–225 (2003).

Faber, S.

J. Nelson, T. Mast, S. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Rep. 90 (W. M. Keck Library, Kamuela, Hawaii, 1985), pp. 5-1–5-44.

Fedrigo, E.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Fowler, J. R.

V. L. Krabbendam, T. A. Sebring, F. B. Ray, J. R. Fowler, L. Stepp, ed., Proc. SPIE3352, 436–445 (1998).

Garfias, F.

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).

Gilmozzi, R.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

S. Strom, L. Stepp, M. Mountain, B. Gregory, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2002).

J. Nelson, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 47–59 (2002).

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Own-Petersen, H. Riewaldt, R. Snel, D. Walker, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2002).

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

L. Montoya, N. Yaitskova, P. Dierickx, K. Dohlen, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 564–573 (2002).

Goncharov, A.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Own-Petersen, H. Riewaldt, R. Snel, D. Walker, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2002).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1986).

Gregory, B.

S. Strom, L. Stepp, M. Mountain, B. Gregory, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2002).

Havas, T. W.

J. N. Aubrun, K. R. Lorell, T. W. Havas, W. C. Henninger, “Performance analysis of the segmented alignment control system for the ten-meter telescope,” Automatica 24, 437–454 (1988).
[CrossRef]

Henninger, W. C.

J. N. Aubrun, K. R. Lorell, T. W. Havas, W. C. Henninger, “Performance analysis of the segmented alignment control system for the ten-meter telescope,” Automatica 24, 437–454 (1988).
[CrossRef]

Hubin, N.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Jochum, L.

F. J. Castro, N. Devaney, L. Jochum, B. Ronquillo, L. Cavaller, P. Dierickx, ed., Proc. SPIE4003, 24–33 (2000).

Kirkman, D.

G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck telescope: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

Koch, F.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Krabbendam, V. L.

V. L. Krabbendam, T. A. Sebring, F. B. Ray, J. R. Fowler, L. Stepp, ed., Proc. SPIE3352, 436–445 (1998).

Langlois, M.

M. Langlois, R. Angel, M. Lloyd-Hard, F. Widi, G. Love, A. Naumov, “High order, reconstructor-free adaptive optics for a 6–8 meter class telescope,” in Beyond Conventional Adaptive Optics, E. Vernet, R. Ragazzoni, S. Esposito, and N. Hubin, eds., ESO Conference Workshop Proceedings No. 58 (European Southern Observatory, Garching bei München, Germany, 2002), pp. 113–120 (2002).

Le Louarn, M.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Lloyd-Hard, M.

M. Langlois, R. Angel, M. Lloyd-Hard, F. Widi, G. Love, A. Naumov, “High order, reconstructor-free adaptive optics for a 6–8 meter class telescope,” in Beyond Conventional Adaptive Optics, E. Vernet, R. Ragazzoni, S. Esposito, and N. Hubin, eds., ESO Conference Workshop Proceedings No. 58 (European Southern Observatory, Garching bei München, Germany, 2002), pp. 113–120 (2002).

S. Esposito, E. Pinna, A. Tozzi, P. Stefenini, N. Devaney, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 214–225 (2003).

N. Yaitskova, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 62–71 (2003).

Lorell, K. R.

J. N. Aubrun, K. R. Lorell, T. W. Havas, W. C. Henninger, “Performance analysis of the segmented alignment control system for the ten-meter telescope,” Automatica 24, 437–454 (1988).
[CrossRef]

Love, G.

M. Langlois, R. Angel, M. Lloyd-Hard, F. Widi, G. Love, A. Naumov, “High order, reconstructor-free adaptive optics for a 6–8 meter class telescope,” in Beyond Conventional Adaptive Optics, E. Vernet, R. Ragazzoni, S. Esposito, and N. Hubin, eds., ESO Conference Workshop Proceedings No. 58 (European Southern Observatory, Garching bei München, Germany, 2002), pp. 113–120 (2002).

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1992), p. 74.

Marchetti, E.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Mast, T.

G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck telescope: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

J. Nelson, T. Mast, S. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Rep. 90 (W. M. Keck Library, Kamuela, Hawaii, 1985), pp. 5-1–5-44.

Michaels, S.

G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck telescope: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

Monnet, G.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Montoya, L.

L. Montoya, N. Yaitskova, P. Dierickx, K. Dohlen, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 564–573 (2002).

N. Yaitskova, L. Montoya, K. Dohlen, P. Dierickx, J. Oschmann, M. Tarenghi, eds., Proc. SPIE5489, 1139–1151 (2004).

Mountain, M.

S. Strom, L. Stepp, M. Mountain, B. Gregory, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2002).

Naumov, A.

M. Langlois, R. Angel, M. Lloyd-Hard, F. Widi, G. Love, A. Naumov, “High order, reconstructor-free adaptive optics for a 6–8 meter class telescope,” in Beyond Conventional Adaptive Optics, E. Vernet, R. Ragazzoni, S. Esposito, and N. Hubin, eds., ESO Conference Workshop Proceedings No. 58 (European Southern Observatory, Garching bei München, Germany, 2002), pp. 113–120 (2002).

Nelson, J.

G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck telescope: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

J. Nelson, T. Mast, S. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Rep. 90 (W. M. Keck Library, Kamuela, Hawaii, 1985), pp. 5-1–5-44.

J. Nelson, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 47–59 (2002).

Noethe, L.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Orlov, V. G.

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).

Oschmann, J.

N. Yaitskova, L. Montoya, K. Dohlen, P. Dierickx, J. Oschmann, M. Tarenghi, eds., Proc. SPIE5489, 1139–1151 (2004).

Own-Petersen, M.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Own-Petersen, H. Riewaldt, R. Snel, D. Walker, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2002).

Pinna, E.

S. Esposito, E. Pinna, A. Tozzi, P. Stefenini, N. Devaney, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 214–225 (2003).

Quattri, M.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Ray, F. B.

V. L. Krabbendam, T. A. Sebring, F. B. Ray, J. R. Fowler, L. Stepp, ed., Proc. SPIE3352, 436–445 (1998).

Riewaldt, H.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Own-Petersen, H. Riewaldt, R. Snel, D. Walker, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2002).

Ronquillo, B.

F. J. Castro, N. Devaney, L. Jochum, B. Ronquillo, L. Cavaller, P. Dierickx, ed., Proc. SPIE4003, 24–33 (2000).

Sanchez, L.

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).

Sarazin, M.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Schumacher, A.

A. SchumacherAchim.Schumacher@iac.es (personal communication).

Schwartz, L.

L. Schwartz, Théorie des Distributions (Hermann, Paris, 1966).

Sebring, T. A.

V. L. Krabbendam, T. A. Sebring, F. B. Ray, J. R. Fowler, L. Stepp, ed., Proc. SPIE3352, 436–445 (1998).

Snel, R.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Own-Petersen, H. Riewaldt, R. Snel, D. Walker, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2002).

Spyromillo, J.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

Stefenini, P.

S. Esposito, E. Pinna, A. Tozzi, P. Stefenini, N. Devaney, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 214–225 (2003).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Stepp, L.

V. L. Krabbendam, T. A. Sebring, F. B. Ray, J. R. Fowler, L. Stepp, ed., Proc. SPIE3352, 436–445 (1998).

S. Strom, L. Stepp, M. Mountain, B. Gregory, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2002).

Strom, S.

S. Strom, L. Stepp, M. Mountain, B. Gregory, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2002).

Tarenghi, M.

N. Yaitskova, L. Montoya, K. Dohlen, P. Dierickx, J. Oschmann, M. Tarenghi, eds., Proc. SPIE5489, 1139–1151 (2004).

Tozzi, A.

S. Esposito, E. Pinna, A. Tozzi, P. Stefenini, N. Devaney, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 214–225 (2003).

Troy, M.

G. Chanan, M. Troy, “Strehl ratio and modulation transfer function for segmented mirror telescope as function of segment phase error,” Appl. Opt. 38, 6642–6647 (1999).
[CrossRef]

G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck telescope: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

Tyson, R. K.

S. Esposito, E. Pinna, A. Tozzi, P. Stefenini, N. Devaney, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 214–225 (2003).

N. Yaitskova, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 62–71 (2003).

Voitsekhovich, V.

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).

Walker, D.

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Own-Petersen, H. Riewaldt, R. Snel, D. Walker, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2002).

Widi, F.

M. Langlois, R. Angel, M. Lloyd-Hard, F. Widi, G. Love, A. Naumov, “High order, reconstructor-free adaptive optics for a 6–8 meter class telescope,” in Beyond Conventional Adaptive Optics, E. Vernet, R. Ragazzoni, S. Esposito, and N. Hubin, eds., ESO Conference Workshop Proceedings No. 58 (European Southern Observatory, Garching bei München, Germany, 2002), pp. 113–120 (2002).

Yaitskova, N.

N. Yaitskova, K. Dohlen, P. Dierickx, “Analytical study of diffraction effects in extremely large segmented telescopes,” J. Opt. Soc. Am. A 20, 1563–1575 (2003).
[CrossRef]

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

L. Montoya, N. Yaitskova, P. Dierickx, K. Dohlen, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 564–573 (2002).

N. Yaitskova, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 62–71 (2003).

N. Yaitskova, L. Montoya, K. Dohlen, P. Dierickx, J. Oschmann, M. Tarenghi, eds., Proc. SPIE5489, 1139–1151 (2004).

Appl. Opt. (1)

G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, D. Kirkman, “Phasing the mirror segments of the Keck telescope: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
[CrossRef]

Appl. Opt. (1)

Automatica (1)

J. N. Aubrun, K. R. Lorell, T. W. Havas, W. C. Henninger, “Performance analysis of the segmented alignment control system for the ten-meter telescope,” Automatica 24, 437–454 (1988).
[CrossRef]

J. Opt. Soc. Am. A (1)

N. Yaitskova, K. Dohlen, P. Dierickx, “Analytical study of diffraction effects in extremely large segmented telescopes,” J. Opt. Soc. Am. A 20, 1563–1575 (2003).
[CrossRef]

Nature (London) (1)

R. Angel, “Ground-based imaging of extrasolar planets using adaptive optics,” Nature (London) 368, 203 (1994).
[CrossRef]

Other (19)

M. Langlois, R. Angel, M. Lloyd-Hard, F. Widi, G. Love, A. Naumov, “High order, reconstructor-free adaptive optics for a 6–8 meter class telescope,” in Beyond Conventional Adaptive Optics, E. Vernet, R. Ragazzoni, S. Esposito, and N. Hubin, eds., ESO Conference Workshop Proceedings No. 58 (European Southern Observatory, Garching bei München, Germany, 2002), pp. 113–120 (2002).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1986).

L. Schwartz, Théorie des Distributions (Hermann, Paris, 1966).

A. SchumacherAchim.Schumacher@iac.es (personal communication).

N. Yaitskova, L. Montoya, K. Dohlen, P. Dierickx, J. Oschmann, M. Tarenghi, eds., Proc. SPIE5489, 1139–1151 (2004).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

K. Dohlen, “Phase masks in astronomy: from the Mach–Zehnder interferometer to coronagraphs,” in Astronomy with High Contrast Imaging II, C. Aime and R. Soummer, eds., EAS Publ. Ser. 12 (European Astronomical Society, 2004), pp. 33–44; http://www2.iap.fr/eas.

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).

S. Esposito, E. Pinna, A. Tozzi, P. Stefenini, N. Devaney, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 214–225 (2003).

N. Yaitskova, R. K. Tyson, M. Lloyd-Hard, eds., Proc. SPIE5169, 62–71 (2003).

L. Montoya, N. Yaitskova, P. Dierickx, K. Dohlen, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 564–573 (2002).

D. Malacara, Optical Shop Testing (Wiley, New York, 1992), p. 74.

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, N. Yaitskova, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 151–170 (2002).

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Own-Petersen, H. Riewaldt, R. Snel, D. Walker, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 214–225 (2002).

J. Nelson, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 47–59 (2002).

S. Strom, L. Stepp, M. Mountain, B. Gregory, J. R. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 116–128 (2002).

J. Nelson, T. Mast, S. Faber, “The design of the Keck Observatory and telescope,” Keck Observatory Rep. 90 (W. M. Keck Library, Kamuela, Hawaii, 1985), pp. 5-1–5-44.

V. L. Krabbendam, T. A. Sebring, F. B. Ray, J. R. Fowler, L. Stepp, ed., Proc. SPIE3352, 436–445 (1998).

F. J. Castro, N. Devaney, L. Jochum, B. Ronquillo, L. Cavaller, P. Dierickx, ed., Proc. SPIE4003, 24–33 (2000).

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Figures (13)

Fig. 1
Fig. 1

Schematic representation of the Mach–Zehnder phasing sensor.

Fig. 2
Fig. 2

Cartesian coordinate system at the boundary between two segments.

Fig. 3
Fig. 3

Mach–Zehnder signal at the edge between two segments due to a λ 4 piston step. The wavelength is 0.5 μm. The signal is shown for two pinhole shapes: Gaussian (solid curve) and top-hat (dashed curve). The oscillation of the signal from the top-hat pinhole will be smoothed out by the atmospheric turbulence, making two signals look identical.

Fig. 4
Fig. 4

(a) Real and (b) imaginary parts of the function J for the signal calculation in the case of the pinhole misalignment. When the pinhole is centered, the imaginary part is zero. The real part is little sensitive to the pinhole misalignment. Curves are shown for λ = 1 μ m and a = 1 .

Fig. 5
Fig. 5

Mach–Zehnder signal in the case of pinhole misalignment and of perfect alignment with the center of the point-spread function. The segment between 0 and 1000 mm is shifted λ 8 up with respect to its two neighbors. In the case of misalignment, the signals near two borders do not have the same intensity.

Fig. 6
Fig. 6

Calibration curves for two opposite values of the pinhole misalignment in comparison with the case of the perfectly aligned pinhole. Regardless of the value of the misalignment, all calibration curves pass through the origin.

Fig. 7
Fig. 7

Typical segment aberration (left column) and corresponding Mach–Zehnder signal (right column) in four particular cases: (a) pure piston error Δ ϕ = λ 4 , (b) two x tilts in the same direction ( δ 2 x = δ 1 x = λ 8 d ) , (c) two y tilts in the opposite directions ( δ 2 y = δ 1 y = λ 8 d ) , (d) two x tilts in the opposite directions ( δ 2 x = δ 1 x = λ 8 d ) . Pinhole size a = 1.5 , and segment size d = 1.6 m . In the right column the gray scale is normalized from 1 to 1 for cases a–c and from 0.02 to 0 for case d.

Fig. 8
Fig. 8

Mach–Zehnder signal in the presence of a gap between segments. The amplitude decreases according to the size of the gap. The signal’s antisymmetry is retained. Curves are shown for the piston step π 2 , λ = 1 μ m , and a = 1 .

Fig. 9
Fig. 9

Relative reduction of the calibration amplitude caused by the intersegment gap as a function of product of gap size (in millimeters) and pinhole size (in arcseconds) for two wavelengths. The loss of intensity in the gap can be partially compensated by the decrease of the pinhole size.

Fig. 10
Fig. 10

Rolled-off segment edge characterized by two main parameters, width η and depth ε, which varies from edge to edge and from segment to segment. They can be measured before the integration of the segment into the telescope and taken into account during the signal analysis.

Fig. 11
Fig. 11

Relative reduction of the calibration amplitude caused by identical rolled-off edges as a function of the product of edge width (in millimeters) and pinhole size (in arc seconds). Curves are presented for four different values of edge depth for λ = 0.5 μ m . As in the case of the gap, the loss of the intensity due to the diffraction on the edge can be partially compensated by the decrease of the pinhole size.

Fig. 12
Fig. 12

Residual phasing error ( ϕ 1 ) due to differences in rolled-off edge profiles: (a) Each point of the curve corresponds to the depth of a left segment edge ε 1 = π + Δ ε and the right segment edge ε 2 = π Δ ε . The difference Δ ε changes along the abscissa axis, so that, in the middle of the graph, ε 1 = ε 2 = π . Three curves correspond to three different pairs of edge width: 2 and 8 mm, 5 and 5 mm, and 8 and 2 mm. (b) Similar to (a) but with the difference in edge widths along the abscissa axis: For each point, η 1 = 5 mm Δ η and η 2 = 5 mm + Δ η . Parameters of the curves are edge depths: 3 π 2 and π 2 , π and π, and π 2 and 3 π 2 . Common parameters are a = 1 and λ = 0.5 μ m .

Fig. 13
Fig. 13

Calibration amplitude as a function of pinhole size in seeing-limit conditions for three gap sizes (solid curves). Note the existence of an optimal pinhole size at approximately 1.5–2 arcsecs. The dashed curves are the calibration amplitude in the diffraction limit condition for the same parameters of the gap. All curves are normalized to the amplitude in diffraction limit condition and a zero gap.

Equations (64)

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u 1 ( w ) = 1 λ U 1 ( ξ ) exp ( i 2 π λ w ξ ) d 2 ξ ,
U 2 ( ξ ) = 1 λ u 1 ( w ) t ( w ) exp ( i 2 π λ w ξ ) d 2 w .
U 2 ( ξ ) = 1 λ 2 U 1 ( ξ ) T ( ξ ξ ) d 2 ξ ,
T ( ξ ) = t ( w ) exp ( i 2 π λ w ξ ) d 2 w .
I 1 ( ξ ) = 1 2 { U 1 ( ξ ) 2 + U 2 ( ξ ) 2 + 2 Re [ U 1 * ( ξ ) U 2 ( ξ ) exp ( i θ ) ] } ,
I 2 ( ξ ) = 1 2 { U 1 ( ξ ) 2 + U 2 ( ξ ) 2 2 Re [ U 1 * ( ξ ) U 2 ( ξ ) exp ( i θ ) ] } ,
S ( ξ ) I 1 ( ξ ) I 2 ( ξ ) = 2 Re [ U 1 * ( ξ ) U 2 ( ξ ) exp ( i θ ) ] .
U 1 ( x ) = exp [ i ϕ ( x ) ] ,
t ( w ) = exp [ ( 2 ln 2 w a ) 2 ] .
U 2 ( x ) = 1 λ u 1 ( w ) exp [ ( w 0.6 a ) 2 ] exp ( 2 π λ w x ) d w u 1 ( 0 ) 1 l π exp [ ( x l ) 2 ] = D St 1 l π exp [ ( x l ) 2 ] ,
U 1 * ( x ) U 2 ( x ) D St 1 l π exp [ i ϕ ( x ) ] ;
S ( x ) = 2 D St l π cos [ θ ϕ ( x ) ] .
S ( x ) 2 D St l π ϕ ( x )
U 1 ( x ) = U 1 ( x ) + n = 1 1 n ! d n U 1 d x n x ( x x ) n .
U 2 ( x ) = U 1 ( x ) + 1 l π n = 1 1 n ! d n U 1 d x n x exp [ ( x l ) 2 ] ( x ) n d x .
U 2 ( x ) = U 1 ( x ) + m = 1 1 2 m ( 2 × 4 × 6 × × 2 m ) l 2 m d 2 m U 1 d x 2 m .
U 2 ( x ) = U 1 ( x ) [ 1 l 2 4 ( d ϕ d x ) 2 + i l 2 4 d 2 ϕ d x 2 ] ,
S ( x ) = [ 2 l 2 2 ( d ϕ d x ) 2 ] cos θ ( l 2 2 d 2 ϕ d x 2 ) sin θ .
l d or a λ 0.6 π d .
U 1 ( ξ ) = { exp [ i ϕ 1 ( ξ ) ] , x < 0 exp [ i ϕ 2 ( ξ ) ] , x > 0 } .
U 1 ( ξ ) = U 1 ( x ) = { exp ( i ϕ 1 ) , x < 0 exp ( i ϕ 2 ) , x > 0 } .
t ( w ) = exp [ ( w 0.6 a ) 2 ] .
T ( ξ ) = π ( 0.6 a ) 2 exp [ ( 0.6 π a λ ξ ) 2 ] = b 2 λ 2 π exp [ ( b ξ ) 2 ] .
b = 0.6 π a λ .
U 2 ( x ) 2 = 0.5 { 2 [ 1 cos ( Δ ϕ ) ] [ 1 Φ 2 ( b x ) ] } ,
S ( x ) = { sin ( Δ ϕ ) sign ( x ) [ 1 Φ ( b x ) ] } sin ( θ ) { 2 [ 1 cos ( Δ ϕ ) ] [ 1 Φ ( b x ) ] } cos ( θ ) .
S ( x ) = sign ( x ) sin ( Δ ϕ ) [ 1 Φ ( b x ) ] ,
Φ ( b x ) = 2 b π 0 x exp ( b 2 x 2 ) d x .
K ( Δ ϕ ) = 0 x 0 S ( x ) d x x 0 0 S ( x ) d x = sin ( Δ ϕ ) { 2 0 x 0 [ 1 Φ ( b x ) ] d x } ,
K ( Δ ϕ ) = A sin ( Δ ϕ + ϕ 1 ) + B .
S ( x ) = Im { [ exp ( i Δ ϕ ) 1 ] J ( x , b , w 0 ) } = sin ( Δ ϕ ) Re [ J ( x , b , w 0 ) ] + [ 1 cos ( Δ ϕ ) ] Im [ J ( x , b , w 0 ) ] ,
J ( x , b , w 0 ) = sign ( x ) exp [ ( w 0 0.6 a ) 2 ] [ 1 Φ ( b x + i w 0 x 0.6 a ) ]
K ( Δ ϕ ) = A sin ( Δ ϕ + ϕ 1 ) A sin ( ϕ 1 ) ,
K ( Δ ϕ ) [ K ( Δ ϕ ) ] = 2 A sin ( ϕ 1 ) cos ( Δ ϕ ) .
tan ( ϕ 1 ) = arg [ 0 x 0 J ( x , b , w 0 ) d x ] ,
ϕ 1 2.8 w 0 x a .
A ( w 0 a ) A ( 0 ) 1 0.98 ( w 0 a ) 2 .
ϕ 1 ( x , y ) = 2 π λ [ ( 2 x + d ) δ 1 x + 2 y δ 1 y ] + ϕ 1 C ,
ϕ 2 ( x , y ) = 2 π λ [ ( 2 x d ) δ 2 x + 2 y δ 2 y ] + ϕ 2 C .
s 1 x = 2 π δ 1 x d λ , s 1 y = 2 π δ 1 y d λ ,
s 2 x = 2 π δ 2 x d λ , s 2 y = 2 π δ 2 y d λ .
S ( x , y ) = { sin [ Δ ϕ ( x , y ) ] Re J 2 ( x ) + Im J 1 ( x ) cos [ Δ ϕ ( x , y ) ] Im J 2 ( x ) , x < 0 sin [ Δ ϕ ( x , y ) ] Re J 1 ( x ) + Im J 2 ( x ) cos [ Δ ϕ ( x , y ) ] Im J 1 ( x ) , x 0 } ,
Δ ϕ ( x , y ) = 2 x d ( s 2 x s 1 x ) + 2 y d ( s 2 y s 1 y ) + ( ϕ 2 C ϕ 1 C ) ( s 1 x + s 2 x ) .
J 1 = sign ( x ) exp [ s 1 x 2 + s 1 y 2 ( d b ) 2 ] [ 1 Φ ( b x + i s 1 x d b ) ] ,
J 2 = sign ( x ) exp [ s 2 x 2 + s 2 y 2 ( d b ) 2 ] [ 1 Φ ( b x + i s 2 x d b ) ] .
S ( x , y ) = sign ( x ) sin ( 4 y d s 2 y ) exp [ ( s 2 y d b ) 2 ] [ 1 Φ ( b x ) ] .
J 1 = exp [ ( s 2 x d b ) 2 ] [ 1 Φ ( b x i s 2 x d b ) ] ,
J 2 = exp [ ( s 2 x d b ) 2 ] [ 1 Φ ( b x + i s 2 x d b ) ] .
S ( x ) = sin ( 4 x d s 2 x ) Re ( J 2 ) + [ 1 + cos ( 4 x d s 2 x ) ] Im ( J 2 ) .
S ( x ) = { sign ( x ) sin ( Δ ϕ ) [ 1 Φ ( b ( x + g 2 ) ) ] , x g 2 0 , x < g 2 } .
K ( Δ ϕ , g ) = A ( g , a , λ ) sin ( Δ ϕ ) .
A ( g , a , λ ) = 1 a 2 a g 1.23 λ [ 1 Φ ( 0.6 π 2 λ x ) ] d x ,
K ( Δ ϕ , η , ε , a , λ ) = A ( a , η , ε , λ ) sin ( Δ ϕ ) .
K ( Δ ϕ ) = A sin ( Δ ϕ ϕ 1 ) .
η 1 = η Δ η , η 2 = η + Δ η ,
ε 1 = ε Δ ε , ε 2 = ε + Δ ε ,
U 1 ( ξ ) B ( ξ ) U 1 ( ξ ) , B ( ξ ) = exp [ i ϕ ( ξ ) ] .
U 2 ( ξ ) = 1 λ 2 U 1 ( ξ ) B ( ξ ) T ( ξ ξ ) d 2 ξ .
U 1 * ( ξ ) U 2 ( ξ ) 1 λ 2 U 1 * ( ξ ) U 1 ( ξ + ξ ) B * ( ξ ) B ( ξ + ξ ) T ( ξ ) d 2 ξ .
B * ( ξ ) B ( ξ + ξ ) = exp [ 3.44 ( ξ r 0 ) 5 3 ] ,
S ( x ) = 2 b 2 π Im { U 1 * ( x ) U 1 ( x + x ) d x exp [ 3.44 ( x 2 + y 2 r 0 2 ) 5 6 ] exp [ b 2 ( x 2 + y 2 ) ] d y } .
S ( x ) sign ( x ) sin ( Δ ϕ ) p seeing 1 [ 1 Φ ( p seeing b x ) ] .
p seeing = 1 + 1.24 ( θ seeing a ) 2 .
A ( θ seeing a ) A diffraction limit [ 1 + 1.24 ( θ seeing a ) 2 ] 3 2 .

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