Abstract

A general formulation is given for the derivation of theoretical temporal power spectra of quantities related to turbulent wave-front phase. These temporal power spectra and their asymptotic power laws and cutoff frequencies are presented for various quantities of interest in the field of interferometry (differential piston), wave-front sensing (Shack–Hartmann and curvature sensor), adaptive optics (Zernike polynomials), and seeing monitoring (differential angle of arrival). We show that the differential piston spectrum has two cutoff frequencies and exhibits a very steep decrease at high frequencies. The curvature sensor is shown to be much less sensitive than the Shack–Hartmann sensor to the low temporal frequencies. A study of the Zernike temporal power spectra shows that their cutoff frequencies increase with the polynomial radial degree. Both single-layer and multilayer plane and spherical waves are considered. The effect of wind direction is also taken into account. We point out the influence of the cone effect on the temporal power spectra when Rayleigh or sodium laser guide stars are used for wave-front sensing. The cone effect results in a temporal decorrelation between natural and laser guide star wave fronts. Finally, we demonstrate that in adaptive optics systems low-order modes require higher servoloop bandwidths than do high-order modes in order for the residual variance to be balanced between the corrected modes. The same conclusion applies to fringe tracking in large telescope interferometers equipped with adaptive optics systems.

© 1995 Optical Society of America

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1994

1993

1992

1991

N. S. Nightingale, D. F. Busher, “Interferometric seeing measurements at the La Palma Observatory,” Mon. Not. R. Astron. Soc. 251, 155–166 (1991).

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

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

1990

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

M. Sarazin, F. Roddier, “The E.S.O. differential image motion monitor,” Astron. Astrophys. 227, 294–303 (1990).

J. Primot, G. Rousset, J.-C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
[CrossRef]

1989

F. Chassat, “Calcul du domaine d’isoplanétisme d’un système d’optique adaptative fonctionnant à travers la turbulence atmosphérique,” J. Opt. (Paris) 20, 13–23 (1989).
[CrossRef]

1988

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

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

1987

1985

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

J.-C. Fontanella, “Analyse de surface d’onde déconvolution et optique active,” J. Opt. (Paris) 16, 257–268 (1985).
[CrossRef]

1983

C. Roddier, F. Roddier, “High angular resolution observations of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 270, L23–L26 (1983).
[CrossRef]

1982

F. Roddier, J.-M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar speckle interferometry,” J. Opt. (Paris) 13, 263–271 (1982).
[CrossRef]

1977

1976

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

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
[CrossRef]

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

1975

A. Labeyrie, “Interference fringes obtained on Vega with two optical telescopes,” Astrophys. J. 196, L71–L75 (1975).
[CrossRef]

D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
[CrossRef]

1971

1970

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

1966

Acton, D. S.

Ameer, G. A.

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

Barclay, H. T.

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

Beckers, J. M.

J. M. Beckers, M. Faucherre, B. Koehler, O. von der Lühe, “A progress report on the implementation of the VLT interferometer,” in Proceedings of European Southern Observatory Symposium on High Resolution Imaging by Interferometry, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 775–787.

Belsher, J.

Boeke, B. R.

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

Boyer, C.

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

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

Browne, S. L.

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

Busher, D. F.

N. S. Nightingale, D. F. Busher, “Interferometric seeing measurements at the La Palma Observatory,” Mon. Not. R. Astron. Soc. 251, 155–166 (1991).

Butts, R. R.

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

Cassaing, F.

S. Robbe, B. Sorrente, F. Cassaing, Y. Rabbia, G. Rousset, L. Damé, P. Cruzalèbes, G. Schumacher, “Active phase stabilization in the I2T: implementation of the ASSI table,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2200, 220–230 (1994).

Chassat, F.

F. Chassat, “Calcul du domaine d’isoplanétisme d’un système d’optique adaptative fonctionnant à travers la turbulence atmosphérique,” J. Opt. (Paris) 20, 13–23 (1989).
[CrossRef]

Clifford, S. F.

S. F. Clifford, “Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence,” J. Opt. Soc. Am. 61, 1285–1292 (1971).
[CrossRef]

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 9–43.
[CrossRef]

Colavita, M. M.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

M. M. Colavita, M. Shao, D. H. Staelin, “Atmospheric phase measurements with the Mark III stellar interferometer,” Appl. Opt. 26, 4106–4112 (1987).
[CrossRef] [PubMed]

Conan, J.-M.

P.-Y. Madec, J.-M. Conan, G. Rousset, “Temporal characterisation of atmospheric wavefront for adaptive optics,” in Proceedings of European Southern Observatory Conference 42 on Progress in Telescope and Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 471–474.

J.-M. Conan, “Etude de la correction partielle en optique adaptative,” Ph.D. dissertation (Paris XI University, Paris, 1994).

J.-M. Conan, P.-Y. Madec, G. Rousset, “Temporal power spectra of turbulent wavefronts,” in Proceedings of the Thirteenth National Solar Observatory Sacramento Peak Summer Workshop on Real Time and Post Facto Solar Image Correction, R. R. Radick, ed. (NSO/SP, Sunspot, N.M., 1992), pp. 6–14.

J.-M. Conan, P.-Y. Madec, G. Rousset, “Image formation in adaptive optics partial correction,” in Proceedings of International Commission for Optics 16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (ICO 16 Secretariat, Garching, Germany, 1991), pp. 181–186.

Cruzalèbes, P.

S. Robbe, B. Sorrente, F. Cassaing, Y. Rabbia, G. Rousset, L. Damé, P. Cruzalèbes, G. Schumacher, “Active phase stabilization in the I2T: implementation of the ASSI table,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2200, 220–230 (1994).

Damé, L.

S. Robbe, B. Sorrente, F. Cassaing, Y. Rabbia, G. Rousset, L. Damé, P. Cruzalèbes, G. Schumacher, “Active phase stabilization in the I2T: implementation of the ASSI table,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2200, 220–230 (1994).

Demerlé, M.

M. Demerlé, P.-Y. Madec, G. Rousset, “Servo-loop analysis for adaptive optics,” in NATO Advanced Science Institutes Series on Adaptive Optics for Astronomy, D. Alloin, J.-M. Mariotti, eds. (Kluwer, Dordrecht, The Netherlands, 1993), pp. 73–88.

Drexler, J. J.

F. D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Waldie, D. B. Soules, J. A. Qualtrough, “Phase structure function measurements with multiple apertures,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 218–223 (1989).
[CrossRef]

D. B. Soules, J. J. Drexler, A. H. Waldie, J. A. Qualtrough, F. D. Eaton, W. A. Peterson, J. R. Hines, “Temporal characteristics of turbulence-induced image motion,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 224–231 (1989).
[CrossRef]

Eaton, F. D.

D. B. Soules, J. J. Drexler, A. H. Waldie, J. A. Qualtrough, F. D. Eaton, W. A. Peterson, J. R. Hines, “Temporal characteristics of turbulence-induced image motion,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 224–231 (1989).
[CrossRef]

F. D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Waldie, D. B. Soules, J. A. Qualtrough, “Phase structure function measurements with multiple apertures,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 218–223 (1989).
[CrossRef]

Faucherre, M.

J. M. Beckers, M. Faucherre, B. Koehler, O. von der Lühe, “A progress report on the implementation of the VLT interferometer,” in Proceedings of European Southern Observatory Symposium on High Resolution Imaging by Interferometry, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 775–787.

Fontanella, J.-C.

J. Primot, G. Rousset, J.-C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
[CrossRef]

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

J.-C. Fontanella, “Analyse de surface d’onde déconvolution et optique active,” J. Opt. (Paris) 16, 257–268 (1985).
[CrossRef]

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

Foy, R.

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

Fried, D. L.

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

D. 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, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
[CrossRef]

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
[CrossRef]

Fugate, R. Q.

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

Gaffard, J.-P.

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G. Rousset, J.-C. Fontanella, P. Kern, F. Rigaut, P. Léna, P. Gigan, F. Rigaut, J.-P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for IR astronomy I: system description,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344 (1990).
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G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

G. Rousset, J.-C. Fontanella, P. Kern, F. Rigaut, P. Léna, P. Gigan, F. Rigaut, J.-P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for IR astronomy I: system description,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344 (1990).
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M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

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M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

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M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

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G. Rousset, J.-C. Fontanella, P. Kern, F. Rigaut, P. Léna, P. Gigan, F. Rigaut, J.-P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for IR astronomy I: system description,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344 (1990).
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G. Rousset, J.-C. Fontanella, P. Kern, F. Rigaut, P. Léna, P. Gigan, F. Rigaut, J.-P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for IR astronomy I: system description,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344 (1990).
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G. Rousset, J.-C. Fontanella, P. Kern, F. Rigaut, P. Léna, P. Gigan, F. Rigaut, J.-P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for IR astronomy I: system description,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344 (1990).
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D. B. Soules, J. J. Drexler, A. H. Waldie, J. A. Qualtrough, F. D. Eaton, W. A. Peterson, J. R. Hines, “Temporal characteristics of turbulence-induced image motion,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 224–231 (1989).
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F. D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Waldie, D. B. Soules, J. A. Qualtrough, “Phase structure function measurements with multiple apertures,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 218–223 (1989).
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D. B. Soules, J. J. Drexler, A. H. Waldie, J. A. Qualtrough, F. D. Eaton, W. A. Peterson, J. R. Hines, “Temporal characteristics of turbulence-induced image motion,” in Propagation Engineering, N. S. Kopeika, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1115, 224–231 (1989).
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G. Rousset, P.-Y. Madec, D. Rabaud, “Adaptive optics partial correction simulation for two telescope interferometry,” in Proceedings of European Southern Observatory Symposium on High Resolution Imaging by Interferometry, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104.

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S. Robbe, B. Sorrente, F. Cassaing, Y. Rabbia, G. Rousset, L. Damé, P. Cruzalèbes, G. Schumacher, “Active phase stabilization in the I2T: implementation of the ASSI table,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2200, 220–230 (1994).

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

G. Rousset, J.-C. Fontanella, P. Kern, F. Rigaut, P. Léna, P. Gigan, F. Rigaut, J.-P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for IR astronomy I: system description,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344 (1990).
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S. Robbe, B. Sorrente, F. Cassaing, Y. Rabbia, G. Rousset, L. Damé, P. Cruzalèbes, G. Schumacher, “Active phase stabilization in the I2T: implementation of the ASSI table,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2200, 220–230 (1994).

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R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
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G. Rousset, J.-C. Fontanella, P. Kern, F. Rigaut, P. Léna, P. Gigan, F. Rigaut, J.-P. Gaffard, C. Boyer, P. Jagourel, F. Merkle, “Adaptive optics prototype system for IR astronomy I: system description,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 336–344 (1990).
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G. Rousset, P.-Y. Madec, D. Rabaud, “Adaptive optics partial correction simulation for two telescope interferometry,” in Proceedings of European Southern Observatory Symposium on High Resolution Imaging by Interferometry, J. M. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104.

J.-M. Conan, P.-Y. Madec, G. Rousset, “Temporal power spectra of turbulent wavefronts,” in Proceedings of the Thirteenth National Solar Observatory Sacramento Peak Summer Workshop on Real Time and Post Facto Solar Image Correction, R. R. Radick, ed. (NSO/SP, Sunspot, N.M., 1992), pp. 6–14.

J.-M. Conan, P.-Y. Madec, G. Rousset, “Image formation in adaptive optics partial correction,” in Proceedings of International Commission for Optics 16 Satellite Conference on Active and Adaptive Optics, F. Merkle, ed. (ICO 16 Secretariat, Garching, Germany, 1991), pp. 181–186.

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S. Robbe, B. Sorrente, F. Cassaing, Y. Rabbia, G. Rousset, L. Damé, P. Cruzalèbes, G. Schumacher, “Active phase stabilization in the I2T: implementation of the ASSI table,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2200, 220–230 (1994).

G. Rousset, P.-Y. Madec, F. Rigaut, “Temporal analysis of turbulent wavefronts sensed by adaptive optics,” in International Commission for Optics Topical Meeting on Atmospheric Volume and Surface Scattering and Propagation (Centro Duplicazione Offset, Florence, Italy, 1991), pp. 77–80.

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M. Sarazin, F. Roddier, “The E.S.O. differential image motion monitor,” Astron. Astrophys. 227, 294–303 (1990).

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S. Robbe, B. Sorrente, F. Cassaing, Y. Rabbia, G. Rousset, L. Damé, P. Cruzalèbes, G. Schumacher, “Active phase stabilization in the I2T: implementation of the ASSI table,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2200, 220–230 (1994).

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M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Solid curve, temporal power spectrum of the piston with V/D = 9 Hz. The asymptotic power laws and the cutoff frequency are indicated; dashed–dotted curve, phase temporal power spectrum (for comparison).

Fig. 2
Fig. 2

Temporal power spectrum of the differential piston for an 18-m s−1 wind along the x axis, 1.8-m telescopes and a 200-m baseline along the y axis. The asymptotic power laws and the two cutoff frequencies νc1 and νc2 are indicated.

Fig. 3
Fig. 3

Angle-of-arrival temporal power spectra: solid curve, angle of arrival along x for a circular aperture (diameter D) and its cutoff frequency νc; dashed curve, angle of arrival along y for a square aperture (size D); V/D = 10 Hz; V along the x axis. The spectra are normalized to have the same variance. The asymptotic power laws are mentioned.

Fig. 4
Fig. 4

Curvature temporal power spectrum on a circular aperture (diameter D); V/D = 10 Hz. The asymptotic power laws and the cutoff frequency are indicated.

Fig. 5
Fig. 5

Eighth Zernike (coma) temporal power spectrum: comparison between single-layer and multilayer spectra integrated over a LASSCA profile. The asymptotic power laws are indicated.

Fig. 6
Fig. 6

Zernike polynomial mean temporal power spectra in a given radial degree n for n = 1, 3, 9. The spectra are normalized to the turbulence variance of one polynomial of the considered radial degree; V/D = 10 Hz. The asymptotic power laws and the cutoff frequencies are indicated.

Fig. 7
Fig. 7

Depiction of spherical-wave propagation through a turbulent layer: apparent wind velocity.

Fig. 8
Fig. 8

Temporal power spectra of the eighth Zernike coefficient for various sources: solid curve, natural star at infinity; dashed curve, sodium-layer laser guide star at 90 km; crosses on solid curve, Rayleigh star at 12 km. Integration is over a LASSCA profile.

Fig. 9
Fig. 9

Diamonds, Zernike polynomial residual variance after correction by an adaptive optics system. Zernike polynomials 2–36 (radial degrees 1–7) are corrected with a 30-Hz bandwidth, assuming that V/D = 10 Hz. The variances are averaged over all the polynomials of a given radial degree. They are calculated for D/r0 = 1, and a (D/r0)5/3 scaling factor should be applied otherwise. The turbulent variances before correction (stars) are also indicated.

Equations (46)

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B G ( ρ ) = G ( r , t ) G ( r + ρ , t ) ,
W G ( f ) = B G ( ρ ) exp ( 2 i π f · ρ ) d ρ ,
G ( r , t ) = M G ( r - ρ ) ϕ ( ρ , t ) d ρ = M G ( r ) * ϕ ( r , t ) .
W G ( f ) = M ˜ G ( f ) 2 W ϕ ( f ) ,
W ϕ ( f ) = 0.033 ( 2 π ) - 2 / 3 ( 2 π / λ ) 2 C N 2 d h f - 11 / 3 ,
G ( r , t + τ ) = G ( r - τ V , t ) .
w G ( ν ) = 1 V - + W G ( ν V , f y ) d f y .
w G ( ν ) = 1 V - + | M ˜ G ( ν V , f y ) | 2 W ϕ ( ν V , f y ) d f y = 0.033 C N 2 d h V ( 2 π ) - 2 / 3 ( 2 π / λ ) 2 × - + | M ˜ G ( ν V , f y ) | 2 [ ( ν / V ) 2 + f y 2 ] - 11 / 6 d f y ,
G ( r ) = ϕ ( r ) .
w ϕ ( ν ) C N 2 d h V ( ν V ) - 8 / 3 .
G ( r ) = ϕ ( r ) x ,             M ˜ G ( f ) = 2 i π f x .
w ϕ / x ( ν ) C N 2 d h V ( ν V ) - 2 / 3 .
G ( r ) = P ( r / D ) * ϕ ( r ) ,
P ( r / D ) = 4 / ( π D 2 ) if r D / 2 , P ( r / D ) = 0 elsewhere .
M ˜ G ( f ) = 2 J 1 ( π D f ) π D f
w piston ( ν ) = 4 V - + [ J 1 ( π D q ) π D q ] 2 W ϕ ( q ) d f y ,
G ( r ) = [ δ ( r + B ) - δ ( r ) ] * P ( r / D ) * ϕ ( r , t ) ,
M ˜ G ( f ) = sin ( π B · f ) 2 J 1 ( π D f ) π D f .
w fringe motion ( ν ) = 4 sin 2 ( π B ν / V ) w piston ( ν ) ,
G ( r ) = P ( r / D ) * ϕ ( r ) x .
M ˜ G ( f ) = ( 2 i π f x ) 2 J 1 ( π D f ) π D f .
w angle of arrival ( ν ) ( ν / V ) 2 w piston ( ν ) ,
G ( r ) = P ( r / D ) * [ 2 x 2 ϕ ( r , t ) + 2 y 2 ϕ ( r , t ) ] ,
M ˜ G ( f ) = - 8 π f J 1 ( π D f ) / D ,
w curvature ( ν ) = 64 π 2 V D 2 - + [ q J 1 ( π D q ) ] 2 W ϕ ( q ) d f y ,
G ( r ) = Z j ( r ) * ϕ ( r , t ) ,
M ˜ G ( f ) = n + 1 2 J n + 1 ( π D f ) π D f × { 2 cos ( m θ ) 2 sin ( m θ ) for m 0 , 1 for m = 0 ,
ν c ~ 0.3 ( n + 1 ) V / D .
G = [ δ ( r + B ) - δ ( r ) ] * P ( r / D ) * x ϕ ( r , t ) ,
w n , m , cos , θ ( ν ) = cos 2 ( m θ ) w n , m , cos , θ = 0 ( ν ) + sin 2 ( m θ ) w n , m , sin , θ = 0 ( ν ) , w n , m , sin , θ ( ν ) = sin 2 ( m θ ) w n , m , cos , θ = 0 ( ν ) + cos 2 ( m θ ) w n , m , sin , θ = 0 ( ν ) ,
w θ ( ν ) = cos 2 ( θ ) w θ = 0 ( ν ) + sin 2 ( θ ) w θ = π / 2 ( ν ) ,
d w G ( ν , h ) = C N ( h ) 2 d h V ( h ) F G [ ν / V ( h ) ] ,
F G [ ν / V ( h ) ] = 0.033 ( 2 π ) - 2 / 3 ( 2 π / λ ) 2 × - + | M ˜ G [ ν V ( h ) , f y ] | 2 × { [ ν / V ( h ) ] 2 + f y 2 } - 11 / 6 d f y ,
d w G ( ν , h ) = C N ( h ) 2 d h V ( h ) F G [ ν / V ( h ) , θ ( h ) ] .
w G ( ν ) = 0 h max C N ( h ) 2 d h V ( h ) F G [ ν / V ( h ) , θ ( h ) ] .
W ϕ ( f ) = 0.033 ( 2 π ) - 2 / 3 ( 2 π / λ ) 2 ( z / L ) 5 / 3 C N 2 d z f - 11 / 3 .
d w G ( ν , z ) = ( z / L ) 8 / 3 C N ( z ) 2 d z V ( z ) F G [ ν z V ( z ) L ] ,
w G ( ν ) = 0 L ( z / L ) 8 / 3 C N ( z ) 2 d z V ( z ) F G [ ν z V L , θ ( z ) ] .
σ 2 = - + T ( ν ) 2 w G ( ν ) d ν .
w G ( ν ) = R G ( τ ) exp ( 2 i π ν τ ) d τ ,
R G ( τ ) = G ( r , t ) G ( r , t + τ ) = B G ( V τ ) .
w G ( ν ) = W G ( f ) exp ( - 2 i π f · V τ ) exp ( 2 i π ν τ ) d f d τ .
w G ( ν ) = W G ( f ) 1 V δ ( f x - ν V ) d f = 1 V W G ( ν V , f y ) d f y .
w G ( ν ) = 0 L d w G ( ν , z ) = C N 2 V 0 L ( z / L ) 8 / 3 F G ( ν z V L ) d z ,
F G ( ν z V L ) = - + ( n + 1 ) 4 J n + 1 ( π D q ) 2 ( π D q ) 2 × q - 11 / 6 { 2 cos ( m θ ) 2 2 sin ( m θ ) 2 1 } d f y
w G ( ν ) = L C N 2 V ( ν V ) - 11 / 3 0 ν / V Z 8 / 3 F G ( Z ) d Z .

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