Abstract

We present an analysis of point-spread functions for segmented mirrors affected by random tip–tilt errors on each segment. In addition to Strehl ratio evaluation, this analysis considers key characteristics such as the intensity and the location of speckles and secondary peaks and the relative energy distribution between these features. We develop a method to describe the shape of a nonaveraged point-spread function and deduce the final expressions for ensemble-averaged characteristics. Based on Keck-type hexagonal segmentation geometry, our study is extended to an arbitrary number of segments, and we describe qualitatively the transition from the case of a mirror with few segments to that of a mirror with several hundred segments—prototype of the next generation of Extremely Large Telescopes.

© 2002 Optical Society of America

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References

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  1. A. Baranne, G. Lemaitre, “Combinaisons optiques pour les très grands télescopes: le concept TEMOS,” C. R. Acad. Sci. Ser. II 305, 445–450 (1987).
  2. D. Burgarella, K. Dohlen, M. Ferrari, F. Zamkotsian, “Large petal telescope, NG-CFHT study report,” 2001. Available at http://www.cfht.hawaii.edu/News/Projects/NGCFHT/Concepts.html .
  3. J. R. Kuhn, G. Moretto, R. Coulter, P. Baudoz, J. E. Graves, D. Jewitt, R. Joseph, G. Kudritzki, G. Luppino, R. McLaren, D. Mickey, M. Northcott, F. Roddier, C. Roddier, C. Shelton, A. Stockton, A. Tokunaga, J. Tonry, B. Tully, R. Wainscoat, “A high dynamic range replacement concept for the Canada–France–Hawaii Telescope,” 2001. Available at http://www.cfht.hawaii.edu/News/Projects/NGCFHT/Concepts.html .
  4. J. Nelson, T. Mast, S. Faber, “The design of the Keck Observatory and telescope,” (W. M. Keck Library, Kamuela, Hawaii, 1985), pp. 5-1–5-44.
  5. V. L. Krabbendam, T. A. Sebring, F. B. Ray, J. R. Flower, “Development and performance of Hobby–Eberly Telescope 11 meter segmented mirror,” in Advanced Technology Optical/IR Telescopes VI, L. Stepp, ed., Proc. SPIE3352, 436–445 (1998).
    [CrossRef]
  6. B. Stobie, K. Mairing, D. Buckley, “Design of the Southern African Large Telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 355–362 (2000).
    [CrossRef]
  7. J. Nelson, T. Mast, “Giant optical devices,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 1–11.
  8. T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, H. Riewaldt, “The proposed 50 m Swedish Extremely LargeTelescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 72–82.
  9. P. Dierickx, R. Gilmozzi, “OWL concept overview,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings NO. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 43–52.
  10. 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]
  11. G. Chanan, M. Troy, E. Sirco, “Phase discontinuity sensing: a method for phasing segmented mirrors in infrared,” Appl. Opt. 38, 704–713 (1999).
    [CrossRef]
  12. K. Dohlen, F. Decortiat, F. Fresneau, P. Lanzoni, “A dual-wavelength, random phase-shift interferometer for phasing large segmented primary,” in Advanced Technology Optical/IR Telescopes VI, L. Stepp, ed., Proc. SPIE3352, 551–559 (1998).
    [CrossRef]
  13. S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, “Curvature equation for a segmented telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).
    [CrossRef]
  14. S. Esposito, N. Devaney, “Segmented mirror co-phasing using pyramid sensor,” presented at the Workshop Beyond Conventional Adaptive Optics, Venice, Italy, May 7–10, 2001.
  15. F. Marchis, S. Cuevas, “Optical tolerance of active telescope architectures for adaptive optics,” Rev. Mex. Astron. Astrofis. 35, 31–44 (1999).
  16. N. Mehta, C. Allen, “Segmented mirror alignment with far-field optimization,” Appl. Opt. 32, 2664–2673 (1993).
    [CrossRef] [PubMed]
  17. 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]
  18. G. Zeider, E. Montgomery, “Diffraction effect with segmented aperture,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 799–809 (1998).
    [CrossRef]
  19. G. Zeider, “Image-based alignment of large segmented telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 241–249 (2000).
    [CrossRef]
  20. N. Yaitskova, K. Dohlen, “Simulation of imaging performance for extremely large segmented telescopes,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 279–290 (2000).
    [CrossRef]
  21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1986).
  22. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  23. M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968).

1999 (3)

1998 (1)

1993 (1)

1987 (1)

A. Baranne, G. Lemaitre, “Combinaisons optiques pour les très grands télescopes: le concept TEMOS,” C. R. Acad. Sci. Ser. II 305, 445–450 (1987).

Allen, C.

Andersen, T.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, H. Riewaldt, “The proposed 50 m Swedish Extremely LargeTelescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 72–82.

Ardeberg, A.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, H. Riewaldt, “The proposed 50 m Swedish Extremely LargeTelescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 72–82.

Baranne, A.

A. Baranne, G. Lemaitre, “Combinaisons optiques pour les très grands télescopes: le concept TEMOS,” C. R. Acad. Sci. Ser. II 305, 445–450 (1987).

Beckers, J.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, H. Riewaldt, “The proposed 50 m Swedish Extremely LargeTelescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 72–82.

Buckley, D.

B. Stobie, K. Mairing, D. Buckley, “Design of the Southern African Large Telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 355–362 (2000).
[CrossRef]

Chanan, G.

Cuevas, S.

F. Marchis, S. Cuevas, “Optical tolerance of active telescope architectures for adaptive optics,” Rev. Mex. Astron. Astrofis. 35, 31–44 (1999).

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, “Curvature equation for a segmented telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).
[CrossRef]

Decortiat, F.

K. Dohlen, F. Decortiat, F. Fresneau, P. Lanzoni, “A dual-wavelength, random phase-shift interferometer for phasing large segmented primary,” in Advanced Technology Optical/IR Telescopes VI, L. Stepp, ed., Proc. SPIE3352, 551–559 (1998).
[CrossRef]

Dekens, F.

Devaney, N.

S. Esposito, N. Devaney, “Segmented mirror co-phasing using pyramid sensor,” presented at the Workshop Beyond Conventional Adaptive Optics, Venice, Italy, May 7–10, 2001.

Dierickx, P.

P. Dierickx, R. Gilmozzi, “OWL concept overview,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings NO. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 43–52.

Dohlen, K.

K. Dohlen, F. Decortiat, F. Fresneau, P. Lanzoni, “A dual-wavelength, random phase-shift interferometer for phasing large segmented primary,” in Advanced Technology Optical/IR Telescopes VI, L. Stepp, ed., Proc. SPIE3352, 551–559 (1998).
[CrossRef]

N. Yaitskova, K. Dohlen, “Simulation of imaging performance for extremely large segmented telescopes,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 279–290 (2000).
[CrossRef]

Esposito, S.

S. Esposito, N. Devaney, “Segmented mirror co-phasing using pyramid sensor,” presented at the Workshop Beyond Conventional Adaptive Optics, Venice, Italy, May 7–10, 2001.

Faber, S.

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

Flicker, R.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, H. Riewaldt, “The proposed 50 m Swedish Extremely LargeTelescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 72–82.

Flower, J. R.

V. L. Krabbendam, T. A. Sebring, F. B. Ray, J. R. Flower, “Development and performance of Hobby–Eberly Telescope 11 meter segmented mirror,” in Advanced Technology Optical/IR Telescopes VI, L. Stepp, ed., Proc. SPIE3352, 436–445 (1998).
[CrossRef]

Fresneau, F.

K. Dohlen, F. Decortiat, F. Fresneau, P. Lanzoni, “A dual-wavelength, random phase-shift interferometer for phasing large segmented primary,” in Advanced Technology Optical/IR Telescopes VI, L. Stepp, ed., Proc. SPIE3352, 551–559 (1998).
[CrossRef]

Garfias, F.

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, “Curvature equation for a segmented telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).
[CrossRef]

Gilmozzi, R.

P. Dierickx, R. Gilmozzi, “OWL concept overview,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings NO. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 43–52.

Gontcharov, A.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, H. Riewaldt, “The proposed 50 m Swedish Extremely LargeTelescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 72–82.

Goodman, J. W.

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

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Jessen, N. C.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, H. Riewaldt, “The proposed 50 m Swedish Extremely LargeTelescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 72–82.

Kirkman, D.

Krabbendam, V. L.

V. L. Krabbendam, T. A. Sebring, F. B. Ray, J. R. Flower, “Development and performance of Hobby–Eberly Telescope 11 meter segmented mirror,” in Advanced Technology Optical/IR Telescopes VI, L. Stepp, ed., Proc. SPIE3352, 436–445 (1998).
[CrossRef]

Lanzoni, P.

K. Dohlen, F. Decortiat, F. Fresneau, P. Lanzoni, “A dual-wavelength, random phase-shift interferometer for phasing large segmented primary,” in Advanced Technology Optical/IR Telescopes VI, L. Stepp, ed., Proc. SPIE3352, 551–559 (1998).
[CrossRef]

Lemaitre, G.

A. Baranne, G. Lemaitre, “Combinaisons optiques pour les très grands télescopes: le concept TEMOS,” C. R. Acad. Sci. Ser. II 305, 445–450 (1987).

Mairing, K.

B. Stobie, K. Mairing, D. Buckley, “Design of the Southern African Large Telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 355–362 (2000).
[CrossRef]

Mannery, E.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, H. Riewaldt, “The proposed 50 m Swedish Extremely LargeTelescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 72–82.

Marchis, F.

F. Marchis, S. Cuevas, “Optical tolerance of active telescope architectures for adaptive optics,” Rev. Mex. Astron. Astrofis. 35, 31–44 (1999).

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, “Giant optical devices,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 1–11.

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

Mehta, N.

Michaels, S.

Montgomery, E.

G. Zeider, E. Montgomery, “Diffraction effect with segmented aperture,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 799–809 (1998).
[CrossRef]

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, “Giant optical devices,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 1–11.

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

Orlov, V. G.

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, “Curvature equation for a segmented telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).
[CrossRef]

Owner-Petersen, M.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, H. Riewaldt, “The proposed 50 m Swedish Extremely LargeTelescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 72–82.

Ray, F. B.

V. L. Krabbendam, T. A. Sebring, F. B. Ray, J. R. Flower, “Development and performance of Hobby–Eberly Telescope 11 meter segmented mirror,” in Advanced Technology Optical/IR Telescopes VI, L. Stepp, ed., Proc. SPIE3352, 436–445 (1998).
[CrossRef]

Riewaldt, H.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, H. Riewaldt, “The proposed 50 m Swedish Extremely LargeTelescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 72–82.

Sanchez, L.

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, “Curvature equation for a segmented telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).
[CrossRef]

Sebring, T. A.

V. L. Krabbendam, T. A. Sebring, F. B. Ray, J. R. Flower, “Development and performance of Hobby–Eberly Telescope 11 meter segmented mirror,” in Advanced Technology Optical/IR Telescopes VI, L. Stepp, ed., Proc. SPIE3352, 436–445 (1998).
[CrossRef]

Sirco, E.

Spiegel, M. R.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968).

Stobie, B.

B. Stobie, K. Mairing, D. Buckley, “Design of the Southern African Large Telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 355–362 (2000).
[CrossRef]

Troy, M.

Voitsekhovich, V.

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, “Curvature equation for a segmented telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).
[CrossRef]

Yaitskova, N.

N. Yaitskova, K. Dohlen, “Simulation of imaging performance for extremely large segmented telescopes,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 279–290 (2000).
[CrossRef]

Zeider, G.

G. Zeider, “Image-based alignment of large segmented telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 241–249 (2000).
[CrossRef]

G. Zeider, E. Montgomery, “Diffraction effect with segmented aperture,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 799–809 (1998).
[CrossRef]

Appl. Opt. (4)

C. R. Acad. Sci. Ser. II (1)

A. Baranne, G. Lemaitre, “Combinaisons optiques pour les très grands télescopes: le concept TEMOS,” C. R. Acad. Sci. Ser. II 305, 445–450 (1987).

Rev. Mex. Astron. Astrofis. (1)

F. Marchis, S. Cuevas, “Optical tolerance of active telescope architectures for adaptive optics,” Rev. Mex. Astron. Astrofis. 35, 31–44 (1999).

Other (17)

K. Dohlen, F. Decortiat, F. Fresneau, P. Lanzoni, “A dual-wavelength, random phase-shift interferometer for phasing large segmented primary,” in Advanced Technology Optical/IR Telescopes VI, L. Stepp, ed., Proc. SPIE3352, 551–559 (1998).
[CrossRef]

S. Cuevas, V. G. Orlov, F. Garfias, V. Voitsekhovich, L. Sanchez, “Curvature equation for a segmented telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 291–302 (2000).
[CrossRef]

S. Esposito, N. Devaney, “Segmented mirror co-phasing using pyramid sensor,” presented at the Workshop Beyond Conventional Adaptive Optics, Venice, Italy, May 7–10, 2001.

D. Burgarella, K. Dohlen, M. Ferrari, F. Zamkotsian, “Large petal telescope, NG-CFHT study report,” 2001. Available at http://www.cfht.hawaii.edu/News/Projects/NGCFHT/Concepts.html .

J. R. Kuhn, G. Moretto, R. Coulter, P. Baudoz, J. E. Graves, D. Jewitt, R. Joseph, G. Kudritzki, G. Luppino, R. McLaren, D. Mickey, M. Northcott, F. Roddier, C. Roddier, C. Shelton, A. Stockton, A. Tokunaga, J. Tonry, B. Tully, R. Wainscoat, “A high dynamic range replacement concept for the Canada–France–Hawaii Telescope,” 2001. Available at http://www.cfht.hawaii.edu/News/Projects/NGCFHT/Concepts.html .

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

V. L. Krabbendam, T. A. Sebring, F. B. Ray, J. R. Flower, “Development and performance of Hobby–Eberly Telescope 11 meter segmented mirror,” in Advanced Technology Optical/IR Telescopes VI, L. Stepp, ed., Proc. SPIE3352, 436–445 (1998).
[CrossRef]

B. Stobie, K. Mairing, D. Buckley, “Design of the Southern African Large Telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 355–362 (2000).
[CrossRef]

J. Nelson, T. Mast, “Giant optical devices,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 1–11.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, H. Riewaldt, “The proposed 50 m Swedish Extremely LargeTelescope,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings No. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 72–82.

P. Dierickx, R. Gilmozzi, “OWL concept overview,” in Proceedings of the Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., ESO Conference and Workshop Proceedings NO. 57 (European Southern Observatories, Garching b. München, Germany, 2000), pp. 43–52.

G. Zeider, E. Montgomery, “Diffraction effect with segmented aperture,” in Space Telescopes and Instruments V, P. Bely, J. Breckinridge, eds., Proc. SPIE3356, 799–809 (1998).
[CrossRef]

G. Zeider, “Image-based alignment of large segmented telescope,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 241–249 (2000).
[CrossRef]

N. Yaitskova, K. Dohlen, “Simulation of imaging performance for extremely large segmented telescopes,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 279–290 (2000).
[CrossRef]

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

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968).

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Figures (15)

Fig. 1
Fig. 1

Segmentation geometry for segmentation order M=2. The total number of segments is N=19.

Fig. 2
Fig. 2

Fourier transform of the segmentation grid (solid curves) and point-spread function (PSF) of an individual segment (dashed curve). Multiplication of these two functions gives the PSF for a perfectly phased telescope normalized by the factor (AN/λz)2. Spatial frequency is normalized by the factor 2λz/3d.

Fig. 3
Fig. 3

Segment with tip–tilt error. XY is the plane of the ideal mirror surface, and bx and by are tip and tilt angles, respectively. Different projections are given in (a) and (b).

Fig. 4
Fig. 4

(a) Illustration of the wave-packet representation of the segmentation PSF. Each segment produces a packet of different harmonics, with frequency 2πw  rj/λz and a shift from center equal to Δwj. When added up, they form the PSF of the mirror. (b) One-dimensional PSF for N=10 and σa=3.5 rad. Intensity and spatial frequency are normalized by the factors (AN/λz)2 and 2λz/3d, respectively.  

Fig. 5
Fig. 5

One-dimensional PSF for N=120. (a) σa=3.5 rad and (b) σa=4.5 rad. Normalization of the scales is the same as that in Fig. 4.

Fig. 6
Fig. 6

Fourier transform of the segmentation grid (curve 0) and the modified segment PSF (curves 1, 2, and 3) for different values of tip–tilt standard deviation: σa=0, 2, and 4 rad, respectively, for N=217. Normalization of the scales is the same as that in Fig. 4.

Fig. 7
Fig. 7

Averaged speckled field. Numbers on the curves correspond to the value of standard deviation in radians: σa=0, 1, 2, 3, 4, and 5 rad. Normalization of the scales is the same as that in Fig. 4.

Fig. 8
Fig. 8

Intensity of secondary peaks (curves labeled 1) and maximum intensity of speckled field (curves labeled 2) for different segment numbers plotted versus standard deviation of tip–tilt error in radians. Normalization of intensity is the same as that in Fig. 4.

Fig. 9
Fig. 9

Example of simulated mirror affected by random tip–tilt error. M=6, and N=127.

Fig. 10
Fig. 10

Simulated PSFs for a weakly segmented system (N=7) affected by tip–tilt errors: (a) rms=4 rad, (b) rms=2 rad, (c) rms=1 rad, (d) rms=0 rad. Intensity is represented in a logarithmic gray scale.

Fig. 11
Fig. 11

PSFs for a highly segmented system: (a) N=61, (b) N=127, (c) N=217. The total size of the primary mirror is the same in each case. While the size of the central spot, inversely proportional to the total mirror diameter, is constant, the distance between secondary maxima, inversely proportional to the segment separation, is increasing. Speckle intensity is decreasing, as predicted. Wave-front rms error is in each case equal to 1.4 rad. The gray scale is logarithmic and normalized to the maximum intensity in each case.

Fig. 12
Fig. 12

Strehl ratio as a function of rms for a segmented mirror affected by tip–tilt errors. Twenty realizations of the mirror were generated with a given value of σa. The points represent mean values for rms and Strehl ratio, and the error bars show the fluctuations about the mean. Squares, N=37; upward triangles, N=217; downward triangles, N=817. The solid curves are the theoretical curves for the same number of segments, and the dotted curve is the Marechal approximation.

Fig. 13
Fig. 13

Relative fluctuation of Strehl ratio as a function of average rms for different segment numbers.

Fig. 14
Fig. 14

Fluctuation of rms versus segmentation order. Standard deviation of tip–tilt error is σa=1 rad, i.e., rms0.37 rad.

Fig. 15
Fig. 15

Average intensity of secondary peaks as a function of rms for tip–tilt affected segmented mirror. Normalization of intensity is the same as that in Fig. 4. Twenty realizations of the mirror were generated with a given value of σa. The points represent mean values for rms and Ism, and the error bars show the fluctuations about the mean. Squares, N=37; upward triangles, N=217, downward triangles, N=817. The curves are the theoretical curves for the same number of segments.

Tables (1)

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Table 1 Segmentation Order M and Corresponding Segment Number N for a Hexagonal Segmented Mirror of Diameter D with Segment Flat-to-Flat Width 0.9 m

Equations (52)

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F(x)=j=1Nfj(x-rj)=j=1Nθ(x-rj)exp[iϕj(x-rj)],
θ(ξ)=1insidethesegmentaperture0outsidethesegmentaperture.
U(w)=1λzF(x)expi 2πλzw  xd2x=1λzj=1Nfj(x-rj)expi 2πλzw  xd2x,
U(w)=1λzj=1Nexpi 2πλzw  rjfj(x-rj)×expi 2πλzw  (x-rj)d2x=ANλz1Nj=1Nexpi 2πλzw  rj1Afj(ξ)×expi 2πλzw  ξd2ξ,
A=32 d2,
PSF(w)=|U(w)|2=ANλz21Nj=1Nexpi 2πλzw  rj×1Afj(ξ)expi 2πλzw  ξd2ξ2.
ϕj(ξ)=0,j=1, 2 ,, N.
PSF(w)=ANλz21Nj=1Nexpi 2πλzw  rj2×1Aθ(ξ)expi 2πλzw  ξd2ξ2=ANλz2GF(w)PSFs(w).
I0=PSF(0)=ANλz2.
GF(w)=sin[(3M+1)β+(M+1)3α]×sin[M(β-3α)]N sin(2β)sin(β-3α)+sin[(3M+2)β-M3α]×sin[(M+1)(β+3α)]N sin(2β)sin(β+3α)2,
2β=mπ,β±3α=nπ
wy=mλz/d,wx±3wy=n2λz/d,
t(w)=1Aθ(ξ)expi 2πλzw  ξd2ξ=sin(3α-β)sinc(α/3+β)+sin(3α+β)sinc(α/3-β)23α.
ϕj(ξ)=axjξx/d+ayjξy/d=aj  ξ/d.
PSF(w)=ANλz21Nj=1Nexpi 2πλzw  rj×1Aθ(ξ)expi 2πλz (w+Δwj)  ξd2ξ2=ANλz21Nj=1N×expi 2πλzw  rjt(w+Δwj)2.
PSF(w)=ANλz21N2j=1N|t(w+Δwj)|2+1N2j=1Nk=1kjNexpi 2πλz (rj-rk)w×t(w+Δwj)t*(w+Δwk).
apjaqk=δpqδjkσa2,apj=0,
σa2=2πλz σw2,σw2=Δwpj2.
PSF(w)=ANλz21N2j=1N t2(w+Δwj)+j=1Nk=1kjNexpi 2πλz (rj-rk)  w×t(w+Δwj)t(w+Δwk)=ANλz21N2j=1NPSFs(w, σa)+j=1Nk=1kjNexpi 2πλz (rj-rk)  w×PSFs(w, σa).
PSFs(w, σa)=t2(w+Δwj),
PSFs(w, σa)=t(w+Δwj)t(w+Δwk),jk.
PSFs(w, σa)=|t(w)|2Q(σa, w-w)d2w,
PSFs(w, σa)=t(w)Q(σa, w-w)d2w2,
Ψ(σa, ξ)=exp[iϕj(ξ)]=exp(iaj  ξ /d)
Ψ(σa, ξ)=exp[-σa2ξ2/(2d2)],
Q(σa, w-w)=2πλz2d22πσa2×exp-2πλz2(w-w)2d22σa2=12πσw2exp-(w-w)22σw2.
limσa0 Q(σa, w-w)=δ(w-w),
PSFs(w, 0)=PSFs(w, 0)=PSFs(w, 0);
PSF(w, σa)
=ANλz21NPSFs(w, σa)+1N2PSFs(w, σa)j=1Nk=1kjNexpi 2πλz (rj-rk)  w=ANλz21N [PSFs(w, σa)-PSFs(w, σa)]+PSFs(w, σa) 1N2×j=1Nk=1Nexpi 2πλz (rj-rk)  w=ANλz21N [PSFs(w, σa)-PSFs(w, σa)]+ANλz2PSFs(w, σa)GF(w),
S(σa)=PSF(0, σa)I0=1NPSFs(0, σa)+N-1NPSFs(0, σa).
S(σa)=1N|t(w)|2Q(σa, w)d2w+N-1Nt(w)Q(σa, w)d2w2.
S(σa)=1N1A2θ(η)θ(η-ξ)d2ηΨ(σa, ξ)d2ξ+N-1N1Aθ(ξ)Ψ(σa, ξ)d2ξ2.
Ψ(σa, ξ)1-(ξσa/d)2/2+(ξσa/d)4/8.
S(σa)1N1-(γσa)2+14 (γσa)4(3+γ)+N-1N1-(γσa)2+14 (γσa)4(1+γ)=1-(γσa)2+14 (γσa)41+γ+2N,
γ2=1Aθ(ξ)(ξ/d)2d2ξ,
γ=1γ41Aθ(ξ)(ξ/d)4d2ξ
(rms)2=1ANϕ2(x)d2x=1Nj=1N1Aθ(ξ)ϕj2(ξ)d2ξ=1Nj=1N1Aθ(ξ)aj  ξd2d2ξ=1Aθ(ξ) 1Nj=1Naj  ξd2d2ξN1Aθ(ξ)ξd2d2ξσa2=(γσa)2.
S(σa)1-(rms)2+(rms)441+γ+2N.
Fsp(w, σa)=PSFs(w, σa)-PSFs(w, σa).
fw,σa(w)=t(w)[Q(σa, w-w)]1/2,
gw,σa(w)=[Q(σa, w-w)]1/20foranyσa,w,and w.
PSFs(w, σa)PSFs(w, σa)=|fw,σa(w)|2d2wfw,σa(w)gw,σa(w)d2w2|gw,σa(w)|2d2w=1.
PSFs(w, σa)PSFs(w, σa),
max[Fsp(w, σa)]=Fsp(0, 4)0.1.
speckle2 2.9λzπd (1+0.2σa2)1/2.
Ism(σa)=ANλz2PSFs2λz3d, 0, σa.
Esp(σa)=AN1-1Aθ(ξ)Ψ(2σa, ξ)d2ξ.
Esp(σa)=AN1-t(w)Q(2σa, w)d2w.
Esp(σa)AN(γσa)2=AN(rms)2,
Epeaks(σa)=ANt(w)Q(2σa, w)d2w.
Epeaks(σa)=AN[1-(γσa)2]=AN[1-(rms)2]=ANS(σa);

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