Abstract

The problem of the reconstruction of random phase distortions from Hartmann test data is discussed. A statistical approach allowing calculation of reconstruction errors associated with the discrete character of the data is developed. It is shown that the correct choice of both the experimental scheme and the method of data reduction can significantly improve the results. Some practical recommendations presented allow one to choose the main parameters of the experimental scheme in such a way that the errors of the retrieved phase distortions are minimal. The results obtained can be applied for both atmospheric and adaptive optics.

© 1996 Optical Society of America

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  1. T. I. Balakhovskaya, V. I. Borisenko, E. A. Vitrichenko, K. L. Maslenikov, A. M. Prokhorov, P. Z. Sagdeyev, E. V. Trushin, L. S. Chesalin, “Fast Hartmann test for problems in astronomical adaptive optics,” Dokl. Akad. Nauk SSSR 274, 257–260 (1984).
  2. E. F. Borra, M. Beauchemin, R. Arsenault, R. Lalande, “Optical-shop testing of liquid mirrors,” Publ. Astron. Soc. Pac. 97, 454–464 (1985).
    [CrossRef]
  3. G. A. Chanan, J. E. Nelson, T. S. Mast, “Segment alignment for the Keck Telescope primary mirror,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 466–470 (1986).
    [CrossRef]
  4. V. V. Voitsekhovich, V. B. Gubin, A. V. Mikulich, “An estimation of the parameters of adaptive astronomical systems on the base of experimental dates,” Opt. Atmosferi, 1, 66–70 (1988) (in Russian).
  5. U. Hopp, U. Graser, “Optical quality of the Calar Alto 3.5 m telescope,” Astron. Astrophys. 235, 543–548 (1990).
  6. A. C. Porter, “Hartmann testing with a CCD,” in Proceedings of the Conference on CCD’s in Astronomy (Astronomical Society of the Pacific, San Francisco, 1990), pp. 393–398.
  7. B. Hulburd, D. Sandler, “Segmented mirrors for atmospheric compensation,” Opt. Eng. 29, 1186–1190 (1990).
    [CrossRef]
  8. M. Iye, T. Noguchi, Y. Torii, Y. Mikami, H. Ando, “Evaluation of seeing on a 62-cm mirror,” Publ. Astron. Soc. Pac. 103, 712–722 (1991).
    [CrossRef]
  9. P. Y. Mades, J. M. Conan, G. Rousset, “Temporal characterization of atmospheric wavefront for adaptive optics,” in Proceedings in the ESO-42 Conference on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 471–474.
  10. W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.
  11. H. Li, H. Xian, W. Jiang, “Atmospheric turbulence parameter measurement using Hartmann–Shack wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 21–25.
  12. D. J. Robertson, B. L. Ellebroek, R. J. McGonegal, C. M. Mountain, J. Oschmann, S. M. Pompea, “Adaptive optics modeling for the Gemini 8-m telescopes project,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 121–126.
  13. A. Wirth, T. L. Bruno, A. J. Jankevics, “Applying Hartmann wavefront sensing technology to precision optical testing of the Hubble Space Telescope corrector,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 325–330.
  14. D. Colucci, M. Lloyd-Hart, D. Wittman, R. Angel, A. Ghez, B. McLeod, “A reflective Shack–Hartmann wave-front sensor for adaptive optics,” Publ. Astron. Soc. 106, 1104–1110 (1994).
    [CrossRef]
  15. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,”J. Opt. Soc. Am. 71, 989–992 (1981).
    [CrossRef]
  16. R. J. Noll, “Zernike polynomials and atmospheric turbulence,”J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  17. V. I. Tatarski, “The effects of the turbulent atmosphere on wave propagation,” National Science Foundation Rep. TT-68-50464 (National Science Foundation, Washington, D.C., 1968), Chap. 4, pp. 385–392.
  18. J. Herrmann, “Least-squares wave-front errors of minimum norm,”J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  19. N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction error of Shack–Hartmann wavefront sensors,” Publ. Astron. Soc. Pac. 106, 182–188 (1994).
    [CrossRef]
  20. J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
    [CrossRef]
  21. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for long and very short exposures,”J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  22. V. V. Voitsekhovich, “Phase-retrieval problem and orthogonal expansions: curvature sensing,” J. Opt. Soc. Am. A 12, 2194–2202 (1995).
    [CrossRef]
  23. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1988), Vol. 2, pp. 229 and 745.

1995 (1)

1994 (2)

D. Colucci, M. Lloyd-Hart, D. Wittman, R. Angel, A. Ghez, B. McLeod, “A reflective Shack–Hartmann wave-front sensor for adaptive optics,” Publ. Astron. Soc. 106, 1104–1110 (1994).
[CrossRef]

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction error of Shack–Hartmann wavefront sensors,” Publ. Astron. Soc. Pac. 106, 182–188 (1994).
[CrossRef]

1991 (1)

M. Iye, T. Noguchi, Y. Torii, Y. Mikami, H. Ando, “Evaluation of seeing on a 62-cm mirror,” Publ. Astron. Soc. Pac. 103, 712–722 (1991).
[CrossRef]

1990 (3)

U. Hopp, U. Graser, “Optical quality of the Calar Alto 3.5 m telescope,” Astron. Astrophys. 235, 543–548 (1990).

B. Hulburd, D. Sandler, “Segmented mirrors for atmospheric compensation,” Opt. Eng. 29, 1186–1190 (1990).
[CrossRef]

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
[CrossRef]

1988 (1)

V. V. Voitsekhovich, V. B. Gubin, A. V. Mikulich, “An estimation of the parameters of adaptive astronomical systems on the base of experimental dates,” Opt. Atmosferi, 1, 66–70 (1988) (in Russian).

1985 (1)

E. F. Borra, M. Beauchemin, R. Arsenault, R. Lalande, “Optical-shop testing of liquid mirrors,” Publ. Astron. Soc. Pac. 97, 454–464 (1985).
[CrossRef]

1984 (1)

T. I. Balakhovskaya, V. I. Borisenko, E. A. Vitrichenko, K. L. Maslenikov, A. M. Prokhorov, P. Z. Sagdeyev, E. V. Trushin, L. S. Chesalin, “Fast Hartmann test for problems in astronomical adaptive optics,” Dokl. Akad. Nauk SSSR 274, 257–260 (1984).

1981 (1)

1980 (1)

1976 (1)

1966 (1)

Ando, H.

M. Iye, T. Noguchi, Y. Torii, Y. Mikami, H. Ando, “Evaluation of seeing on a 62-cm mirror,” Publ. Astron. Soc. Pac. 103, 712–722 (1991).
[CrossRef]

Angel, R.

D. Colucci, M. Lloyd-Hart, D. Wittman, R. Angel, A. Ghez, B. McLeod, “A reflective Shack–Hartmann wave-front sensor for adaptive optics,” Publ. Astron. Soc. 106, 1104–1110 (1994).
[CrossRef]

Arsenault, R.

E. F. Borra, M. Beauchemin, R. Arsenault, R. Lalande, “Optical-shop testing of liquid mirrors,” Publ. Astron. Soc. Pac. 97, 454–464 (1985).
[CrossRef]

Balakhovskaya, T. I.

T. I. Balakhovskaya, V. I. Borisenko, E. A. Vitrichenko, K. L. Maslenikov, A. M. Prokhorov, P. Z. Sagdeyev, E. V. Trushin, L. S. Chesalin, “Fast Hartmann test for problems in astronomical adaptive optics,” Dokl. Akad. Nauk SSSR 274, 257–260 (1984).

Beauchemin, M.

E. F. Borra, M. Beauchemin, R. Arsenault, R. Lalande, “Optical-shop testing of liquid mirrors,” Publ. Astron. Soc. Pac. 97, 454–464 (1985).
[CrossRef]

Borisenko, V. I.

T. I. Balakhovskaya, V. I. Borisenko, E. A. Vitrichenko, K. L. Maslenikov, A. M. Prokhorov, P. Z. Sagdeyev, E. V. Trushin, L. S. Chesalin, “Fast Hartmann test for problems in astronomical adaptive optics,” Dokl. Akad. Nauk SSSR 274, 257–260 (1984).

Borra, E. F.

E. F. Borra, M. Beauchemin, R. Arsenault, R. Lalande, “Optical-shop testing of liquid mirrors,” Publ. Astron. Soc. Pac. 97, 454–464 (1985).
[CrossRef]

Bruno, T. L.

A. Wirth, T. L. Bruno, A. J. Jankevics, “Applying Hartmann wavefront sensing technology to precision optical testing of the Hubble Space Telescope corrector,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 325–330.

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1988), Vol. 2, pp. 229 and 745.

Chanan, G. A.

G. A. Chanan, J. E. Nelson, T. S. Mast, “Segment alignment for the Keck Telescope primary mirror,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 466–470 (1986).
[CrossRef]

Chesalin, L. S.

T. I. Balakhovskaya, V. I. Borisenko, E. A. Vitrichenko, K. L. Maslenikov, A. M. Prokhorov, P. Z. Sagdeyev, E. V. Trushin, L. S. Chesalin, “Fast Hartmann test for problems in astronomical adaptive optics,” Dokl. Akad. Nauk SSSR 274, 257–260 (1984).

Colucci, D.

D. Colucci, M. Lloyd-Hart, D. Wittman, R. Angel, A. Ghez, B. McLeod, “A reflective Shack–Hartmann wave-front sensor for adaptive optics,” Publ. Astron. Soc. 106, 1104–1110 (1994).
[CrossRef]

Conan, J. M.

P. Y. Mades, J. M. Conan, G. Rousset, “Temporal characterization of atmospheric wavefront for adaptive optics,” in Proceedings in the ESO-42 Conference on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 471–474.

Ellebroek, B. L.

D. J. Robertson, B. L. Ellebroek, R. J. McGonegal, C. M. Mountain, J. Oschmann, S. M. Pompea, “Adaptive optics modeling for the Gemini 8-m telescopes project,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 121–126.

Fontanella, J. C.

Fried, D. L.

Ghez, A.

D. Colucci, M. Lloyd-Hart, D. Wittman, R. Angel, A. Ghez, B. McLeod, “A reflective Shack–Hartmann wave-front sensor for adaptive optics,” Publ. Astron. Soc. 106, 1104–1110 (1994).
[CrossRef]

Graser, U.

U. Hopp, U. Graser, “Optical quality of the Calar Alto 3.5 m telescope,” Astron. Astrophys. 235, 543–548 (1990).

Guan, C.

W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.

Gubin, V. B.

V. V. Voitsekhovich, V. B. Gubin, A. V. Mikulich, “An estimation of the parameters of adaptive astronomical systems on the base of experimental dates,” Opt. Atmosferi, 1, 66–70 (1988) (in Russian).

Herrmann, J.

Hopp, U.

U. Hopp, U. Graser, “Optical quality of the Calar Alto 3.5 m telescope,” Astron. Astrophys. 235, 543–548 (1990).

Huang, S.

W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.

Hulburd, B.

B. Hulburd, D. Sandler, “Segmented mirrors for atmospheric compensation,” Opt. Eng. 29, 1186–1190 (1990).
[CrossRef]

Iye, M.

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction error of Shack–Hartmann wavefront sensors,” Publ. Astron. Soc. Pac. 106, 182–188 (1994).
[CrossRef]

M. Iye, T. Noguchi, Y. Torii, Y. Mikami, H. Ando, “Evaluation of seeing on a 62-cm mirror,” Publ. Astron. Soc. Pac. 103, 712–722 (1991).
[CrossRef]

Jankevics, A. J.

A. Wirth, T. L. Bruno, A. J. Jankevics, “Applying Hartmann wavefront sensing technology to precision optical testing of the Hubble Space Telescope corrector,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 325–330.

Jiang, W.

H. Li, H. Xian, W. Jiang, “Atmospheric turbulence parameter measurement using Hartmann–Shack wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 21–25.

W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.

Lalande, R.

E. F. Borra, M. Beauchemin, R. Arsenault, R. Lalande, “Optical-shop testing of liquid mirrors,” Publ. Astron. Soc. Pac. 97, 454–464 (1985).
[CrossRef]

Li, H.

W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.

H. Li, H. Xian, W. Jiang, “Atmospheric turbulence parameter measurement using Hartmann–Shack wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 21–25.

Li, M.

W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.

Ling, N.

W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.

Liu, C.

W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.

Lloyd-Hart, M.

D. Colucci, M. Lloyd-Hart, D. Wittman, R. Angel, A. Ghez, B. McLeod, “A reflective Shack–Hartmann wave-front sensor for adaptive optics,” Publ. Astron. Soc. 106, 1104–1110 (1994).
[CrossRef]

Mades, P. Y.

P. Y. Mades, J. M. Conan, G. Rousset, “Temporal characterization of atmospheric wavefront for adaptive optics,” in Proceedings in the ESO-42 Conference on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 471–474.

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1988), Vol. 2, pp. 229 and 745.

Maslenikov, K. L.

T. I. Balakhovskaya, V. I. Borisenko, E. A. Vitrichenko, K. L. Maslenikov, A. M. Prokhorov, P. Z. Sagdeyev, E. V. Trushin, L. S. Chesalin, “Fast Hartmann test for problems in astronomical adaptive optics,” Dokl. Akad. Nauk SSSR 274, 257–260 (1984).

Mast, T. S.

G. A. Chanan, J. E. Nelson, T. S. Mast, “Segment alignment for the Keck Telescope primary mirror,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 466–470 (1986).
[CrossRef]

McGonegal, R. J.

D. J. Robertson, B. L. Ellebroek, R. J. McGonegal, C. M. Mountain, J. Oschmann, S. M. Pompea, “Adaptive optics modeling for the Gemini 8-m telescopes project,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 121–126.

McLeod, B.

D. Colucci, M. Lloyd-Hart, D. Wittman, R. Angel, A. Ghez, B. McLeod, “A reflective Shack–Hartmann wave-front sensor for adaptive optics,” Publ. Astron. Soc. 106, 1104–1110 (1994).
[CrossRef]

Mikami, Y.

M. Iye, T. Noguchi, Y. Torii, Y. Mikami, H. Ando, “Evaluation of seeing on a 62-cm mirror,” Publ. Astron. Soc. Pac. 103, 712–722 (1991).
[CrossRef]

Mikulich, A. V.

V. V. Voitsekhovich, V. B. Gubin, A. V. Mikulich, “An estimation of the parameters of adaptive astronomical systems on the base of experimental dates,” Opt. Atmosferi, 1, 66–70 (1988) (in Russian).

Mountain, C. M.

D. J. Robertson, B. L. Ellebroek, R. J. McGonegal, C. M. Mountain, J. Oschmann, S. M. Pompea, “Adaptive optics modeling for the Gemini 8-m telescopes project,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 121–126.

Nelson, J. E.

G. A. Chanan, J. E. Nelson, T. S. Mast, “Segment alignment for the Keck Telescope primary mirror,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 466–470 (1986).
[CrossRef]

Noguchi, T.

M. Iye, T. Noguchi, Y. Torii, Y. Mikami, H. Ando, “Evaluation of seeing on a 62-cm mirror,” Publ. Astron. Soc. Pac. 103, 712–722 (1991).
[CrossRef]

Noll, R. J.

Oschmann, J.

D. J. Robertson, B. L. Ellebroek, R. J. McGonegal, C. M. Mountain, J. Oschmann, S. M. Pompea, “Adaptive optics modeling for the Gemini 8-m telescopes project,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 121–126.

Pompea, S. M.

D. J. Robertson, B. L. Ellebroek, R. J. McGonegal, C. M. Mountain, J. Oschmann, S. M. Pompea, “Adaptive optics modeling for the Gemini 8-m telescopes project,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 121–126.

Porter, A. C.

A. C. Porter, “Hartmann testing with a CCD,” in Proceedings of the Conference on CCD’s in Astronomy (Astronomical Society of the Pacific, San Francisco, 1990), pp. 393–398.

Primot, J.

Prokhorov, A. M.

T. I. Balakhovskaya, V. I. Borisenko, E. A. Vitrichenko, K. L. Maslenikov, A. M. Prokhorov, P. Z. Sagdeyev, E. V. Trushin, L. S. Chesalin, “Fast Hartmann test for problems in astronomical adaptive optics,” Dokl. Akad. Nauk SSSR 274, 257–260 (1984).

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1988), Vol. 2, pp. 229 and 745.

Robertson, D. J.

D. J. Robertson, B. L. Ellebroek, R. J. McGonegal, C. M. Mountain, J. Oschmann, S. M. Pompea, “Adaptive optics modeling for the Gemini 8-m telescopes project,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 121–126.

Rong, Z.

W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.

Rousset, G.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
[CrossRef]

P. Y. Mades, J. M. Conan, G. Rousset, “Temporal characterization of atmospheric wavefront for adaptive optics,” in Proceedings in the ESO-42 Conference on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 471–474.

Sagdeyev, P. Z.

T. I. Balakhovskaya, V. I. Borisenko, E. A. Vitrichenko, K. L. Maslenikov, A. M. Prokhorov, P. Z. Sagdeyev, E. V. Trushin, L. S. Chesalin, “Fast Hartmann test for problems in astronomical adaptive optics,” Dokl. Akad. Nauk SSSR 274, 257–260 (1984).

Sandler, D.

B. Hulburd, D. Sandler, “Segmented mirrors for atmospheric compensation,” Opt. Eng. 29, 1186–1190 (1990).
[CrossRef]

Takato, N.

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction error of Shack–Hartmann wavefront sensors,” Publ. Astron. Soc. Pac. 106, 182–188 (1994).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, “The effects of the turbulent atmosphere on wave propagation,” National Science Foundation Rep. TT-68-50464 (National Science Foundation, Washington, D.C., 1968), Chap. 4, pp. 385–392.

Torii, Y.

M. Iye, T. Noguchi, Y. Torii, Y. Mikami, H. Ando, “Evaluation of seeing on a 62-cm mirror,” Publ. Astron. Soc. Pac. 103, 712–722 (1991).
[CrossRef]

Trushin, E. V.

T. I. Balakhovskaya, V. I. Borisenko, E. A. Vitrichenko, K. L. Maslenikov, A. M. Prokhorov, P. Z. Sagdeyev, E. V. Trushin, L. S. Chesalin, “Fast Hartmann test for problems in astronomical adaptive optics,” Dokl. Akad. Nauk SSSR 274, 257–260 (1984).

Vitrichenko, E. A.

T. I. Balakhovskaya, V. I. Borisenko, E. A. Vitrichenko, K. L. Maslenikov, A. M. Prokhorov, P. Z. Sagdeyev, E. V. Trushin, L. S. Chesalin, “Fast Hartmann test for problems in astronomical adaptive optics,” Dokl. Akad. Nauk SSSR 274, 257–260 (1984).

Voitsekhovich, V. V.

V. V. Voitsekhovich, “Phase-retrieval problem and orthogonal expansions: curvature sensing,” J. Opt. Soc. Am. A 12, 2194–2202 (1995).
[CrossRef]

V. V. Voitsekhovich, V. B. Gubin, A. V. Mikulich, “An estimation of the parameters of adaptive astronomical systems on the base of experimental dates,” Opt. Atmosferi, 1, 66–70 (1988) (in Russian).

Wang, C.

W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.

Wirth, A.

A. Wirth, T. L. Bruno, A. J. Jankevics, “Applying Hartmann wavefront sensing technology to precision optical testing of the Hubble Space Telescope corrector,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 325–330.

Wittman, D.

D. Colucci, M. Lloyd-Hart, D. Wittman, R. Angel, A. Ghez, B. McLeod, “A reflective Shack–Hartmann wave-front sensor for adaptive optics,” Publ. Astron. Soc. 106, 1104–1110 (1994).
[CrossRef]

Wu, X.

W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.

Xian, H.

W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.

H. Li, H. Xian, W. Jiang, “Atmospheric turbulence parameter measurement using Hartmann–Shack wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 21–25.

Yamaguchi, I.

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Astron. Astrophys. (1)

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Opt. Eng. (1)

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W. Jiang, H. Li, C. Liu, X. Wu, S. Huang, H. Xian, Z. Rong, C. Wang, M. Li, N. Ling, C. Guan, “A 37 element adaptive optics system with H-S wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 127–135.

H. Li, H. Xian, W. Jiang, “Atmospheric turbulence parameter measurement using Hartmann–Shack wavefront sensor,” in Proceedings of the ICO-16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (International Commission for Optics, 16 Secretariat, Garching, Germany, 1993), pp. 21–25.

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Figures (10)

Fig. 1
Fig. 1

Notation used for the calculations.

Fig. 2
Fig. 2

Errors of coefficients for the square [Fig. 3(a)] and for the polar [Fig. 3(b)] arrangement of subpupils.

Fig. 3
Fig. 3

Arrangement of subpupils: (a) the square arrangement of 32 subpupils, r/R = 0.147; (b) the polar arrangement of 31 subpupils, r/R = 0.137, 0.163.

Fig. 4
Fig. 4

Dependence of the errors of coefficients on the normalized subpupil radius for the square arrangement [Fig. 3(a)]. Digits with arrows point to the numbers of coefficients.

Fig. 5
Fig. 5

Errors of coefficients for the square [Fig. 6(a)] and for the hexagonal [Fig. 6(b)] arrangement of subpupils.

Fig. 6
Fig. 6

Arrangement of subpupils: (a) the square arrangement of 48 subpupils, r/R = 0.119, 0.098; (b) the hexagonal arrangement of 48 subpupils, r/R = 0.121. The central circle shows the zone of obscuration. The obscuration ratio is 0.25.

Fig. 7
Fig. 7

Two-mask scheme.

Fig. 8
Fig. 8

Arrangement of subpupils for the two-mask scheme: (a) the first mask, 18 subpupils, r/R = 0.196; (b) the second mask, 12 subpupils, r/R = 0.196. The total number of subpupils is 30. The central circle shows the zone of obscuration. The obscuration ratio is 0.25.

Fig. 9
Fig. 9

Errors of coefficients for the hexagonal arrangement [Fig. 6(b)] and for the two-mask scheme (Fig. 8).

Fig. 10
Fig. 10

Errors of coefficients for the two-mask scheme (Fig. 8) in the case of conventional and averaged Zernike polynomials.

Equations (24)

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S ( ρ ) = i = 2 a i Z i ( ρ / R ) .
Z i even ( ρ ) = 2 ( n + 1 ) R n m ( ρ ) cos ( m ϕ ) , m 0 , Z i odd ( ρ ) = 2 ( n + 1 ) R n m ( ρ ) cos ( m ϕ ) , m 0 , Z i ( ρ ) = ( n + 1 ) R n 0 ( ρ ) , m = 0 ,
a i = 1 π R 2 G R d 2 ρ Z i ( ρ / R ) S ( ρ ) .
S e ( ρ ) = i = 2 N a i e Z i ( ρ / R ) ,
ρ s 1 π r s G s d 2 ρ S ( ρ ) , s = 1,2 , , M ,
s = 1 M [ ( i = 2 N a i e Z i x f x ) 2 + ( i = 2 N a i e Z i y f y ) 2 ] = min ,
f x ( ρ s ) = 1 π r s G s d 2 ρ S ( ρ ) / x , f y ( ρ s ) = 1 π r s G s d 2 ρ S ( ρ ) / y .
i = 2 N a i e b i j = c j , j = 2,3 , , N ,
b i j = s = 1 M ( Z i x Z j x + Z i y Z j y ) , i , j = 2,3 , , N , c j = s = 1 M ( Z j x f x + Z j y f y ) , j = 2,3 , , N .
σ i = ( a i a i e ) 2 a i 2 = a i 2 2 a i a i e + a i e a i e a i 2 ,
a i 2 = 1 π 2 R 4 G R d 2 ρ 1 d 2 ρ 2 Z i ( ρ 1 / R ) Z i ( ρ 2 / R ) B S ( ρ 1 , ρ 2 ) .
i = 2 N a k a i e b i j = a k c j , j , k = 2,3 , , N .
i = 2 N a i e a i e b i l 2 + 2 i = 2 N 1 j = i + 1 N a i e a j e b i l b j l = c l 2 , l = 2 , 3 , , N , i = 2 N a i e a i e b i k b i l + i = 2 N 1 j = i + 1 N a i e a j e ( b i k b j l + b i l b j k ) = c k c l , k = 2,3 , , N 1 , l = k + 1 , k + 2 , , N .
a k c j = 1 π R 2 s = 1 M [ Z j s x G R d 2 ρ Z k ( ρ R ) S ( ρ ) f x s + Z j s y G R d 2 ρ Z k ( ρ R ) S ( ρ ) f y s , c k c l = s = 1 M t = 1 M { z k s x z l t d x f x s f x t + [ Z k s x Z l y t + Z k s y Z l t x ] f z s f y t + Z k s y Z l t y f y s f y t } .
S ( ρ ) f x s = 1 π r s 2 G s d 2 r 1 x 1 B S ( r 1 , ρ ) , S ( ρ ) f y s = 1 π r s 2 G s d 2 r 1 y 1 B S ( r 1 , ρ ) , f x s f x t = 1 π 2 r s 2 r t 2 G s d 2 r 1 G t d 2 r 2 2 x 1 x 2 B S ( r 1 , r 2 ) , f y s f y t = 1 π 2 r s 2 r t 2 G s d 2 r 1 G t d 2 r 2 2 y 1 y 2 B S ( r 1 , r 2 ) , f x s f y t = 1 π 2 r s 2 r t 2 G s d 2 r 1 G t d 2 r 2 2 x 1 y 2 B S ( r 1 , r 2 ) ,
D S ( ρ 1 , ρ 2 ) = 6.88 ( | ρ 1 ρ 2 | / r 0 ) 5 / 3 ,
B S ( ρ 1 , ρ 2 ) = 0.49 r 0 d 2 κ κ 11 / 3 exp [ i κ ( ρ 1 ρ 2 ) ] .
a i 2 = 0.487 ( n + 1 ) ( R r 0 ) 5 / 3 Γ ( 14 / 3 ) Γ ( n 5 / 6 ) Γ ( n + 23 / 6 ) [ Γ ( 17 / 6 ) ] 2 ,
for a k c j , a k c j = 0.487 ( 1 ) ( n m ) / 2 ( 2 δ 0 ) ( n + 1 ) × ( R r 0 ) 5 / 3 s = 1 M { Ψ m + 1 ( r s R , p s R ) f m + 1 ( p s R ) + Ψ m 1 ( r s R , p s R ) × f m 1 ( p s R ) , m 0 , Ψ 1 ( r s R , p s R ) f 1 ( p s R ) , m = 0 ,
f m + 1 ( p s R ) = { Z j x ( p s R ) cos ( m + 1 ) ϕ + Z j y ( p s R ) × sin ( m + 1 ) ϕ , k is even , Z j x ( p s R ) sin ( m + 1 ) ϕ Z j y ( p s R ) , × cos ( m + 1 ) ϕ , k is odd , f m 1 ( p s R ) = { Z j x ( p s R ) cos ( m 1 ) ϕ + Z j y ( p s R ) × sin ( m 1 ) ϕ , k is even , Z j x ( p s R ) sin ( m 1 ) ϕ Z j y ( p s R ) , × cos ( m 1 ) ϕ , k is odd , f 1 ( p s R ) = 2 Z j ρ ( p s R ) , Ψ μ ( r s R , p s R ) = ( p s R ) μ Γ ( n + μ 2 1 3 ) Γ ( μ + 1 ) Γ ( n μ 2 + 4 3 ) × F 4 ( μ n 2 4 3 , n + μ 2 1 3 ; 2 , μ + 1 ; r s 2 / R 2 , p s 2 / R 2 ) ;
and for c k c l , c k c l | s t = 0.487 ( R r 0 ) 5 / 3 s = 1 M t = 1 M [ Z k x ( p s R ) Z l x ( p t R ) + Z k y ( p s R ) Z l y ( p t R ) ] Ω 0 ( r s p s t , r t p s t ) + [ Z k y ( p s R ) Z l y ( p t R ) Z k x ( p s R ) Z l x ( p t R ) ] Ω 2 ( r s p s t , r t p s t ) × cos 2 δ [ Z k x ( p s R ) Z l y ( p l R ) + Z k y ( p s R ) Z l x ( p t R ) ] × Ω 2 ( r s p s t , r t p s t ) sin 2 δ , Ω μ ( r s p s t , r t p s t ) = ( R p s t ) 1 / 3 Γ ( μ / 2 + 1 / 6 ) ) Γ ( μ / 2 + 5 / 6 ) × F 4 ( μ / 2 + 1 / 6 , μ / 2 + 1 / 6 ; 2,2 : r s 2 / p s t 2 , r t 2 / p s t 2 , ) 2 δ = arccos { [ p s 2 cos 2 ϕ s + p t 2 cos 2 ϕ t 2 p s p t cos ( ϕ s + ϕ t ) ] / p s t 2 } ,
c k c l | s t = 0.487 ( R r 0 ) 5 / 3 Γ ( 8 / 3 ) Γ ( 1 / 6 ) Γ ( 17 / 6 ) [ Γ ( 11 / 6 ) ] 2 × s = 1 M ( r s R ) 1 / 3 [ Z k x ( p s R ) Z l x ( p s R ) + Z k y ( p s R ) Z l y ( p s R ) ] .
ρ c 2 r 1 / 3 , l 0 r L 0 ,
Z i x ( ρ s ) 1 π r s 2 G s d 2 ρ Z i x ( ρ ) , Z i y ( ρ s ) 1 π r s 2 G s d 2 ρ Z i y ( ρ ) , s = 1 , 2 , , M .

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