Abstract

An approach is presented for evaluating the performance achieved by a closed-loop adaptive-optics system that is employed with an astronomical telescope. This method applies to systems incorporating one or several guide stars, a wave-front reconstruction algorithm that is equivalent to a matrix multiply, and one or several deformable mirrors that are optically conjugate to different ranges. System performance is evaluated in terms of residual mean-square phase distortion and the associated optical transfer function. This evaluation accounts for the effects of the atmospheric turbulence Cn2(h) and wind profiles, the wave-front sensor and deformable-mirror fitting error, the sensor noise, the control-system bandwidth, and the net anisoplanatism for a given constellation of natural and/or laser guide stars. Optimal wave-front reconstruction algorithms are derived that minimize the telescope’s field-of-view-averaged residual mean-square phase distortion. Numerical results are presented for adaptive-optics configurations incorporating a single guide star and a single deformable mirror, multiple guide stars and a single deformable mirror, or multiple guide stars and two deformable mirrors.

© 1994 Optical Society of America

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References

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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. C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
    [CrossRef]
  3. J. F. Belsher, D. L. Fried, “Expected antenna gain when correcting tilt-free wavefronts,” (Optical Sciences Company, Placentia, Calif., 1984).
  4. M. Welsh, “Imaging performance analysis of adaptive telescopes using laser guide stars,” Appl. Opt. 30, 5021–5030 (1991).
    [CrossRef] [PubMed]
  5. D. L. Fried, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52–61 (1982).
    [CrossRef]
  6. R. Foy, A. Labeyrie, “Feasibility of adaptive telescopes using laser probe,” Astron. Astrophys. 152, 129–131 (1985).
  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]
  8. J. M. Beckers, “Increasing the size of the isoplanatic patch within multiconjugate adaptive optics,” in Proceedings of European Southern Observatory Conference and Workshop on Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed., Vol. 30 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1988), pp. 693–703.
  9. D. Johnston, B. Welsh, “Estimating contributions of turbulence layers to total wave-front phase aberration,” in Atmospheric Propagation and Remote Sensing, A. Kohnle, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1688, 510–521 (1992).
    [CrossRef]
  10. 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]
  11. R. Hudgin, “Wave-front compensation error due to finite corrector-element size,”J. Opt. Soc. Am. 67, 393–396 (1977).
    [CrossRef]
  12. E. P. Wallner, “Optimal wave-front correction using slope measurement,”J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  13. 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]
  14. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,”J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  15. J. Herrmann, “Least-squares wave-front errors of minimum norm,”J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  16. D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front compensation systems,”J. Opt. Soc. Am. 66, 193–206 (1976).
    [CrossRef]
  17. G. A. Tyler, “Turbulence-induced adaptive-optics performance evaluation: degradation in the time domain,” J. Opt. Soc. Am. A 1, 251–262 (1984).
    [CrossRef]
  18. Equations (2.1) and (2.2) imply that all modes of the wave-front-distortion profile must be compensated at the same control bandwidth. This simplification corresponds to the limitations of the existing closed-loop adaptive-optics systems with which we are familiar (see Ref. 1). More general approaches are possible but are not considered here.
  19. This objective represents a departure from previous studies of reconstruction algorithms for multiconjugate systems (see Ref. 9), which have instead developed reconstructors to estimate the contributions of individual atmospheric-turbulence layers to the total wave-front-distortion profile.
  20. Associated solutions for Λ and γare Λ = RQ(I− R−1AS−1G) (GTs−1G)−1and γ= A. These solutions are not unique, since the matrix (QT− I) is singular.
  21. The square root R1/2of a symmetric, positive-definite matrix Ris the quantity OTΛ1/2O, where the rows of the unitary matrix Oare the eigenvectors of Rand Λ is a diagonal matrix formed from the corresponding (positive) eigenvalues.
  22. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  23. R. J. Sasiela, “A unified approach to electromagnetic wave propagation in turbulence and the evaluation of multi-parameter integrals,” (Massachusetts Institute of Technology, Cambridge, Mass., 1988).
  24. B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics systems using slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
    [CrossRef]
  25. F. Olver, “Bessel functions of integer order,” in Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, eds. (Dover, New York, 1973), pp. 358–389.
  26. D. L. Fried, “Limited resolution looking down through the atmosphere,”J. Opt. Soc. Am. 56, 1380–1384 (1966).
    [CrossRef]
  27. R. R. Beland, J. H. Brown, R. E. Good, E. A. Murphy, “Optical turbulence characterization of AMOS,” (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1989).
  28. D. P. Greenwood, “Bandwidth specifications for adaptive optics systems,”J. Opt. Soc. Am. 67, 390–392 (1977).
    [CrossRef]
  29. J. Bahcall, R. Soniera, Astrophys. J. Suppl. 47, 357 (1981).
    [CrossRef]
  30. Y. Luke, “Integrals of Bessel functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, eds. (Dover, New York, 1973), pp. 479–494.
  31. F. Oberhettinger, “Hypergeometric functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, eds. (Dover, New York, 1973), pp. 555–566.
  32. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

1991 (5)

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]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (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]

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]

M. Welsh, “Imaging performance analysis of adaptive telescopes using laser guide stars,” Appl. Opt. 30, 5021–5030 (1991).
[CrossRef] [PubMed]

1989 (1)

1985 (1)

R. Foy, A. Labeyrie, “Feasibility of adaptive telescopes using laser probe,” Astron. Astrophys. 152, 129–131 (1985).

1984 (1)

1983 (1)

1982 (1)

1981 (1)

J. Bahcall, R. Soniera, Astrophys. J. Suppl. 47, 357 (1981).
[CrossRef]

1980 (1)

1978 (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1977 (4)

1976 (1)

1966 (1)

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]

Bahcall, J.

J. Bahcall, R. Soniera, Astrophys. J. Suppl. 47, 357 (1981).
[CrossRef]

Barclay, H. T.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Beckers, J. M.

J. M. Beckers, “Increasing the size of the isoplanatic patch within multiconjugate adaptive optics,” in Proceedings of European Southern Observatory Conference and Workshop on Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed., Vol. 30 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1988), pp. 693–703.

Beland, R. R.

R. R. Beland, J. H. Brown, R. E. Good, E. A. Murphy, “Optical turbulence characterization of AMOS,” (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1989).

Belsher, J. F.

J. F. Belsher, D. L. Fried, “Expected antenna gain when correcting tilt-free wavefronts,” (Optical Sciences Company, Placentia, Calif., 1984).

Boeke, B. R.

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]

Brown, J. H.

R. R. Beland, J. H. Brown, R. E. Good, E. A. Murphy, “Optical turbulence characterization of AMOS,” (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1989).

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]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

Foy, R.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescopes using laser probe,” Astron. Astrophys. 152, 129–131 (1985).

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, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52–61 (1982).
[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]

D. L. Fried, “Limited resolution looking down through the atmosphere,”J. Opt. Soc. Am. 56, 1380–1384 (1966).
[CrossRef]

J. F. Belsher, D. L. Fried, “Expected antenna gain when correcting tilt-free wavefronts,” (Optical Sciences Company, Placentia, Calif., 1984).

Fugate, R. Q.

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]

Gardner, C. S.

Good, R. E.

R. R. Beland, J. H. Brown, R. E. Good, E. A. Murphy, “Optical turbulence characterization of AMOS,” (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1989).

Graves, J. E.

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]

Greenwood, D. P.

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Herrmann, J.

Hudgin, R.

Hudgin, R. H.

Johnston, D.

D. Johnston, B. Welsh, “Estimating contributions of turbulence layers to total wave-front phase aberration,” in Atmospheric Propagation and Remote Sensing, A. Kohnle, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1688, 510–521 (1992).
[CrossRef]

Labeyrie, A.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescopes using laser probe,” Astron. Astrophys. 152, 129–131 (1985).

Luke, Y.

Y. Luke, “Integrals of Bessel functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, eds. (Dover, New York, 1973), pp. 479–494.

Murphy, D. V.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Murphy, E. A.

R. R. Beland, J. H. Brown, R. E. Good, E. A. Murphy, “Optical turbulence characterization of AMOS,” (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1989).

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]

Oberhettinger, F.

F. Oberhettinger, “Hypergeometric functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, eds. (Dover, New York, 1973), pp. 555–566.

Olver, F.

F. Olver, “Bessel functions of integer order,” in Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, eds. (Dover, New York, 1973), pp. 358–389.

Page, D. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

Primmerman, C. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (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, F.

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]

Ruane, R. E.

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.

R. J. Sasiela, “A unified approach to electromagnetic wave propagation in turbulence and the evaluation of multi-parameter integrals,” (Massachusetts Institute of Technology, Cambridge, Mass., 1988).

Soniera, R.

J. Bahcall, R. Soniera, Astrophys. J. Suppl. 47, 357 (1981).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

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]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

Wallner, E. P.

Welsh, B.

D. Johnston, B. Welsh, “Estimating contributions of turbulence layers to total wave-front phase aberration,” in Atmospheric Propagation and Remote Sensing, A. Kohnle, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1688, 510–521 (1992).
[CrossRef]

Welsh, B. M.

Welsh, 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]

Zollars, B. G.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (1)

R. Foy, A. Labeyrie, “Feasibility of adaptive telescopes using laser probe,” Astron. Astrophys. 152, 129–131 (1985).

Astrophys. J. Suppl. (1)

J. Bahcall, R. Soniera, Astrophys. J. Suppl. 47, 357 (1981).
[CrossRef]

J. Opt. Soc. Am. (9)

J. Opt. Soc. Am. A (3)

Nature (London) (2)

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]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Proc. IEEE (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[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 (13)

R. J. Sasiela, “A unified approach to electromagnetic wave propagation in turbulence and the evaluation of multi-parameter integrals,” (Massachusetts Institute of Technology, Cambridge, Mass., 1988).

F. Olver, “Bessel functions of integer order,” in Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, eds. (Dover, New York, 1973), pp. 358–389.

Y. Luke, “Integrals of Bessel functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, eds. (Dover, New York, 1973), pp. 479–494.

F. Oberhettinger, “Hypergeometric functions,” in Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, eds. (Dover, New York, 1973), pp. 555–566.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

R. R. Beland, J. H. Brown, R. E. Good, E. A. Murphy, “Optical turbulence characterization of AMOS,” (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1989).

J. F. Belsher, D. L. Fried, “Expected antenna gain when correcting tilt-free wavefronts,” (Optical Sciences Company, Placentia, Calif., 1984).

J. M. Beckers, “Increasing the size of the isoplanatic patch within multiconjugate adaptive optics,” in Proceedings of European Southern Observatory Conference and Workshop on Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed., Vol. 30 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1988), pp. 693–703.

D. Johnston, B. Welsh, “Estimating contributions of turbulence layers to total wave-front phase aberration,” in Atmospheric Propagation and Remote Sensing, A. Kohnle, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1688, 510–521 (1992).
[CrossRef]

Equations (2.1) and (2.2) imply that all modes of the wave-front-distortion profile must be compensated at the same control bandwidth. This simplification corresponds to the limitations of the existing closed-loop adaptive-optics systems with which we are familiar (see Ref. 1). More general approaches are possible but are not considered here.

This objective represents a departure from previous studies of reconstruction algorithms for multiconjugate systems (see Ref. 9), which have instead developed reconstructors to estimate the contributions of individual atmospheric-turbulence layers to the total wave-front-distortion profile.

Associated solutions for Λ and γare Λ = RQ(I− R−1AS−1G) (GTs−1G)−1and γ= A. These solutions are not unique, since the matrix (QT− I) is singular.

The square root R1/2of a symmetric, positive-definite matrix Ris the quantity OTΛ1/2O, where the rows of the unitary matrix Oare the eigenvectors of Rand Λ is a diagonal matrix formed from the corresponding (positive) eigenvalues.

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Figures (20)

Fig. 1
Fig. 1

Unfolded, foreshortened optical schematic of a multiconjugate adaptive-optics system with two guide stars and two deformable mirrors. CP1 and CP2 are the atmospheric layers that are conjugate to the deformable-mirror locations DM1 and DM2. WFS’s, wave-front sensors.

Fig. 2
Fig. 2

Adaptive-optics system control-loop dynamics.

Fig. 3
Fig. 3

Sample atmospheric turbulence Cn2(h) profile. This profile is derived from U.S. Air Force Geophysics Laboratory thermosonde data recorded on December 11, 1985. Integrating this profile yields the parameters r0 = 0.285 m and θ0 = 18.6 μrad for a wavelength of 0.5 μm and a zenith angle of 0°.

Fig. 4
Fig. 4

Atmospheric-wind-speed profile. This profile was recorded simultaneously with Fig. 3. The associated Greenwood frequency is 19.7 Hz at λ = 0.5 μm.

Fig. 5
Fig. 5

Actuator–subaperture geometries evaluated for fitting-error coefficients. This figure illustrates (a) the Fried and (b) the Hudgin actuator–subaperture geometries with eight subapertures/aperture diameter. The large open circles represent the telescope aperture, and the actuator locations are indicated by the dots. Wave-front-sensor subapertures and the gradient components that are measured are indicated by the small squares and vectors. The edge subapertures are truncated by the boundary of the telescope aperture. The small gaps between the subapertures appear only for purposes of illustration.

Fig. 6
Fig. 6

Fitting-error results for minimal-variance reconstructors with and without closed-loop constraints. Here D is the telescope-aperture diameter, L is the width of a subaperture, and the mean-square phase error resulting from fitting error is cF(D/r0)5/3.

Fig. 7
Fig. 7

Mean-square tilt-included wave-front reconstruction error versus D/L and wave-front-sensor noise level. Here D is the telescope-aperture diameter, L is the width of a subaperture, the sensor noise is expressed in terms of rms waves of phase-difference measurement accuracy, and 2 is the mean-square residual phase error resulting from both noise and fitting error.

Fig. 8
Fig. 8

Mean-square tilt-removed wave-front reconstruction error versus D/L and wave-front-sensor noise level. This figure is identical to Fig. 7, except that full-aperture wave-front tilt is not included in calculating the phase variance 2.

Fig. 9
Fig. 9

Short-exposure OTF’s resulting from anisoplanatism and fitting error at a 0.5-μm wavelength for D = 4 m, L = 0.25 m, ψ = 0°, and the atmospheric profiles shown in Figs. 3 and 4. These results assume zero wave-front-sensor measurement noise and an infinite wave-front control-loop bandwidth.

Fig. 10
Fig. 10

Effect of noise and finite servo bandwidth on the short-exposure OTF for a natural on-axis guide star. These curves plot the ratios between short-exposure OTF’s, including noise and finite-bandwidth effects and the noise-free, infinite-bandwidth OTF that is plotted in Fig. 9. Note that the wave-front-sensor (WFS) noise level is specified at a wave-front-sensor sampling rate that is ten times larger than the control-loop bandwidth.

Fig. 11
Fig. 11

Effect of noise and finite servo bandwidth on the short-exposure OTF for a resonant sodium guide star at z = 90 km. These curves assume a guide star in the mesospheric sodium layer but are otherwise similar to Fig. 10. WFS, wave-front sensor.

Fig. 12
Fig. 12

Effect of noise and finite servo bandwidth on the short-exposure OTF for a Rayleigh-backscatter guide star at z = 20 km. These curves assume a guide-star altitude of 20 km but are otherwise similar to Fig. 11. WFS, wave-front sensor.

Fig. 13
Fig. 13

Short-exposure OTF’s resulting from anisoplanatism and fitting error for three multiple-guide-star constellations. These results are again for the parameter values D = 4 m, L = 0.25 m, ψ = 0° and the atmospheric profiles shown in Figs. 3 and 4. These results also assume zero wave-front-sensor measurement noise and an infinite control-loop bandwidth. NSA is the number of subapertures for the sodium-guide-star wave-front sensor. The Rayleigh-guide-star altitude is 20 km. Each guide star is sensed over a quadrant of the telescope aperture for the case of four Rayleigh guide stars.

Fig. 14
Fig. 14

Effect of guide-star offset on the long-exposure OTF for a single natural guide star. These results are again for the parameter values D = 4 m, L = 0.25 m, ψ = 0°, and the atmospheric profiles shown in Figs. 3 and 4. These results also assume zero wave-front-sensor measurement noise and an infinite control-loop bandwidth.

Fig. 15
Fig. 15

Effect of tracking-guide-star offset on the long-exposure OTF for an on-axis resonant sodium guide star at z = 90 km. These curves are for the case of a mesospheric-sodium-layer guide star but are otherwise similar to Fig. 14.

Fig. 16
Fig. 16

Effect of tracking-guide-star offset on long-exposure OTF for an on-axis Rayleigh-backscatter guide star at z = 20 km. These curves are for the case of a 20-km guide-star altitude but are otherwise similar to Fig. 14.

Fig. 17
Fig. 17

FOV and guide-star geometries for multiconjugate adaptive-optics calculations. The telescope FOV that is to be compensated is a square that is 100 μrad in width. The five circles indicate the directions of natural and/or laser guide stars used for wave-front sensing. The wave-front reconstruction algorithm is selected to minimize the weighted sum of residual mean-square phase errors at nine points in the FOV, as indicated by the points and the weights.

Fig. 18
Fig. 18

Long-exposure OTF’s for a multiconjugate configuration with five natural guide stars. These results assume the parameter values D = 3 m, L = 0.25 m, ψ = 0°, and λ = 0.5 μm and the atmospheric profiles illustrated in Figs. 3 and 4. Deformable mirrors are located conjugate to altitudes of 0 and 5 km, and the interactuator spacing is 0.25 m for both mirrors. These results also assume zero wave-front-sensor measurement noise and an infinite control-loop bandwidth.

Fig. 19
Fig. 19

Long-exposure OTF’s for a multiconjugate configuration with one on-axis natural guide star and four mesospheric-sodium guide stars. This figure is similar to Fig. 18, except that the four natural guide stars at the corners of the telescope’s FOV have been replaced by guide stars in the mesospheric sodium layer.

Fig. 20
Fig. 20

Discrete weights for x-wave-front slope measurements on (a) unobscured and (b) partially obscured wave-front-sensor subapertures. The phase values at the nine points are summed with the indicated weights to approximate the average x-wavefront tilt over the continuous subaperture. This approximation is exact for any wave-front quadratic in both x and y over the square that is bounded by the four corner points.

Tables (4)

Tables Icon

Table 1 Short-Exposure Strehl Ratio versus Control-Loop Bandwidth (f) and Sensor Noise Level at λ = 0.5 μm for a 4-m-Aperture Telescope with 0.25-m Subapertures under the Atmospheric Conditions Given in Figs. 3 and 4

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Table 2 Relative Reductions in Long-Exposure Strehl Ratio Resulting from Finite Servo Bandwidth (f) and Sensor Noise Level for a Multiconjugate Adaptive-Optics Systema

Tables Icon

Table 3 Gaussian Quadrature Weights, Altitudes, and Wind Speeds for Altitude Integrations Weighted by Cn2(z)

Tables Icon

Table 4 Discrete Weights for Temporal Integrals Weighted by k[exp(−)]

Equations (93)

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e ( t ) = M [ y ( t ) - G c ( t ) ] ,
d c d t = k e ( t ) ,
( x , θ ) = ϕ ( x , θ ) - i c i r i ( x , θ ) ,
[ f , g ] = d θ W F ( θ ) d x W A ( x ) f ( x , θ ) g ( x , θ ) .
d x W A ( x ) = 1 ,
d θ W F ( θ ) = 1.
ϕ ˜ ( x , θ ) = ϕ ( x , θ ) - i f i ( x ) d x W A ( x ) f i ( x ) ϕ ( x , θ ) .
d x W A ( x ) f i ( x ) f j ( x ) = { 1 if i = j 0 otherwise .
ϕ ˜ = P ϕ .
( H c ) ( x , θ ) = i c i r i ( x , θ ) .
[ c , c ] [ P H c , P H c ] .
[ c , c ] = c T R c ,
R i j = [ P r i , P r j ] = d θ W F ( θ ) d x W A ( x ) r ˜ i ( x , θ ) r ˜ j ( x , θ ) .
= ϕ - H c ,
2 = [ P , P ] .
c ( t ) = 0 d τ k exp ( - k τ M G ) M y ( t - τ ) .
M G = Q ,
Q M = M ,
exp ( - k τ M G ) M = exp ( - k τ Q ) M = [ i = 0 ( - k τ Q ) i i ! ] M = Q M i = 0 ( - k τ ) i i ! = M exp ( - k τ ) .
c ( t ) = M s ( t ) ,
s ( t ) = 0 d τ k exp ( - k τ ) y ( t - τ ) .
2 = [ P ( ϕ - H M s ) , P ( ϕ - H M s ) ] = [ P ϕ , P ϕ ] - 2 i , j [ P ϕ , P r i ] M i j s j + i , j i , j R i i M i j M i j s j s j
2 = 0 2 - 2 i , j M i j A i j + i , j i , j R i i M i j M i j S j j ,
0 2 = [ P ϕ , P ϕ ] ,
A i j = [ P ϕ , P r i ] s j ,
S i j = s i s j .
- A i j + i , j M i j R i i S j j = j λ j j G j j + i γ i j ( Q - I ) i i ,
- A + R M S = Λ G T + ( Q T - I ) Γ .
M = Q [ R - 1 A S - 1 + ( I - R - 1 A S - 1 G ) ( G T S - 1 G ) - 1 G T S - 1 ]
2 = 0 2 - tr ( M A T + A M T - M S M T R ) .
tr ( U V T ) = tr ( V T U ) ,
2 = 0 2 - tr ( B R ) ,
B = M A T R - 1 + R - 1 A M T - M S M T .
R 1 / 2 B R 1 / 2 = O T Λ O .
O O T = O T O = I ,
Λ = diag ( λ 1 , , λ n ) .
2 = 0 2 - tr ( B R ) = 0 2 - tr ( O T Λ O ) = 0 2 - tr ( O O T Λ ) = 0 2 - tr ( Λ )
l i = { 0 if λ i 0 1 otherwise ,
L = diag ( l 1 , , l n ) ,
Q * = R - 1 / 2 O T L O R 1 / 2 .
M * = Q * M .
Q * 2 = Q * ,
Q * T R = R Q * .
2 = 0 2 - tr ( M * A T + A M * T - M * S M * T R ) = 0 2 - tr [ O T L O R 1 / 2 ( M A T R - 1 + R - 1 A M T - M S M T ) R 1 / 2 ] = 0 2 - tr ( O T L O O T Λ O ) = 0 2 - tr ( L Λ )
ϕ ˜ ( x , θ , t ) = ϕ ( x , θ , t ) - i f i ( x ) d x W A ( x ) f i ( x ) ϕ ( x , θ , t ) .
ϕ ( x , θ , t ) = 2 π λ 0 z 0 d z n ( x + z θ , z , t ) .
n ( x , z , t ) = n 0 ( x - t v , z ) ,
Φ n ( κ , z ) n ^ 0 ( κ , z ) 2 = 9.69 × 10 - 3 C n 2 ( z ) κ - 11 / 3 ,
s ( t ) = 0 d τ k exp ( - k τ ) y ( t - τ ) ,
y i ( t ) = d x W i s ( x ) ϕ i ( x , t ) + α i ( t ) ,
α i ( t ) α i ( t ) = δ i i δ ( t - t ) P i .
ϕ i ( x , t ) = 2 π λ 0 z i d z n [ x + ( z z i ) ( p i - x ) , z , t ] ,
u = 0 d τ w ( τ ) { d x v ( X ) ( 2 π λ ) × 0 z i d ζ n [ x + ( ζ z i ) ( p - x ) , ζ , t - τ ] + α ( t - τ ) } .
w ( τ ) = { δ ( τ ) if u = ϕ ˜ ( x , θ , t ) k exp ( - k τ ) if u = s i ( t ) ,
v ( x ) = { W A ( x ) [ δ ( x - x ) - i f i ( x ) f i ( x ) ] if u = ϕ ˜ ( x , θ , t ) W i s ( x ) if u = s i ( t ) ,
p = { z 0 θ if u = ϕ ˜ ( x , θ , t ) p i if u = s i ( t ) ,
α ( τ ) = { 0 if u = ϕ ˜ ( x , θ , t ) α i ( τ ) if u = s i ( t ) .
d x v ( x ) = 0.
r i ( x , θ ) = h i ( x + d i θ ) .
R i j = [ P r i , P r j ] = d θ W F ( θ ) d x W A ( x ) × [ h i ( x + d i θ ) - k f k ( x ) d x W A ( x ) f k ( x ) × h i ( x + d i θ ) ] × [ h i ( x + d i θ ) - k f k ( x ) d x W A ( x ) f k ( x ) × h i ( x + d i θ ) ] ,
G i j = 2 π λ d x W j s ( x ) h i [ x + ( d i z j ) ( p j - x ) ] .
0 2 = [ P ϕ , P ϕ ] = d θ W F ( θ ) d x W A ( x ) [ ϕ ˜ ( x , θ , t ) ] 2 ,
A i j = [ P ϕ , P r i ] s j ( t ) = d θ W F ( θ ) d x W A ( x ) ϕ ˜ ( x , θ , t ) s j ( t ) × [ h i ( x + d i θ ) - k f k ( x ) d x W A ( x ) f k ( x ) × h i ( x + d i θ ) ] ,
S i j = s i ( t ) s j ( t ) .
u i u j = ( 2 π λ ) 2 d x 1 d x 2 v i ( x 1 ) v j ( x 2 ) × 0 0 d τ 1 d τ 2 w i ( τ 1 ) w j ( τ 2 ) × 0 z i 0 z i d ζ 1 d ζ 2 n [ x 1 + ( ζ 1 z i ) , ζ 1 , t - τ 1 ] × n [ x 2 + ( ζ 2 z j ) , ζ 2 , t - τ 2 ] + 0 0 d τ 1 d τ 2 w i ( τ 1 ) w j ( τ 2 ) α i ( t - τ 1 ) α j ( t - τ 2 ) .
u i u j = 9.69 × 10 - 3 ( 2 π λ ) 2 d x 1 d x 2 v i ( x 1 ) v j ( x 2 ) × 0 0 d τ 1 d τ 2 w i ( τ 1 ) w j ( τ 2 ) 0 min ( z i , z j ) d ζ C n 2 ( ζ ) × { d κ κ - 11 / 3 exp ( - 2 π i κ · Δ ) × exp [ - 2 π i ( τ 1 - τ 2 ) κ · v ] + c } + δ i j P i 0 d τ w i 2 ( τ ) .
Δ = x 1 - x 2 + ( ζ z i ) ( p i - x 1 ) - ( ζ z j ) ( p j - x 2 ) .
c = - 9.69 × 10 - 3 2 π 0 d κ κ - 8 / 3 J 0 ( 2 π τ 1 - τ 2 κ v )
u i u j = 2 π 9.69 × 10 - 3 ( 2 π λ ) 2 d x 1 d x 2 v i ( x 1 ) v j ( x 2 ) × - d δ [ 0 d τ w i ( τ + δ / 2 ) w j ( τ - δ / 2 ] × 0 min ( z i , z j ) d ζ C n 2 ( ζ ) 0 d κ κ - 8 / 3 J 0 ( 2 π δ κ v ) × [ J 0 ( 2 π κ Δ ) - 1 ] + δ i j P i 0 d τ w i 2 ( τ ) .
1 = r 0 - 5 / 3 [ 2.91 6.88 ( 2 π λ ) 2 0 d ζ C n 2 ( ζ ) ] - 1 ,
u i u j = 0.97 ( D r 0 ) 5 / 3 [ 0 d ζ C n 2 ( ζ ) ] - 1 d x 1 d x 2 v i ( x 1 ) v j ( x 2 ) × - d δ [ 0 d τ w i ( τ + δ / 2 ) w j ( τ - δ / 2 ) ] × 0 min ( z i , z j ) d ζ C n 2 ( ζ ) f ( 2 δ v D , 2 Δ D ) + δ i j P i 0 d τ w i 2 ( τ ) ,
f ( a , b ) = 0 d ν ν - 8 / 3 J 0 ( a ν ) [ J 0 ( b ν ) - 1 ] .
OTF ( κ , θ ) = d x W A ( x ) W A ( x - λ κ ) exp [ - ½ D ( x , x - λ κ , θ ) ] d x W A 2 ( x ) ,
D ( x , x , θ ) = 2 ( x , θ ) + 2 ( x , θ ) - 2 ( x , θ ) ( x , θ ) ,
( x , θ ) ( x , θ ) = [ ϕ ˜ ( x , θ ) - i r ˜ i ( x , θ ) j M i j s j ] × [ ϕ ˜ ( x , θ ) - i r ˜ i ( x , θ ) j M i j s j ] = ϕ ˜ ( x , θ ) ϕ ˜ ( x , θ ) - i r ˜ i ( x , θ ) j M i j ϕ ˜ ( x , θ ) s j - i r ˜ i ( x , θ ) j M i j ϕ ˜ ( x , θ ) s j + i i r ˜ i ( x , θ ) r ˜ i ( x , θ ) j j M i j M i j s j s j .
2 = c F ( L / r 0 ) 5 / 3 ,
2 / ( L / r 0 ) 5 / 3 = { 0.305 ( unconstrained estimator , Fried geometry ) 0.325 ( constrained estimator , Fried geometry ) 0.350 ( unconstrained estimator , Hudgin geometry ) 0.365 ( constrained estimator , Hudgin geometry ) .
u i u j ( L / r 0 ) 5 / 3 = 0.97 ( D L ) 5 / 3 [ 0 d ζ C n 2 ( ζ ) ] - 1 × d x 1 d x 2 v i ( x 1 ) v j ( x 2 ) × - d δ [ 0 d τ w i ( τ + δ / 2 ) w j ( τ - δ / 2 ) ] × 0 min ( z i , z j ) d ζ C n 2 ( ζ ) f ( 2 δ v D , 2 Δ D ) + δ i j P i 0 d τ w i 2 ( τ ) / ( L / r 0 ) 5 / 3 .
σ N 2 = ( C 1 + C 2 ln N S ) σ PD 2 ,
x T R 1 / 2 O T Λ O R 1 / 2 x = x T ( R M A T + A M T R + R M S M T R ) x = 0.
O T L O R 1 / 2 x = 0 ,
Q * x = 0.
f ( a , b ) = 0 d ν ν - 8 / 3 J 0 ( a ν ) [ J 0 ( b ν ) - 1 ] .
f ( a , b ) = - 3 5 ν - 5 / 3 J 0 ( a ν ) [ J 0 ( b ν ) - 1 ] 0 + 3 5 0 d ν ν - 5 / 3 [ a J 1 ( a ν ) - a J 1 ( a ν ) J 0 ( b ν ) - b J 1 ( b ν ) J 0 ( a ν ) ] .
f ( a , b ) = 3 5 [ a g ( 0 , a ) - a g ( b , a ) - b g ( a , b ) ] ,
g ( a , b ) = 0 d ν ν - 5 / 3 J 0 ( a ν ) J 1 ( b ν ) .
g ( a , b ) = { a 2 / 3 2 5 / 3 Γ ( 1 / 6 ) Γ ( 11 / 6 ) ( b a ) 2 / 3 F 2 1 [ 1 6 , - 5 6 ; 1 ; ( a b ) 2 ] if a b a 2 / 3 2 5 / 3 Γ ( 1 / 6 ) Γ ( 5 / 6 ) ( b a ) F 2 1 [ 1 6 , 1 6 ; 2 ; ( b a ) 2 ] otherwise ,
F 2 1 ( a , b ; c ; z ) = Γ ( c ) Γ ( a ) Γ ( b ) n = 0 Γ ( a + n ) Γ ( b + n ) Γ ( c + n ) z n n !
0 d z C n 2 ( z ) f ( z ) i = 0 M c i f ( z i ) .
0 d z C n 2 ( z ) z m = i = 0 M c i z i m
0 d τ w ( τ ) f ( t - τ ) i = 0 M w ( i Δ τ ) c i Δ τ f ( t - i Δ τ ) .
e 2 = [ 0 d τ ϕ ( x , θ , t - τ ) w ( τ ) - i = 0 M ϕ ( x , θ , t - i Δ τ ) w ( i Δ τ ) c i Δ τ ]
d x v ( x ) f ( x ) i , j v i j f ( i Δ x , j Δ x ) .

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