Abstract

The two-dimensional (2D) spatial covariance of the angle-of-arrival (AA) fluctuations is often used to investigate the properties of wave fronts corrugated by the atmosphere for high-angular-resolution techniques. Theoretical series expansions of this covariance are presented. The fast convergence of these series reduces the calculation time of the covariance done by numerical integration. The 2D covariance is a nonradial function. A physical interpretation of this anisotropy is proposed. The spatiotemporal correlation of the AA is deduced from the covariance assuming the “frozen-flow” hypothesis. The impact of the anisotropy on the evaluation of the number of predominant turbulent layers and on the corresponding winds is investigated, and an analysis of temporal correlations is performed. A simple theoretical approximation of the decorrelation time of the AA is given, which is found to be in agreement with experimental results.

© 2000 Optical Society of America

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    [CrossRef]

1999

1998

R. Avila, J. Vernin, S. Cuevas, “Turbulence profiles with generalized Scidar at San Pedro Màrtir Observatory and isoplanatism studies,” Publ. Astron. Soc. Pac. 110, 1106–1116 (1998).
[CrossRef]

M. Schöck, E. Spillar, “Measuring wind speeds and turbulence with a wave-front sensor,” Opt. Lett. 23, 150–152 (1998).
[CrossRef]

1997

1996

E. Gendron, P. Lena, “Single layer atmospheric turbulence demonstrated by adaptive optics observations,” Astrophys. Space Sci. 239, 221–228 (1996).
[CrossRef]

1995

V. Voitsekhovich, S. Cuevas, “Adaptive optics and the outer scale of turbulence,” J. Opt. Soc. Am. A 12, 2523–2531 (1995).
[CrossRef]

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wavefront outer scale at the O.C.A.,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

1994

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. I. The instrument and first results of angle of arrival measurements,” Astron. Astrophys. Suppl. Ser. 108, 173–180 (1994).

1993

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

1992

J. Borgnino, F. Martin, A. Ziad, “Effect of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91, 267–279 (1992).
[CrossRef]

M. Jorgenson, G. Aitken, “Prediction of atmospherically induced wave-front degradations,” Opt. Lett. 17, 466–468 (1992).
[CrossRef] [PubMed]

1991

D. 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]

F. Rigaut, G. Rousset, P. Kern, J. Fontanella, J. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

1990

1982

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

F. Roddier, J. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[CrossRef]

1971

1970

1967

J. Strohbehn, S. Clifford, “Polarization and angle-of-arrival fluctuations for a plane wave propagated through a turbulent medium,” IEEE Trans. Antennas Propag. AP-15, 416–422 (1967).
[CrossRef]

1966

1953

H. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Agabi, A.

R. Avila, A. Ziad, J. Borgnino, F. Martin, A. Agabi, A. Tokovinin, “Theoretical spatio-temporal analysis of angle of arrival induced by atmospheric turbulence as observed with the Grating Scale Monitor experiment,” J. Opt. Soc. Am. A 14, 3070–3082 (1997).
[CrossRef]

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wavefront outer scale at the O.C.A.,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. I. The instrument and first results of angle of arrival measurements,” Astron. Astrophys. Suppl. Ser. 108, 173–180 (1994).

A. Agabi, “GSM: une expérience dédiée à la mesure des paramètres de cohérence des fronts d’onde en Haute Résolution Angulaire,” Ph.D. thesis (Université de Nice-Sophia Antipolis, Nice, France, 1994).

Aitken, G.

Avila, R.

R. Avila, J. Vernin, S. Cuevas, “Turbulence profiles with generalized Scidar at San Pedro Màrtir Observatory and isoplanatism studies,” Publ. Astron. Soc. Pac. 110, 1106–1116 (1998).
[CrossRef]

R. Avila, A. Ziad, J. Borgnino, F. Martin, A. Agabi, A. Tokovinin, “Theoretical spatio-temporal analysis of angle of arrival induced by atmospheric turbulence as observed with the Grating Scale Monitor experiment,” J. Opt. Soc. Am. A 14, 3070–3082 (1997).
[CrossRef]

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, “Report on G.S.M. measurement campaign at La Silla,” (European Southern Observatory, Garching, Germany, 1998).

R. Conan, A. Ziad, R. Avila, A. Tokovinin, F. Martin, J. Borgnino, “Spatio-temporal analysis of the wave front with the GSM,” in Topical Meeting on Astronomy with Adaptive Optics, Present Results and Future Programs, D. Bonaccini, ed. (European Southern Observatory, Garching, Germany, 1998), pp. 133–142.

A. Tokovinin, A. Ziad, F. Martin, R. Avila, J. Borgnino, R. Conan, M. Sarazin, “Wavefront outer scale monitoring at La Silla ,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

Babcock, H.

H. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Borgnino, J.

R. Avila, A. Ziad, J. Borgnino, F. Martin, A. Agabi, A. Tokovinin, “Theoretical spatio-temporal analysis of angle of arrival induced by atmospheric turbulence as observed with the Grating Scale Monitor experiment,” J. Opt. Soc. Am. A 14, 3070–3082 (1997).
[CrossRef]

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wavefront outer scale at the O.C.A.,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. I. The instrument and first results of angle of arrival measurements,” Astron. Astrophys. Suppl. Ser. 108, 173–180 (1994).

J. Borgnino, F. Martin, A. Ziad, “Effect of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91, 267–279 (1992).
[CrossRef]

J. Borgnino, “Estimation of the spatial coherence outer scale relevant to long baseline interferometry and imaging in optical astronomy,” Appl. Opt. 29, 1863–1865 (1990).
[CrossRef] [PubMed]

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, “Report on G.S.M. measurement campaign at La Silla,” (European Southern Observatory, Garching, Germany, 1998).

R. Conan, A. Ziad, R. Avila, A. Tokovinin, F. Martin, J. Borgnino, “Spatio-temporal analysis of the wave front with the GSM,” in Topical Meeting on Astronomy with Adaptive Optics, Present Results and Future Programs, D. Bonaccini, ed. (European Southern Observatory, Garching, Germany, 1998), pp. 133–142.

A. Tokovinin, A. Ziad, F. Martin, R. Avila, J. Borgnino, R. Conan, M. Sarazin, “Wavefront outer scale monitoring at La Silla ,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

Chassat, F.

F. Chassat, “Propagation optique à travers la turbulence atmosphérique. Etude modale de l’anisoplanétisme et application à l’optique adaptative,” Ph.D. thesis (Université Paris XI Orsay, Paris, France, 1992).

Clifford, S.

J. Strohbehn, S. Clifford, “Polarization and angle-of-arrival fluctuations for a plane wave propagated through a turbulent medium,” IEEE Trans. Antennas Propag. AP-15, 416–422 (1967).
[CrossRef]

Conan, R.

A. Tokovinin, A. Ziad, F. Martin, R. Avila, J. Borgnino, R. Conan, M. Sarazin, “Wavefront outer scale monitoring at La Silla ,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, “Report on G.S.M. measurement campaign at La Silla,” (European Southern Observatory, Garching, Germany, 1998).

R. Conan, A. Ziad, R. Avila, A. Tokovinin, F. Martin, J. Borgnino, “Spatio-temporal analysis of the wave front with the GSM,” in Topical Meeting on Astronomy with Adaptive Optics, Present Results and Future Programs, D. Bonaccini, ed. (European Southern Observatory, Garching, Germany, 1998), pp. 133–142.

Consortini, A.

Cuevas, S.

R. Avila, J. Vernin, S. Cuevas, “Turbulence profiles with generalized Scidar at San Pedro Màrtir Observatory and isoplanatism studies,” Publ. Astron. Soc. Pac. 110, 1106–1116 (1998).
[CrossRef]

V. Voitsekhovich, S. Cuevas, “Adaptive optics and the outer scale of turbulence,” J. Opt. Soc. Am. A 12, 2523–2531 (1995).
[CrossRef]

Ellerbroek, B.

Fontanella, J.

F. Rigaut, G. Rousset, P. Kern, J. Fontanella, J. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Fried, D.

Gaffard, J.

F. Rigaut, G. Rousset, P. Kern, J. Fontanella, J. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Gendron, E.

E. Gendron, P. Lena, “Single layer atmospheric turbulence demonstrated by adaptive optics observations,” Astrophys. Space Sci. 239, 221–228 (1996).
[CrossRef]

Gilli, J.

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

F. Roddier, J. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[CrossRef]

Jorgenson, M.

Kern, P.

F. Rigaut, G. Rousset, P. Kern, J. Fontanella, J. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Lena, P.

E. Gendron, P. Lena, “Single layer atmospheric turbulence demonstrated by adaptive optics observations,” Astrophys. Space Sci. 239, 221–228 (1996).
[CrossRef]

F. Rigaut, G. Rousset, P. Kern, J. Fontanella, J. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Lund, G.

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

Lutomirski, R.

Martin, F.

R. Avila, A. Ziad, J. Borgnino, F. Martin, A. Agabi, A. Tokovinin, “Theoretical spatio-temporal analysis of angle of arrival induced by atmospheric turbulence as observed with the Grating Scale Monitor experiment,” J. Opt. Soc. Am. A 14, 3070–3082 (1997).
[CrossRef]

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wavefront outer scale at the O.C.A.,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. I. The instrument and first results of angle of arrival measurements,” Astron. Astrophys. Suppl. Ser. 108, 173–180 (1994).

J. Borgnino, F. Martin, A. Ziad, “Effect of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91, 267–279 (1992).
[CrossRef]

R. Conan, A. Ziad, R. Avila, A. Tokovinin, F. Martin, J. Borgnino, “Spatio-temporal analysis of the wave front with the GSM,” in Topical Meeting on Astronomy with Adaptive Optics, Present Results and Future Programs, D. Bonaccini, ed. (European Southern Observatory, Garching, Germany, 1998), pp. 133–142.

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, “Report on G.S.M. measurement campaign at La Silla,” (European Southern Observatory, Garching, Germany, 1998).

A. Tokovinin, A. Ziad, F. Martin, R. Avila, J. Borgnino, R. Conan, M. Sarazin, “Wavefront outer scale monitoring at La Silla ,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

Merkle, F.

F. Rigaut, G. Rousset, P. Kern, J. Fontanella, J. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Rigaut, F.

F. Rigaut, G. Rousset, P. Kern, J. Fontanella, J. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Roddier, F.

F. Roddier, J. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[CrossRef]

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

F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” in Progress in Optics Vol. XIX, E. Wolf, ed. (Elsevier, New York, 1981).

Ronchi, L.

Rousset, G.

F. Rigaut, G. Rousset, P. Kern, J. Fontanella, J. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Sarazin, M.

A. Tokovinin, A. Ziad, F. Martin, R. Avila, J. Borgnino, R. Conan, M. Sarazin, “Wavefront outer scale monitoring at La Silla ,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

Sasiela, R.

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

R. Sasiela, Electromagnetic Wave Propagation in Turbulence. Evaluation and Application of Mellin Transforms (Springer-Verlag, New York, 1994).

Schöck, M.

Shelton, J.

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

Spillar, E.

Stefanutti, L.

Strohbehn, J.

J. Strohbehn, S. Clifford, “Polarization and angle-of-arrival fluctuations for a plane wave propagated through a turbulent medium,” IEEE Trans. Antennas Propag. AP-15, 416–422 (1967).
[CrossRef]

Tatarski, V.

V. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

Tokovinin, A.

R. Avila, A. Ziad, J. Borgnino, F. Martin, A. Agabi, A. Tokovinin, “Theoretical spatio-temporal analysis of angle of arrival induced by atmospheric turbulence as observed with the Grating Scale Monitor experiment,” J. Opt. Soc. Am. A 14, 3070–3082 (1997).
[CrossRef]

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wavefront outer scale at the O.C.A.,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. I. The instrument and first results of angle of arrival measurements,” Astron. Astrophys. Suppl. Ser. 108, 173–180 (1994).

A. Tokovinin, A. Ziad, F. Martin, R. Avila, J. Borgnino, R. Conan, M. Sarazin, “Wavefront outer scale monitoring at La Silla ,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

R. Conan, A. Ziad, R. Avila, A. Tokovinin, F. Martin, J. Borgnino, “Spatio-temporal analysis of the wave front with the GSM,” in Topical Meeting on Astronomy with Adaptive Optics, Present Results and Future Programs, D. Bonaccini, ed. (European Southern Observatory, Garching, Germany, 1998), pp. 133–142.

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, “Report on G.S.M. measurement campaign at La Silla,” (European Southern Observatory, Garching, Germany, 1998).

Tyler, G.

Vernin, J.

R. Avila, J. Vernin, S. Cuevas, “Turbulence profiles with generalized Scidar at San Pedro Màrtir Observatory and isoplanatism studies,” Publ. Astron. Soc. Pac. 110, 1106–1116 (1998).
[CrossRef]

F. Roddier, J. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[CrossRef]

Voitsekhovich, V.

Winker, D.

Yura, H.

Ziad, A.

R. Avila, A. Ziad, J. Borgnino, F. Martin, A. Agabi, A. Tokovinin, “Theoretical spatio-temporal analysis of angle of arrival induced by atmospheric turbulence as observed with the Grating Scale Monitor experiment,” J. Opt. Soc. Am. A 14, 3070–3082 (1997).
[CrossRef]

A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wavefront outer scale at the O.C.A.,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

F. Martin, A. Tokovinin, A. Agabi, J. Borgnino, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. I. The instrument and first results of angle of arrival measurements,” Astron. Astrophys. Suppl. Ser. 108, 173–180 (1994).

J. Borgnino, F. Martin, A. Ziad, “Effect of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91, 267–279 (1992).
[CrossRef]

R. Conan, A. Ziad, R. Avila, A. Tokovinin, F. Martin, J. Borgnino, “Spatio-temporal analysis of the wave front with the GSM,” in Topical Meeting on Astronomy with Adaptive Optics, Present Results and Future Programs, D. Bonaccini, ed. (European Southern Observatory, Garching, Germany, 1998), pp. 133–142.

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, “Report on G.S.M. measurement campaign at La Silla,” (European Southern Observatory, Garching, Germany, 1998).

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A. Tokovinin, A. Ziad, F. Martin, R. Avila, J. Borgnino, R. Conan, M. Sarazin, “Wavefront outer scale monitoring at La Silla ,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

Appl. Opt.

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Astron. Astrophys. Suppl. Ser.

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A. Agabi, J. Borgnino, F. Martin, A. Tokovinin, A. Ziad, “G.S.M.: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wavefront outer scale at the O.C.A.,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

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[CrossRef]

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[CrossRef]

J. Opt. (Paris)

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[CrossRef]

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[CrossRef]

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[CrossRef]

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A. Tokovinin, A. Ziad, F. Martin, R. Avila, J. Borgnino, R. Conan, M. Sarazin, “Wavefront outer scale monitoring at La Silla ,” in Adaptive Optical System Technologies, D. Bonaccini, R. Tyson, eds., Proc. SPIE3353, 1155–1162 (1998).
[CrossRef]

R. Conan, A. Ziad, R. Avila, A. Tokovinin, F. Martin, J. Borgnino, “Spatio-temporal analysis of the wave front with the GSM,” in Topical Meeting on Astronomy with Adaptive Optics, Present Results and Future Programs, D. Bonaccini, ed. (European Southern Observatory, Garching, Germany, 1998), pp. 133–142.

A. Ziad, “Estimation des échelles limites de cohérence spatiale des fronts d’onde et optimisation des observations à Haute Résolution Angulaire en Astronomie,” Ph.D. thesis (Université de Nice-Sophia Antipolis, Nice, France, 1993).

F. Martin, A. Tokovinin, A. Ziad, R. Conan, J. Borgnino, R. Avila, “Report on G.S.M. measurement campaign at La Silla,” (European Southern Observatory, Garching, Germany, 1998).

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Figures (9)

Fig. 1
Fig. 1

Schematic layout of the baseline r, the wind v, and the direction u of measurement of the AA fluctuations.

Fig. 2
Fig. 2

Relative error between the results obtained from numerical computation of the integral expression and of series expansions with the corresponding computation times.

Fig. 3
Fig. 3

(a) 2D spatial covariance, (b) isolevel of the covariance, (c) longitudinal and transverse covariances of the AA fluctuations for f0=0.04 m-1, r0=0.1 m, and D=0.1 m.

Fig. 4
Fig. 4

Ray-tracing scheme of a plane wave incoming on a lens and the projections of the rays in particular planes. These planes contain the X and Y components of the rays crossing the X and Y paths. For reasons of readability, only a few rays are drawn.

Fig. 5
Fig. 5

Temporal cross correlations for different directions of the wind, η=0°, 45°, and 90°, and for baseline orientations equal to γ=0°, 45°, and 90° and the corresponding cut lines on the spatial covariance of the AA fluctuations.

Fig. 6
Fig. 6

Delay corresponding to the maxima of the correlations as a function of the wind direction for different orientations of the baseline. Isotropy corresponds to the delay computed with Eq. (4.5), and anisotropy corresponds to the position of the maximum of the spatiotemporal correlation of Eq. (4.2) (v=5 m s-1 and r=1 m).

Fig. 7
Fig. 7

Coefficients Aν (circles) of the power (r/D)ν and relative error (r, γ) (diamonds) on the value of the covariance.

Fig. 8
Fig. 8

Decorrelation time of the AA expressed in units of D/v versus Df0 for different directions of the wind.

Fig. 9
Fig. 9

Theoretical (circles) and experimental (crosses) decorrelation times τ0 at La Silla on September 18, 1997.

Tables (1)

Tables Icon

Table 1 Series Ranges of Applicability for Most Rapid Convergence

Equations (99)

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Wϕ(r),u(f )=λ2(fu)2Wϕ(f ),
Cϕ(r),u(r)=λ2df(fu)2Wϕ(f )G(f )exp(2πifr),
Cϕ(r),u(r)=πλ20+df f3Wϕ(f )G(f )(J0(2πrf )-cos{2[γ-arg(u)]}J2(2πrf )).
Cϕ(r),arg(u)=0(r)=Cα(r, γ)=πλ20+d f3Wϕ(f )G(f )×[J0(2πrf )-cos(2γ)J2(2πrf )].
G(f )=2J1(πDf )πDf2.
Wϕ(f )=0.0229r0-5/3(f2+f02)-11/6,
Cα(r, γ)=0.0719λ2r0-5/30+df f3(f2+f02)-11/6×2J1(πDf )πDf2[J0(2πrf )-cos(2γ)J2(2πrf )].
r<1πf0,D<1πf0,andr<D
C(r, γ)=8.8578×10-4λ2r0-5/3f0-11/3D-4n=0+m=0+(-1)n+mn!m!(πDf0)4(πDf0)2n(πrf0)2m×Γ3/2+n, n+m+2, -n-m-1/63+n, 2+n, 1+m+(πDf0)11/3rD8/3rD2n(πrf0)2m×Γ3/2+n, m+11/6, -n-m-4/33/2-n, 1/2-n, n+m+7/3+(πDf0)11/3(πDf0)2nrD2m×Γm-n+1/6, 4/3-m+n, n+11/617/6-m+n, 11/6-m+n, 1+m-cos(2γ)(πDf0)6rD2(πDf0)2n(πrf0)2m×Γ3/2+n, n+m+3, -n-m-7/63+n, 2+n, 3+m+(πDf0)11/3rD8/3rD2n(πrf0)2m
×Γ3/2+n, m+11/6, -n-m-1/33/2-n, 1/2-n, n+m+10/3+(πDf0)11/3rD2(πDf0)2nrD2m
×Γm-n+7/6, 1/3-m+n, n+11/611/6-m+n, 5/6-m+n, 3+m,
r<1πf0, D<1πf0, and r<D
Cα(r, γ)=8.8578×10-4λ2r0-5/3f0-11/3D-4n=0+m=0+(-1)n+mn!m!×(πDf0)11/3Dr1/3Dr2n(πrf0)2mΓ3/2+n, m+11/6, n-m+1/63+n, 2+n, m-n+5/6+(πDf0)4(πDf0)2n(πrf0)2mΓ3/2+n, n+m+2, -n-m-1/63+n, 2+n, 1+m-cos(2γ)(πDf0)11/3Dr1/3Dr2n(πrf0)2mΓ3/2+n, m+11/6, n-m+7/63+n, 2+n, m-n+11/6+(πDf0)6rD2(πDf0)2n(πrf0)2mΓ3/2+n, n+m+3, -n-m-7/63+n, 2+n, 3+m,
Γa1 ,, ai ,, akb1 ,, bj ,, bl=i=1kΓ(ai)j=1lΓ(bj).
(r, 0)=100 |Cα(r, 0)integ-Cα(r, 0)series|Cα(r, 0)integ.
Cα(r, γ)=0.0719λ2r0-5/30+df f3(f2+f02)-11/6×[J0(2πrf )-cos(2γ)J2(2πrf )].
Cα(r, γ)=0.0858λ2r0-5/3f01/3{[23cos2(γ)+sin2(γ)]×(2πrf0)-1/6K1/6(2πrf0)-cos2(γ)(2πrf0)5/6K5/6(2πrf0)},
Cα(r, τ)=Cα,r(τ)=α(ξ, t)α(ξ+r, t+τ),
Cα,r(τ)=α(ξ, t)α(ξ+r-τv, t)
=Cα(r-τv)
=Cα(r)*δ(r-τv),
Cα,r(τ)=Cα(rx, ry)*δ(rx-τvx, ry-τvy).
Δτ=r cos(η-γ)v.
φ(ν)=2πνΔτ.
Wα,r(ν)=0.0228λ2r0-5/31vexp(2iπΔτν)×-+dqνvcos(η)-q sin(η)2×νv2+q2+1L02-11/6×2 J1πDf[(ν/v)2+q2]1/2πD[(ν/v)2+q2]1/22×exp[2iπqr sin(γ-η)].
Cα(rκ, η)=σα2κ,
Cα(r, η)=iAνirDνi.
Cα(r, γ)=0.0589λ2r0-5/3D-1/3×0.776rD8/3[1.75+cos(2γ)]-rD2[2+cos(2γ)][1.101-2.642(πDf0)2+ 2.572(πDf0)7/3+0.890(πDf0)13/3]+ 2.882-3.001(πDf0)1/3+1.628(πDf0)2-1.286(πDf0)7/3(for r<D),
Cα(r, γ)=0.0589λ2r0-5/3D-1/3×-3.001(πDf0)1/3-1.286(πDf0)7/3+rD-1/3[5-cos(2γ)][0.411+0.188(πDf0)2](for r>D).
σα2=0.0589λ2r0-5/3D-1/3n=0+(-1)nn!×(πDf0)2n+1/3Γ3/2+n, -n-1/63+n+(πDf0)2nΓn+1/6, 4/3+n17/6+n,
σα2=0.0589λ2r0-5/3D-1/3(πDf0)1/3Γ3/2, -1/63+Γ1/6, 4/317/6-(πDf0)7/3Γ5/2, -7/64-(πDf0)2Γ-5/6, 7/323/6
=0.0589λ2r0-5/3D-1/3×[-3.001(πDf0)1/3+2.882-1.286(πDf0)7/3+1.628(πDf0)2]
=0.1697λ2r0-5/3D-1/3{1-1.041(πDf0)1/3+0.565(πDf0)2-0.446(πDf0)7/3}.
rκD-1/3=F(κ, Df0)5-cos(2γκ),
F(κ, Df0)=(1-κ-1)[3.001(πDf0)1/3+1.286(πDf0)7/3]+κ-1[2.882+1.628(πDf0)2]0.411+0.188(πDf0)2,
τ0α=DvF(κ, Df0)5-cos(2η)-3.
τ0α=DvF(κ, 0)5-cos(2η)-3=Dv5-cos(2η)7κ-13.
τ0β=DvF(κ, Df0)5+cos(2η)-3.
R=τ0ατ0β=5-cos(2η)5+cos(2η)3.
η=12arccos5 1-R1/31+R1/3 or
π-12arccos5 1-R1/31+R1/3,
v=1000D{F(κ, Df0)[(τ0α)1/3+(τ0β)1/3]}-3.
v=v(h)Cn2(h)dhCn2(h)dh.
Cα(r, γ)=0.0719λ2r0-5/3f0-11/32πD2[I1-cos(2γ)I2],
I1=0+f2J12(af )J0bf-1f0f-2+1-11/6dff,
I2=0+f2J12(af )J2bf-1f0f-2+1-11/6dff,
ff0-2+1-11/6f0-s2Γ[-s/2, 11/6+s/2]Γ(11/6),
-113<Re(s)<0,
f2J12(af )a-(s+2)×12π Γs/2+2, -1/2-s/21-s/2, -s/2,
-4<Re(s)<-1,
J0(f-1)2-s-1Γ-s/21+s/2,
-32<Re(s)<0,
J2(f-1)2-s-1Γ-s/2+12+s/2,
-32<Re(s)<2,
I1=a-22πΓ(11/6)1(2πi)2Csds(af0)-2sCtdt(2ab)-2t×Γs+t+2, -1/2-s-t, -s, 11/6+s, -t1-s-t, -s-t, 1+t,
I2=a-22πΓ(11/6)1(2πi)2Csds(af0)-2sCtdt(2ab)-2t×Γs+t+2, -1/2-s-t, -s, 11/6+s, -t1-s-t, -s-t, 2+t.
(1)s+t+2=-n,-s=-m,
(2)s+t+2=-n,11/6+s=-m,
(3)s+t+2=-n,-t=-m,
(4)-1/2-s-t=-n,-s=-m,
(5)-1/2-s-t=-n,11/6+s=-m,
(6)-1/2-s-t=-n,-t=-m,
(7)-s=-n,-t=-m,
(8)11/6+s=-n,-t=-m,
(1)s+t+2=-n,-s=-m,
(2)s+t+2=-n,11/6+s=-m,
(3)s+t+2=-n,-t+1=-m,
(4)-1/2-s-t=-n,-s=-m,
(5)-1/2-s-t=-n,11/6+s=-m,
(6)-1/2-s-t=-n,-t+1=-m,
(7)-s=-n,-t+1=-m,
(8)11/6+s=-n,-t+1=-m.
(1)1πrf02mDr2n(n!)0(m!)2,
(2)(πrf0)2mDr2n(n!)0(m!)-2,
(3)(πrf0)2m(πDf0)2n(n!)-2(m!)-2,
(4)1πrf02mrD2n(n!)0(m!)2,
(5)(πrf0)2mrD2n(n!)0(m!)-2,
(6)(πrf0)2m1πDf02n(n!)2(m!)-2,
(7)rD2m1πDf02n(n!)2(m!)0,
(8)rD2m(πDf0)2n(n!)-2(m!)0.
C(r, γ)=0.0719λ2r0-5/3f0-11/3[I1-cos(2γ)I2],
I1=0+f4J0(2πfr)f0f-2+1-11/6dff,
I2=0+f4J2(2πfr)f0f-2+1-11/6dff.
f4J0(2πfr)(2πr)-(s+4)2s+3Γs/2+2-s/2+1,
-4<Re(s)<-52,
f4J2(2πfr)(2πr)-(s+4)2s+3Γs/2+3-s/2,
-6<Re(s)<-52
I1=4Γ(11/6)(2πr)412πi×CΓs/2+1/12, -s/2+23/12, s/2-1/12-s/2+11/12×2πrf02-sds,
I2=4Γ(11/6)(2πr)412πiCΓ[s/2+3, s/2+11/6]×2πrf02-sds,
I1=4Γ(11/6)(2πr)42πrf0223/612πi×1112CΓ[s/2+1/12, s/2-1/12]2πrf02-sds
-12CsΓ(s/2+1/12, s/2-1/12)2πrf02-sds].
I1=4Γ(11/6)(2πr)42πrf0223/6×113 K1/6(2πrf0)+2x dK1/6(x)dxx=2πrf0.
I1=4Γ(11/6)(2πr)4 (πrf0)23/6×103 K1/6(2πrf0)-4πrf0K5/6(2πrf0).
I2=4Γ(11/6)(2πr)42πrf0229/612πiCΓ[s/2+7/12,s/2-7/12]2πrf02-sds
=16Γ(11/6)(2πr)42πrf0229/6K7/6(2πrf0)
=16Γ(11/6)(2πr)42πrf0229/6×K5/6(2πrf0)+13 (2πrf0)-1K1/6(2πrf0).
C(r, γ)=0.0719×21/6Γ(11/6) λ2r0-5/3f01/323cos2(γ)+sin2(γ)(2πrf0)-1/6K1/6(2πrf0)-cos2(γ)(2πrf0)5/6K5/6(2πrf0).
Cα(r, γ)=8.8578×10-4λ2r0-5/3f0-11/3D-4(πDf0)11/3-rD20/3[1.3+cos(2γ)]Γ5/2, 17/6, -7/31/2, -1/2, 16/3(πDf0)2+rD14/3107+cos(2γ)Γ5/2, 11/6, -4/31/2, -1/2, 13/3+Γ3/2, 17/6, -4/33/2, 1/2, 13/3(πDf0)2+rD4cos(2γ)Γ13/6, -2/3, 11/65/6, -1/6, 4-Γ7/6, 1/3, 17/611/6, 5/6, 4(πDf0)2+Γ3/2, 4, -13/63, 2, 4(πDf0)13/3-Γ5/2, 5, -19/64, 3, 4(πDf0)19/3-rD8/3[1.75+cos(2γ)]Γ3/2, 11/6, -1/33/2, 1/2, 10/3-rD2[2+cos(2γ)]Γ7/6, 1/35/6, 3-Γ1/6, 4/311/6, 3(πDf0)2+Γ3/2, -7/62, 3(πDf0)7/3-Γ5/2, -13/63, 3(πDf0)13/3+Γ1/6, 4/317/6+Γ3/2, -1/63(πDf0)1/3-Γ-5/6, 7/323/6(πDf0)2-Γ5/2, -7/64(πDf0)7/3.
Cα(r, γ)=8.8578×10-4λ2r0-5/3f0-11/3D-4(πDf0)11/3rD4cos(2γ)Γ3/2, 4, -13/63, 2, 4(πDf0)13/3-Γ5/2, 5, -19/64, 3, 4(πDf0)19/3rD2[2+cos(2γ)]Γ3/2, 3, -7/63, 2, 3(πDf0)7/3-Γ5/2, 4,-13/64, 3, 3(πDf0)13/3+rD5/3[2.2+cos(2γ)]Γ3/2, 17/6, 1/63, 2, 17/6(πDf0)2+Γ3/2, 2, -1/63, 2(πDf0)1/3-Γ5/2, 3,-7/64, 3(πDf0)7/3+rD-1/3[5-cos(2γ)]Γ3/2, 11/6, 7/63, 2, 11/6+Γ5/2, 17/6, 7/64, 3, 11/6(πDf0)2-rD-7/3[1+7 cos(2γ)]Γ5/2, 11/6, 7/64, 3, -1/6.

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