Abstract

The performance of a closed-loop adaptive-optics system may in principle be improved by selection of distinct and independently optimized control bandwidths for separate components, or modes, of the wave-front-distortion profile. We describe a method for synthesizing and optimizing a multiple-bandwidth adaptive-optics control system from performance estimates previously derived for single-bandwidth control systems operating over a range of bandwidths. The approach is applicable to adaptive-optics systems that use either one or several wave-front sensing beacons and also to systems that include multiple deformable mirrors for atmospheric-turbulence compensation across an extended field of view. Numerical results are presented for the case of an atmospheric-turbulence profile consisting of a single translating phase screen with Kolmogorov statistics, a Shack–Hartmann wave-front sensor with from 8 to 16 subapertures across the aperture of the telescope, and a continuous-face-sheet deformable mirror with actuators conjugate to the corners of the wave-front-sensor subapertures. The use of multiple control bandwidths significantly relaxes the wave-front-sensor noise level that is permitted for the adaptive-optics system to operate near the performance limit imposed by fitting error. Nearly all of this reduction is already achieved through the use of a control system that uses only two distinct bandwidths, one of which is the zero bandwidth.

© 1994 Optical Society of America

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  1. R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
    [CrossRef]
  2. F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
    [CrossRef]
  3. R. Q. Fugate, B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek, W. J. Lange, A. C. Slavin, W. J. Wild, D. M. Winker, J. M. Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, M. D. Oliker, D. W. Swindle, R. A. Cleis, “Two generations of laser-guide-star adaptive-optics experiments at the Starfire Optical Range,” J. Opt. Soc. Am. A 11, 310–324 (1994).
    [CrossRef]
  4. V. E. Zuez, V. P. Lukin, “Dynamic characteristics of optical adaptive systems,” Appl. Opt. 26, 139–144 (1987).
    [CrossRef]
  5. C. Boyer, E. Gendron, P. Y. Madec, “Adaptive optics for high resolution imagery: control algorithms for optimized modal corrections,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1780, 943–957 (1992).
  6. E. Gendron, “Modal control optimization in an adaptive optics system,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.
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  9. G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.
  10. F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on the 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).
  11. G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F. Rigaut, J. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for infrared astronomy,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344, 1990.
    [CrossRef]
  12. E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
    [CrossRef]
  13. F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10, 957–965 (1993).
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  18. D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front compensation systems,”J. Opt. Soc. Am. 66, 193–206 (1976).
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  24. B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
    [CrossRef]
  25. R. C. Fisher, An Introduction to Linear Algebra (Dickenson, Encino, Calif., 1970).
  26. Equation (2.27) requires that the matrix Sbe positive definite. Sis positive semidefinite by construction, and any eigenvector of Swith a zero eigenvalue defines a wave-front-sensor measurement mode that must be identically zero. Such measurement modes, in the cases in which they exist at all, cannot contribute to the wave-front estimate. When necessary we may replace the measurement vector sby its projection onto the subspace orthogonal to all such modes and redefine Mand Gas their restrictions to this subspace.
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    [CrossRef]
  28. The case of a fixed, known wind direction requires a slight modification to the formulas developed in Ref. 22 for the matrices Aand S. Ensemble averaging over the direction of the wind in Eq. (3.22) of Ref. 22 now has no effect, and the ray separation vector Δmust be replaced by Δ+ (τ1− τ2)vfor the remainder of the derivation. The velocity vof the random wind must also be set to zero. The resulting formula for a general element of the matrix Aor Sis identical to Eq. (3.27) of Ref. 22, except that the term f(2δv/D, 2Δ/D) is replaced by f(0, 2|δv+ Δ|/D). The function f(a, b) remains as defined by Eq. (3.28) of Ref. 22.

1994 (4)

1993 (1)

1991 (4)

B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[CrossRef]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on the 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

1990 (1)

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

1987 (1)

1984 (1)

1983 (1)

1980 (1)

1978 (1)

1977 (2)

1976 (2)

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Baum, G.

Beuzit, J. L.

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

Boeke, B. R.

R. Q. Fugate, B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek, W. J. Lange, A. C. Slavin, W. J. Wild, D. M. Winker, J. M. Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, M. D. Oliker, D. W. Swindle, R. A. Cleis, “Two generations of laser-guide-star adaptive-optics experiments at the Starfire Optical Range,” J. Opt. Soc. Am. A 11, 310–324 (1994).
[CrossRef]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Boyer, C.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F. Rigaut, J. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for infrared astronomy,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344, 1990.
[CrossRef]

C. Boyer, E. Gendron, P. Y. Madec, “Adaptive optics for high resolution imagery: control algorithms for optimized modal corrections,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1780, 943–957 (1992).

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Cleis, R. A.

Cuby, J.

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

Ellerbroek, B. L.

Fisher, R. C.

R. C. Fisher, An Introduction to Linear Algebra (Dickenson, Encino, Calif., 1970).

Fontanella, J.

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F. Rigaut, J. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for infrared astronomy,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344, 1990.
[CrossRef]

Fontanella, J. C.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on the 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Fried, D. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase difference measurements,”J. Opt. Soc. Am. 67, 370–375 (1977).
[CrossRef]

D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front compensation systems,”J. Opt. Soc. Am. 66, 193–206 (1976).
[CrossRef]

Fugate, R. Q.

R. Q. Fugate, B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek, W. J. Lange, A. C. Slavin, W. J. Wild, D. M. Winker, J. M. Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, M. D. Oliker, D. W. Swindle, R. A. Cleis, “Two generations of laser-guide-star adaptive-optics experiments at the Starfire Optical Range,” J. Opt. Soc. Am. A 11, 310–324 (1994).
[CrossRef]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Gaffard, J.

G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F. Rigaut, J. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for infrared astronomy,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344, 1990.
[CrossRef]

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

Gaffard, J. P.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on the 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

Gardner, C. S.

Gendron, E.

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

C. Boyer, E. Gendron, P. Y. Madec, “Adaptive optics for high resolution imagery: control algorithms for optimized modal corrections,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1780, 943–957 (1992).

E. Gendron, “Modal control optimization in an adaptive optics system,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

Gigan, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F. Rigaut, J. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for infrared astronomy,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344, 1990.
[CrossRef]

Golub, G. H.

G. H. Golub, C. Van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, Baltimore, Md., 1989).

Graves, J. E.

Greenwood, D. P.

Herrmann, J.

Higgins, C. H.

Hubin, N.

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

Hudgin, R. H.

Jagourel, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F. Rigaut, J. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for infrared astronomy,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344, 1990.
[CrossRef]

Jelonek, M. P.

Kern, P.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on the 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F. Rigaut, J. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for infrared astronomy,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344, 1990.
[CrossRef]

Lange, W. J.

Lena, P.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on the 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

Lena, P. J.

G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F. Rigaut, J. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for infrared astronomy,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344, 1990.
[CrossRef]

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

Lukin, V. P.

Madec, P. Y.

C. Boyer, E. Gendron, P. Y. Madec, “Adaptive optics for high resolution imagery: control algorithms for optimized modal corrections,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1780, 943–957 (1992).

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

Markey, J. K.

McKenna, D. L.

Merkle, F.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on the 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F. Rigaut, J. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for infrared astronomy,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344, 1990.
[CrossRef]

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

Moroney, J. F.

Noll, R. J.

Northcott, M.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Northcott, M. J.

Oliker, M. D.

Parenti, R. R.

Ribak, E. N.

Richard, J.

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

Richard, J. C.

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

Rigaut, F.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on the 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F. Rigaut, J. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for infrared astronomy,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344, 1990.
[CrossRef]

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Roddier, D.

Roddier, F.

Rousset, G.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on the 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F. Rigaut, J. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for infrared astronomy,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344, 1990.
[CrossRef]

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

Ruane, R. E.

R. Q. Fugate, B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek, W. J. Lange, A. C. Slavin, W. J. Wild, D. M. Winker, J. M. Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, M. D. Oliker, D. W. Swindle, R. A. Cleis, “Two generations of laser-guide-star adaptive-optics experiments at the Starfire Optical Range,” J. Opt. Soc. Am. A 11, 310–324 (1994).
[CrossRef]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Sasiela, R. J.

Schwartz, C.

Slavin, A. C.

Spinhirne, J. M.

Swindle, D. W.

Tyler, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

G. A. Tyler, “Turbulence-induced adaptive-optics performance evaluation: degradation in the time domain,” J. Opt. Soc. Am. A 1, 251–262 (1984).
[CrossRef]

Van Loan, C.

G. H. Golub, C. Van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, Baltimore, Md., 1989).

Vittot, M.

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

Wallner, E. P.

Wang, J. Y.

Welsh, B. M.

Wild, W. J.

Winker, D. M.

Wopat, L. M.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Wynia, J. M.

Zuez, V. E.

Appl. Opt. (1)

Astron. Astrophys. (2)

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on the 3.6 m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (7)

B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[CrossRef]

G. A. Tyler, “Turbulence-induced adaptive-optics performance evaluation: degradation in the time domain,” J. Opt. Soc. Am. A 1, 251–262 (1984).
[CrossRef]

B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10, 957–965 (1993).
[CrossRef]

R. R. Parenti, R. J. Sasiela, “Laser-guide-star systems for astronomical applications,” J. Opt. Soc. Am. A 11, 288–3091994.
[CrossRef]

R. Q. Fugate, B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek, W. J. Lange, A. C. Slavin, W. J. Wild, D. M. Winker, J. M. Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, M. D. Oliker, D. W. Swindle, R. A. Cleis, “Two generations of laser-guide-star adaptive-optics experiments at the Starfire Optical Range,” J. Opt. Soc. Am. A 11, 310–324 (1994).
[CrossRef]

C. Schwartz, G. Baum, E. N. Ribak, “Turbulence-degraded wave fronts as fractal surfaces,” J. Opt. Soc. Am. A 11, 444–451 (1994).
[CrossRef]

Nature (London) (1)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Other (9)

C. Boyer, E. Gendron, P. Y. Madec, “Adaptive optics for high resolution imagery: control algorithms for optimized modal corrections,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1780, 943–957 (1992).

E. Gendron, “Modal control optimization in an adaptive optics system,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, C. Boyer, P. Y. Madec, P. Gigan, J. C. Richard, M. Vittot, J. P. Gaffard, F. Rigaut, P. Lena, “The Come-On-Plus adaptive optics system: results and performance,” presented at the International Commission for Optics–16 Satellite Conference, August 2–5, 1993, Garching, Germany.

G. Rousset, J. Fontanella, P. Kern, P. J. Lena, P. Gigan, F. Rigaut, J. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for infrared astronomy,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344, 1990.
[CrossRef]

E. Gendron, J. Cuby, F. Rigaut, P. J. Lena, J. Fontanella, G. Rousset, J. Gaffard, C. Boyer, J. Richard, M. Vittot, F. Merkle, N. Hubin, “Come-On-Plus project: an upgrade of the Come-On adaptive optics prototype system,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1542, 297–307 (1991).
[CrossRef]

G. H. Golub, C. Van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, Baltimore, Md., 1989).

The case of a fixed, known wind direction requires a slight modification to the formulas developed in Ref. 22 for the matrices Aand S. Ensemble averaging over the direction of the wind in Eq. (3.22) of Ref. 22 now has no effect, and the ray separation vector Δmust be replaced by Δ+ (τ1− τ2)vfor the remainder of the derivation. The velocity vof the random wind must also be set to zero. The resulting formula for a general element of the matrix Aor Sis identical to Eq. (3.27) of Ref. 22, except that the term f(2δv/D, 2Δ/D) is replaced by f(0, 2|δv+ Δ|/D). The function f(a, b) remains as defined by Eq. (3.28) of Ref. 22.

R. C. Fisher, An Introduction to Linear Algebra (Dickenson, Encino, Calif., 1970).

Equation (2.27) requires that the matrix Sbe positive definite. Sis positive semidefinite by construction, and any eigenvector of Swith a zero eigenvalue defines a wave-front-sensor measurement mode that must be identically zero. Such measurement modes, in the cases in which they exist at all, cannot contribute to the wave-front estimate. When necessary we may replace the measurement vector sby its projection onto the subspace orthogonal to all such modes and redefine Mand Gas their restrictions to this subspace.

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Figures (6)

Fig. 1
Fig. 1

Adaptive-optics system control-loop dynamics: (a) an adaptive-optics control system in which all wave-front modes are compensated at the same servo bandwidth; (b) a system that uses different bandwidths and estimation algorithms for distinct, orthogonal subspaces of wave-front modes. Only two separate bandwidths are illustrated for the sake of concreteness, but the approach may be generalized to any number of distinct control bandwidths.

Fig. 2
Fig. 2

Performance of an adaptive-optics control system with use of a single control bandwidth, as a function of normalized noise level and normalized servo lag. These results were obtained with use of the results described in Subsection 2.A. The results in this figure assume D/d = 8 and an unknown and random direction for the wind. The normalized residual phase variance is σ2/(d/r0)5/3, the normalized servo lag is (fgr0)/(fd), and the normalized root-mean-square (RMS) wave-front-sensor noise is defined as σn/[2π(d/r0)4/3]. The factor 2π is included in this definition so that the noise level is expressed in terms of waves when d/r0 = 1. WFS, wave-front-sensor.

Fig. 3
Fig. 3

Performance of an adaptive-optics control system with use of reduced-range single-bandwidth control as a function of normalized noise level and servo lag. These results correspond to those illustrated in Fig. 2, except that they assume the use of a reduced-range single-bandwidth control system as described in Subsection 2.D. WFS, wave-front-sensor.

Fig. 4
Fig. 4

Performance comparison of single-bandwidth control, reduced-range single-bandwidth control, and multiple-bandwidth control for D/d = 8 and a random wind direction. The multiple-bandwidth control results are taken from Table 2. The single-bandwidth and the reduced-range single-bandwidth results are the minima of the curves plotted in Figs. 2 and 3, respectively. The definitions for normalized residual phase variance and normalized RMS wave-front-sensor noise are as in Fig. 2. WFS, wave-front-sensor.

Fig. 5
Fig. 5

Performance comparison of single-bandwidth control, reduced-range single-bandwidth control, and multiple-bandwidth control for different normalized aperture diameters and wind models. These plots are similar to those of Fig. 4, except that each subfigure describes results for a different normalized aperture diameter (D/d) and wind condition (known or unknown direction). (a) D/d = 12, random wind direction; (b) D/d = 16, random wind direction; (c) D/d = 8, known wind direction; (d) D/d = 12, known wind direction; (e) D/d = 16, known wind direction. WFS, wave-front-sensor.

Fig. 6
Fig. 6

Modal distribution of wave-front errors before (x) and after (+) compensation by the multiple-bandwidth control algorithm. These plots are for the normalized aperture diameter D/d = 8 and a normalized wave-front-sensor noise level of 0.1 wave at a sampling rate of ten times the Greenwood frequency. (a) Unknown wind direction, (b) known wind direction.

Tables (3)

Tables Icon

Table 1 Dimensionality of the Adaptive-Optics Systems Considered

Tables Icon

Table 2 Performance of a Multiple-Bandwidth Adaptive-Optics Control System as a Function of Wave-Front-Sensor Noise with D/d = 8 and a Random Wind Directiona

Tables Icon

Table 3 Distribution of Modal Control Bandwidths for the Multiple-Bandwidth Control Results Given in Table 2a

Equations (94)

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σ 2 ( f ) = σ n 2 ( f ) + σ s 2 ( f ) ,
σ 2 = i = 1 N σ i 2 ( f i ) ,
σ i 2 ( f i ) = ( σ i 2 ) n ( f i ) + ( σ i 2 ) s ( f i ) ,
e = M ( y - G c ) ,
d c d t = k e .
d r W A ( r ) ϕ ( r , θ ) 0 ,
Δ ϕ ( r , θ ) = i c i h i ( r , θ ) ,
Δ ϕ = H c .
[ f , g ] = d r d θ W A ( r ) W F ( θ ) f ( r , θ ) g ( r , θ ) d r d θ W A ( r ) W F ( θ ) ,
σ 2 = [ ϕ - Δ ϕ , ϕ - Δ ϕ ] ,
[ H c , H c ] = c T c ,
[ h i , h j ] = δ i j ,
e ( i ) = M ( i ) ( y - G c ) ,
d c ( i ) d t = k ( i ) P ( i ) e ( i ) ,
c = i c ( i ) .
P ( i ) P ( i ) = P ( i ) ,
[ P ( i ) ] T = P ( i ) ,
P ( i ) P ( j ) = 0             for i j ,
i P ( i ) = I .
c ( t ) = 0 d τ k exp ( - k τ M G ) M y ( t - τ ) .
M G = I
c ( t ) = M s ( t ) ,
s ( t ) = 0 d τ k exp ( - k τ ) y ( t - τ ) .
σ 2 = [ ϕ - H M s , ϕ - H M s ] .
σ 2 = [ ϕ , ϕ ] - 2 i , j [ ϕ , h i ] s j M i j + i , j , i , j [ h i , h i ] M i j M i j s j s j .
σ 0 2 = [ ϕ , ϕ ] ,
A i j = [ ϕ , h i ] s j ,
S j j = s j s j
σ 2 = σ 0 2 - 2 i , j A i j M i j + i , j , j M i j M i j S j j = σ 0 2 - tr ( A M T ) - tr ( M A T ) + tr ( M S M T ) ,
M = A S - 1 + ( I - A S - 1 G ) ( G T S - 1 G ) - 1 G T S - 1 .
σ 2 = σ 0 2 - tr ( A S - 1 A T ) + tr [ ( I - A S - 1 G ) ( G T S - 1 G ) - 1 ( I - G T S - 1 A T ) ] .
M ( i ) G = I
d d t [ P ( i ) c ] = P ( i ) d c d t = j P ( i ) d c ( j ) d t ,
d d t [ P ( i ) c ] = k ( i ) P ( i ) M ( i ) y - k ( i ) P ( i ) c ,
P ( i ) c ( t ) = 0 d τ k ( i ) exp [ - k ( i ) τ ] P ( i ) M ( i ) y ( t - τ ) .
s ( i ) ( t ) = 0 d τ k ( i ) exp [ - k ( i ) τ ] y ( t - τ )
P ( i ) c ( t ) = P ( i ) M ( i ) s ( i ) ( t ) .
c ( t ) = i P ( i ) M ( i ) s ( i ) ( t ) ,
σ 2 = [ ϕ - i H P ( i ) M ( i ) s ( i ) , ϕ - i H P ( i ) M ( i ) s ( i ) ] ,
σ 2 = σ 0 2 - 2 i , j , k [ ϕ , h j ] s k ( i ) [ P ( i ) M ( i ) ] j k + i , j , k , i , j , k [ h j , h j ] [ P ( i ) M ( i ) ] j k [ P ( i ) M ( i ) ] j k s k ( i ) s k ( i ) .
A j k ( i ) = [ ϕ , h j ] s k ( i ) ,
S k k ( i i ) = s k ( i ) s k ( i ) ,
σ 2 = σ 0 2 - 2 i , j , k A j k ( i ) [ P ( i ) M ( i ) ] j k + i , i , j , k , k [ P ( i ) M ( i ) ] j k [ P ( i ) M ( i ) ] j k S k k ( i i )
σ 2 = σ 0 2 - i tr { P ( i ) M ( i ) [ A ( i ) ] T } - i tr { A ( i ) [ M ( i ) ] T [ P ( i ) ] T } + i , i tr { P ( i ) M ( i ) S ( i i ) [ M ( i ) ] T [ P ( i ) ] T } .
tr ( M N ) = tr ( N M )
σ 2 = σ 0 2 - i tr ( P ( i ) { M ( i ) [ A ( i ) ] T + A ( i ) [ M ( i ) ] T - M ( i ) S ( i i ) [ M ( i ) ] T } P ( i ) ) .
M ( i ) = M ( i ) [ A ( i ) ] T + A ( i ) [ M ( i ) ] T - M ( i ) S ( i i ) [ M ( i ) ] T
σ 2 = σ 0 2 - i tr ( P ( i ) M ( i ) P ( i ) ) .
M j k ( i ) = [ h j , ϕ ] [ h k , ϕ ] - [ h j , ϕ - H M ( i ) s ( i ) ] × [ h k , ϕ - H M ( i ) s ( i ) ] .
σ * 2 = min { σ 0 2 - i tr [ P ( i ) M ( i ) P ( i ) ] : [ P ( 1 ) , , P ( n c ) ] P } = σ 0 2 - max { i tr [ P ( i ) M ( i ) P ( i ) ] : [ P ( 1 ) , , P ( n c ) ] P } ,
P ( i ) = U Λ ( i ) U T ,
U U T = U T U = I ,
Λ ( i ) = diag [ λ 1 ( i ) , , λ n ( i ) ] ,
λ j ( i ) = 0 or 1 ,
i Λ ( i ) = I .
σ * 2 = σ 0 2 - max { i tr [ Λ ( i ) U T M ( i ) U ) : U unitary , [ Λ ( 1 ) , , Λ ( n c ) ] L } ,
max { i tr [ Λ ( i ) U T M ( i ) U ] : [ Λ ( 1 ) , , Λ ( n c ) ] L } = j = 1 n max i [ U T M ( i ) U ] j j ,
λ j ( i ) = { 1 if [ U T M ( i ) U ] j j = max k [ U T M ( k ) U ] j j 0 otherwise .
σ * 2 = σ 0 2 - max { j = 1 n max i [ U T M ( i ) U ] j j : U unitary } ,
g j ( U , k ) = [ U T M ( k ) U ] j j
σ 2 = σ 0 2 - j = 1 n max 0 k K [ U T M ( k ) U ] j j ,
max 0 k K [ U T M ( k ) U ] j j max i [ U T M ( k i ) U ] j j + / n
max U j = 1 N max 0 k K [ U T M ( k ) U ] j j max U j = 1 N max i [ U T M ( k i ) U ] j j + .
n c = 2 ,
k ( 1 ) = k ,
k ( 2 ) = 0.
s ( 2 ) = 0 ,
A ( 2 ) = 0 ,
S ( 22 ) = 0 ,
M ( 2 ) = 0 ,
σ * 2 = σ 0 2 - max { j = 1 n max [ ( U T M U ) j j , 0 ] : U unitary } .
M = U 0 D U 0 T ,
σ * 2 = σ 0 2 - max { j = 1 n max [ ( U T U 0 D U 0 T U ) j j , 0 ] : U unitary } .
σ * 2 = σ 0 2 - j = 1 n max ( d j , 0 ) .
ϕ ( r ) - ϕ ( r + r ) 2 = 6.88 ( r r 0 ) 5 / 3 ,
ϕ ( r , t ) = ϕ ( r - t v , 0 ) ,
f g = 0.423 v / r 0 .
σ 0 2 = ( d r 0 ) 5 / 3 1.0299 ( D d ) 5 / 3 ,
G = G ( D d ) ,
A = ( d r 0 ) 5 / 3 A 0 ( D d , f g r 0 f d ) ,
S = ( d r 0 ) 5 / 3 S 0 ( D d , f g r 0 f d ) + σ n 2 ( f f g ) S 1 ( D d ) = ( d r 0 ) 5 / 3 [ S 0 ( D d , f g r 0 f d ) + σ n 2 ( d / r 0 ) 8 / 3 ( f g r 0 f d ) - 1 S 1 ( D d ) ] .
σ 2 ( d / r 0 ) 5 / 3 = e 2 ( D d , f g r 0 f d , σ n ( d / r 0 ) 4 / 3 ) .
f ( U ) = i = 1 n max 1 k n c { [ U T M ( k ) U ] i i } .
f ( U ) = i = 1 r u i T F u i + i = r + 1 n u i T G u i ,
U U T = U 1 U 1 T + U 2 U 2 T = I .
tr ( M N ) = tr ( N M ) ,             tr ( M + N ) = tr ( M ) + tr ( N ) ,
f ( U ) = tr ( U 1 T F U 1 ) + tr ( U 2 T G U 2 ) = tr ( F U 1 U 1 T ) + tr ( G U 2 U 2 T ) = tr ( F U 1 U 1 T ) + tr [ G ( I - U 1 U 1 T ) ] = tr ( F U 1 U 1 T ) + tr ( G - G U 1 U 1 T ) = tr ( F U 1 U 1 T - G U 1 U 1 T ) + tr ( G ) = tr [ ( F - G ) U 1 U 1 T ] + tr ( G ) = tr [ U 1 T ( F - G ) U 1 ] + tr ( G ) .
λ 1 ( A ) λ n ( A ) .
i = 1 r q i T A q i = tr ( Q r T A Q r ) i = 1 r λ i ( A ) .
λ k + 1 [ A ( k + 1 ) ] λ k [ A ( k ) ] λ k [ A ( k + 1 ) ] λ 1 [ A ( k ) ] λ 1 [ A ( k + 1 ) ] .
λ r [ A ( r ) ] λ r [ A ( n ) ] , , λ 1 [ A ( r ) ] λ 1 [ A ( n ) ] .
max U unitary f ( U ) = max U unitary i = 1 n max { ( U T M U ) i i , 0 } = λ i ( M ) > 0 λ i ( M ) .
i = 1 n max { ( U T M U ) i i , 0 } λ i ( M ) > 0 λ i ( M ) .
i = 1 r ( U T M U ) i i = tr ( U r T M U r ) i = 1 r λ i ( M ) λ i ( M ) > 0 λ i ( M ) ,

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