Abstract

We present an analytical derivation of the on-axis performance of adaptive optics systems using a given number of guide stars of arbitrary altitude, distributed at arbitrary angular positions in the sky. The expressions of the residual error are given for cases of both continuous and discrete turbulent atmospheric profiles. Assuming Shack–Hartmann wavefront sensing with circular apertures, we demonstrate that the error is formally described by integrals of products of three Bessel functions. We compare the performance of adaptive optics correction when using natural, sodium, or Rayleigh laser guide stars. For small diameter class telescopes (5m), we show that a small number of Rayleigh beacons can provide similar performance to that of a single sodium laser, for a lower overall cost of the instrument. For bigger apertures, using Rayleigh stars may not be such a suitable alternative because of the too severe cone effect that drastically degrades the quality of the correction.

© 2013 Optical Society of America

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2013

2012

P. Wizinowich, “Progress in laser guide star adaptive optics and lessons learned,” Proc. SPIE 8447, 84470D (2012).
[CrossRef]

S. Ströbele, P. La Penna, R. Arsenault, R. D. Conzelmann, B. Delabre, M. Duchateau, R. Dorn, E. Fedrigo, N. Hubin, J. Quentin, P. Jolley, M. Kiekebusch, J. P. Kirchbauer, B. Klein, J. Kolb, H. Kuntschner, M. Le Louarn, J. L. Lizon, P.-Y. Madec, L. Pettazzi, C. Soenke, S. Tordo, J. Vernet, and R. Muradore, “GALACSI system design and analysis,” Proc. SPIE 8447, 844737 (2012).
[CrossRef]

R. Conan, F. Bennet, A. H. Bouchez, M. A. van Dam, B. Espeland, W. Gardhouse, C. d’Orgeville, S. Parcell, P. Piatrou, I. Price, F. Rigaut, G. Trancho, and K. Uhlendorf, “The Giant Magellan Telescope laser tomography adaptive optics system,” Proc. SPIE 8447, 84473P (2012).
[CrossRef]

2011

2010

T. Fusco, S. Meimon, Y. Clenet, M. Cohen, H. Schnetler, J. Paufique, V. Michau, J.-P. Amans, D. Gratadour, C. Petit, C. Robert, P. Jagourel, E. Gendron, G. Rousset, J.-M. Conan, and N. Hubin, “ATLAS: the E-ELT laser tomographic adaptive optics system,” Proc. SPIE 7736, 77360D (2010).
[CrossRef]

D. Bonaccini Calia, Y. Feng, W. Hackenberg, R. Holzlöhner, L. Taylor, and S. Lewis, “Laser development for sodium laser guide stars at ESO,” The Messenger 139, 12–19 (2010).

A. Costille, C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, and T. Fusco, “Wide field adaptive optics laboratory demonstration with closed-loop tomographic control,” J. Opt. Soc. Am. A 27, 469–483 (2010).
[CrossRef]

2009

2007

L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24, 569–577 (2007).
[CrossRef]

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

2006

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[CrossRef]

H. Shu, L. Luo, G. Han, and J.-L. Coatrieux, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23, 1960–1966 (2006).
[CrossRef]

S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
[CrossRef]

2005

N. Hubin, R. Arsenault, R. Conzelmann, B. Delabre, M. Le Louarn, S. Stroebele, and R. Stuik, “Ground layer adaptive optics,” C.R. Physique 6, 1099–1109 (2005).
[CrossRef]

2003

2002

J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945 (2002).
[CrossRef]

M. Schöck, R. Foy, M. Tallon, L. Noethe, and J.-P. Pique, “Performance analysis of polychromatic laser guide stars used for wavefront tilt sensing,” Mon. Not. R. Astron. Soc. 337, 910–920 (2002).
[CrossRef]

E. Viard, M. Le Louarn, and N. Hubin, “Adaptive optics with four laser guide stars: correction of the cone effect in large telescopes,” Appl. Opt. 41, 11–20 (2002).
[CrossRef]

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
[CrossRef]

2001

W. Min and S. Yi, “Turbulence-induced Zernike aberrations of optical wavefronts in partial adaptive compensation,” J. Mod. Opt. 48, 1559–1567 (2001).

A. Tokovinin, M. Le Louarn, E. Viard, N. Hubin, and R. Conan, “Optimized modal tomography in adaptive optics,” Astron. Astrophys. 378, 710–721 (2001).
[CrossRef]

2000

M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, “New challenges for adaptive optics: extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535–544 (2000).
[CrossRef]

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, and G. Rousset, “Optimal phase reconstruction in large field of view: application to multiconjugate adaptive optics systems,” Proc. SPIE 4125, 65–76 (2000).
[CrossRef]

1999

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

1997

C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, and K. E. Waltjen, “Image improvement from a sodium-layer laser guide star adaptive optics system,” Science 277, 1649–1652 (1997).
[CrossRef]

1996

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

1995

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

1994

1993

R. J. Sasiela and J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
[CrossRef]

1991

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

1990

M. Tallon and R. Foy, “Adaptive telescope with laser probe—isoplanatism and cone effect,” Astron. Astrophys. 235, 549–557 (1990).

G. A. Tyler, “Analysis of propagation through turbulence—evaluation of an integral involving the product of three Bessel functions,” J. Opt. Soc. Am. A 7, 1218–1223 (1990).
[CrossRef]

N. A. Roddier, “Atmospheric wavefront simulation and Zernike polynomials,” Proc. SPIE 1237, 668–679 (1990).
[CrossRef]

1989

1985

R. Foy and A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

H. T. Yura and M. T. Tavis, “Centroid anisoplanatism,” J. Opt. Soc. Am. A 2, 765–773 (1985).
[CrossRef]

1983

1981

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

1977

1976

1973

1965

1941

A. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ numbers,” Doklady Akademiia Nauk SSSR 30, 301–305 (1941).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1970).

Allaert, E.

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
[CrossRef]

Amans, J.-P.

T. Fusco, S. Meimon, Y. Clenet, M. Cohen, H. Schnetler, J. Paufique, V. Michau, J.-P. Amans, D. Gratadour, C. Petit, C. Robert, P. Jagourel, E. Gendron, G. Rousset, J.-M. Conan, and N. Hubin, “ATLAS: the E-ELT laser tomographic adaptive optics system,” Proc. SPIE 7736, 77360D (2010).
[CrossRef]

Ameer, G. A.

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

An, J.

C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, and K. E. Waltjen, “Image improvement from a sodium-layer laser guide star adaptive optics system,” Science 277, 1649–1652 (1997).
[CrossRef]

Ares, J.

Arines, J.

Arsenault, R.

S. Ströbele, P. La Penna, R. Arsenault, R. D. Conzelmann, B. Delabre, M. Duchateau, R. Dorn, E. Fedrigo, N. Hubin, J. Quentin, P. Jolley, M. Kiekebusch, J. P. Kirchbauer, B. Klein, J. Kolb, H. Kuntschner, M. Le Louarn, J. L. Lizon, P.-Y. Madec, L. Pettazzi, C. Soenke, S. Tordo, J. Vernet, and R. Muradore, “GALACSI system design and analysis,” Proc. SPIE 8447, 844737 (2012).
[CrossRef]

N. Hubin, R. Arsenault, R. Conzelmann, B. Delabre, M. Le Louarn, S. Stroebele, and R. Stuik, “Ground layer adaptive optics,” C.R. Physique 6, 1099–1109 (2005).
[CrossRef]

Avicola, K.

C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, and K. E. Waltjen, “Image improvement from a sodium-layer laser guide star adaptive optics system,” Science 277, 1649–1652 (1997).
[CrossRef]

Bará, S.

Béchet, C.

C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in Adaptative Optics for Extremely Large TelescopesY. Clénet, J.-M. Conan, Th. Fusco, and G. Rousset, eds. (EDP Sciences, 2010), p. 03004.

Beeman, B. V.

C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, and K. E. Waltjen, “Image improvement from a sodium-layer laser guide star adaptive optics system,” Science 277, 1649–1652 (1997).
[CrossRef]

Belsher, J. F.

Bennet, F.

R. Conan, F. Bennet, A. H. Bouchez, M. A. van Dam, B. Espeland, W. Gardhouse, C. d’Orgeville, S. Parcell, P. Piatrou, I. Price, F. Rigaut, G. Trancho, and K. Uhlendorf, “The Giant Magellan Telescope laser tomography adaptive optics system,” Proc. SPIE 8447, 84473P (2012).
[CrossRef]

Bissinger, H. D.

C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, and K. E. Waltjen, “Image improvement from a sodium-layer laser guide star adaptive optics system,” Science 277, 1649–1652 (1997).
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Boeke, B. R.

Bonaccini, D.

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
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D. Bonaccini Calia, Y. Feng, W. Hackenberg, R. Holzlöhner, L. Taylor, and S. Lewis, “Laser development for sodium laser guide stars at ESO,” The Messenger 139, 12–19 (2010).

Bouchez, A. H.

R. Conan, F. Bennet, A. H. Bouchez, M. A. van Dam, B. Espeland, W. Gardhouse, C. d’Orgeville, S. Parcell, P. Piatrou, I. Price, F. Rigaut, G. Trancho, and K. Uhlendorf, “The Giant Magellan Telescope laser tomography adaptive optics system,” Proc. SPIE 8447, 84473P (2012).
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M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
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C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, and K. E. Waltjen, “Image improvement from a sodium-layer laser guide star adaptive optics system,” Science 277, 1649–1652 (1997).
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Brunetto, E.

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
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Bufton, J. L.

Buzzoni, B.

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
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Campbell, C. E.

Campbell, R. D.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
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M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
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Cleis, R. A.

Clenet, Y.

T. Fusco, S. Meimon, Y. Clenet, M. Cohen, H. Schnetler, J. Paufique, V. Michau, J.-P. Amans, D. Gratadour, C. Petit, C. Robert, P. Jagourel, E. Gendron, G. Rousset, J.-M. Conan, and N. Hubin, “ATLAS: the E-ELT laser tomographic adaptive optics system,” Proc. SPIE 7736, 77360D (2010).
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Coatrieux, J.-L.

Cohen, M.

T. Fusco, S. Meimon, Y. Clenet, M. Cohen, H. Schnetler, J. Paufique, V. Michau, J.-P. Amans, D. Gratadour, C. Petit, C. Robert, P. Jagourel, E. Gendron, G. Rousset, J.-M. Conan, and N. Hubin, “ATLAS: the E-ELT laser tomographic adaptive optics system,” Proc. SPIE 7736, 77360D (2010).
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Conan, J.-M.

T. Fusco, S. Meimon, Y. Clenet, M. Cohen, H. Schnetler, J. Paufique, V. Michau, J.-P. Amans, D. Gratadour, C. Petit, C. Robert, P. Jagourel, E. Gendron, G. Rousset, J.-M. Conan, and N. Hubin, “ATLAS: the E-ELT laser tomographic adaptive optics system,” Proc. SPIE 7736, 77360D (2010).
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A. Costille, C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, and T. Fusco, “Wide field adaptive optics laboratory demonstration with closed-loop tomographic control,” J. Opt. Soc. Am. A 27, 469–483 (2010).
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B. Neichel, T. Fusco, and J.-M. Conan, “Tomographic reconstruction for wide-field adaptive optics systems: Fourier domain analysis and fundamental limitations,” J. Opt. Soc. Am. A 26, 219–235 (2009).
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T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, and G. Rousset, “Optimal phase reconstruction in large field of view: application to multiconjugate adaptive optics systems,” Proc. SPIE 4125, 65–76 (2000).
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J.-M. Conan, “'Etude de la correction partielle en optique adaptative,” Ph.D. thesis (Université Paris XI Orsay, 1994).

Conan, R.

R. Conan, F. Bennet, A. H. Bouchez, M. A. van Dam, B. Espeland, W. Gardhouse, C. d’Orgeville, S. Parcell, P. Piatrou, I. Price, F. Rigaut, G. Trancho, and K. Uhlendorf, “The Giant Magellan Telescope laser tomography adaptive optics system,” Proc. SPIE 8447, 84473P (2012).
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A. Tokovinin, M. Le Louarn, E. Viard, N. Hubin, and R. Conan, “Optimized modal tomography in adaptive optics,” Astron. Astrophys. 378, 710–721 (2001).
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Conzelmann, R.

N. Hubin, R. Arsenault, R. Conzelmann, B. Delabre, M. Le Louarn, S. Stroebele, and R. Stuik, “Ground layer adaptive optics,” C.R. Physique 6, 1099–1109 (2005).
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Conzelmann, R. D.

S. Ströbele, P. La Penna, R. Arsenault, R. D. Conzelmann, B. Delabre, M. Duchateau, R. Dorn, E. Fedrigo, N. Hubin, J. Quentin, P. Jolley, M. Kiekebusch, J. P. Kirchbauer, B. Klein, J. Kolb, H. Kuntschner, M. Le Louarn, J. L. Lizon, P.-Y. Madec, L. Pettazzi, C. Soenke, S. Tordo, J. Vernet, and R. Muradore, “GALACSI system design and analysis,” Proc. SPIE 8447, 844737 (2012).
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Costille, A.

Cullum, M. J.

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
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R. Conan, F. Bennet, A. H. Bouchez, M. A. van Dam, B. Espeland, W. Gardhouse, C. d’Orgeville, S. Parcell, P. Piatrou, I. Price, F. Rigaut, G. Trancho, and K. Uhlendorf, “The Giant Magellan Telescope laser tomography adaptive optics system,” Proc. SPIE 8447, 84473P (2012).
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S. Ströbele, P. La Penna, R. Arsenault, R. D. Conzelmann, B. Delabre, M. Duchateau, R. Dorn, E. Fedrigo, N. Hubin, J. Quentin, P. Jolley, M. Kiekebusch, J. P. Kirchbauer, B. Klein, J. Kolb, H. Kuntschner, M. Le Louarn, J. L. Lizon, P.-Y. Madec, L. Pettazzi, C. Soenke, S. Tordo, J. Vernet, and R. Muradore, “GALACSI system design and analysis,” Proc. SPIE 8447, 844737 (2012).
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N. Hubin, R. Arsenault, R. Conzelmann, B. Delabre, M. Le Louarn, S. Stroebele, and R. Stuik, “Ground layer adaptive optics,” C.R. Physique 6, 1099–1109 (2005).
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Di Chirico, C.

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
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D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
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S. Ströbele, P. La Penna, R. Arsenault, R. D. Conzelmann, B. Delabre, M. Duchateau, R. Dorn, E. Fedrigo, N. Hubin, J. Quentin, P. Jolley, M. Kiekebusch, J. P. Kirchbauer, B. Klein, J. Kolb, H. Kuntschner, M. Le Louarn, J. L. Lizon, P.-Y. Madec, L. Pettazzi, C. Soenke, S. Tordo, J. Vernet, and R. Muradore, “GALACSI system design and analysis,” Proc. SPIE 8447, 844737 (2012).
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R. Conan, F. Bennet, A. H. Bouchez, M. A. van Dam, B. Espeland, W. Gardhouse, C. d’Orgeville, S. Parcell, P. Piatrou, I. Price, F. Rigaut, G. Trancho, and K. Uhlendorf, “The Giant Magellan Telescope laser tomography adaptive optics system,” Proc. SPIE 8447, 84473P (2012).
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S. Ströbele, P. La Penna, R. Arsenault, R. D. Conzelmann, B. Delabre, M. Duchateau, R. Dorn, E. Fedrigo, N. Hubin, J. Quentin, P. Jolley, M. Kiekebusch, J. P. Kirchbauer, B. Klein, J. Kolb, H. Kuntschner, M. Le Louarn, J. L. Lizon, P.-Y. Madec, L. Pettazzi, C. Soenke, S. Tordo, J. Vernet, and R. Muradore, “GALACSI system design and analysis,” Proc. SPIE 8447, 844737 (2012).
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Foy, R.

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C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, and K. E. Waltjen, “Image improvement from a sodium-layer laser guide star adaptive optics system,” Science 277, 1649–1652 (1997).
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Fusco, T.

T. Fusco, S. Meimon, Y. Clenet, M. Cohen, H. Schnetler, J. Paufique, V. Michau, J.-P. Amans, D. Gratadour, C. Petit, C. Robert, P. Jagourel, E. Gendron, G. Rousset, J.-M. Conan, and N. Hubin, “ATLAS: the E-ELT laser tomographic adaptive optics system,” Proc. SPIE 7736, 77360D (2010).
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A. Costille, C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, and T. Fusco, “Wide field adaptive optics laboratory demonstration with closed-loop tomographic control,” J. Opt. Soc. Am. A 27, 469–483 (2010).
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B. Neichel, T. Fusco, and J.-M. Conan, “Tomographic reconstruction for wide-field adaptive optics systems: Fourier domain analysis and fundamental limitations,” J. Opt. Soc. Am. A 26, 219–235 (2009).
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T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, and G. Rousset, “Optimal phase reconstruction in large field of view: application to multiconjugate adaptive optics systems,” Proc. SPIE 4125, 65–76 (2000).
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Gardhouse, W.

R. Conan, F. Bennet, A. H. Bouchez, M. A. van Dam, B. Espeland, W. Gardhouse, C. d’Orgeville, S. Parcell, P. Piatrou, I. Price, F. Rigaut, G. Trancho, and K. Uhlendorf, “The Giant Magellan Telescope laser tomography adaptive optics system,” Proc. SPIE 8447, 84473P (2012).
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C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, and K. E. Waltjen, “Image improvement from a sodium-layer laser guide star adaptive optics system,” Science 277, 1649–1652 (1997).
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Gendron, E.

T. Fusco, S. Meimon, Y. Clenet, M. Cohen, H. Schnetler, J. Paufique, V. Michau, J.-P. Amans, D. Gratadour, C. Petit, C. Robert, P. Jagourel, E. Gendron, G. Rousset, J.-M. Conan, and N. Hubin, “ATLAS: the E-ELT laser tomographic adaptive optics system,” Proc. SPIE 7736, 77360D (2010).
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T. Fusco, S. Meimon, Y. Clenet, M. Cohen, H. Schnetler, J. Paufique, V. Michau, J.-P. Amans, D. Gratadour, C. Petit, C. Robert, P. Jagourel, E. Gendron, G. Rousset, J.-M. Conan, and N. Hubin, “ATLAS: the E-ELT laser tomographic adaptive optics system,” Proc. SPIE 7736, 77360D (2010).
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D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
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Hackenberg, W.

D. Bonaccini Calia, Y. Feng, W. Hackenberg, R. Holzlöhner, L. Taylor, and S. Lewis, “Laser development for sodium laser guide stars at ESO,” The Messenger 139, 12–19 (2010).

Hackenberg, W. K.

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
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M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
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Higgins, C. H.

Holzlöhner, R.

D. Bonaccini Calia, Y. Feng, W. Hackenberg, R. Holzlöhner, L. Taylor, and S. Lewis, “Laser development for sodium laser guide stars at ESO,” The Messenger 139, 12–19 (2010).

Hu, P. H.

Hubin, N.

S. Ströbele, P. La Penna, R. Arsenault, R. D. Conzelmann, B. Delabre, M. Duchateau, R. Dorn, E. Fedrigo, N. Hubin, J. Quentin, P. Jolley, M. Kiekebusch, J. P. Kirchbauer, B. Klein, J. Kolb, H. Kuntschner, M. Le Louarn, J. L. Lizon, P.-Y. Madec, L. Pettazzi, C. Soenke, S. Tordo, J. Vernet, and R. Muradore, “GALACSI system design and analysis,” Proc. SPIE 8447, 844737 (2012).
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T. Fusco, S. Meimon, Y. Clenet, M. Cohen, H. Schnetler, J. Paufique, V. Michau, J.-P. Amans, D. Gratadour, C. Petit, C. Robert, P. Jagourel, E. Gendron, G. Rousset, J.-M. Conan, and N. Hubin, “ATLAS: the E-ELT laser tomographic adaptive optics system,” Proc. SPIE 7736, 77360D (2010).
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N. Hubin, R. Arsenault, R. Conzelmann, B. Delabre, M. Le Louarn, S. Stroebele, and R. Stuik, “Ground layer adaptive optics,” C.R. Physique 6, 1099–1109 (2005).
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E. Viard, M. Le Louarn, and N. Hubin, “Adaptive optics with four laser guide stars: correction of the cone effect in large telescopes,” Appl. Opt. 41, 11–20 (2002).
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A. Tokovinin, M. Le Louarn, E. Viard, N. Hubin, and R. Conan, “Optimized modal tomography in adaptive optics,” Astron. Astrophys. 378, 710–721 (2001).
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M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, “New challenges for adaptive optics: extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535–544 (2000).
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Jagourel, P.

T. Fusco, S. Meimon, Y. Clenet, M. Cohen, H. Schnetler, J. Paufique, V. Michau, J.-P. Amans, D. Gratadour, C. Petit, C. Robert, P. Jagourel, E. Gendron, G. Rousset, J.-M. Conan, and N. Hubin, “ATLAS: the E-ELT laser tomographic adaptive optics system,” Proc. SPIE 7736, 77360D (2010).
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Johansson, E. M.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
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Jolley, P.

S. Ströbele, P. La Penna, R. Arsenault, R. D. Conzelmann, B. Delabre, M. Duchateau, R. Dorn, E. Fedrigo, N. Hubin, J. Quentin, P. Jolley, M. Kiekebusch, J. P. Kirchbauer, B. Klein, J. Kolb, H. Kuntschner, M. Le Louarn, J. L. Lizon, P.-Y. Madec, L. Pettazzi, C. Soenke, S. Tordo, J. Vernet, and R. Muradore, “GALACSI system design and analysis,” Proc. SPIE 8447, 844737 (2012).
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Kanz, K.

C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, and K. E. Waltjen, “Image improvement from a sodium-layer laser guide star adaptive optics system,” Science 277, 1649–1652 (1997).
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Kiekebusch, M.

S. Ströbele, P. La Penna, R. Arsenault, R. D. Conzelmann, B. Delabre, M. Duchateau, R. Dorn, E. Fedrigo, N. Hubin, J. Quentin, P. Jolley, M. Kiekebusch, J. P. Kirchbauer, B. Klein, J. Kolb, H. Kuntschner, M. Le Louarn, J. L. Lizon, P.-Y. Madec, L. Pettazzi, C. Soenke, S. Tordo, J. Vernet, and R. Muradore, “GALACSI system design and analysis,” Proc. SPIE 8447, 844737 (2012).
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N. Hubin, R. Arsenault, R. Conzelmann, B. Delabre, M. Le Louarn, S. Stroebele, and R. Stuik, “Ground layer adaptive optics,” C.R. Physique 6, 1099–1109 (2005).
[CrossRef]

Stuik, R.

N. Hubin, R. Arsenault, R. Conzelmann, B. Delabre, M. Le Louarn, S. Stroebele, and R. Stuik, “Ground layer adaptive optics,” C.R. Physique 6, 1099–1109 (2005).
[CrossRef]

Summers, D. M.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[CrossRef]

Swindle, D. W.

Tallon, M.

M. Schöck, R. Foy, M. Tallon, L. Noethe, and J.-P. Pique, “Performance analysis of polychromatic laser guide stars used for wavefront tilt sensing,” Mon. Not. R. Astron. Soc. 337, 910–920 (2002).
[CrossRef]

M. Tallon and R. Foy, “Adaptive telescope with laser probe—isoplanatism and cone effect,” Astron. Astrophys. 235, 549–557 (1990).

C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in Adaptative Optics for Extremely Large TelescopesY. Clénet, J.-M. Conan, Th. Fusco, and G. Rousset, eds. (EDP Sciences, 2010), p. 03004.

Tallon-Bosc, I.

C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in Adaptative Optics for Extremely Large TelescopesY. Clénet, J.-M. Conan, Th. Fusco, and G. Rousset, eds. (EDP Sciences, 2010), p. 03004.

Tamai, R.

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
[CrossRef]

Tapia, M.

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
[CrossRef]

Tarenghi, M.

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in Turbulent Medium (McGraw-Hill, 1961).

Tatulli, E.

Tavis, M. T.

Taylor, L.

D. Bonaccini Calia, Y. Feng, W. Hackenberg, R. Holzlöhner, L. Taylor, and S. Lewis, “Laser development for sodium laser guide stars at ESO,” The Messenger 139, 12–19 (2010).

Thiébaut, É.

C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in Adaptative Optics for Extremely Large TelescopesY. Clénet, J.-M. Conan, Th. Fusco, and G. Rousset, eds. (EDP Sciences, 2010), p. 03004.

Tokovinin, A.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

A. Tokovinin, M. Le Louarn, E. Viard, N. Hubin, and R. Conan, “Optimized modal tomography in adaptive optics,” Astron. Astrophys. 378, 710–721 (2001).
[CrossRef]

M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, “New challenges for adaptive optics: extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535–544 (2000).
[CrossRef]

Tordo, S.

S. Ströbele, P. La Penna, R. Arsenault, R. D. Conzelmann, B. Delabre, M. Duchateau, R. Dorn, E. Fedrigo, N. Hubin, J. Quentin, P. Jolley, M. Kiekebusch, J. P. Kirchbauer, B. Klein, J. Kolb, H. Kuntschner, M. Le Louarn, J. L. Lizon, P.-Y. Madec, L. Pettazzi, C. Soenke, S. Tordo, J. Vernet, and R. Muradore, “GALACSI system design and analysis,” Proc. SPIE 8447, 844737 (2012).
[CrossRef]

Trancho, G.

R. Conan, F. Bennet, A. H. Bouchez, M. A. van Dam, B. Espeland, W. Gardhouse, C. d’Orgeville, S. Parcell, P. Piatrou, I. Price, F. Rigaut, G. Trancho, and K. Uhlendorf, “The Giant Magellan Telescope laser tomography adaptive optics system,” Proc. SPIE 8447, 84473P (2012).
[CrossRef]

Tyler, G. A.

G. A. Tyler, “Rapid evaluation of d0: the effective diameter of a laser-guide-star adaptive-optics system,” J. Opt. Soc. Am. A 11, 325–338 (1994).
[CrossRef]

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

G. A. Tyler, “Analysis of propagation through turbulence—evaluation of an integral involving the product of three Bessel functions,” J. Opt. Soc. Am. A 7, 1218–1223 (1990).
[CrossRef]

Uhlendorf, K.

R. Conan, F. Bennet, A. H. Bouchez, M. A. van Dam, B. Espeland, W. Gardhouse, C. d’Orgeville, S. Parcell, P. Piatrou, I. Price, F. Rigaut, G. Trancho, and K. Uhlendorf, “The Giant Magellan Telescope laser tomography adaptive optics system,” Proc. SPIE 8447, 84473P (2012).
[CrossRef]

Unsbo, P.

van Dam, M. A.

R. Conan, F. Bennet, A. H. Bouchez, M. A. van Dam, B. Espeland, W. Gardhouse, C. d’Orgeville, S. Parcell, P. Piatrou, I. Price, F. Rigaut, G. Trancho, and K. Uhlendorf, “The Giant Magellan Telescope laser tomography adaptive optics system,” Proc. SPIE 8447, 84473P (2012).
[CrossRef]

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[CrossRef]

Van Kersteren, A.

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
[CrossRef]

Véran, J. P.

J. P. Véran, “Estimation de la réponse impulsionnelle et restauration d'image en optique adaptative. Application au système d’optique du télescope Canada-France-Hawaii,” Ph.D. thesis (École Nationale Supérieure des Télécommunications, 1997).

Vernet, J.

S. Ströbele, P. La Penna, R. Arsenault, R. D. Conzelmann, B. Delabre, M. Duchateau, R. Dorn, E. Fedrigo, N. Hubin, J. Quentin, P. Jolley, M. Kiekebusch, J. P. Kirchbauer, B. Klein, J. Kolb, H. Kuntschner, M. Le Louarn, J. L. Lizon, P.-Y. Madec, L. Pettazzi, C. Soenke, S. Tordo, J. Vernet, and R. Muradore, “GALACSI system design and analysis,” Proc. SPIE 8447, 844737 (2012).
[CrossRef]

Viard, E.

E. Viard, M. Le Louarn, and N. Hubin, “Adaptive optics with four laser guide stars: correction of the cone effect in large telescopes,” Appl. Opt. 41, 11–20 (2002).
[CrossRef]

A. Tokovinin, M. Le Louarn, E. Viard, N. Hubin, and R. Conan, “Optimized modal tomography in adaptive optics,” Astron. Astrophys. 378, 710–721 (2001).
[CrossRef]

Voziakova, O.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

Wallner, E. P.

Waltjen, K. E.

C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, and K. E. Waltjen, “Image improvement from a sodium-layer laser guide star adaptive optics system,” Science 277, 1649–1652 (1997).
[CrossRef]

Wild, W. J.

Winker, D. M.

Wizinowich, P.

P. Wizinowich, “Progress in laser guide star adaptive optics and lessons learned,” Proc. SPIE 8447, 84470D (2012).
[CrossRef]

Wizinowich, P. L.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[CrossRef]

Wopat, L. M.

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

Wynia, J. M.

Yi, S.

W. Min and S. Yi, “Turbulence-induced Zernike aberrations of optical wavefronts in partial adaptive compensation,” J. Mod. Opt. 48, 1559–1567 (2001).

Yura, H. T.

Zwillinger, D.

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, 7th ed. (Elsevier Academic, 2007).

Appl. Opt.

Astron. Astrophys.

M. Tallon and R. Foy, “Adaptive telescope with laser probe—isoplanatism and cone effect,” Astron. Astrophys. 235, 549–557 (1990).

R. Foy and A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

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

A. Tokovinin, M. Le Louarn, E. Viard, N. Hubin, and R. Conan, “Optimized modal tomography in adaptive optics,” Astron. Astrophys. 378, 710–721 (2001).
[CrossRef]

R. Ragazzoni, E. Marchetti, and F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

C.R. Physique

N. Hubin, R. Arsenault, R. Conzelmann, B. Delabre, M. Le Louarn, S. Stroebele, and R. Stuik, “Ground layer adaptive optics,” C.R. Physique 6, 1099–1109 (2005).
[CrossRef]

Doklady Akademiia Nauk SSSR

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J. Math. Phys.

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[CrossRef]

J. Mod. Opt.

W. Min and S. Yi, “Turbulence-induced Zernike aberrations of optical wavefronts in partial adaptive compensation,” J. Mod. Opt. 48, 1559–1567 (2001).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945 (2002).
[CrossRef]

C. E. Campbell, “Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,” J. Opt. Soc. Am. A 20, 209–217 (2003).
[CrossRef]

H. Shu, L. Luo, G. Han, and J.-L. Coatrieux, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23, 1960–1966 (2006).
[CrossRef]

S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
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L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24, 569–577 (2007).
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B. Neichel, T. Fusco, and J.-M. Conan, “Tomographic reconstruction for wide-field adaptive optics systems: Fourier domain analysis and fundamental limitations,” J. Opt. Soc. Am. A 26, 219–235 (2009).
[CrossRef]

A. Costille, C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, and T. Fusco, “Wide field adaptive optics laboratory demonstration with closed-loop tomographic control,” J. Opt. Soc. Am. A 27, 469–483 (2010).
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G. Molodij, “Wavefront propagation in turbulence: an unified approach to the derivation of angular correlation functions,” J. Opt. Soc. Am. A 28, 1732–1740 (2011).
[CrossRef]

E. Tatulli, “Transformation of Zernike coefficients: a Fourier-based method for scaled, translated, and rotated wavefront apertures,” J. Opt. Soc. Am. A 30, 726–732 (2013).
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D. L. Fried and J. F. Belsher, “Analysis of fundamental limits to artificial-guide-star adaptive-optics-system performance for astronomical imaging,” J. Opt. Soc. Am. A 11, 277–287 (1994).
[CrossRef]

R. R. Parenti and R. J. Sasiela, “Laser guide-star systems for astronomical applications,” J. Opt. Soc. Am. A 11, 288–309 (1994).
[CrossRef]

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

G. A. Tyler, “Rapid evaluation of d0: the effective diameter of a laser-guide-star adaptive-optics system,” J. Opt. Soc. Am. A 11, 325–338 (1994).
[CrossRef]

H. T. Yura and M. T. Tavis, “Centroid anisoplanatism,” J. Opt. Soc. Am. A 2, 765–773 (1985).
[CrossRef]

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

G. A. Tyler, “Analysis of propagation through turbulence—evaluation of an integral involving the product of three Bessel functions,” J. Opt. Soc. Am. A 7, 1218–1223 (1990).
[CrossRef]

Mon. Not. R. Astron. Soc.

V. Kornilov, A. Tokovinin, N. Shatsky, O. Voziakova, S. Potanin, and B. Safonov, “Combined MASS-DIMM instruments for atmospheric turbulence studies,” Mon. Not. R. Astron. Soc. 382, 1268–1278 (2007).
[CrossRef]

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

M. Schöck, R. Foy, M. Tallon, L. Noethe, and J.-P. Pique, “Performance analysis of polychromatic laser guide stars used for wavefront tilt sensing,” Mon. Not. R. Astron. Soc. 337, 910–920 (2002).
[CrossRef]

M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, “New challenges for adaptive optics: extremely large telescopes,” Mon. Not. R. Astron. Soc. 317, 535–544 (2000).
[CrossRef]

Nature

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

Proc. SPIE

S. Ströbele, P. La Penna, R. Arsenault, R. D. Conzelmann, B. Delabre, M. Duchateau, R. Dorn, E. Fedrigo, N. Hubin, J. Quentin, P. Jolley, M. Kiekebusch, J. P. Kirchbauer, B. Klein, J. Kolb, H. Kuntschner, M. Le Louarn, J. L. Lizon, P.-Y. Madec, L. Pettazzi, C. Soenke, S. Tordo, J. Vernet, and R. Muradore, “GALACSI system design and analysis,” Proc. SPIE 8447, 844737 (2012).
[CrossRef]

R. Conan, F. Bennet, A. H. Bouchez, M. A. van Dam, B. Espeland, W. Gardhouse, C. d’Orgeville, S. Parcell, P. Piatrou, I. Price, F. Rigaut, G. Trancho, and K. Uhlendorf, “The Giant Magellan Telescope laser tomography adaptive optics system,” Proc. SPIE 8447, 84473P (2012).
[CrossRef]

T. Fusco, S. Meimon, Y. Clenet, M. Cohen, H. Schnetler, J. Paufique, V. Michau, J.-P. Amans, D. Gratadour, C. Petit, C. Robert, P. Jagourel, E. Gendron, G. Rousset, J.-M. Conan, and N. Hubin, “ATLAS: the E-ELT laser tomographic adaptive optics system,” Proc. SPIE 7736, 77360D (2010).
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P. Wizinowich, “Progress in laser guide star adaptive optics and lessons learned,” Proc. SPIE 8447, 84470D (2012).
[CrossRef]

T. Fusco, J.-M. Conan, V. Michau, L. M. Mugnier, and G. Rousset, “Optimal phase reconstruction in large field of view: application to multiconjugate adaptive optics systems,” Proc. SPIE 4125, 65–76 (2000).
[CrossRef]

D. Bonaccini, W. K. Hackenberg, M. J. Cullum, E. Brunetto, T. Ott, M. Quattri, E. Allaert, M. Dimmler, M. Tarenghi, A. Van Kersteren, C. Di Chirico, B. Buzzoni, P. Gray, R. Tamai, and M. Tapia, “ESO VLT laser guide star facility,” Proc. SPIE 4494, 276–289 (2002).
[CrossRef]

N. A. Roddier, “Atmospheric wavefront simulation and Zernike polynomials,” Proc. SPIE 1237, 668–679 (1990).
[CrossRef]

Prog. Opt.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

Publ. Astron. Soc. Pac.

M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D. Campbell, J. C. Y. Chin, S. K. Hartman, R. E. Lafon, P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: performance characterization,” Publ. Astron. Soc. Pac. 118, 310–318 (2006).
[CrossRef]

Science

C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, and K. E. Waltjen, “Image improvement from a sodium-layer laser guide star adaptive optics system,” Science 277, 1649–1652 (1997).
[CrossRef]

The Messenger

D. Bonaccini Calia, Y. Feng, W. Hackenberg, R. Holzlöhner, L. Taylor, and S. Lewis, “Laser development for sodium laser guide stars at ESO,” The Messenger 139, 12–19 (2010).

Other

D. Sandler, “Laser beacon adaptive optics systems,” Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge, 1999), p. 331.

V. I. Tatarskii, Wave Propagation in Turbulent Medium (McGraw-Hill, 1961).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, Springer Series on Wave Phenomena (Springer, 1994).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1970).

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, 7th ed. (Elsevier Academic, 2007).

G. Rousset, “Wave-front sensors,” Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge, 1999), p. 91.

J.-M. Conan, “'Etude de la correction partielle en optique adaptative,” Ph.D. thesis (Université Paris XI Orsay, 1994).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998), p. 448.

J. P. Véran, “Estimation de la réponse impulsionnelle et restauration d'image en optique adaptative. Application au système d’optique du télescope Canada-France-Hawaii,” Ph.D. thesis (École Nationale Supérieure des Télécommunications, 1997).

T. Fusco, “Optique adaptative et traitement d’images pour l’astronomie: de nouveaux enjeux et de nouvelles solutions,” Ph.D. thesis (Office national d’études et de recherches aérospatiales, 2000).

C. Béchet, M. Le Louarn, R. Clare, M. Tallon, I. Tallon-Bosc, and É. Thiébaut, “Closed-loop ground layer adaptive optics simulations with elongated spots: impact of modeling noise correlations,” in Adaptative Optics for Extremely Large TelescopesY. Clénet, J.-M. Conan, Th. Fusco, and G. Rousset, eds. (EDP Sciences, 2010), p. 03004.

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Figures (7)

Fig. 1.
Fig. 1.

Left: sketch of LGS AO observations. The angular location of the laser spot is αp, and its height is L. For a given altitude h, the laser beacon light crosses a turbulence portion of radius ζ(h)R, with ζ(h)=(Lh/h). Right: representation of the Shack–Hartmann (SH) subapertures, in polar coordinates. The kth subaperture is located at a normalized radius ρk and an angle θk.

Fig. 2.
Fig. 2.

Same as Fig. 1 in the case of discrete turbulent layers. The location of the LGS defines for each layer a so-called metapupil of radius the maximum between R and R+h(|αmax|(R/L)).

Fig. 3.
Fig. 3.

Top left: residual error as a function of the turbulent layer altitude for both L=90km and L=15km, with D/r0=1. The methods used to compute the error are the matrix approach (solid line), the general approach (dash–dotted line), and through simulations (symbols+statistical dispersion). Top right: residual error as a function of the turbulence strength, for various (single layers and combination of three layers) atmospheric profiles. Bottom: K-band Strehl ratio as a function of the number of subapertures, for different GS altitude (left) and AO control loop time delays (right).

Fig. 4.
Fig. 4.

Top: residual error as a function of the number of subapertures (left) and the angular separation between the science and guide stars (right). Bottom: residual error as a function of the estimated turbulence layer altitude (left) and LGS angular position (right). The correct values have been set to h=5km and α=0.5D/L. Different control matrices are considered: the SVD method [Eq. (51), dashed line], MMSE NGS [Eq. (52), dash–dotted line], and MMSE LGS [Eq. (53), solid line].

Fig. 5.
Fig. 5.

K-band SR as a function of LGS circle radius, for different numbers of LGSs, in cases of both high altitude (L=90km, left) and low altitude (L=15km, right) guide stars. The dash–dotted line displays the corresponding single LGS case, whereas the dashed line illustrates the single NGS case. The dotted vertical line shows the LGS angle corresponding to the edge of the telescope. Observations with 2, 3, and 6 LGSs are considered, as indicated on the plots.

Fig. 6.
Fig. 6.

Left: input Hufnagel Cn2(h) profile. Middle: residual error as a function of the number of equivalent layers, for different numbers of laser spots at L=90km, as indicated on the curves. Right: same as previously, but for L=15km.

Fig. 7.
Fig. 7.

K-band performance of one sodium (solid lines) and three/six Rayleigh (dash–dotted/dashed lines) laser stars as a function of the photon flux. In the sodium case the star indicates the expected return flux for a 15 W laser. The filled star in the case of the VLT shows the current operating point according to Wizinowich [53]. For the Rayleigh case, the stars show the minimum power required for reaching the saturation regime, varying from 5 to 20 W.

Tables (1)

Tables Icon

Table 1. Evaluation of Integrals in Terms of Bessel Functions

Equations (121)

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H2J(s,n1,n2,a,b)=0xsJn12(ax)Jn2(bx)dx,
H3J(s,n1,n2,n3,a,b,c)=0xsJn1(ax)Jn2(bx)Jn3(cx)dx,
Φ(Rρ)=j=3ϕjZj(ρ)
Tr{Cov(ϕ)}=σϕ2=0.135(Dr0)53,
r0=[0.033(2π)23(2πλ)20.0230Cn2(h)dh]35.
BΦlgs(Rρ)=Φlgs(R[ρ1+ρ],αp)Φlgs(Rρ1,αp)=(2πλ)20Ln(Rζ(h)[ρ1+ρ],αp)dh0Ln(Rζ(h)ρ1,αp)dh
=(2πλ)20LBΔnh(Rζ(h)ρ)dh,
WΦlgs(κ)=(2πλ)20L1[Rζ(h)]2WΔnh(κRζ(h))dh.
WΔnh(κ)=0.033(2π)23|κ|113Cn2(h)=(λ2π)20.023r053|κ|113Cn2(h)0Cn2(h)dh,
WΦlgs(κ)=0.023(Rr0)53|κ|1130L[ζ(h)]53Cn2(h)dh0Cn2(h)dh.
Φlgs(Rρ,αp)=j=1ϕjlgs(αp)Zj(ρ).
BΦlgsΦ(Rρ,Rρ1,αp,τ)=Φ(R[ρ1+ρ],t)Φlgs(Rρ1,αp,t+τ)=(2πλ)20Ln(R[ρ1+ρ])dh0Ln(Rζ(h)ρ1+hαp+τv(h))dh
=(2πλ)20LBΔnh(Rρ1[1ζ(h)]+Rρhαpτv(h))dh.
v(h)=5+30exp[(h9.4)24.82],
t0=0.314r0v¯.
skx,y(αp)=λ2πAssubapkΦlgs(r,αp)x,yd2r=λR2πAssubapkx,y[Φlgs(Rρ,αp)]d2ρ.
s=Dϕlgs.
Dkjx,y=λR2πAssubapkZj(ρ)x,yd2ρ=λR2πAsΠsk(RRsρ)Zj(ρ)x,yd2ρ,
Π^sk(κ)=Πs(RRs[ρρk])exp2iπρ·κd2ρ=[RsR]J1(2πRs|κ|R)|κ|exp2iπρk·κ.
Dkjx=λ2πRssn,m[β|m|1x(θk)H3J(0,1,n+1,|m|1,Rs/R,1,ρk)β|m|+1x(θk)H3J(0,1,n+1,|m|+1,Rs/R,1,ρk)],
Dkjy=λ2πRssn,m[β|m|1y(θk)H3J(0,1,n+1,|m|1,Rs/R,1,ρk)+β|m|+1y(θk)H3J(0,1,n+1,|m|+1,Rs/R,1,ρk)],
sn,m=i|m|(1)3n2n+1{2ifm01ifm=0;
β|m|±1,kx,y(θk)={cos([|m|±1]θk),sin([|m|±1]θk)ifm0sin([|m|±1]θk),cos([|m|±1]θk)ifm<0.
s^x=sx(λπR)ϕ1lgs,s^y=sy(λπR)ϕ2lgs,
σres2=Πp(ρ)|Φ(Rρ)Φ˜(Rρ)|2d2ρ,
ϕ˜=M(s^+ϵ),
σres2=ϕϕ˜2atm,ϵ=ϕM(s^+ϵ)2atm,ϵ,
σres2=Tr{(ϕM(s^+ϵ))(ϕM(s^+ϵ))Tatm,ϵ}=Tr{ϕϕT+Ms^s^TMTϕs^TMTMs^ϕT+MϵϵTMT}=Tr{Cov(ϕ)+MCov(s^)MT2Cov(ϕ,s^)MT+MCov(ϵ)MT}.
Csxx=skx(αp)slx(αq)+(λπR)2ϕ1lgs(αp)ϕ1lgs(αq)(λπR)[skx(αp)ϕ1lgs(αq)+slx(αq)ϕ1lgs(αp)],
Csyy=sky(αp)sly(αq)+(λπR)2ϕ2lgs(αp)ϕ2lgs(αq)(λπR)[sky(αp)ϕ2lgs(αq)+sly(αq)ϕ2lgs(αp)],
Csxy=skx(αp)sly(αq)+(λπR)2ϕ1lgs(αp)ϕ2lgs(αq)(λπR)[skx(αp)ϕ2lgs(αq)+sly(αq)ϕ1lgs(αp)].
{skx(αp)slx(αq)sky(αp)sly(αq)skx(αp)sly(αq)}=0.04930Cn2(h)dh(Dr0)53(λRs)2×0Ldh[ζ(h)]53·Cn2(h)[{110}HJ2(8/3,1,0,Rs/R,ρklpq(h)){cos(2θklpq(h))cos(2θklpq(h))sin(2θklpq(h))}HJ2(8/3,1,2,Rs/R,ρklpq(h))],
{ϕ1lgs(αp)ϕ1lgs(αq)ϕ2lgs(αp)ϕ2lgs(αq)ϕ1lgs(αp)ϕ2lgs(αq)}=7.7910Cn2(h)dh(Dr0)53×0Ldh[ζ(h)]53·Cn2(h)[{110}HJ2(14/3,2,0,1,ρpq(h)){cos(2θpq)cos(2θpq)sin(2θpq)}HJ2(14/3,2,2,1,ρpq(h))],
{skx(αp)ϕ1lgs(αq)sky(αp)ϕ2lgs(αq)skx(αp)ϕ2lgs(αq)sky(αp)ϕ1lgs(αq)}=0.6200Cn2(h)dh(Dr0)53(λRs)×0Ldh[ζ(h)]53·Cn2(h)[{1100}HJ3(11/3,1,2,0,Rs/R,1,ρkpq(h)){cos(2θkpq)cos(2θkpq)sin(2θkpq)sin(2θkpq)}HJ3(11/3,1,2,2,Rs/R,1,ρkpq(h))],
Csϕx=skx(αp)ϕj(λπR)ϕ1lgs(αp)ϕj,
Csϕy=sky(αp)ϕj(λπR)ϕ2lgs(αp)ϕj.
{skx(αp)ϕjsky(αp)ϕj}=sn,m0.3100Cn2(h)dh(Dr0)53(λRs)0LdhCn2(h)[{β|m|1x(θkp(h))β|m|1y(θkp(h))}HJ3(11/3,1,n+1,|m|1,ζ(h)Rs/R,1,ρkp(h))+{β|m|+1x(θkp(h))β|m|+1y(θkp(h))}HJ3(11/3,1,n+1,|m|+1,ζ(h)Rs/R,1,ρkp(h)],
{ϕ1lgs(αp)ϕjϕ2lgs(αp)ϕj}=sn,m3.9860Cn2(h)dh(Dr0)530Ldh[ζ(h)]1Cn2(h)[{β|m|1x(θp)β|m|1y(θp)}HJ3(14/3,2,n+1,|m|1,ζ(h),1,ρp(h))+{β|m|+1x(θp)β|m|+1y(θp)}HJ3(14/3,2,n+1,|m|+1,ζ(h),1,ρp(h)],
σp2=(π2)21Nph(XTXD)2,
σd2=(π3)2σe2Nph2(4XT2XD)2,
Φ(Rρ)=j=4k=1Nelϕj(hk)Zj(ρ),
Φlgs(Rρ,αp)=j=1k=1Nelϕjlgs(αp,hk)Zj(ρ),
RM(h)={Rif|αmax|RLR+h(|αmax|RL)if|αmax|>RL.
ϕ(hk)=Whk·ϕM(hk),
ϕlgs(αp,hk)=Lhkαp·ϕM(hk).
ϕ=k=1NelWhk·ϕM(hk),
s=Dk=1NelLhkα·ϕM(hk),
Cov(ϕ)=k=1NelWhkCov(ϕhkM)WhkT,
Cov(s)=Dk=1NelLhkαCov(ϕhkM)[Lhkα]TDT,
Cov(ϕ,s)=Dk=1NelLhkαCov(ϕhkM)WhkT=k=1NelWhkCov(ϕhkM)[Lhkα]TDT,
Msvd=[DT·D]1·DT.
Mngs=Cov(ϕ)DT[DCov(ϕ)DT+Cov(ϵ)]1.
Mlgs=k=1NelWhkCov(ϕhkM)[Lhkα]TDT[Dk=1NelLhkαCov(ϕhkM)[Lhkα]TDT+Cov(ϵ)]1.
Δs2=(s^Dϕ˜)2=(s^DM[s^+ϵ])2.
Mlgscl=k=1NelLhkαCov(ϕhkM)[Lhkα]TDT[Dk=1NelLhkαCov(ϕhkM)[Lhkα]TDT+Cov(ϵ)]1.
hk=hmin(k)hmax(k)hCn2(h)dhhmin(k)hmax(k)Cn2(h)dh,
Cn2(hk)=hmin(k)hmax(k)Cn2(h)dhΔh.
02πcos(mγ)exp(iycos(γθk))dγ={2π(1)|m|2cos(mθk)J|m|(y)ifmeven2iπ(1)|m|12cos(mθk)J|m|(y)ifmodd,
02πsin(mγ)exp(iycos(γθk))dγ={2π(1)|m|2sin(mθk)J|m|(y)ifmeven2iπ(1)|m|12sin(mθk)J|m|(y)ifmodd.
Znm(ρ,θ)=Zj(ρ,θ)=n+1Rnm(ρ){2cos(|m|θ)ifm>02sin(|m|θ)ifm<01ifm=0,
Rnm(ρ)=s=0(n|m|)/2(1)s(ns)!s![(n+|m|)/2s]![(n|m|)/2s]!ρn2s.
Πp(ρ)Zj(ρ)Zk(ρ)d2ρ={1ifj=k0ifjk
ϕj=Πp(ρ)Zj(ρ)Φ(Rρ)d2ρ.
Qj(κ,γ)=(1)nn+1Jn+1(2πκ)πκ{(1)(n|m|)/2i|m|2cos(|m|γ)ifm>0(1)(n|m|)/2i|m|2sin(|m|γ)ifm<0(1)n/2ifm=0.
Dkjx,y=λR2πAsπ[Πp(ρ)Zj(ρ)]x,yΠsk(RRsρ)d2ρ.
Dkjx,y=λR2πAsπ·2iπκx,yQj(κ)Π^sk(κ)d2κ.
Dkjx,y=iλRπ2π2πAs[RsR]κx,yQj(κ)J1(2πRs|κ|R)|κ|exp2iπρk·κd2κ=iλRsκx,yQj(κ)J1(2πRs|κ|R)|κ|exp2iπρk·κd2κ.
[DkjxDkjy]=(1)niλπRsn+10dκJ1(2πRsRκ)Jn+1(2πκ)×02πdγ[cos(γ)sin(γ)]exp2iπρkκcos(γθk){(1)(n|m|)/2i|m|2cos(|m|γ)ifm>0(1)(n|m|)/2i|m|2sin(|m|γ)ifm<0(1)n/2ifm=0.
02πdγcos(γ)cos(|m|γ)expiycos(γθk)=π[cos([|m|1]θk)J|m|1(y)cos([|m|+1]θk)J|m|+1(y)]×{i(1)|m|22if|m|even(1)|m|12if|m|odd
02πdγcos(γ)sin(|m|γ)expiycos(γθk)=π[sin([|m|1]θk)J|m|1(y)sin([|m|+1]θk)J|m|+1(y)]×{i(1)|m|22if|m|even(1)|m|12if|m|odd
02πdγcos(γ)expiycos(γθk)=2iπcos(θk)J1(y)
02πdγsin(γ)cos(|m|γ)expiycos(γθk)=π[sin([|m|1]θk)J|m|1(y)+sin([|m|+1]θk)J|m|+1(y)]×{i(1)|m|2if|m|even(1)|m|+12if|m|odd
02πdγsin(γ)sin(|m|γ)expiycos(γθk)=π[cos([|m|1]θk)J|m|1(y)+cos([|m|+1]θk)J|m|+1(y)]×{i(1)|m|2if|m|even(1)|m|+12if|m|odd
02πdγsin(γ)expiycos(γθk)=2iπsin(θk)J1(y)
Dkjx=λRssn,m0dκJ1(2πRsRκ)Jn+1(2πκ)[β|m|1,kxJ|m|1(2πρkκ)β|m|+1,kxJ|m|+1(2πρkκ)],
Dkjy=λRssn,m0dκJ1(2πRsRκ)Jn+1(2πκ)[β|m|1,kyJ|m|1(2πρkκ)+β|m|+1,kyJ|m|+1(2πρkκ)],
sn,m=i|m|(1)3n2n+1{2ifm01ifm=0;β|m|±1,kx={cos([|m|±1]θk)ifm0sin([|m|±1]θk)ifm0.
skx(αp)slx(αq)=(λR2πAs)2subap(k,l)x1[Φlgs(Rρ1,αp)]x2[Φlgs(Rρ2,αq)]d2ρ1d2ρ2.
skx(αp)slx(αq)=(λR2πAs)2×Πsk(RRsρ1)Πsl(RRsρ2)2x1x2BΦlgs(R[ρ1ρ2],Δαqp)d2ρ1d2ρ2,
=(λR22πAs)2d2ρ1d2ρ2Πsk(RRsρ1)Πsl(RRsρ2)×0L[ζ(h)]22BΔnhx1x2(ζ(h)R[ρ1ρ2]+hΔαqp)dh,
skx(αp)slx(αq)=(λAs)2d2κΠ^sk(κ)Π^sl*(κ)κx2×0Ldh1[ζ(h)]2WΔnh(κRζ(h))exp2iπhΔαqpRζ(h)·κ.
skx(αp)slx(αq)=0.0232530Cn2(h)dh(λRAs)2(Dr0)53×d2κΠ^sk(κ)Π^sl*(κ)κx2|κ|113×0Ldh[ζ(h)]53Cn2(h)exp2iπhΔαqpRζ(h)·κ.
skx(αp)slx(αq)=0.023π22530Cn2(h)dh(λRs)2(Dr0)530Ldh[ζ(h)]53Cn2(h)×0dκ[J1(2πRsRκ)]2κ83×02πdγcos2(γ)exp2iπρklpq(h)κcos(γθklpq(h)),
02πdγcos2(γ)exp2iπρklpq(h)κcos(γθklpq(h))=π[J0(2πρklpq(h)κ)cos(2θklpq(h))J2(2πρklpq(h)κ)].
skx(αp)slx(αq)=0.023π2530Cn2(h)dh(λRs)2(Dr0)530Ldh[ζ(h)]53Cn2(h)×0dκ[J1(2πRsRκ)]2κ83[J0(2πρklpq(h)κ)cos(2θklpq(h))J2(2πρklpq(h)κ)].
sky(αp)sly(αq)=0.0232530Cn2(h)dh(λRAs)2(Dr0)53×d2κΠ^sk(κ)Π^sl*(κ)κy2|κ|113×0Ldhζ(h)53Cn2(h)exp2iπhΔαqpRζ(h)·κ
skx(αp)sly(αq)=0.0232530Cn2(h)dh(λRAs)2(Dr0)53×d2κΠ^sk(κ)Π^sl*(κ)κxκy|κ|113×0Ldhζ(h)53Cn2(h)exp2iπhΔαqpRζ(h)·κ,
sky(αp)sly(αq)=0.023π2530Cn2(h)dh(λRs)2(Dr0)530Ldh[ζ(h)]53Cn2(h)×0dκ[J1(2πRsRκ)]2κ83[J0(2πρklpq(h)κ)+cos(2θklpq(h))J2(2πρklpq(h)κ)],
skx(αp)sly(αq)=0.023π2530Cn2(h)dh(λRs)2(Dr0)530Ldh[ζ(h)]53Cn2(h)×0dκ[J1(2πRsRκ)]2κ83[sin(2θklpq(h))J2(2πρklpq(h)κ)].
ϕ1lgs(αp)ϕ1lgs(αq)=πp(ρ1)Z1(ρ1)πp(ρ2)Z1(ρ2)BΦlgs(R[ρ1ρ2],Δαqp)d2ρ1d2ρ2.
ϕ1lgs(αp)ϕ1lgs(αq)=1R2d2κQ1(κ)Q1*(κ)×0Ldh[ζ(h)]2WΔnh(κRζ(h))exp2iπhΔαqpRζ(h)·κ
=0.0232530Cn2(h)dh(Dr0)53d2κQ1(κ)Q1*(κ)|κ|113×0Ldh[ζ(h)]53Cn2(h)exp2iπhΔαqpRζ(h)·κ.
ϕ1lgs(αp)ϕ1lgs(αq)=4×0.023π22530Cn2(h)dh(Dr0)530Ldh[ζ(h)]53Cn2(h)×0dκ[J2(2πκ)]2κ143×02πdγcos2(γ)exp2iπρpq(h)κcos(γθpq),
ϕ1lgs(αp)ϕ1lgs(αq)=4×0.023π2530Cn2(h)dh(Dr0)530Ldh[ζ(h)]53Cn2(h)×0dκ[J2(2πκ)]2κ143[J0(2πρpq(h)κ)cos(2θpq(h))J2(2πρpq(h)κ)].
ϕ2lgs(αp)ϕ2lgs(αq)=4×0.023π2530Cn2(h)dh(Dr0)530Ldh[ζ(h)]53Cn2(h)×0dκ[J2(2πκ)]2κ143[J0(2πρpq(h)κ)+cos(2θpq(h))J2(2πρpq(h)κ)],
ϕ1lgs(αp)ϕ2lgs(αq)=4×0.023π2530Cn2(h)dh(Dr0)530Ldh[ζ(h)]53Cn2(h)×0dκ[J2(2πκ)]2κ143[sin(2θpq(h))J2(2πρpq(h)κ)].
skx(αp)ϕ1lgs(αq)=(λR2πAs)×Πsk(RRsρ1)πp(ρ2)Z1(ρ2)x1[BΦlgs(R[ρ1ρ2],Δαqp)]d2ρ1d2ρ2
=(λR22πAs)d2ρ1d2ρ2Πsk(RRsρ1)πp(ρ2)Z1(ρ2)×0L[ζ(h)]BΔnhx1(ζ(h)R[ρ1ρ2]+hΔαqp)dh,
skx(αp)ϕ1lgs(αq)=i(λRAs)d2κΠ^sk(κ)Q1*(κ)κx×0Ldh1[ζ(h)]2WΔnh(κRζ(h))exp2iπhΔαqpRζ(h)·κ,
=i0.0232530Cn2(h)dh(λRAs)(Dr0)53×d2κΠ^sk(κ)Q1*(κ)κx|κ|113×0Ldh[ζ(h)]53Cn2(h)exp2iπhΔαqpRζ(h)·κ.
skx(αp)ϕ1lgs(αq)=2×0.023π22530Cn2(h)dh(λRs)(Dr0)530Ldh[ζ(h)]53Cn2(h)×0dκJ1(2πRsRκ)J2(2πκ)κ113×02πdγcos2(γ)exp2iπρkpq(h)κcos(γθkpq),
skx(αp)ϕ1lgs(αq)=2×0.023π2530Cn2(h)dh(λRs)(Dr0)530Ldh[ζ(h)]53Cn2(h)×0dκJ1(2πRsRκ)J2(2πκ)κ113[J0(2πρkpq(h)κ)cos(2θkpq(h))J2(2πρkpq(h)κ)].
sky(αp)ϕ2lgs(αq)=i0.0232530Cn2(h)dh(λRAs)(Dr0)53×d2κΠ^sk(κ)Q2*(κ)κy|κ|113×0Ldh[ζ(h)]53Cn2(h)exp2iπhΔαqpRζ(h)·κ,
skx(αp)ϕ2lgs(αq)=i0.0232530Cn2(h)dh(λRAs)(Dr0)53×d2κΠ^sk(κ)Q2*(κ)κx|κ|113×0Ldh[ζ(h)]53Cn2(h)exp2iπhΔαqpRζ(h)·κ,
sky(αp)ϕ1lgs(αq)=i0.0232530Cn2(h)dh(λRAs)(Dr0)53×d2κΠ^sk(κ)Q1*(κ)κy|κ|113×0Ldh[ζ(h)]53Cn2(h)exp2iπhΔαqpRζ(h).κ,
sky(αp)ϕ2lgs(αq)=2×0.023π2530Cn2(h)dh(λRs)(Dr0)530Ldh[ζ(h)]53Cn2(h)×0dκJ1(2πRsRκ)J2(2πκ)κ113[J0(2πρkpq(h)κ)+cos(2θkpq(h))J2(2πρkpq(h)κ)],
skx(αp)ϕ2lgs(αq)=sky(αp)ϕ1lgs(αq)=2×0.023π2530Cn2(h)dh(λRs)(Dr0)53×0Ldh[ζ(h)]53Cn2(h)0dκJ1(2πRsRκ)J2(2πκ)κ113[sin(2θkpq(h))J2(2πρkpq(h)κ)].
skx(αp)ϕj=(λR2πAs)d2ρ1d2ρ2Πsk(RRsρ1)πp(ρ2)Zj(ρ2)×x1[ϕlgs(Rρ1,αp)]ϕ(Rρ2),
=(λR22πAs)d2ρ1d2ρ2Πsk(RRsρ1)πp(ρ2)Zj(ρ2)×0L[ζ(h)]BΔnhx1(R[ζ(h)ρ1ρ2]+hαp)dh.
skx(αp)ϕj=i(λRAs)d2κ0LdhΠ^sk*(ζ(h)κ)Qj(κ)κx[ζ(h)]WΔnh(κR)exp2iπhαpR·κ=i0.0232530Cn2(h)dh(λRAs)(Dr0)53×d2κ0LdhΠ^sk*(ζ(h)κ)Qj(κ)κx|κ|113[ζ(h)]Cn2(h)exp2iπhαpR·κ.
skx(αp)ϕj=i(1)m2sn,m0.023π22530Cn2(h)dh(λRs)(Dr0)530LdhCn2(h)×0dκJ1(2πζ(h)RsRκ)J2(2πκ)κ113×02πdγcos(γ){cos(|m|γ)sin(|m|γ)1}exp2iπρkp(h)κcos(γθkp(h)),
skx(αp)ϕj=sn,m0.023π2530Cn2(h)dh(λRs)(Dr0)530LdhCn2(h)×0dκJ1(2πζ(h)RsRκ)J2(2πκ)κ113×[β|m|1,kx(θkp(h))J|m|1(2πρkp(h)κ)β|m|+1,kx(θkp(h))J|m|+1((2πρkp(h)κ)].
sky(αp)ϕj=i0.0232530Cn2(h)dh(λRAs)(Dr0)53×d2κ0LdhΠ^sk*(ζ(h)κ)Qj(κ)κy|κ|113[ζ(h)]Cn2(h)exp2iπhαpR·κ,
sky(αp)ϕj=sn,m0.023π2530Cn2(h)dh(λRs)(Dr0)530LdhCn2(h)×0dκJ1(2πζ(h)RsRκ)J2(2πκ)κ113×[β|m|1,ky(θkp(h))J|m|1(2πρkp(h)κ)+β|m|+1,ky(θkp(h))J|m|+1((2πρkp(h)κ)].
ϕ1lgsϕj=d2ρ1d2ρ2πp(ρ1)Z1(ρ1)πp(ρ2)Zj(ρ2)ϕlgs(Rρ1,αp)ϕ(Rρ2)=d2ρ1d2ρ2πp(ρ1)Z1(ρ1)πp(ρ2)Zj(ρ2)0LdhBΔnh(R[ζ(h)ρ1ρ2]+hαp),
ϕ1lgsϕj=1R2d2κ0LdhQ1*(ζ(h)κ)Qj(κ)WΔnh(κR)exp2iπhαpR·κ,
=0.0232530Cn2(h)dh(Dr0)53×d2κ0LdhQ1*(ζ(h)κ)Qj(κ)|κ|113Cn2(h)exp2iπhαpR·κ.
ϕ1lgsϕj=i(1)m2sn,m4×0.023π22530Cn2(h)dh(Dr0)530Ldh[ζ(h)]1Cn2(h)×0dκJ2(2πζ(h)κ)Jn+1(2πκ)κ113×02πdγcos(γ){cos(|m|γ)sin(|m|γ)1}exp2iπρp(h)κcos(γθp),
ϕ1lgsϕj=sn,m2×0.023π2530Cn2(h)dh(Dr0)530Ldh[ζ(h)]1Cn2(h)×0dκJ2(2πζ(h)κ)Jn+1(2πκ)κ143×[β|m|1,kx(θp)J|m|1(2πρp(h)κ)β|m|+1,kx(θp)J|m|+1((2πρp(h)κ)].
ϕ2lgsϕj=0.0232530Cn2(h)dh(Dr0)53×d2κ0LdhQ2*(ζ(h)κ)Qj(κ)|κ|113Cn2(h)exp2iπhαpR·κ,
=sn,m2×0.023π2530Cn2(h)dh(Dr0)530Ldh[ζ(h)]1Cn2(h)×0dκJ2(2πζ(h)κ)Jn+1(2πκ)κ143×[β|m|1,ky(θp)J|m|1(2πρp(h)κ)+β|m|+1,kx(θp)J|m|+1((2πρp(h)κ)].

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