Abstract

With computational techniques developed in the investigation of high-energy laser-beam-control systems, a set of concise analytic models describing the essential properties of a laser-guide-star phase-conjugation system has been assembled. With the aid of these models an optimization strategy for mating adaptive optics to a 4-m-class optical telescope is evolved, and it is shown that such a system might be expected to improve the effective atmospheric seeing conditions by nearly a factor of 10 within the isoplanatic patch of the turbulence probe. For operation at visible wavelengths, a compensation system having ~300 actuators and a closed-loop bandwidth of 20 Hz is recommended. All the key hardware components have already been built and tested, with the exception of a suitable laser source for high-repetition-rate illumination of the Earth’s sodium layer.

© 1994 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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1994

1993

1992

B. G. Zollars, “Atmospheric-turbulence compensation experiments using synthetic beacons,” Lincoln Lab. J. 5, 67–92 (1992).

1991

R. A. Humphreys, C. A. Primmerman, L. C. Bradley, J. Herrmann, “Atmospheric-turbulence measurements using a synthetic beacon in the mesospheric sodium layer,” Opt. Lett. 16, 1367–1369 (1991).
[CrossRef] [PubMed]

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]

D. V. Murphy, C. A. Primmerman, D. A. Page, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 16, 1797–1799 (1991).
[CrossRef] [PubMed]

F. Roddier, M. Northcott, J. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (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, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

B. M. Welsh, L. A. Thompson, “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]

1990

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

J. C. Twichell, B. E. Burke, R. K. Reich, W. H. McGonagle, C. M. Huang, M. W. Bautz, J. P. Doty, G. R. Ricker, R. W. Mountain, V. S. Dolat, “Advanced CCD imager technology for use from 1 to 10000 Å,” Rev. Sci. Instrum. 61, 2744–2746 (1990).
[CrossRef]

1989

1988

1987

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
[CrossRef]

1985

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

1982

1980

1979

1978

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

1977

1976

1973

1966

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, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Arnold, B.

P. Johnson, R. Trissel, L. Cuellar, B. Arnold, D. Sandler, “Real time wavefront reconstruction for a 512 subaperture adaptive optical system,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1543, 460–471 (1991).
[CrossRef]

Azouit, M.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[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]

D. V. Murphy, C. A. Primmerman, D. A. Page, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 16, 1797–1799 (1991).
[CrossRef] [PubMed]

Bautz, M. W.

J. C. Twichell, B. E. Burke, R. K. Reich, W. H. McGonagle, C. M. Huang, M. W. Bautz, J. P. Doty, G. R. Ricker, R. W. Mountain, V. S. Dolat, “Advanced CCD imager technology for use from 1 to 10000 Å,” Rev. Sci. Instrum. 61, 2744–2746 (1990).
[CrossRef]

Beland, S.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Belsher, J. F.

J. F. Belsher, D. L. Fried, “Adaptive optics mirror fitting error: analysis and results,” (Optical Sciences Company, Placentia, Calif., 1983).

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, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Bradley, L. C.

Brailove, A. A.

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, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Bufton, J. L.

Burke, B. E.

J. C. Twichell, B. E. Burke, R. K. Reich, W. H. McGonagle, C. M. Huang, M. W. Bautz, J. P. Doty, G. R. Ricker, R. W. Mountain, V. S. Dolat, “Advanced CCD imager technology for use from 1 to 10000 Å,” Rev. Sci. Instrum. 61, 2744–2746 (1990).
[CrossRef]

Caccia, J. L.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Cowie, L.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Cowley, D.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Cuellar, L.

P. Johnson, R. Trissel, L. Cuellar, B. Arnold, D. Sandler, “Real time wavefront reconstruction for a 512 subaperture adaptive optical system,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1543, 460–471 (1991).
[CrossRef]

Dolat, V. S.

J. C. Twichell, B. E. Burke, R. K. Reich, W. H. McGonagle, C. M. Huang, M. W. Bautz, J. P. Doty, G. R. Ricker, R. W. Mountain, V. S. Dolat, “Advanced CCD imager technology for use from 1 to 10000 Å,” Rev. Sci. Instrum. 61, 2744–2746 (1990).
[CrossRef]

Doty, J. P.

J. C. Twichell, B. E. Burke, R. K. Reich, W. H. McGonagle, C. M. Huang, M. W. Bautz, J. P. Doty, G. R. Ricker, R. W. Mountain, V. S. Dolat, “Advanced CCD imager technology for use from 1 to 10000 Å,” Rev. Sci. Instrum. 61, 2744–2746 (1990).
[CrossRef]

Everson, J. H.

J. H. Everson, “New developments in deformable mirror surface devices,” in Adaptive Optical Components I, S. Holly, L. James, eds., Proc. Soc. Photo-Opt. Instrum. Eng.141, 11–15 (1978).
[CrossRef]

Feinleib, J.

J. Feinleib, proposal 82-P4 (Adaptive Optics Associates, Cambridge, Mass., 1982).

Foy, R.

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

R. Foy, M. Tallon, “ATLAS experiment to test the laser probe technique for wavefront measurements,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 174–183 (1989).
[CrossRef]

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, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

G. A. Tyler, D. L. Fried, “Image-position error associated with a quadrant detector,”J. Opt. Soc. Am. 72, 804–808 (1982).
[CrossRef]

D. L. Fried, “Least-square 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-compensative systems,”J. Opt. Soc. Am. 66, 193–206 (1976).
[CrossRef]

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

J. F. Belsher, D. L. Fried, “Adaptive optics mirror fitting error: analysis and results,” (Optical Sciences Company, Placentia, Calif., 1983).

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, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Gallahan, G. M.

J. E. Harvey, G. M. Gallahan, “Wavefront error compensation capabilities of multi-actuator deformable mirrors,” in Adaptive Optical Components I, S. Holly, L. James, eds., Proc. Soc. Photo-Opt. Instrum. Eng.141, 50–57 (1978).
[CrossRef]

Gardner, C. S.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Nonlinear resonant absorption effects on the design of resonance fluorescence lidars and laser guide stars,” Appl. Opt. 28, 4141–4153 (1989).
[CrossRef] [PubMed]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
[CrossRef]

B. M. Welsh, C. S. Gardner, L. A. Thompson, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
[CrossRef]

L. A. Thompson, C. S. Gardner, “Excimer laser guide star techniques for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 184–190 (1989).
[CrossRef]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Sodium laser guide star technique for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 191–202 (1989).
[CrossRef]

Graves, J.

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

Graves, J. E.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Greenwood, D. P.

Hardy, J.

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

Harvey, J. E.

J. E. Harvey, G. M. Gallahan, “Wavefront error compensation capabilities of multi-actuator deformable mirrors,” in Adaptive Optical Components I, S. Holly, L. James, eds., Proc. Soc. Photo-Opt. Instrum. Eng.141, 50–57 (1978).
[CrossRef]

Herrmann, J.

Hill, S.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Huang, C. M.

J. C. Twichell, B. E. Burke, R. K. Reich, W. H. McGonagle, C. M. Huang, M. W. Bautz, J. P. Doty, G. R. Ricker, R. W. Mountain, V. S. Dolat, “Advanced CCD imager technology for use from 1 to 10000 Å,” Rev. Sci. Instrum. 61, 2744–2746 (1990).
[CrossRef]

Hudgin, R. H.

Hufnagel, R. E.

R. E. Hufnagel, “Variations of atmospheric turbulence,” in Digest of Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974), paper WA1.

Humphreys, R. A.

Jeys, T. H.

Johnson, P.

P. Johnson, R. Trissel, L. Cuellar, B. Arnold, D. Sandler, “Real time wavefront reconstruction for a 512 subaperture adaptive optical system,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1543, 460–471 (1991).
[CrossRef]

Kolmogorov, A.

A. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ [sic] numbers,” in Turbulence, Classic Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Interscience, New York, 1961), pp. 151–155.

Labeyrie, A.

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

Limburg, E.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

McGonagle, W. H.

J. C. Twichell, B. E. Burke, R. K. Reich, W. H. McGonagle, C. M. Huang, M. W. Bautz, J. P. Doty, G. R. Ricker, R. W. Mountain, V. S. Dolat, “Advanced CCD imager technology for use from 1 to 10000 Å,” Rev. Sci. Instrum. 61, 2744–2746 (1990).
[CrossRef]

McKenna, D.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Measures, R. M.

R. M. Measures, Laser Remote Sensing Fundamentals and Applications (Wiley, New York, 1984), Chap. 7, p. 241.

Miller, M. G.

M. G. Miller, P. L. Zieske, “Turbulence environmental characterization,” (Rome Air Development Center, Griffiss Air Force Base, N.Y., 1979).

Mooney, J. G.

R. J. Sasiela, J. G. Mooney, “An optical phase reconstructor based on using a multiplier-accumulator approach,” in Adaptive Optics, J. E. Ludman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.551, 170–176 (1985).
[CrossRef]

Mooradian, A.

Mountain, R. W.

J. C. Twichell, B. E. Burke, R. K. Reich, W. H. McGonagle, C. M. Huang, M. W. Bautz, J. P. Doty, G. R. Ricker, R. W. Mountain, V. S. Dolat, “Advanced CCD imager technology for use from 1 to 10000 Å,” Rev. Sci. Instrum. 61, 2744–2746 (1990).
[CrossRef]

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]

D. V. Murphy, C. A. Primmerman, D. A. Page, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 16, 1797–1799 (1991).
[CrossRef] [PubMed]

Northcott, M.

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

Page, D. A.

D. V. Murphy, C. A. Primmerman, D. A. Page, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 16, 1797–1799 (1991).
[CrossRef] [PubMed]

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]

Poularikas, A. D.

A. D. Poularikas, S. Seely, Signals and Systems (PSW Boston, Mass., 1985), Chap. 6.

Primmerman, C. A.

Reich, R. K.

J. C. Twichell, B. E. Burke, R. K. Reich, W. H. McGonagle, C. M. Huang, M. W. Bautz, J. P. Doty, G. R. Ricker, R. W. Mountain, V. S. Dolat, “Advanced CCD imager technology for use from 1 to 10000 Å,” Rev. Sci. Instrum. 61, 2744–2746 (1990).
[CrossRef]

Ricker, G. R.

J. C. Twichell, B. E. Burke, R. K. Reich, W. H. McGonagle, C. M. Huang, M. W. Bautz, J. P. Doty, G. R. Ricker, R. W. Mountain, V. S. Dolat, “Advanced CCD imager technology for use from 1 to 10000 Å,” Rev. Sci. Instrum. 61, 2744–2746 (1990).
[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, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Roddier, C.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Roddier, F.

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

F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
[CrossRef] [PubMed]

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[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, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Salmon, D.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Sandler, D.

P. Johnson, R. Trissel, L. Cuellar, B. Arnold, D. Sandler, “Real time wavefront reconstruction for a 512 subaperture adaptive optical system,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1543, 460–471 (1991).
[CrossRef]

Sasiela, R. J.

Seely, S.

A. D. Poularikas, S. Seely, Signals and Systems (PSW Boston, Mass., 1985), Chap. 6.

Shelton, J. D.

Songaila, A.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Tallon, M.

R. Foy, M. Tallon, “ATLAS experiment to test the laser probe technique for wavefront measurements,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 174–183 (1989).
[CrossRef]

Thompson, L. A.

B. M. Welsh, L. A. Thompson, “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]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
[CrossRef]

L. A. Thompson, C. S. Gardner, “Excimer laser guide star techniques for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 184–190 (1989).
[CrossRef]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Sodium laser guide star technique for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 191–202 (1989).
[CrossRef]

B. M. Welsh, C. S. Gardner, L. A. Thompson, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
[CrossRef]

Trissel, R.

P. Johnson, R. Trissel, L. Cuellar, B. Arnold, D. Sandler, “Real time wavefront reconstruction for a 512 subaperture adaptive optical system,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1543, 460–471 (1991).
[CrossRef]

Twichell, J. C.

J. C. Twichell, B. E. Burke, R. K. Reich, W. H. McGonagle, C. M. Huang, M. W. Bautz, J. P. Doty, G. R. Ricker, R. W. Mountain, V. S. Dolat, “Advanced CCD imager technology for use from 1 to 10000 Å,” Rev. Sci. Instrum. 61, 2744–2746 (1990).
[CrossRef]

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, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

G. A. Tyler, D. L. Fried, “Image-position error associated with a quadrant detector,”J. Opt. Soc. Am. 72, 804–808 (1982).
[CrossRef]

G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” (Optical Sciences Company, Placentia, Calif., 1988).

Vernin, V.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

Welsh, B. M.

B. M. Welsh, L. A. Thompson, “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]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Nonlinear resonant absorption effects on the design of resonance fluorescence lidars and laser guide stars,” Appl. Opt. 28, 4141–4153 (1989).
[CrossRef] [PubMed]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[CrossRef]

B. M. Welsh, C. S. Gardner, L. A. Thompson, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
[CrossRef]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Sodium laser guide star technique for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 191–202 (1989).
[CrossRef]

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, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Yura, H. T.

Zieske, P. L.

M. G. Miller, P. L. Zieske, “Turbulence environmental characterization,” (Rome Air Development Center, Griffiss Air Force Base, N.Y., 1979).

Zollars, B. G.

B. G. Zollars, “Atmospheric-turbulence compensation experiments using synthetic beacons,” Lincoln Lab. J. 5, 67–92 (1992).

D. V. Murphy, C. A. Primmerman, D. A. Page, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 16, 1797–1799 (1991).
[CrossRef] [PubMed]

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.

Astron. Astrophys.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Lincoln Lab. J.

B. G. Zollars, “Atmospheric-turbulence compensation experiments using synthetic beacons,” Lincoln Lab. J. 5, 67–92 (1992).

Nature (London)

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]

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, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
[CrossRef]

Opt. Lett.

Proc. IEEE

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

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Publ. Astron. Soc. Pac.

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

Rev. Sci. Instrum.

J. C. Twichell, B. E. Burke, R. K. Reich, W. H. McGonagle, C. M. Huang, M. W. Bautz, J. P. Doty, G. R. Ricker, R. W. Mountain, V. S. Dolat, “Advanced CCD imager technology for use from 1 to 10000 Å,” Rev. Sci. Instrum. 61, 2744–2746 (1990).
[CrossRef]

Other

J. Feinleib, proposal 82-P4 (Adaptive Optics Associates, Cambridge, Mass., 1982).

A. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ [sic] numbers,” in Turbulence, Classic Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Interscience, New York, 1961), pp. 151–155.

M. G. Miller, P. L. Zieske, “Turbulence environmental characterization,” (Rome Air Development Center, Griffiss Air Force Base, N.Y., 1979).

R. E. Hufnagel, “Variations of atmospheric turbulence,” in Digest of Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974), paper WA1.

F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, V. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 485–491 (1990).
[CrossRef]

R. Foy, M. Tallon, “ATLAS experiment to test the laser probe technique for wavefront measurements,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 174–183 (1989).
[CrossRef]

L. A. Thompson, C. S. Gardner, “Excimer laser guide star techniques for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 184–190 (1989).
[CrossRef]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Sodium laser guide star technique for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 191–202 (1989).
[CrossRef]

B. M. Welsh, C. S. Gardner, L. A. Thompson, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
[CrossRef]

A. D. Poularikas, S. Seely, Signals and Systems (PSW Boston, Mass., 1985), Chap. 6.

J. F. Belsher, D. L. Fried, “Adaptive optics mirror fitting error: analysis and results,” (Optical Sciences Company, Placentia, Calif., 1983).

J. H. Everson, “New developments in deformable mirror surface devices,” in Adaptive Optical Components I, S. Holly, L. James, eds., Proc. Soc. Photo-Opt. Instrum. Eng.141, 11–15 (1978).
[CrossRef]

J. E. Harvey, G. M. Gallahan, “Wavefront error compensation capabilities of multi-actuator deformable mirrors,” in Adaptive Optical Components I, S. Holly, L. James, eds., Proc. Soc. Photo-Opt. Instrum. Eng.141, 50–57 (1978).
[CrossRef]

R. J. Sasiela, J. G. Mooney, “An optical phase reconstructor based on using a multiplier-accumulator approach,” in Adaptive Optics, J. E. Ludman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.551, 170–176 (1985).
[CrossRef]

P. Johnson, R. Trissel, L. Cuellar, B. Arnold, D. Sandler, “Real time wavefront reconstruction for a 512 subaperture adaptive optical system,” in Active and Adaptive Optical Components, M. A. Ealey, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1543, 460–471 (1991).
[CrossRef]

G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” (Optical Sciences Company, Placentia, Calif., 1988).

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

W. L. Wolfe, G. J. Zissis, eds., The Infrared Handbook (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1985), Chap. 3, pp. 20–22.

E. D. Hinkley, ed., Laser Monitoring of the Atmosphere (Springer-Verlag, Berlin, 1976), Chap. 4, p. 76.

R. M. Measures, Laser Remote Sensing Fundamentals and Applications (Wiley, New York, 1984), Chap. 7, p. 241.

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Figures (24)

Fig. 1
Fig. 1

Comparison of the HV-21 model that is typically used for calculations of daytime turbulence (see also Section 1 of Appendix A) and the modified HV model that was developed by the authors to represent nighttime turbulence conditions at good seeing locations.

Fig. 2
Fig. 2

Bufton wind-velocity profile.

Fig. 3
Fig. 3

Essential components of an adaptive-optics phase-compensation system. Error signals that are derived from the phase difference between the incoming wave front and the surface figure applied by the compensation system are generated by the phase sensor and are subsequently applied to the deformable mirror. A separate tilt sensor is used to drive the tracking mirror.

Fig. 4
Fig. 4

Schematic description of a Hartmann wave-front sensor. The incoming wave front is divided into a matrix of subapertures by a lenslet array, which produces a set of focused spots on a detector array. The displacement of the spots from their local null position provides a measure of the local phase gradient.

Fig. 5
Fig. 5

Performance comparison of the two major types of Hartmann-sensor camera. This calculation was performed for a phase sensor having two pixels per subaperture subarray and a subaperture dimension that is smaller than r0.

Fig. 6
Fig. 6

Cross section of a typical deformable mirror with a flexible face sheet and a two-dimensional array of actuator stacks. The actuators are cemented to the face sheet, so that the surface can be deformed by either extension or contraction of the actuator dimension.

Fig. 7
Fig. 7

Servo-loop diagram for a digital control system with a boxcar-averaging device and a digital accumulator. The error signal at the wave-front sensor is averaged for a period of τd. The accumulator provides the deformable mirror with an impulse drive signal at the end of each integration interval, which is subsequently held until the next drive signal.

Fig. 8
Fig. 8

Turbulence transfer function for a quasi-cw closed-loop compensation system as a function of loop gain.

Fig. 9
Fig. 9

Noise transfer function for a quasi-cw closed-loop compensation system as a function of loop gain.

Fig. 10
Fig. 10

Power spectrum for turbulence-induced figure error. This model incorporates atmospheric moments that were developed from the modified HV turbulence profile and the Bufton wind profile.

Fig. 11
Fig. 11

Power spectrum for turbulence-induced tilt error, given in units of square radians of single-axis tilt per Hz. This model incorporates atmospheric moments that were developed from the modified HV turbulence profile and the Bufton wind profile.

Fig. 12
Fig. 12

Comparison of short-exposure beam profiles near the transition to strong phase distortion. This simulation was performed for an aperture diameter of 4 m and an r0 of 16 cm. (a) Represents a ds/r0 of 1.8, for which a Strehl of 0.2 was obtained. (b) Corresponds to a ds/r0 of 2.5 and a Strehl of 0.03. [The vertical scale for (b) has been multiplied by a factor of 5.]

Fig. 13
Fig. 13

Comparison of the predicted short-exposure resolution with far-field beam-profile measurements derived from a ray-trace simulation in which Kolmogorov phase screens were compensated with a zonal-type adaptive element having a variable actuator spacing. Each simulation result represents the average over five independent screens that were each adjusted to produce an r0 of 16 cm over a 4-m aperture.

Fig. 14
Fig. 14

Illustration of the two major sources of phase error that are introduced by laser-guide-star sources. The vertical rays represent light originating from a distant source that accumulates phase error as they travel through the atmosphere. The beacon rays follow slightly different paths to the receiver and are unable to sample the turbulence above the beacon or to sample the turbulence below the beacon properly.

Fig. 15
Fig. 15

Comparison of phase-variance contributions resulting from turbulence lying above and below the synthetic beacon. The shape of the low-altitude curve is complicated by the inclusion of a larger fraction of the atmosphere as the beacon is raised.

Fig. 16
Fig. 16

Multiple-beacon sampling geometry. Focal anisoplanatism is reduced by positioning a beacon over the center of each mirror section. Stitching errors occur when the relative source positions cannot be accurately determined and result in low-spatial-frequency figure distortions.

Fig. 17
Fig. 17

Figure error that is expected for the combination of un-sampled upper-altitude turbulence and aperture-section stitching with multiple beacons.

Fig. 18
Fig. 18

Resolution as a function of angular displacement between the target object and the fiducial star. This function is approximately linear between 50 and 250 μrad.

Fig. 19
Fig. 19

Optimization analysis of the phase-compensation system parameters as a function of figure variance. A total throughput value of 20% has been assumed in this calculation.

Fig. 20
Fig. 20

Performance analysis of the optimized figure- and tilt-compensation systems as a function of irradiance (referenced to the top of the atmosphere). A total throughput value of 20% has been assumed for both systems.

Fig. 21
Fig. 21

Comparison of stellar densities for the galactic pole, the galactic equator, and the average density as a function of visual magnitude.

Fig. 22
Fig. 22

Beacon-laser specifications for resonance backscatter from the Earth’s sodium layer. A two-way throughput of 4% has been assumed in this calculation.

Fig. 23
Fig. 23

Comparison of the corrected- and uncorrected-beam profiles for a design criteria that restricts the tilt jitter to twice the diffraction-limited beam diameter and achieves a signal-to-background ratio of 10. This plot represents the point-spread-function comparison for operation at 0.55 μm.

Fig. 24
Fig. 24

Error propagator for section tilts that are associated with a multiple-beacon system. The computed values can be approximated by the function 0.11(D/Ds)5/3.

Tables (3)

Tables Icon

Table 1 Summary of Turbulence Parameters for Imaging Systems Operating at 0.55 μm

Tables Icon

Table 2 Baseline System Parameters for Optimization Study

Tables Icon

Table 3 Summary of Compensation-System Design Specifications and Operational Utility for a Factor of 5 Improvement in the Long-Exposure Strehl for a 4-m Telescope

Equations (131)

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

D n ( r 1 , r 2 ) [ n ( r 1 ) - n ( r 2 ) ] 2 ,
D n ( r ) = C n 2 ( h ) r 2 / 3 .
D ϕ ( r ) = 2.91 k 2 r 5 / 3 sec ( ζ ) C n 2 ( h ) d h ,
D ϕ ( r ) = 6.88 ( r / r 0 ) 5 / 3 ,
r 0 = [ 0.423 k 2 sec ( ζ ) C n 2 ( h ) d h ] - 3 / 5 .
μ n = C n 2 ( h ) h n d h ,
r 0 = [ 0.423 k 2 sec ( ζ ) μ 0 ] - 3 / 5 .
θ 0 = [ 2.91 k 2 sec 8 / 3 ( ζ ) C n 2 ( h ) h 5 / 3 d h ] - 3 / 5 = [ 2.91 k 2 sec 8 / 3 ( ζ ) μ 5 / 3 ] - 3 / 5 .
v n = C n 2 ( h ) v n ( h ) d h ,
τ 0 = [ 2.91 k 2 sec ( ζ ) C n 2 ( h ) v 5 / 3 ( h ) d h ] - 3 / 5 = [ 2.91 k 2 sec ( ζ ) v 5 / 3 ] - 3 / 5 .
θ s = 1.22 λ r 0 ,
C n 2 ( h ) = 8.16 × 10 - 54 h 10 exp ( - h / 1000 ) + 3.02 × 10 - 17 × exp ( - h / 1500 ) + 1.90 × 10 - 15 exp ( - h / 100 ) .
v ( h ) = 5 + 30 exp { - [ ( h - 9400 ) / 4800 ] 2 } .
g = 2 π [ d s 2 λ f Γ ( 0 ) ] - Γ L + Γ R Γ L + Γ R             ( rad / subaperture ) ,
Γ ( x ) = ( d s λ f ) [ sin ( π d s x / λ f ) π d s x / λ f ] 2
d spot f λ d s { 1 + ( d s r 0 ) 2 [ 1 - 0.37 ( r 0 d s ) 1 / 3 ] } 1 / 2 f λ d s [ 1 + ( d s r 0 ) 2 ] 1 / 2 .
g = π [ 1 + ( d s r 0 ) 2 ] 1 / 2 - Γ L + Γ R Γ L + Γ R .
( σ g 2 ) pn 2 π 2 [ 1 + d s / r 0 ) 2 ] N pe ,
( σ g 2 ) sn 2 { 2 π G e N pe [ 1 + ( d s r 0 ) 2 ] 1 / 2 } 2 N rms 2 = 8 π 2 [ 1 + ( d s / r 0 ) 2 ] N rms 2 ( G e N pe ) 2 ,
N pe = ( 2 π h c k p ) η τ d d s 2 I p ,
( σ fig 2 ) noise π h c ( k c 2 / k p ) [ 1 + ( k p / k c ) 12 / 5 ( d s / r 0 ) 2 ] η τ d d s 2 I p photon noise + 2 ( h c k c ) 2 [ 1 + ( k p / k c ) 12 / 5 ( d s / r 0 ) 2 ] N rms 2 ( G e η τ d d s 2 I p ) 2 sensor noise .
2 ϕ = div ( g ) .
g = A ϕ ,
ϕ = [ ( A t A ) - 1 A ] g = B g .
( σ fig 2 ) fit 0.274 ( D / r 0 ) 5 / 3 N a - 5 / 6 .
( σ fig 2 ) fit 0.34 ( d s / r 0 ) 5 / 3 .
( σ fig 2 ) fit 0.5 ( d s / r 0 ) 5 / 3 .
( σ fig 2 ) servo = 0 T f ( f ) F f ( f ) d f turbulence - induced figure error + 0 T fn ( f ) F fn ( f ) d f figure noise             ( rad 2 of phase ) ,
( σ tilt 2 ) servo = 0 T t ( f ) F t ( f ) d f turbulence - induced tilt error + 0 T tn ( f ) F tn ( f ) d f tilt noise             ( rad 2 of tilt ) .
H s ( f ) = sin ( π f τ d ) π f τ d exp ( - i π f τ d ) = 1 - exp ( - i 2 π f τ d ) 2 π f τ d ,
H a ( f ) = g l 2 π f τ d ,
H m ( f ) = 1 i 2 π f τ m + 1 .
T ϕ ( f ) = | 1 1 + H s ( f ) H a ( f ) | 2 = ( 2 π f τ d ) 2 4 sin 2 ( π f τ d ) [ g 1 2 / ( 2 π f τ d ) 2 - g l ] + ( 2 π f τ d ) 2 .
T n ( f ) = | H s ( f ) H a ( f ) 1 + H s ( f ) H a ( f ) | 2 = 1 1 - ( 2 π f τ d ) 2 / g l + ( 2 π f τ d ) 4 / 4 g l 2 sin 2 ( π f τ d ) .
T ϕ ( f ) ( 2 π f τ d ) 2 + ( 2 π f τ d ) 4 / 12 1 + ( 2 π f τ d ) 4 / 12 ,
T n ( f ) 1 1 + ( 2 π f τ d ) 4 / 12 .
F f ( f ) = { 0.132 sec ( ζ ) k c 2 D 4 μ 0 12 / 5 v 5 / 3 - 7 / 5 f 4 / 3 f 0.705 D - 1 μ 0 - 3 / 5 v 5 / 3 3 / 5 0.0326 sec ( ζ ) k c 2 v 5 / 3 f - 8 / 3 f 0.705 D - 1 μ 0 - 3 / 5 v 5 / 3 3 / 5 .
F t ( f ) = { 1.60 sec ( ζ ) v - 1 / 3 f - 2 / 3 f 0.445 D - 1 v - 1 / 3 - 1 / 5 v 14 / 3 1 / 5 0.028 sec ( ζ ) D - 5 v 14 / 3 f - 17 / 3 f 0.445 D - 1 v - 1 / 3 - 1 / 5 v 14 / 3 1 / 5 ,
H avg ( f ) = [ sin ( π f τ d ) π f τ d ] 2 ,
F f n ( f ) = 2 τ d ( σ fig 2 ) noise = 2 π h c ( k c 2 / k p ) [ 1 + ( k p / k c ) 12 / 5 ( d s / r 0 ) 2 ] η d s 2 I p + 2 ( h c k c ) 2 [ 1 + ( k p / k c ) 12 / 5 ( d s / r 0 ) 2 ] N rms 2 ( G e η d s 2 I p ) 2 τ d ( rad 2 of phase / Hz ) .
F t n ( f ) = 8 h c [ 1 + ( k t / k c ) 12 / 5 ( D / r 0 ) 2 ] k t η D 4 I t + 32 ( h c ) 2 [ 1 + ( k t / k c ) 12 / 5 ( D / r 0 ) 2 ] N rms 2 π 2 D 6 ( G e η I t ) 2 τ dt ( rad 2 of tilt / Hz ) .
0 T f ( f ) F f ( f ) d f 0.0326 sec ( ζ ) k c 2 v 5 / 3 × 0 ( 2 π f τ d ) 2 + ( 2 π f τ d ) 4 / 12 1 + ( 2 π f τ d ) 4 / 12 f - 8 / 3 d f 2.80 sec ( ζ ) k c 2 v 5 / 3 τ d 5 / 3             ( rad 2 of phase ) .
( σ fig 2 ) servo 0.962 ( τ d / τ 0 ) 5 / 3 + 2 π h c ( k c 2 / k p ) [ 1 + ( k p / k c ) 12 / 5 ( d s / r 0 ) 2 ] 3 η τ d d s 2 I p + 2 ( h c k c ) 2 [ 1 + ( k p / k c ) 12 / 5 ( d s / r 0 ) 2 ] N rms 2 3 ( G e η τ d d s 2 I p ) 2 ( rad 2 of phase ) .
0 T t ( f ) F t ( f ) d f 1.60 sec ( ζ ) v - 1 / 3 × 0 0.445 D - 1 v - 1 / 3 - 1 / 5 v 14 / 3 1 / 5 ( 2 π f τ dt ) 2 f - 2 / 3 d f 4.09 sec ( ζ ) D - 7 / 3 v - 1 / 3 8 / 15 v 14 / 3 7 / 15 τ dt 2 ( rad 2 of tilt ) ,
( σ tilt 2 ) servo 4.09 sec ( ζ ) D - 7 / 3 v - 1 / 3 8 / 15 v 14 / 3 7 / 15 τ dt 2 + 8 h c [ 1 + ( k t / k c ) 12 / 5 ( D / r 0 ) 2 ] 3 k t η τ dt D 4 I t + 32 ( h c ) 2 [ 1 + ( k t / k c ) 12 / 5 ( D / r 0 ) 2 ] N rms 2 3 π 2 D 6 ( G e η τ dt I t ) 2 ( rad 2 of tilt ) .
σ total 2 = i σ i 2 .
Strehl exp ( - σ 2 ) ,
Strehl 1 1 + [ D / ρ 0 ] 2 .
Strehl SE exp ( - σ fig 2 ) + 1 - exp ( - σ fig 2 ) 1 + ( D / ρ 0 SE ) 2 ,
resolution SE 1.22 λ c D × ( exp ( - 2 σ fig 2 ) + { [ 1 - exp ( - σ fig 2 ) ] 2 / [ 1 + ( D / ρ 0 SE ) 2 ] } ) 1 / 2 Strehl SE .
Strehl LE exp ( - σ fig 2 ) 1 + 4.94 ( D / λ c ) 2 σ tilt 2 + 1 - exp ( - σ fig 2 ) 1 + ( D / ρ 0 LE ) 2
resolution LE 1.22 ( λ c D ) ( { exp ( - 2 σ fig 2 ) / [ 1 + 4.94 ( D / λ c ) 2 σ tilt 2 ] } + { [ 1 - exp ( - σ fig 2 ) ] 2 / [ 1 + ( D / ρ 0 LE ) 2 ] } ) 1 / 2 Strehl LE ,
ρ 0 LE r 0
ρ 0 SE r 0 [ 1 + 0.37 ( r 0 / D ) 1 / 3 ] .
σ 2 = 0.207 k c 2 0 C n 2 ( z ) [ f ( κ ) g ( κ , z ) d κ ] d z ,
f ( κ ) = κ - 11 / 3 .
g ( κ ) fig = 1 - [ 2 J 1 ( κ D / 2 ) κ D / 2 ] 2 - [ 4 J 2 ( κ D / 2 ) κ D / 2 ] 2 ,
σ fig 2 = 1.30 k c 2 0 C n 2 ( z ) d z × 0 κ - 8 / 3 { 1 - [ 2 J 1 ( κ D / 2 ) κ D / 2 ] 2 - [ 4 J 2 ( κ D / 2 ) κ D / 2 ] 2 } d κ = 0.057 sec ( ζ ) k c 2 D 5 / 3 μ 0 = 0.134 ( D / r 0 ) 5 / 3 .
μ n ( H ) = H h n C n 2 ( h ) d h ,
σ upper 2 = 0.057 D 5 / 3 k c 2 sec ( ζ ) μ 0 ( H ) .
σ lower 2 = 1.30 k c 2 sec ( ζ ) 0 d z C n 2 ( z ) 0 H d κ κ - 8 / 3 × ( 2 { 1 - J 1 [ κ D ( H - z ) / 2 H ] κ D ( H - z ) / 2 H } - { 2 J 1 ( κ D / 2 ) κ D / 2 - 2 J 1 [ κ D ( H - z ) / 2 H ] κ D ( H - z ) / 2 H } 2 - { 4 J 2 ( k D / 2 ) κ D / 2 - 4 J 2 [ κ D ( H - z ) / 2 H ] κ D ( H - z ) / 2 H } 2 ) .
σ lower 2 D 5 / 3 k c 2 sec ( ζ ) × [ 0.500 μ 5 / 3 ( H ) H 5 / 3 - 0.452 μ 2 ( H ) H 2 + ] .
σ lower 2 N b - 5 / 6 D 5 / 3 k c 2 × sec ( ζ ) [ 0.500 μ 5 / 3 ( H ) H 5 / 3 - 0.452 μ 2 ( H ) H 2 + ] .
g ( κ ) tp = [ 4 J 2 ( κ D s / 2 ) κ D s / 2 ] 2
σ tp 2 D s 5 / 3 k c 2 sec ( ζ ) [ 0.368 μ 2 ( H ) H 2 + ] .
σ stitch 2 D 5 / 3 k c 2 sec ( ζ ) [ 0.040 μ 2 ( H ) H 2 + ] ,
θ 0 = [ 2.91 k c 2 sec 8 / 3 ( ζ ) μ 5 / 3 ] - 3 / 5 ,
( σ fig 2 ) bo = ( θ / θ 0 ) 5 / 3 ,
g ( κ , z ) ta = [ 8 ( k c D ) 2 ] [ 4 J 2 ( κ D / 2 ) κ D / 2 ] 2 [ 2 - 2 J 0 ( κ θ z ) ] ,
( σ tilt 2 ) bo sec ( ζ ) D - 1 / 3 × { 5.34 μ 2 ( H c ) [ θ sec ( ζ ) D ] 2 + 6.08 μ 0 ( H c ) } ,
resolution LE = 1.22 ( λ c D ) [ 1 + 4.94 ( D λ c ) 2 ( σ tilt 2 ) bo ] 1 / 2 1.22 ( λ c D ) { 1 + [ θ ( θ 0 ) tilt ] 2 } 1 / 2 ,
( θ 0 ) tilt = [ 0.668 k c 2 sec 3 ( ζ ) μ 2 D - 1 / 3 ] - 1 / 2 .
( σ fig 2 ) sys 0.5 ( d s / r 0 ) 5 / 3 fitting error + 0.962 ( τ d / τ 0 ) 5 / 3 finite - bandwidth error + 2 ( h c k c ) 2 [ 1 + ( k p / k c ) 12 / 5 ( d s / r 0 ) 2 ] N rms 2 3 ( γ p η τ d d s 2 I p ) 2 sensor noise ( rad 2 of phase ) ,
d s / r 0 = 0.811 [ ( σ fig 2 ) sys ] 3 / 5 ,
τ d / τ 0 = 0.548 [ ( σ fig 2 ) sys ] 3 / 5 ,
I p = 3.39 h c k c - 1 / 5 k p 6 / 5 N rms γ p η τ 0 r 0 2 [ ( σ fig 2 ) sys ] 17 / 10 × { 1 + 1.52 ( k c k p ) 12 / 5 [ ( σ fig 2 ) sys ] - 6 / 5 } 1 / 2 .
( N pe ) ps N rms = 7.68 ( k p k c ) 1 / 5 [ ( σ fig 2 ) sys ] 1 / 10 × { 1 + 1.52 ( k c k p ) 12 / 5 [ ( σ fig 2 ) sys ] 6 / 5 } 1 / 2 ,
( σ tilt 2 ) sys 0.202 ( λ c D ) 2 [ τ dt ( τ 0 ) tilt ] 2 finite - bandwidth error + 8 h c k c - 12 / 5 k t 7 / 5 3 γ t η τ dt r 0 2 D 2 I t photon noise             ( rad 2 of single - axis tilt ) ,
( τ 0 ) tilt = [ 0.512 k c 2 sec ( ζ ) v - 1 / 3 8 / 15 v 14 / 3 7 / 15 D - 1 / 3 ] - 1 / 2 .
τ dt ( τ 0 ) tilt = 0.784 [ ( σ tilt 2 ) sys ( 0.61 λ c / D ) 2 ] 1 / 2
I t = 19.5 h c k c - 17 / 5 k t 7 / 5 γ t η ( τ 0 ) tilt r 0 2 D 3 [ ( σ tilt 2 ) sys ] 3 / 2 ,
( N pe ) tr = 1.71 ( k t k c ) 2 / 5 ( D r 0 ) 2 [ ( σ tilt 2 ) sys ( 0.61 λ c / D ) 2 ] - 1 ,
( σ fig 2 ) sys 2.05 ( h c ) 10 / 17 k c - 2 / 17 k p 12 / 17 ( N rms / γ p η τ 0 r 0 2 I p ) 10 / 17             ( rad 2 of phase ) ,
( σ tilt 2 ) sys 7.25 ( h c ) 2 / 3 k c - 34 / 15 k t 14 / 15 [ γ t η ( τ 0 ) tilt I t ] - 2 / 3 r 0 - 4 / 3 D - 2             ( rad 2 of single - axis tilt ) .
Strehl LE exp ( - σ fig 2 ) 1 + 4.94 ( D / λ c ) 2 σ tilt 2 ,
resolution LE 1.22 ( λ c D ) [ 1 + 4.94 ( D λ c ) 2 σ tilt 2 ] 1 / 2 .
m v = - 2.5 log ( I ) - 21.2
fractional sky coverage = π ϑ 2 × average star density ,
average density = 1.45 exp ( 0.96 m v )             ( stars / rad 2 ) ,
density at pole = 1.27 × 10 - 4 m v 8.2             ( stars / rad 2 ) ,
density at equator = 3.97 exp ( m v )             ( stars / rad 2 ) .
average density = 2.02 × 10 - 9 I - 1.04             ( stars / rad 2 ) .
ϑ fig = [ ( σ fig 2 ) bo ] 3 / 5 θ 0 .
ϑ tilt = 1.36 [ ( σ tilt 2 ) bo ( 0.61 λ c / D ) 2 ] 1 / 2 ( θ 0 ) tilt .
I p P l γ l C s [ d σ ( π ) / d Ω ] [ sec ( ζ ) H s ] 2 ,
P l 3.39 [ sec ( ζ ) H s ] 2 h c k c - 1 / 5 k p 6 / 5 N rms γ l γ p η τ 0 r 0 2 C s [ d σ ( π ) / d Ω ] [ ( σ fig 2 ) sys ] 17 / 10 × { 1 + 1.52 ( k c k p ) 12 / 5 [ ( σ fig 2 ) sys ] - 6 / 5 } 1 / 2 .
Strehl LE 1 1 + ( D / r 0 ) 2 .
ρ 0 SE r 0 [ 1 + 0.37 ( r 0 / D ) 1 / 3 ]
Strehl LE exp { - [ ( σ fig 2 ) sys + ( σ fig 2 ) fa + ( σ fig 2 ) bo ] } 1 + 4.94 ( D λ c ) 2 [ ( σ tilt 2 ) sys + ( σ tilt 2 ) bo ] ,
σ tilt 2 = ( σ tilt 2 ) fb + ( σ tilt 2 ) pn + ( σ tilt 2 ) bo
σ tilt 2 0.607 ( λ c / D ) 2 ,
σ fig 2 = ( σ fig 2 ) fit error + ( σ fig 2 ) fb + ( σ fig 2 ) sn + [ ( σ fig 2 ) fa + ( σ fig 2 ) bo ] ,
σ fig 2 ln [ 0.1 ( D / r 0 ) 2 1 + 4.94 ( D / λ c ) 2 σ tilt 2 ] .
C n 2 ( h ) = 0 0 m < h < 19 m = 4.008 × 10 - 13 h - 1.054 19 m < h < 230 m = 1.300 × 10 - 15 230 m < h < 850 m = 6.352 × 10 - 7 h - 2.966 850 m < h < 7000 m = 6.209 × 10 - 16 h - 0.6229 7000 m < h < 20 , 000 m .
C n 2 ( h ) = 5.94 × 10 - 53 ( w / 27 ) 2 h 10 exp ( - h / 1000 ) = + 2.7 × 10 - 16 exp ( - h / 1500 ) + A exp ( - h / 100 ) .
v ( h ) = v g + 30 exp { - [ ( h - 9400 ) / 4800 ] 2 } ,
linear dynamic range 2 π ( n - 1 )             ( rad / subaperture ) .
D ϕ ( d s ) = 6.88 ( d s / r 0 ) 5 / 3 ,
g = 2 π [ 1 + ( d s r 0 ) 2 ] 1 / 2 i = 1 n ( i - n 2 - 1 2 ) Γ i i = 1 n Γ i             ( rad / subaperture ) ,
σ g 2 i = 1 n { 2 π ( 2 i - n - 1 ) G e N pe [ 1 + ( d s r 0 ) 2 ] 1 / 2 } 2 N rms 2 ,
σ g 2 = 4 π 2 n ( n - 1 ) ( n + 1 ) 3 ( G e N pe ) 2 [ 1 + ( d s r 0 ) 2 ] N rms 2 .
( σ fig 2 ) noise 2 π 2 [ 1 + ( d s / r 0 ) 2 ] N pe photon noise + 4 π 2 n ( n - 1 ) ( n + 1 ) 3 ( G e N pe ) 2 [ 1 + ( d s r 0 ) 2 ] N rms 2 sensor noise .
F ϕ ( f ) = - 1 2 W ϕ ( r , f ) T ϕ ( r ) d r ,
W ϕ ( r , f ) = 2.079 ( f / f 0 ) - 8 / 3 f 0 - 1 ( D / r 0 ) 5 / 3 sin 2 ( f r / f 0 D ) ,
v = ( v 5 / 3 μ 0 ) 3 / 5 .
T f ( r ) = 8 π 2 D 2 × { cos - 1 ( r / D ) - ( r / D ) [ 1 - ( r / D ) 2 ] 1 / 2 - π 0 r D 0 D r .
F f f - 8 / 3 { 1 π 0 1 r ( f 2 r 2 - f 4 r 4 3 ) [ cos - 1 ( r ) - r ( 1 - r 2 ) 1 / 2 ] d r - 0 1 / 2 r ( f 2 r 2 - f 4 r 4 3 ) d r f f 0 D / r 1 π 0 1 r ( 1 2 ) [ cos - 1 ( r ) - r ( 1 - r 2 ) 1 / 2 ] d r - 0 1 / 2 r ( 1 2 ) d r f f 0 D / r ,
0 1 r cos - 1 ( r ) d r = π 8 , 0 1 r 3 cos - 1 ( r ) d r = 3 π 64 , 0 1 r 5 cos - 1 ( r ) d r = 5 π 192 , 0 1 r 2 ( 1 - r 2 ) 1 / 2 d r = π 16 , 0 1 r 4 ( 1 - r 2 ) 1 / 2 d r = π 32 , 0 1 r 6 ( 1 - r 2 ) 1 / 2 d r = 5 π 256 .
F f ( f ) = { 0.132 sec ( ζ ) k p 2 D 4 μ 0 12 / 5 v 5 / 3 - 7 / 5 f 4 / 3 f 0.705 D - 1 μ 0 - 3 / 5 v 5 / 3 3 / 5 0.0326 sec ( ζ ) k p 2 v 5 / 3 f - 8 / 3 f 0.705 D - 1 μ 0 - 3 / 5 v 5 / 3 3 / 5 .
0 F f ( f ) d f = 0.06 sec ( ζ ) k p 2 D 5 / 3 μ 0 = 0.14 ( D / r 0 ) 5 / 3 ,
ϕ = [ ( A t A ) - 1 A ] g = B g ,
g = A ϕ .
error propagator = 1 N ϕ tr [ ( A t A ) - 1 ] ,
error propagator = 1 N ϕ tr [ ( A t C t - 1 A ) - 1 ] ,
C t ( d ) = - 0.7986 cos ( 2 θ d ) ( D s d ) 1 / 3 × F 3 2 [ 5 2 , 1 6 , - 5 6 ; 5 , 3 ; ( D s d ) 2 ] + 1.331 cos 2 ( θ d ) ( D s d ) 1 / 3 × F 3 2 [ 5 2 , 1 6 , 1 6 ; 5 , 3 ; ( D s d ) 2 ] ,             d > D s ,
error propagator 0.11 ( D / D s ) 5 / 3
σ 2 = a x m + b x - n ,
x = ( n b m a ) 1 / ( m + n ) .
σ 2 = [ ( n m ) m / ( m + n ) + ( n m ) - n / ( m + n ) ] a n / ( m + n ) n m / ( m + n ) .
σ 2 = ( n m + n ) σ 2 + ( m m + n ) σ 2 ,
a x m = ( n m + n ) σ 2             and             b x - n = ( m m + n ) σ 2

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