Abstract

A fundamental discovery in segmented-mirror active optics is described: symmetrizing the geometry of the sensor–actuator array provides a computationally effective symmetrization of the mathematical description for the figure control of a Keck-type telescope mirror. The author establishes a universal mathematical control model and provides an efficient algorithm to solve this model equation set. This model can be applied to multifarious Keck-type mirror configurations with a similar sensor–actuator geometry design, no matter what kind of outline shapes and how many segments they have. With the underlying symmetry, a further extension of this algorithm is possible without increasing the number of parameters to be estimated for the recently proposed extremely large telescopes, such as the 30-m California Extremely Large Telescope (CELT) and the 100-m OverWhelmingly Large (OWL) telescope. Moreover, careful choice of boundary conditions in conjunction with the proper choice of minimization algorithm yields results that exceed the performance of the current existing techniques given by Nelson and Mast [Appl. Opt. 21, 2631 (1982)]. This method allows noise performance analysis. Several computer simulation models for application of this algorithm are given for the Keck 10-m Telescope and the Large Sky Area Multi-Object Fiber Spectroscopic Telescope’s (LAMOST’s) MA (Reflecting Schmidt plate) and MB (spherical primary mirror).

© 2001 Optical Society of America

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References

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  1. T. S. Mast, J. E. Nelson, “Figure control for a fully segmented telescope mirror,” Appl. Opt. 21, 2631–2641 (1982).
    [CrossRef] [PubMed]
  2. R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
    [CrossRef]
  3. G. H. Golub, C. Reinsch, “Singular value decomposition and least squares solutions,” Numer. Math. 14, 403–420 (1970).
    [CrossRef]
  4. R. Gilmozzi, B. Delabre, P. Diericky, N. Hubin, F. Koch, G. Monnet, M. Quatteri, F. Rigaut, R. N. Wilson, “The future of filled aperture telescopes: is a 100 m feasible?” in Advanced Technology Optical/IR Telescope VI, L. M. Stepp, ed., Proc. SPIE3352, 778–791 (1998).
    [CrossRef]
  5. G. Chanan, J. Nelson, C. Ohara, E. Sirko, “Design issues for the active control system of the California Extremely Large Telescope (CELT),” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.
  6. S. Wang, D. Su, Y. Chu, X. Cui, Y. Wang, “Special configuration of a very large Schmidt telescope for extensive astronomical spectroscopic observation,” Appl. Opt. 35, 5155–5161 (1996).
    [CrossRef] [PubMed]
  7. W. Zou, “The figure control of large segmented mirror telescope,” Master’s thesis (Chinese Academy of Sciences, Nanjing, China, 1996).
  8. W. Zou, Z. Zhang, “Generalized wave-front reconstruction algorithm applied in a Shack–Hartmann test,” Appl. Opt. 39, 250–268 (2000).
    [CrossRef]
  9. J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Chinese ed. (Springer-Verlag, New York, 1994).
  10. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flanery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1998), pp. 278–283.
  11. D. Su, Y. Wang, “Automatic correction of aberration in astro-optical system,” Acta Astron. Sinica 15, 51–60 (1974).
  12. D. Su, “A merit function composed by spot-diagrams and the damp least square method,” Opt. Instrum. Tech. No. 4, 1–12 (1980).
  13. D. Su, W. Zou, Z. Zhang, Y. Qu, S. Yang, L. Wang, Y. Rao, “Experiment system of segmented-mirror active optics,” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

2000 (1)

1996 (1)

1982 (1)

1980 (1)

D. Su, “A merit function composed by spot-diagrams and the damp least square method,” Opt. Instrum. Tech. No. 4, 1–12 (1980).

1974 (1)

D. Su, Y. Wang, “Automatic correction of aberration in astro-optical system,” Acta Astron. Sinica 15, 51–60 (1974).

1970 (1)

G. H. Golub, C. Reinsch, “Singular value decomposition and least squares solutions,” Numer. Math. 14, 403–420 (1970).
[CrossRef]

Andrede, S.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Arthur, A. A.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Biocca, A.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Bulirsch, R.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Chinese ed. (Springer-Verlag, New York, 1994).

Chanan, G.

G. Chanan, J. Nelson, C. Ohara, E. Sirko, “Design issues for the active control system of the California Extremely Large Telescope (CELT),” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

Chu, Y.

Cohen, R. W.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Cui, X.

Delabre, B.

R. Gilmozzi, B. Delabre, P. Diericky, N. Hubin, F. Koch, G. Monnet, M. Quatteri, F. Rigaut, R. N. Wilson, “The future of filled aperture telescopes: is a 100 m feasible?” in Advanced Technology Optical/IR Telescope VI, L. M. Stepp, ed., Proc. SPIE3352, 778–791 (1998).
[CrossRef]

Diericky, P.

R. Gilmozzi, B. Delabre, P. Diericky, N. Hubin, F. Koch, G. Monnet, M. Quatteri, F. Rigaut, R. N. Wilson, “The future of filled aperture telescopes: is a 100 m feasible?” in Advanced Technology Optical/IR Telescope VI, L. M. Stepp, ed., Proc. SPIE3352, 778–791 (1998).
[CrossRef]

Flanery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flanery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1998), pp. 278–283.

Franck, J.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Fuertes, J. M.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Gabor, G.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Gilmozzi, R.

R. Gilmozzi, B. Delabre, P. Diericky, N. Hubin, F. Koch, G. Monnet, M. Quatteri, F. Rigaut, R. N. Wilson, “The future of filled aperture telescopes: is a 100 m feasible?” in Advanced Technology Optical/IR Telescope VI, L. M. Stepp, ed., Proc. SPIE3352, 778–791 (1998).
[CrossRef]

Golub, G. H.

G. H. Golub, C. Reinsch, “Singular value decomposition and least squares solutions,” Numer. Math. 14, 403–420 (1970).
[CrossRef]

Hubin, N.

R. Gilmozzi, B. Delabre, P. Diericky, N. Hubin, F. Koch, G. Monnet, M. Quatteri, F. Rigaut, R. N. Wilson, “The future of filled aperture telescopes: is a 100 m feasible?” in Advanced Technology Optical/IR Telescope VI, L. M. Stepp, ed., Proc. SPIE3352, 778–791 (1998).
[CrossRef]

Jared, R. C.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Koch, F.

R. Gilmozzi, B. Delabre, P. Diericky, N. Hubin, F. Koch, G. Monnet, M. Quatteri, F. Rigaut, R. N. Wilson, “The future of filled aperture telescopes: is a 100 m feasible?” in Advanced Technology Optical/IR Telescope VI, L. M. Stepp, ed., Proc. SPIE3352, 778–791 (1998).
[CrossRef]

Llacer, J.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Mast, T.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Mast, T. S.

Meng, J.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Merrick, T.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Minor, R.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Monnet, G.

R. Gilmozzi, B. Delabre, P. Diericky, N. Hubin, F. Koch, G. Monnet, M. Quatteri, F. Rigaut, R. N. Wilson, “The future of filled aperture telescopes: is a 100 m feasible?” in Advanced Technology Optical/IR Telescope VI, L. M. Stepp, ed., Proc. SPIE3352, 778–791 (1998).
[CrossRef]

Nelson, J.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

G. Chanan, J. Nelson, C. Ohara, E. Sirko, “Design issues for the active control system of the California Extremely Large Telescope (CELT),” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

Nelson, J. E.

Ohara, C.

G. Chanan, J. Nelson, C. Ohara, E. Sirko, “Design issues for the active control system of the California Extremely Large Telescope (CELT),” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

Orayani, M.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flanery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1998), pp. 278–283.

Qu, Y.

D. Su, W. Zou, Z. Zhang, Y. Qu, S. Yang, L. Wang, Y. Rao, “Experiment system of segmented-mirror active optics,” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

Quatteri, M.

R. Gilmozzi, B. Delabre, P. Diericky, N. Hubin, F. Koch, G. Monnet, M. Quatteri, F. Rigaut, R. N. Wilson, “The future of filled aperture telescopes: is a 100 m feasible?” in Advanced Technology Optical/IR Telescope VI, L. M. Stepp, ed., Proc. SPIE3352, 778–791 (1998).
[CrossRef]

Rao, Y.

D. Su, W. Zou, Z. Zhang, Y. Qu, S. Yang, L. Wang, Y. Rao, “Experiment system of segmented-mirror active optics,” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

Reinsch, C.

G. H. Golub, C. Reinsch, “Singular value decomposition and least squares solutions,” Numer. Math. 14, 403–420 (1970).
[CrossRef]

Rigaut, F.

R. Gilmozzi, B. Delabre, P. Diericky, N. Hubin, F. Koch, G. Monnet, M. Quatteri, F. Rigaut, R. N. Wilson, “The future of filled aperture telescopes: is a 100 m feasible?” in Advanced Technology Optical/IR Telescope VI, L. M. Stepp, ed., Proc. SPIE3352, 778–791 (1998).
[CrossRef]

Salz, P.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Schaefer, B.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Sirko, E.

G. Chanan, J. Nelson, C. Ohara, E. Sirko, “Design issues for the active control system of the California Extremely Large Telescope (CELT),” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

Stoer, J.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Chinese ed. (Springer-Verlag, New York, 1994).

Su, D.

S. Wang, D. Su, Y. Chu, X. Cui, Y. Wang, “Special configuration of a very large Schmidt telescope for extensive astronomical spectroscopic observation,” Appl. Opt. 35, 5155–5161 (1996).
[CrossRef] [PubMed]

D. Su, “A merit function composed by spot-diagrams and the damp least square method,” Opt. Instrum. Tech. No. 4, 1–12 (1980).

D. Su, Y. Wang, “Automatic correction of aberration in astro-optical system,” Acta Astron. Sinica 15, 51–60 (1974).

D. Su, W. Zou, Z. Zhang, Y. Qu, S. Yang, L. Wang, Y. Rao, “Experiment system of segmented-mirror active optics,” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flanery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1998), pp. 278–283.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flanery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1998), pp. 278–283.

Wang, L.

D. Su, W. Zou, Z. Zhang, Y. Qu, S. Yang, L. Wang, Y. Rao, “Experiment system of segmented-mirror active optics,” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

Wang, S.

Wang, Y.

Wilson, R. N.

R. Gilmozzi, B. Delabre, P. Diericky, N. Hubin, F. Koch, G. Monnet, M. Quatteri, F. Rigaut, R. N. Wilson, “The future of filled aperture telescopes: is a 100 m feasible?” in Advanced Technology Optical/IR Telescope VI, L. M. Stepp, ed., Proc. SPIE3352, 778–791 (1998).
[CrossRef]

Witebsky, C.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

Yang, S.

D. Su, W. Zou, Z. Zhang, Y. Qu, S. Yang, L. Wang, Y. Rao, “Experiment system of segmented-mirror active optics,” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

Zhang, Z.

W. Zou, Z. Zhang, “Generalized wave-front reconstruction algorithm applied in a Shack–Hartmann test,” Appl. Opt. 39, 250–268 (2000).
[CrossRef]

D. Su, W. Zou, Z. Zhang, Y. Qu, S. Yang, L. Wang, Y. Rao, “Experiment system of segmented-mirror active optics,” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

Zou, W.

W. Zou, Z. Zhang, “Generalized wave-front reconstruction algorithm applied in a Shack–Hartmann test,” Appl. Opt. 39, 250–268 (2000).
[CrossRef]

W. Zou, “The figure control of large segmented mirror telescope,” Master’s thesis (Chinese Academy of Sciences, Nanjing, China, 1996).

D. Su, W. Zou, Z. Zhang, Y. Qu, S. Yang, L. Wang, Y. Rao, “Experiment system of segmented-mirror active optics,” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

Acta Astron. Sinica (1)

D. Su, Y. Wang, “Automatic correction of aberration in astro-optical system,” Acta Astron. Sinica 15, 51–60 (1974).

Appl. Opt. (3)

Numer. Math. (1)

G. H. Golub, C. Reinsch, “Singular value decomposition and least squares solutions,” Numer. Math. 14, 403–420 (1970).
[CrossRef]

Opt. Instrum. Tech. No. 4 (1)

D. Su, “A merit function composed by spot-diagrams and the damp least square method,” Opt. Instrum. Tech. No. 4, 1–12 (1980).

Other (7)

D. Su, W. Zou, Z. Zhang, Y. Qu, S. Yang, L. Wang, Y. Rao, “Experiment system of segmented-mirror active optics,” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

W. Zou, “The figure control of large segmented mirror telescope,” Master’s thesis (Chinese Academy of Sciences, Nanjing, China, 1996).

R. Gilmozzi, B. Delabre, P. Diericky, N. Hubin, F. Koch, G. Monnet, M. Quatteri, F. Rigaut, R. N. Wilson, “The future of filled aperture telescopes: is a 100 m feasible?” in Advanced Technology Optical/IR Telescope VI, L. M. Stepp, ed., Proc. SPIE3352, 778–791 (1998).
[CrossRef]

G. Chanan, J. Nelson, C. Ohara, E. Sirko, “Design issues for the active control system of the California Extremely Large Telescope (CELT),” presented at the Astronomical Telescopes and Instrumentation 2000 meeting, Munich, Germany, March 27–31, 2000.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Chinese ed. (Springer-Verlag, New York, 1994).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flanery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1998), pp. 278–283.

R. C. Jared, A. A. Arthur, S. Andrede, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. SPIE1236, 996–1008 (1990).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Configuration of the primary mirror of the Keck 10-m telescope showing sensor and actuator locations.

Fig. 2
Fig. 2

Possible sensor–actuator geometrical relationships in two adjacent segments.

Fig. 3
Fig. 3

Universal schematic of a segmented mirror configuration for figure control. The central hexagonal region is Ω0, and the outer parallelogram region is Ω1. The actuator position distribution scheme is a proportional spacing array, which is different from that of the Keck 10-m telescope.

Fig. 4
Fig. 4

Typical image spot diagram of the Keck 10-m primary mirror as given by Mast and Nelson,1 which was generated by 50-nm rms displacement sensor noise.

Fig. 5
Fig. 5

(a) Superposition of 100 spot diagrams of the Keck primary mirror with four actuators locked: p70=p71=p72=p76=0. The image’s rms radius is 0.0794 arc sec. (b) Superposition of 100 spot diagrams of the Keck primary with all the virtual actuators constrained to zero. The image rms radius is 0.0277 arc sec.

Fig. 6
Fig. 6

(a) Image spot diagram of the Keck primary mirror generated by the MN LS method with fiducial actuators p70=p71=p72=0.0. (b) Corresponding three-dimensional surface plot. (c) Superposition of 100 image spot diagrams. The image rms radius is 0.0499 arc sec, the 80% energy radius is 0.061 arcsec, and the rms surface error is 170.62 nm. (d) Same as (c) but here the three fiducial actuators are p19=p65=p138=0. The image rms radius is 0.04 arc sec and the rms surface error is 119.28 nm.

Fig. 7
Fig. 7

Superposition of 100 image spot diagrams of the Keck primary mirror generated by the DLS method with three fiducial actuators p70=p71=p72=0.0. The damping factor is 10-6. The image rms radius is 0.0482 arc sec.

Fig. 8
Fig. 8

Configuration of MA and its extended domain Ω1 for the application of the generalized figure-control algorithm in the LAMOST.

Fig. 9
Fig. 9

(a) Superposition of 100 image spot diagrams of the LAMOST mirror MA generated by the LS method with four actuators locked: p70=p71=p72=p76=0.0. The rms surface error is 268.87 nm, the image rms radius is 0.107 arc sec, and the 80% energy radius is 0.126 arc sec. (b) Superposition of 100 image spot diagrams of the LAMOST mirror MA generated by the MN LS method with three fiducial actuators, p70=p71=p72=0.0. The image rms radius is 0.074 arc sec and the 80% energy radius is 0.091 arc sec.

Fig. 10
Fig. 10

(a) Superposition of 200 image spot diagrams of the LAMOST mirror MB generated by the MN LS method with three fiducial actuators, p70=p71=p72=0.0. The image rms radius is 0.051 arc sec and the rms surface error is 183.14 nm. (b) Same as (a) but with the three fiducial actuators replaced by p19=p65=p138=0. The image rms radius is 0.0401 arc sec, and the rms surface error is 117.95 nm.  

Tables (1)

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Table 1 Possible Geometrical Sensor–Actuator Position Permutations and the Corresponding Linear Coefficients in Eq. (2.9)

Equations (74)

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S=12x2-x1y2-y1x3-x1y3-y1.
x-x1y-y1,z-z1x2-x1,y2-y1,z2-z1x3-x1,y3-y1,z3-z1=0.
z=ξ-x1,η-y1,z1ξ-x2,η-y2,z2ξ-x3,η-y3,z3x2-x1,y2-y1x3-x1,y3-y1,
z=1S0 (S1z1+S2z2+S3z3),
S1+S2+S3=S0.
a-23 cda+23 c.
zs=L1z1+L2z2+L3z3,
zs=L1z1+L3z2+L2z3,
L1=13-ac,L2=13+a-d2c,
L3=13+a+d2c.
zs=R1z1+R2z2+R3z3,
zs=R1z1+R3z+R2z3,
R1=13+ac,R2=13-a+d2c,
R3=13-a-d2c.
s=C1z1+C2z2+C3z3+C4z4+C5z5+C6z6,
EEE P1P2P7=S1S2S7
A1P=SI,
E=R3R2R1-L3-L2-L1R3R2R1-L3-L2-L1R3R2R1-L3-L2-L16×21.
B1C1B1C1B1C1 P1P2P7=S8S9  S13
A2P=SII,
B1= L1L2L3L1L2L3L1L2L37×21,
C=-R1-R2-R3-R1-R2-R3-R1-R2-R37×21.
B2C2B2C2B2C2 P1P2P7=S14S15  S19
A3P=SIII,
B2= 000R2R1R3R2R1R3R2R1R36×21,
C2= -L2-L1-L3-L2-L1-L3-L2-L1-L30006×21.
A1A2A3P=SISIISIII,
A0P=S0.
A0P=S0.
AP=S,
A=A0A0240×147,S=S0S0240×1.
ATAP=ATS,
ATA=A0TA0+A0TA0147×147,
ATS=A0TS0+A0TS0147×1.
R(1)=(R3R2R1),L(1)=(-L3-L2-L1),
 
R(2)=(-R1-R2-R3),L(2)=(L1L2L3),
 
R(3)=(R2R1R3),L(3)=(-L2-L1-L3),
 
A0TA0=D1FFTD2FFTD2FFTD3147×147,
D1=d1eeTd2eeTd2eeTd321×21,
D2=d4eeTd5eeTd5eeTd621×21,
D3=d7eeTd8eeTd8eeTd921×21,
F=f1f2f1f2f121×21,
d1=L(2)TL(2)+R(1)TR(1),
d2=L(1)TL(1)+L(2)TL(2)+R(1)TR(1)+R(3)TR(3),
d3=L(1)TL(1)+L(2)TL(2)+R(3)TR(3),
d4=L(2)TL(2)+L(3)TL(3)+R(1)TR(1)+R(2)TR(2),
d5=L(1)TL(1)+L(2)TL(2)+L(3)TL(3)+R(1)TR(1)+R(2)TR(2)+R(3)TR(3),
d6=L(1)TL(1)+L(2)TL(2)+R(2)TR(2)+R(3)TR(3),
d7=L(3)TL(3)+R(1)TR(1)+R(2)TR(2),
d8=L(1)TL(1)+L(3)TL(3)+R(1)TR(1)+R(2)TR(2),
d9=L(1)TL(1)+R(2)TR(2),
e=R(1)TL(1),
f1=L(2)TR(2),
f2=R(3)TL(3).
ATA=A0TA0+A0TA0.
A0TS0=A1TSI+A2TSII+A3TSIII,
A0TS0=ETS1+B1TS8+B2TS14ETS2+B1TS9+C1TS8+B2TS15+C2TS14ETS3+B1TS10+C1TS9+B2TS16+C2TS15ETS4+B1TS11+C1TS10+B2TS17+C2TS16ETS5+B1TS12+C1TS11+B2TS18+C2TS17ETS6+B1TS13+C1TS12+B2TS19+C2TS18ETS7+C1TS13+C2TS19147×1.
ETS2+B1TS9+C1TS8+B2TS15+C2TS14=R(1)Ts7+R(2)Ts43+L(2)Ts50+L(3)Ts85L(1)Ts7+R(1)Ts8+R(2)Ts44+L(2)Ts51+L(3)Ts86+R(3)Ts91L(1)Ts8+R(1)Ts9+R(2)Ts45+L(2)Ts52+L(3)Ts87+R(3)Ts92L(1)Ts9+R(1)Ts10+R(2)Ts46+L(2)Ts53+L(3)Ts88+R(3)Ts93L(1)Ts10+R(1)Ts11+R(2)Ts47+L(2)Ts54+L(3)Ts89+R(3)Ts94L(1)Ts11+R(1)Ts12+R(2)Ts48+L(2)Ts55+L(3)Ts90+R(3)Ts95L1(T)s12+R(2)Ts49+L(2)Ts56+R(3)Ts9621×1.
ATS=A0TS+A0TS0.
(ATA+δ)ΔP=ATΔS+δ(Pphase-P0),
Ψ=i=1npi2=i=1n(aip1+bi)2.
p1=-aibi/i=1nai2.
X2=XTX=i=1m|xi|21/2
lub2(A)=maxx0XTATAXXTX=ρ(ATA)=lub2(ATA),
(A+ΔA)T(A+ΔA)(P+ΔP)=(A+ΔA)T(S+ΔS).
 
ΔP2[cond(ATA)]1/2lub2(A) ΔS2,
cond(ATA)=lub2(ATA)lub2(ATA)-1.
ΔS2=nsσs,
ΔP2=npσp,
σpnsnp[cond(ATA)]1/2lub2(A) σs.

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