Abstract

Future telescopes with diameters greater than 10 m, usually referred to as extremely large telescopes (ELTs), will employ segmented mirrors made up of hundreds or even thousands of segments, with tight constraints on the piston errors between individual segments. The 10-m Keck telescopes are routinely phased with the narrow-band phasing technique. This is a variation of the Shack-Hartmann wave-front sensor in which the signal is the correlation between individual subimages and simulated images. We have investigated the applicability of this technique to ELTs, and in the process we have developed what to our knowledge is a new algorithm in which each subimage provides on its own a piston-dependent value. We also discuss an alternative algorithm to resolve the λ ambiguity that allows detection of problematic cases, and a modification of the singular-value-decomposition procedure used to phase the whole mirror, using weightings on individual measurement errors. By means of simulations we show that the modified technique shows improved performance and that it can work with sufficient precision on telescopes as large as 100 m.

© 2002 Optical Society of America

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References

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  1. C. D. Bello, N. Devaney, J. Castro, “The effect of piston errors on the image quality of ground-based segmented mirror telescopes,” Rev. Mex. Astron. Astrofis. 36, 57–66 (2000).
  2. G. W. Zeiders, E. E. Montgomery, “Diffraction effects with segmented apertures,” in Space Telescopes and Instruments V, P. Y. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 799–809 (1998).
    [CrossRef]
  3. 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]
  4. N. Yaitskova, K. Dohlen (both of Observatoire de Marseille, 2 Place Le Verrier, 13248 Marseille cedex 4, France) are preparing a manuscript to be called “Theoretical and computational study of the image quality in extremely large segmented telescopes.”
  5. D. Gavel, “The effect of the Keck telescope’s segmented primary on the performance of the Keck adaptive optics system,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 144–150 (1997).
    [CrossRef]
  6. C. D. Bello Figueroa, “Improving the image quality of large segmented mirror telescopes,” Ph.D. dissertation (Instituto de Astrofı́sica de Canarias, La Laguna, Tenerife, Spain, 2000).
  7. N. Devaney, L. Cavaller, L. Jochum, C. D. Bello, J. Castro, “Guacamole: the GTC guiding, acquisition and calibration module,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 146–153 (2000).
    [CrossRef]
  8. G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Mast, D. Kirkman, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt. 37, 140–155 (1998).
    [CrossRef]
  9. G. Chanan, C. Ohara, M. Troy, “Phasing the mirror segments of the Keck telescopes II: the narrow-band phasing algorithm,” Appl. Opt. 39, 4706–4714 (2000).
    [CrossRef]
  10. M. Löfdahl, H. Eriksson, “An algorithm for resolving 2π ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40, 2019–2058 (2001).
  11. B. Lasker, C. Sturch, B. McLean, J. Russel, H. Jenker, M. Shara, “The Guide Star Catalog. I. Astronomical foundations and image processing,” Astron. J. 99, 2019–2058 (1990).
    [CrossRef]
  12. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, New York, 1992), Chap. 2.
  13. G. Chanan, M. Troy, C. Ohara, “Phasing the primary mirror segments of the Keck telescopes: a comparison of different techniques,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 188–201 (2000).
    [CrossRef]

2001

M. Löfdahl, H. Eriksson, “An algorithm for resolving 2π ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40, 2019–2058 (2001).

2000

C. D. Bello, N. Devaney, J. Castro, “The effect of piston errors on the image quality of ground-based segmented mirror telescopes,” Rev. Mex. Astron. Astrofis. 36, 57–66 (2000).

G. Chanan, C. Ohara, M. Troy, “Phasing the mirror segments of the Keck telescopes II: the narrow-band phasing algorithm,” Appl. Opt. 39, 4706–4714 (2000).
[CrossRef]

1998

1990

B. Lasker, C. Sturch, B. McLean, J. Russel, H. Jenker, M. Shara, “The Guide Star Catalog. I. Astronomical foundations and image processing,” Astron. J. 99, 2019–2058 (1990).
[CrossRef]

Bello, C. D.

C. D. Bello, N. Devaney, J. Castro, “The effect of piston errors on the image quality of ground-based segmented mirror telescopes,” Rev. Mex. Astron. Astrofis. 36, 57–66 (2000).

N. Devaney, L. Cavaller, L. Jochum, C. D. Bello, J. Castro, “Guacamole: the GTC guiding, acquisition and calibration module,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 146–153 (2000).
[CrossRef]

Bello Figueroa, C. D.

C. D. Bello Figueroa, “Improving the image quality of large segmented mirror telescopes,” Ph.D. dissertation (Instituto de Astrofı́sica de Canarias, La Laguna, Tenerife, Spain, 2000).

Castro, J.

C. D. Bello, N. Devaney, J. Castro, “The effect of piston errors on the image quality of ground-based segmented mirror telescopes,” Rev. Mex. Astron. Astrofis. 36, 57–66 (2000).

N. Devaney, L. Cavaller, L. Jochum, C. D. Bello, J. Castro, “Guacamole: the GTC guiding, acquisition and calibration module,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 146–153 (2000).
[CrossRef]

Cavaller, L.

N. Devaney, L. Cavaller, L. Jochum, C. D. Bello, J. Castro, “Guacamole: the GTC guiding, acquisition and calibration module,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 146–153 (2000).
[CrossRef]

Chanan, G.

Dekens, F.

Devaney, N.

C. D. Bello, N. Devaney, J. Castro, “The effect of piston errors on the image quality of ground-based segmented mirror telescopes,” Rev. Mex. Astron. Astrofis. 36, 57–66 (2000).

N. Devaney, L. Cavaller, L. Jochum, C. D. Bello, J. Castro, “Guacamole: the GTC guiding, acquisition and calibration module,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 146–153 (2000).
[CrossRef]

Dohlen, K.

N. Yaitskova, K. Dohlen (both of Observatoire de Marseille, 2 Place Le Verrier, 13248 Marseille cedex 4, France) are preparing a manuscript to be called “Theoretical and computational study of the image quality in extremely large segmented telescopes.”

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]

Eriksson, H.

M. Löfdahl, H. Eriksson, “An algorithm for resolving 2π ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40, 2019–2058 (2001).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, New York, 1992), Chap. 2.

Gavel, D.

D. Gavel, “The effect of the Keck telescope’s segmented primary on the performance of the Keck adaptive optics system,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 144–150 (1997).
[CrossRef]

Jenker, H.

B. Lasker, C. Sturch, B. McLean, J. Russel, H. Jenker, M. Shara, “The Guide Star Catalog. I. Astronomical foundations and image processing,” Astron. J. 99, 2019–2058 (1990).
[CrossRef]

Jochum, L.

N. Devaney, L. Cavaller, L. Jochum, C. D. Bello, J. Castro, “Guacamole: the GTC guiding, acquisition and calibration module,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 146–153 (2000).
[CrossRef]

Kirkman, D.

Lasker, B.

B. Lasker, C. Sturch, B. McLean, J. Russel, H. Jenker, M. Shara, “The Guide Star Catalog. I. Astronomical foundations and image processing,” Astron. J. 99, 2019–2058 (1990).
[CrossRef]

Löfdahl, M.

M. Löfdahl, H. Eriksson, “An algorithm for resolving 2π ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40, 2019–2058 (2001).

Mast, J.

McLean, B.

B. Lasker, C. Sturch, B. McLean, J. Russel, H. Jenker, M. Shara, “The Guide Star Catalog. I. Astronomical foundations and image processing,” Astron. J. 99, 2019–2058 (1990).
[CrossRef]

Michaels, S.

Montgomery, E. E.

G. W. Zeiders, E. E. Montgomery, “Diffraction effects with segmented apertures,” in Space Telescopes and Instruments V, P. Y. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 799–809 (1998).
[CrossRef]

Ohara, C.

G. Chanan, C. Ohara, M. Troy, “Phasing the mirror segments of the Keck telescopes II: the narrow-band phasing algorithm,” Appl. Opt. 39, 4706–4714 (2000).
[CrossRef]

G. Chanan, M. Troy, C. Ohara, “Phasing the primary mirror segments of the Keck telescopes: a comparison of different techniques,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 188–201 (2000).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, New York, 1992), Chap. 2.

Russel, J.

B. Lasker, C. Sturch, B. McLean, J. Russel, H. Jenker, M. Shara, “The Guide Star Catalog. I. Astronomical foundations and image processing,” Astron. J. 99, 2019–2058 (1990).
[CrossRef]

Shara, M.

B. Lasker, C. Sturch, B. McLean, J. Russel, H. Jenker, M. Shara, “The Guide Star Catalog. I. Astronomical foundations and image processing,” Astron. J. 99, 2019–2058 (1990).
[CrossRef]

Sturch, C.

B. Lasker, C. Sturch, B. McLean, J. Russel, H. Jenker, M. Shara, “The Guide Star Catalog. I. Astronomical foundations and image processing,” Astron. J. 99, 2019–2058 (1990).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, New York, 1992), Chap. 2.

Troy, M.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, New York, 1992), Chap. 2.

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]

N. Yaitskova, K. Dohlen (both of Observatoire de Marseille, 2 Place Le Verrier, 13248 Marseille cedex 4, France) are preparing a manuscript to be called “Theoretical and computational study of the image quality in extremely large segmented telescopes.”

Zeiders, G. W.

G. W. Zeiders, E. E. Montgomery, “Diffraction effects with segmented apertures,” in Space Telescopes and Instruments V, P. Y. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 799–809 (1998).
[CrossRef]

Appl. Opt.

Astron. J.

B. Lasker, C. Sturch, B. McLean, J. Russel, H. Jenker, M. Shara, “The Guide Star Catalog. I. Astronomical foundations and image processing,” Astron. J. 99, 2019–2058 (1990).
[CrossRef]

Opt. Eng.

M. Löfdahl, H. Eriksson, “An algorithm for resolving 2π ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40, 2019–2058 (2001).

Rev. Mex. Astron. Astrofis.

C. D. Bello, N. Devaney, J. Castro, “The effect of piston errors on the image quality of ground-based segmented mirror telescopes,” Rev. Mex. Astron. Astrofis. 36, 57–66 (2000).

Other

G. W. Zeiders, E. E. Montgomery, “Diffraction effects with segmented apertures,” in Space Telescopes and Instruments V, P. Y. Bely, J. B. Breckinridge, eds., Proc. SPIE3356, 799–809 (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]

N. Yaitskova, K. Dohlen (both of Observatoire de Marseille, 2 Place Le Verrier, 13248 Marseille cedex 4, France) are preparing a manuscript to be called “Theoretical and computational study of the image quality in extremely large segmented telescopes.”

D. Gavel, “The effect of the Keck telescope’s segmented primary on the performance of the Keck adaptive optics system,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 144–150 (1997).
[CrossRef]

C. D. Bello Figueroa, “Improving the image quality of large segmented mirror telescopes,” Ph.D. dissertation (Instituto de Astrofı́sica de Canarias, La Laguna, Tenerife, Spain, 2000).

N. Devaney, L. Cavaller, L. Jochum, C. D. Bello, J. Castro, “Guacamole: the GTC guiding, acquisition and calibration module,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 146–153 (2000).
[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, New York, 1992), Chap. 2.

G. Chanan, M. Troy, C. Ohara, “Phasing the primary mirror segments of the Keck telescopes: a comparison of different techniques,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 188–201 (2000).
[CrossRef]

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Figures (19)

Fig. 1
Fig. 1

Simulated diffraction patterns for a split squared subaperture with wave-front step δ between the two halves given by kδ = 0, 2π/6, 4π/6, … , 10π/6.

Fig. 2
Fig. 2

x projection of images of Fig. 1.

Fig. 3
Fig. 3

Peak ratio calibration curve (λ = 650 nm). WF, wave front.

Fig. 4
Fig. 4

Measurement precision of the Keck (diamonds) and the peak ratio (triangles) technique under perfect conditions.

Fig. 5
Fig. 5

One-dimensional diffraction patterns at various seeing conditions r 0 (500 nm) = ∞, 16 cm, 6 cm.

Fig. 6
Fig. 6

Peak ratio calibration curves at various seeing conditions down to r 0 (500 nm) = 8 cm (λ = 650 nm).

Fig. 7
Fig. 7

One-dimensional diffraction patterns at various photon noise levels N γ = 2000, 500, 200.

Fig. 8
Fig. 8

Measurement precision of the peak ratio technique with high spatial-frequency wave-front aberrations resulting from segment polishing with an amplitude of 10, 20, and 30 nm (wave front). Note that one example aberration structure is used; each segment will have its own characteristic aberration. Triangles indicate data without segment figure errors, higher polishing errors produce higher measurement errors.

Fig. 12
Fig. 12

Measurement precision of the peak ratio technique under various seeing conditions, with calibration data obtained for best seeing conditions. Triangles, data with perfect seeing conditions.

Fig. 9
Fig. 9

Performance of the peak ratio method on a whole mirror. The original piston values are normally distributed with a rms of 20 nm (not shown). In this simulation a seeing of 20 cm and 5000 photons per subaperture were assumed. The peak ratio method without phase plate (left) resulted in a rms piston error of 19.5 nm and a rms piston-step measurement error of 15.4 nm. Using a phase plate (right) resulted in a rms piston error of 4.5 nm and a rms piston-step measurement error of 4.9 nm.

Fig. 10
Fig. 10

Measurement precision of the Keck technique under the same polishing conditions as in Fig. 8. Triangles represent data without segment figure errors; higher polishing errors produce higher measurement errors.

Fig. 11
Fig. 11

Measurement precision of the Keck technique under various seeing conditions. Diamonds, data with perfect seeing conditions.

Fig. 13
Fig. 13

Measurement precision of the peak ratio technique under various seeing conditions, with the calibration data corresponding to the actual seeing condition. Data with perfect seeing conditions are marked.

Fig. 14
Fig. 14

Rms measurement precision of the peak ratio technique under various photon noise conditions (λ = 650 nm).

Fig. 15
Fig. 15

Segmented mirror of the Keck/GTC type (36 segments) before (left) and after (right) one phasing iteration. In this simulation perfect conditions were assumed. The original piston values are uniformly distributed with a range of 6530 nm and a rms of 1990 nm. Two measurements at different wavelengths were performed (λ = 650, 850 nm). The resulting rms piston error is 0.27 nm, and the rms piston-step measurement error is 0.31 nm.

Fig. 16
Fig. 16

Segmented ELT mirror (1002 segments) before (top) and after (bottom) one phasing iteration. In this simulation perfect conditions were assumed. The original piston values are uniformly distributed with a range of 3900 nm and a rms of 1180 nm. Two measurements at different wavelengths were performed (λ = 650, 850 nm). The resulting rms piston error is 0.46 nm, and the rms piston-step measurement error is 0.33 nm.

Fig. 17
Fig. 17

Segmented ELT mirror (846 segments) before (left) and after (right) one phasing iteration. In this simulation perfect conditions were assumed. The original piston values are uniformly distributed with a range of 4000 nm and a rms of 1125 nm. Two measurements at different wavelengths were performed (λ = 650, 850 nm). The resulting rms piston error is 2.0 nm, and the rms piston-step measurement error is 0.4 nm. The smooth residual is an example of the propagation of errors for which the sensitivity of the technique is low.

Fig. 18
Fig. 18

Segmented mirror (120 segments) before (left) and after (right) one phasing iteration. In this simulation a seeing of r 0 = 24 cm and an exposure time yielding 20,000 photons per subaperture were assumed. The original piston values are uniformly distributed with a range of 3800 nm and a rms of 1153 nm. Two measurements at different wavelengths were performed (λ = 650, 850 nm). The resulting rms piston error is 4.5 nm, even though the rms piston-step measurement error is 231 nm. This is because in two measurements the λ-ambiguity algorithm yielded a wrong result. This did not affect the overall measurement, because the ambiguity was detected and a higher error was automatically assigned. The rms piston-step measurement error excluding these two cases is 3.7 nm.

Fig. 19
Fig. 19

Computation time needed for the SVD algorithm as a function of matrix size for our current, unoptimized code. Crosses represent data for mirrors with 180, 396, 612, 810, and 1002 segments.

Tables (1)

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Table 1 Limiting Stellar Magnitude and the Corresponding Required Accessible Field (FOV) to Find such a Star with a Probability of 90% for the Peak Ratio Method at Different Photon Levels at the North and the South Galactic Polesa

Equations (11)

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

S1-σ21-1N,
S1-σθ2γ,
PS=x1+kλ1=x2+jλ2, k, jI.
|x1+kλ1| <λMax, |x2+jλ2| <λMax,
PS=x1±σx1+kλ1=x2±σx2+jλ2 |x1+kλ1|+σx1<λMax, |x2+jλ2|+σx2<λMax.
n-ηtA10-0.4MΔλ,
P=1-exp-r2νπ/36002,
Piston1i-Piston2i=PSi, i=1,  , Nsubapertures,
j=1NsegmentsPiston=0.
Piston1i-Piston2iσPSi=PSiσPSi, i=i,  , Nsubapertures.
σ2=0.348λ/d1/3λ/r05/3,

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