Abstract

A method is presented for sensing atmospheric wave-front tilt from a laser guide star (LGS) by observing a laser beacon with auxiliary telescopes. The analysis is performed with a LGS scatter model and Zernike polynomial expansion of wave-front distortions. It is shown that integration of the LGS image over its angular extent and the position of the auxiliary telescope in an array reduce the tilt sensing error associated with the contribution from the downward path. This allows us to single out only the wave-front tilt of the transmitted beam on the uplink path that corresponds to the tilt for the scientific object. The tilt angular correlation is analyzed in the atmosphere with a finite turbulence outer scale. The tilt correlation angle depends on the angular size of the telescope and the outer scale of turbulence. The tilt sensing error increases with the auxiliary telescope diameter, suggesting that an auxiliary telescope must be small. The Strehl ratio associated with the contribution from the downward path is in the range from 0.1 to 0.9 when the relative telescope diameter D/ r 0 varies from 4 to 93 and the turbulence outer scale is in the 10–150-m range. Tilt correction increases the Strehl ratio compared with the uncorrected image for all the system parameters and seeing conditions considered. The method discussed gives a higher performance than the conventional technique, which uses an off-axis natural guide star. A scheme for measuring tilt with a beam projected from a small aperture is described. This scheme allows us to avoid phosphorescence of the main optical train for a sodium LGS.

© 2000 Optical Society of America

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1999 (1)

1998 (2)

M. R. Whiteley, B. M. Welsh, M. C. Roggemann, “Incorporating higher-order modal measurements in tilt estimation: natural and laser guide star applications,” Appl. Opt. 37, 8287–8296 (1998).
[CrossRef]

A. Ghedina, R. Ragazzoni, A. Baruffolo, “Isokinetic patch measurements on the edge of the Moon,” Astron. Astrophys. Suppl. Ser. 130, 561–566 (1998).
[CrossRef]

1996 (4)

1995 (3)

N. Takato, I. Yamaguchi, “Spatial correlation of Zernike phase-expansion coefficients for atmospheric turbulence with finite outer scale,” J. Opt. Soc. Am. A 12, 958–963 (1995).
[CrossRef]

R. Foy, A. Migus, F. Biraben, G. Grynberg, P. R. McCullough, M. Tallon, “The polychromatic artificial Sodium star: a new concept for correcting the atmospheric tilt,” Astron. Astrophys. 111, 569–578 (1995).

R. Ragazzoni, S. Esposito, E. Marchetti, “Auxiliary telescopes for absolute tip–tilt determination of a laser guide star,” Mon. Not. R. Astron. Soc. 276, L76–L78 (1995).

1994 (5)

1993 (1)

1991 (3)

D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (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]

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 wave-front distortion using scattering light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

1989 (2)

F. Chassat, “Calcul du domain d’isoplanet̀isme d’un system̀ d’optique adaptative fonctionnat à travers la turbulence atmospher̀ique,” J. Opt. (Paris) 20, 13–23 (1989).
[CrossRef]

P. H. Hu, J. Stone, T. Stanley, “Application of Zernike polynomials to atmospheric propagation problems,” J. Opt. Soc. Am. A 6, 1595–1608 (1989).
[CrossRef]

1987 (1)

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 (1)

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

1980 (1)

1979 (1)

1976 (1)

1974 (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 632–637 (1974).

Agahi, A.

A. Ziad, J. Borgnino, F. Martin, A. Agahi, “Experimental estimation of the spatial-coherence outer scale from a wave-front statistical analysis,” Astron. Astrophys. 282, 1021–1033 (1994).

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 wave-front distortion using scattering light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Angel, J. R. P.

Avicula, K.

Avita, R.

A. A. Tokovinin, A. Zaid, F. Martin, R. Avita, J. Borgnino, R. Cohan, M. Sazarin, “Wave-front outer scale monitoring at La Silla,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

Banakh, V. A.

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, Dedham, Mass., 1987).

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]

Baruffolo, A.

A. Ghedina, R. Ragazzoni, A. Baruffolo, “Isokinetic patch measurements on the edge of the Moon,” Astron. Astrophys. Suppl. Ser. 130, 561–566 (1998).
[CrossRef]

Belen’kii, M. S.

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Experimental validation of a technique to measure tilt from a laser guide star,” Opt. Lett. 24, 637–639 (1999).
[CrossRef]

M. S. Belen’kii, “Tilt sensing technique with a small aperture beam and related physical phenomena,” (U.S. Air Force Office of Scientific Research, Bolling AFB, Washington, D.C., 1996).

M. S. Belen’kii, “Experimental study of the tilt angular correlation and the effect of stratospheric turbulence on star image motion,” (U.S. Air Force Office of Scientific Research, Bolling AFB, Washington, D.C., 1996).

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Measurements of tilt angular anisoplanatism,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 481–487 (1997).
[CrossRef]

M. S. Belen’kii, “Fundamental limitation in adaptive optics: how to eliminate it? A full aperture tilt measurement technique with a laser guide star,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 321–323 (1994).
[CrossRef]

M. S. Belen’kii, “Full aperture tilt measurement technique with a laser guide star,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 289–300 (1995).
[CrossRef]

M. S. Belen’kii, “Tilt angular correlation and tilt sensing techniques with a laser guide star,” in Optics in Atmospheric Propagation, Adaptive Systems, and Lidar Techniques for Remote Sensing, A. D. Devir, A. Kohnle, C. Werner, eds., Proc. SPIE2956, 206–217 (1996).
[CrossRef]

M. S. Belen’kii, “Principle of equivalency of the phase difference and off-axis tilt sensing technique with a laser guide star,” in Image Propagation through the Atmosphere, J. C. Dainty, L. R. Bissonnette, eds., Proc. SPIE2828, 280–292 (1996).
[CrossRef]

M. S. Belen’kii, “Multiple aperture averaging technique for measuring full aperture tilt with a laser guide star,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 101–112 (1997).
[CrossRef]

Biraben, F.

R. Foy, A. Migus, F. Biraben, G. Grynberg, P. R. McCullough, M. Tallon, “The polychromatic artificial Sodium star: a new concept for correcting the atmospheric tilt,” Astron. Astrophys. 111, 569–578 (1995).

Bissinger, H. D.

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 wave-front distortion using scattering light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Borgnino, J.

A. Ziad, J. Borgnino, F. Martin, A. Agahi, “Experimental estimation of the spatial-coherence outer scale from a wave-front statistical analysis,” Astron. Astrophys. 282, 1021–1033 (1994).

A. A. Tokovinin, A. Zaid, F. Martin, R. Avita, J. Borgnino, R. Cohan, M. Sazarin, “Wave-front outer scale monitoring at La Silla,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).

Brase, J. M.

Brown, J. M.

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Experimental validation of a technique to measure tilt from a laser guide star,” Opt. Lett. 24, 637–639 (1999).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Measurements of tilt angular anisoplanatism,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 481–487 (1997).
[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, “Measurements of atmospheric wave-front distortion using scattering light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 632–637 (1974).

Chassat, F.

F. Chassat, “Calcul du domain d’isoplanet̀isme d’un system̀ d’optique adaptative fonctionnat à travers la turbulence atmospher̀ique,” J. Opt. (Paris) 20, 13–23 (1989).
[CrossRef]

Cleis, R. A.

Cohan, R.

A. A. Tokovinin, A. Zaid, F. Martin, R. Avita, J. Borgnino, R. Cohan, M. Sazarin, “Wave-front outer scale monitoring at La Silla,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

Duff, J.

Ellerbroek, B. L.

R. Racine, B. L. Ellerbroek, “Profiles of nighttime turbulence above Mauna Kea and isoplanatism extension in adaptive optics,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 248–257 (1995).
[CrossRef]

Esposito, S.

S. Esposito, A. Riccardi, R. Ragazzoni, “Focus anisoplanatism effects on tip–tilt compensation for adaptive optics with use of a sodium laser beacon as a tracking reference,” J. Opt. Soc. Am. A 13, 1916–1923 (1996).
[CrossRef]

R. Ragazzoni, S. Esposito, E. Marchetti, “Auxiliary telescopes for absolute tip–tilt determination of a laser guide star,” Mon. Not. R. Astron. Soc. 276, L76–L78 (1995).

Foy, R.

R. Foy, A. Migus, F. Biraben, G. Grynberg, P. R. McCullough, M. Tallon, “The polychromatic artificial Sodium star: a new concept for correcting the atmospheric tilt,” Astron. Astrophys. 111, 569–578 (1995).

R. Foy, A. Labeyrie, “Feasibility of adaptive telescopes with 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, “Measurements of atmospheric wave-front distortion using scattering light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

D. L. Fried, “Varieties of isoplanatism,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. SPIE75, 20–29 (1976).
[CrossRef]

Friedman, H. W.

Fugate, R. Q.

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Experimental validation of a technique to measure tilt from a laser guide star,” Opt. Lett. 24, 637–639 (1999).
[CrossRef]

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. C. Slavin, R. E. Ruane, R. A. Cleis, “Characterization of artificial guide stars generated in the mesospheric sodium layer with a sum-frequency laser,” J. Opt. Soc. Am. A 11, 806–812 (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, “Measurements of atmospheric wave-front distortion using scattering light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Measurements of tilt angular anisoplanatism,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 481–487 (1997).
[CrossRef]

Gardner, C. S.

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]

Gavel, D. T.

Ghedina, A.

A. Ghedina, R. Ragazzoni, A. Baruffolo, “Isokinetic patch measurements on the edge of the Moon,” Astron. Astrophys. Suppl. Ser. 130, 561–566 (1998).
[CrossRef]

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 632–637 (1974).

Grynberg, G.

R. Foy, A. Migus, F. Biraben, G. Grynberg, P. R. McCullough, M. Tallon, “The polychromatic artificial Sodium star: a new concept for correcting the atmospheric tilt,” Astron. Astrophys. 111, 569–578 (1995).

Horton, J. A.

Hu, P. H.

Jelonek, M. P.

Karis, S. J.

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Experimental validation of a technique to measure tilt from a laser guide star,” Opt. Lett. 24, 637–639 (1999).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Measurements of tilt angular anisoplanatism,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 481–487 (1997).
[CrossRef]

Kiefer, R.

Labeyrie, A.

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

Lange, W. J.

Lloyd-Hart, M.

Ma, S.

Marchetti, E.

R. Ragazzoni, S. Esposito, E. Marchetti, “Auxiliary telescopes for absolute tip–tilt determination of a laser guide star,” Mon. Not. R. Astron. Soc. 276, L76–L78 (1995).

Martin, F.

A. Ziad, J. Borgnino, F. Martin, A. Agahi, “Experimental estimation of the spatial-coherence outer scale from a wave-front statistical analysis,” Astron. Astrophys. 282, 1021–1033 (1994).

A. A. Tokovinin, A. Zaid, F. Martin, R. Avita, J. Borgnino, R. Cohan, M. Sazarin, “Wave-front outer scale monitoring at La Silla,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

Max, C. E.

McCarthy, D.

McCullough, P. R.

R. Foy, A. Migus, F. Biraben, G. Grynberg, P. R. McCullough, M. Tallon, “The polychromatic artificial Sodium star: a new concept for correcting the atmospheric tilt,” Astron. Astrophys. 111, 569–578 (1995).

Migus, A.

R. Foy, A. Migus, F. Biraben, G. Grynberg, P. R. McCullough, M. Tallon, “The polychromatic artificial Sodium star: a new concept for correcting the atmospheric tilt,” Astron. Astrophys. 111, 569–578 (1995).

Mills, S. P.

Mironov, V. L.

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, Dedham, Mass., 1987).

Morris, J. R.

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]

Neymann, C. R.

Noll, R. J.

Olivier, S. S.

Page, D. A.

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[CrossRef]

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]

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 632–637 (1974).

Racine, R.

R. Racine, B. L. Ellerbroek, “Profiles of nighttime turbulence above Mauna Kea and isoplanatism extension in adaptive optics,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 248–257 (1995).
[CrossRef]

Ragazzoni, R.

A. Ghedina, R. Ragazzoni, A. Baruffolo, “Isokinetic patch measurements on the edge of the Moon,” Astron. Astrophys. Suppl. Ser. 130, 561–566 (1998).
[CrossRef]

R. Ragazzoni, “Absolute tip–tilt determination with laser beacons,” Astron. Astrophys. 305, L13–L16 (1996).

R. Ragazzoni, “Propagation delay of a laser beacon as a tool to retrieve absolute tilt measurements,” Astrophys. J. 465, L73–L75 (1996).
[CrossRef]

S. Esposito, A. Riccardi, R. Ragazzoni, “Focus anisoplanatism effects on tip–tilt compensation for adaptive optics with use of a sodium laser beacon as a tracking reference,” J. Opt. Soc. Am. A 13, 1916–1923 (1996).
[CrossRef]

R. Ragazzoni, S. Esposito, E. Marchetti, “Auxiliary telescopes for absolute tip–tilt determination of a laser guide star,” Mon. Not. R. Astron. Soc. 276, L76–L78 (1995).

Rapp, R. W.

Riccardi, A.

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 wave-front distortion using scattering light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Roggemann, M. C.

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M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. C. Slavin, R. E. Ruane, R. A. Cleis, “Characterization of artificial guide stars generated in the mesospheric sodium layer with a sum-frequency laser,” J. Opt. Soc. Am. A 11, 806–812 (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, “Measurements of atmospheric wave-front distortion using scattering light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

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Sandler, D. G.

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Sazarin, M.

A. A. Tokovinin, A. Zaid, F. Martin, R. Avita, J. Borgnino, R. Cohan, M. Sazarin, “Wave-front outer scale monitoring at La Silla,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

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Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 632–637 (1974).

Slavin, A. C.

Stahl, S.

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Tallon, M.

R. Foy, A. Migus, F. Biraben, G. Grynberg, P. R. McCullough, M. Tallon, “The polychromatic artificial Sodium star: a new concept for correcting the atmospheric tilt,” Astron. Astrophys. 111, 569–578 (1995).

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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]

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A. A. Tokovinin, A. Zaid, F. Martin, R. Avita, J. Borgnino, R. Cohan, M. Sazarin, “Wave-front outer scale monitoring at La Silla,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[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 wave-front distortion using scattering light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

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M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).

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 wave-front distortion using scattering light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Yamaguchi, I.

Zaid, A.

A. A. Tokovinin, A. Zaid, F. Martin, R. Avita, J. Borgnino, R. Cohan, M. Sazarin, “Wave-front outer scale monitoring at La Silla,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
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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. (2)

Astron. Astrophys. (4)

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R. Ragazzoni, “Absolute tip–tilt determination with laser beacons,” Astron. Astrophys. 305, L13–L16 (1996).

R. Foy, A. Migus, F. Biraben, G. Grynberg, P. R. McCullough, M. Tallon, “The polychromatic artificial Sodium star: a new concept for correcting the atmospheric tilt,” Astron. Astrophys. 111, 569–578 (1995).

Astron. Astrophys. Suppl. Ser. (1)

A. Ghedina, R. Ragazzoni, A. Baruffolo, “Isokinetic patch measurements on the edge of the Moon,” Astron. Astrophys. Suppl. Ser. 130, 561–566 (1998).
[CrossRef]

Astrophys. J. (1)

R. Ragazzoni, “Propagation delay of a laser beacon as a tool to retrieve absolute tilt measurements,” Astrophys. J. 465, L73–L75 (1996).
[CrossRef]

J. Opt. (Paris) (1)

F. Chassat, “Calcul du domain d’isoplanet̀isme d’un system̀ d’optique adaptative fonctionnat à travers la turbulence atmospher̀ique,” J. Opt. (Paris) 20, 13–23 (1989).
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J. Stone, P. H. Hu, S. P. Mills, S. Ma, “Anisoplanatic effects in finite-aperture optical systems,” J. Opt. Soc. Am. A 11, 347–357 (1994).
[CrossRef]

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. C. Slavin, R. E. Ruane, R. A. Cleis, “Characterization of artificial guide stars generated in the mesospheric sodium layer with a sum-frequency laser,” J. Opt. Soc. Am. A 11, 806–812 (1994).
[CrossRef]

C. E. Max, K. Avicula, J. M. Brase, H. W. Friedman, H. D. Bissinger, J. Duff, D. T. Gavel, J. A. Horton, R. Kiefer, J. R. Morris, S. S. Olivier, R. W. Rapp, J. T. Salmon, K. E. Waltien, “Design, layout, and early results of a feasibility experiment for sodium-layer laser-guide-star adaptive optics,” J. Opt. Soc. Am. A 11, 813–824 (1994).
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D. G. Sandler, S. Stahl, J. R. P. Angel, M. Lloyd-Hart, D. McCarthy, “Adaptive optics for diffraction-limited infrared imaging with 8-m telescopes,” J. Opt. Soc. Am. A 11, 925–945 (1994).
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[CrossRef]

S. Esposito, A. Riccardi, R. Ragazzoni, “Focus anisoplanatism effects on tip–tilt compensation for adaptive optics with use of a sodium laser beacon as a tracking reference,” J. Opt. Soc. Am. A 13, 1916–1923 (1996).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

R. Ragazzoni, S. Esposito, E. Marchetti, “Auxiliary telescopes for absolute tip–tilt determination of a laser guide star,” Mon. Not. R. Astron. Soc. 276, L76–L78 (1995).

Nature (London) (3)

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]

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 wave-front distortion using scattering light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 632–637 (1974).

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M. S. Belen’kii, “Multiple aperture averaging technique for measuring full aperture tilt with a laser guide star,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 101–112 (1997).
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R. Racine, B. L. Ellerbroek, “Profiles of nighttime turbulence above Mauna Kea and isoplanatism extension in adaptive optics,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 248–257 (1995).
[CrossRef]

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971).

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M. S. Belen’kii, “Experimental study of the tilt angular correlation and the effect of stratospheric turbulence on star image motion,” (U.S. Air Force Office of Scientific Research, Bolling AFB, Washington, D.C., 1996).

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Measurements of tilt angular anisoplanatism,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 481–487 (1997).
[CrossRef]

A. A. Tokovinin, A. Zaid, F. Martin, R. Avita, J. Borgnino, R. Cohan, M. Sazarin, “Wave-front outer scale monitoring at La Silla,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Schematic configuration of a LGS scheme with an auxiliary telescope: 1, main transmitting telescope; 2, LGS; 3, science object or natural star; 4, auxiliary receiving telescope; θ R , FOV of the receiver; γ, angle between the axes of the transmitter and the receiver; Δ, separation between the main transmitting telescope and the receiver.

Fig. 2
Fig. 2

Angular correlation coefficient for the lateral and the longitudinal tilts on the horizontal and the vertical propagation paths: curves 1, 3, 4, 6, 7, and 8 correspond to the horizontal propagation path; curves 2 and 5 correspond to the vertical path; curves 1, 2, 4, 5, and 6 correspond to a lateral tilt; curves 3, 7, and 8 correspond to a longitudinal tilt. Ratio L 0/D is equal to infinity for curves 1–3; L 0/D = 10 for curves 4, 5, and 7; L 0/D = 3 for curves 6 and 8.

Fig. 3
Fig. 3

Angular correlation coefficient for the lateral and the longitudinal tilts on the vertical propagation path: curves 1–4 correspond to the lateral tilt and curves 5 and 6 correspond to the longitudinal tilt. Ratio L 0/D is equal to infinity for curve 1; L 0/D = 50 for curve 2; L 0/D = 10 for curves 3 and 5; L 0/D = 3 for curves 4 and 6.

Fig. 4
Fig. 4

Dependence of the coefficient of proportionality α of the tilt angular correlation scale on the relative turbulence outer scale L 0/D a .

Fig. 5
Fig. 5

Dependence of the tilt angular correlation coefficient on the angular distance.

Fig. 6
Fig. 6

Heuristic model for the tilt angular correlation.

Fig. 7
Fig. 7

Tilt angular averaging function.

Fig. 8
Fig. 8

Schematic diagram of the tilt-measurement scheme using an array of auxiliary telescopes.

Fig. 9
Fig. 9

Strehl ratio associated with the contribution of the downward path for seeing conditions at the SOR. D = 1.5 m for curves 1, 2, 3, and 4; D = 3.5 m for curves 5 and 6; N = 2 for curves 3 and 4; N = 4 for curves 1, 2, 5, and 6; L 0 = 10 m for curves 1, 3, and 5; L 0 = 150 m for curves 2, 4, and 6.

Fig. 10
Fig. 10

Strehl ratio associated with the contribution of the downward path for seeing conditions at the Asiago Astronomical Observatory: D = 4 m for curves 1, 2, 3, and 4; D = 8 m for curves 5 and 6; N = 2 for curves 3 and 4; N = 4 for curves 1, 2, 5, and 6; L 0 = 10 m for curves 1, 3, and 5; L 0 = 150 m for curves 2, 4, and 6.

Fig. 11
Fig. 11

Dependence of the ratio SR c /SR u on the turbulence outer scale: D = 4 m for curves 1 and 5; D = 8 for curves 2, 3, and 4, N = 2 for curve 5; N = 4 for curves 1, 2, 3, and 4; λ = 0.55 µm for curves 1 and 2; λ = 1.25 µm for curve 3; λ = 2.2 µm for curves 4 and 5.

Fig. 12
Fig. 12

Schematic configuration of the tilt-measurement technique with a beam projected from a small aperture.

Equations (43)

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

Usr=m=1M AmGrm11, rm; L, rPrm11Uirm.
Grm11, rm; L, r=k2πiL-rm11expikr-rm22L-rm11+iSr, rm,  Sr, rm
Uirm= d2r0GL, r0; rm11, rmU0r0,
UΛρ= d2rUsrG0L, r; Λ, ρ·Trexpikr22Λ,
ρLGS= d2ρIΛρρ/P,
φLGS2=4.05a0-1/3+RL-1/3-27/6a02+RL2-1/6secγ0HdξCn2ξ1-ξ/H5/3.
φLGS=φLBa0-φSSRL.
φLGS2=φLB2+φSS2-2φt,r2,
φLGS2=4.05a0-1/31-1.245a0/RL1/3+0.374a0/RL2RL-1/3secγ×0HdξCn2ξ1-ξ/H5/3.
φLGS,yx=-dyIΛx, yy d2ρIΛρΛ.
φ¯LGS,y=2aˆv-1-aˆvaˆvdxφLGS,yx,
φ¯LGS,y2=φLB,y2+φ¯SS,y2,
φLB,y2=2.02 0HdξCn2ξ1-ξ/H2ae-1/3ξ,
ae2ξ=a021-ξ/Fi2+ξ/ka02+4ξ/kρc2,
φ¯SS,y2=2.02 sec γ 0HdξCn2ξ1-ξ/H2av2ξ/H2+RL21-ξ/H2-1/6 2F11/6, 3/2, 2; 1-X1,
X1=ae2ξ/H2+RL21-ξ/H2/av2ξ/H2+RL21-ξ/H2.
ϕnK=0.0096Cn2K2+K0K2-11/6,
bx,yθ=0HdhCn2h0dK2K/Da2+K0K2-11/6 J22KJ02θhDa KJ22θhDa KK0HdhCn2h0dK2K/Da2+K0K2-116 J22KK,
bx,yθ=0dK2K/Da2+K0K2-11/6J22K/K1F21/2; 1, 3/2;-μK216μK12F23/2; 5, 3;-μK20dK2K/Da2+K0K2-11/6 J22K/K.
Cn2h: μ0=0dhCn2h, μ1=0dhCn2hh.
θt=αL0/DaDa/h¯.
φ¯SS,y2=2φPS2θR0θRdθbyθ/θt1-θ/θR,
GAθR/θt-1.
φLGSθ, R, t=φLBt+φSSθ, R, t,
φ˜y=φLB,yt+1θR0θRdθ 1Ni=1N1T0TdtφSS,yθ, Ri, t.
φ˜y2=φLB,y2+σSS,y2,
σSS,y2=φPS2GAGSGT,
GAθR/θt=2θR0θRdθbA,yθ/θt1-θ/θR,
GSN=1N2i=1Nj=1N bS,yRi, Rj,
GTT/tau=2T0TdtbT,yt/tau1-t/T.
GS=1/N.
SRc=11+π22σ1λ/D2,
σ12=σSS,y2+σFA,y2.
φPS2=3.04μ0Da-1/3fL0/Da,
σSS,y2  Da2/3.
φLB=φFT+φLTρ,
φau=φFT+φLTρ-φSS.
φFT=φA-φM.
φ¯y=θR-1-θR/2θR/2dθφyθ.
φ¯y2=θR-2-θR/2θR/2dθ1-θR/2θR/2dθ2φyθ1φyθ2.
φyθ1φyθ2=φPS2byθ-/θt,
φ¯y2=φPS2/θR-θR0dθ-1+θ-/θRbyθ-/θt+0θRdθ-1-θ-/θRbyθ-/θt.
φ¯y2=2φPS2/θR0θRdθ-byθ-/θt1-θ-/θR.

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