Abstract

Theoretical investigations of the statistical properties of the wave front perturbed by atmospheric turbulence are presented. They are deduced from the calculation of the two-dimensional spatial covariance and the temporal cross spectrum of the angle-of-arrival fluctuations with a finite outer scale over a pair of circular pupils as in the case of the grating scale monitor or any other Shack–Hartmann-type sensor. Both calculations lead to integral expressions that are numerically evaluated and hold for any baseline vector in the mean wave-front plane. It is proposed to retrieve the wave-front outer scale L0 from estimations of this two-dimensional spatial covariance, normalized by the angle-of-arrival structure function. To eliminate instrument vibration errors, the covariance and the structure function are estimated from measurements obtained by mechanically independent and mechanically coupled devices, respectively. The angle-of-arrival temporal cross spectrum is calculated for any mean wind velocity vector. It is shown that the baseline component in the mean wind direction affects the phase of the angle-of-arrival temporal cross spectrum, whereas the component in the perpendicular direction affects the modulus. From simultaneous measurements of the phase of the angle-of-arrival temporal cross spectrum obtained with two nonparallel baselines, one can calculate the mean wind speed and direction, which allows estimation of the coherence time for techniques of optical observation at high angular resolution through the atmosphere.

© 1997 Optical Society of America

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    [CrossRef]
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    [CrossRef]

1997 (1)

1996 (2)

G. Boreman, C. Dainty, “Zernike expansions for non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 13, 517–522 (1996).
[CrossRef]

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

1995 (7)

J. Davis, P. R. Lawson, A. J. Booth, W. J. Tango, E. D. Thorvaldson, “Atmospheric path variations for baselines up to 80 m measured withthe Sydney University Stellar Interferometer,” Mon. Not. R. Astron. Soc. 277, L53–L58 (1995).

N. Takato, I. Yamaguchi, “Spatial correlation of Zernike phase-expansion coefficients for atmosphericturbulence with finite outer scale,” J. Opt. Soc. Am. A 12, 957–965 (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).

J.-M. Conan, G. Rousset, P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
[CrossRef]

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

D. F. Buscher, J. T. Armstrong, C. A. Hummel, A. Quirrenbach, D. Mozurkewich, K. J. Johnston, C. S. Denison, M. M. Colavita, M. Shao, “Interferometric seeing measurements on Mt. Wilson: power spectra andouter scales,” Appl. Opt. 34, 1081–1096 (1995).
[CrossRef] [PubMed]

C. E. Coulman, J. Vernin, A. Fuchs, “Optical seeing-mechanism of formation of thin turbulent laminae inthe atmosphere,” Appl. Opt. 34, 5461–5474 (1995).
[CrossRef] [PubMed]

1994 (4)

A. Ziad, J. Borgnino, A. Agabi, F. Martin, “Optimized spectral bandwidth in high angular resolution imaging, effectof a finite spatial-coherence outer scale,” Exp. Astron. 5, 247–268 (1994).
[CrossRef]

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).

F. Delaudier, D. Sidi, M. Crochet, J. Vernin, “Direct evidence of `sheets' in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

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

1993 (1)

1992 (3)

J. Borgnino, F. Martin, A. Ziad, “Effect of a finite outer scale on the covariances of angle-of-arrivalfluctuations,” Opt. Commun. 91, 267–269 (1992).
[CrossRef]

B. Lopez, “How to monitor optimum exposure times for high resolution imaging modes?” Astron. Astrophys. 253, 635–640 (1992).

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projectedperformances of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

1991 (3)

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

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

D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmosphericoptical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991).
[CrossRef]

1990 (1)

1988 (2)

1987 (2)

1986 (1)

1984 (1)

F. Roddier, P. Lena, “Long-baseline Michelson interferometry with ground-based telescopesoperating at optical wavelengths,” J. Opt. (Paris) 15, 171–182 (1984).
[CrossRef]

1982 (1)

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

1977 (1)

1973 (2)

J. L. Bufton, “Comparison of vertical profile turbulence structure with stellar observations,” Appl. Opt. 12, 1785–1793 (1973).
[CrossRef] [PubMed]

C. E. Coulman, “Vertical profiles of small-scale temperature structure in the atmosphere,” Boundary-Layer Meteorol. 4, 169–177 (1973).
[CrossRef]

Agabi, A.

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).

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

A. Ziad, J. Borgnino, A. Agabi, F. Martin, “Optimized spectral bandwidth in high angular resolution imaging, effectof a finite spatial-coherence outer scale,” Exp. Astron. 5, 247–268 (1994).
[CrossRef]

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).

Aime, C.

Armstrong, J. T.

D. F. Buscher, J. T. Armstrong, C. A. Hummel, A. Quirrenbach, D. Mozurkewich, K. J. Johnston, C. S. Denison, M. M. Colavita, M. Shao, “Interferometric seeing measurements on Mt. Wilson: power spectra andouter scales,” Appl. Opt. 34, 1081–1096 (1995).
[CrossRef] [PubMed]

D. F. Buscher, J. T. Armstrong, D. Mozurkewich, C. S. Denison, “Atmospheric fluctuation measurements with the MKIII interferometer and the power spectra of everything,” in High Resolution Imaging by Interferometry II, Proceedings of European Southern Observatory Conference 39, J. M. Beckers, J. M. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1029–1038.

Baldwin, J. E.

C. A. Haniff, J. E. Baldwin, P. J. Warner, T. R. Scott, “Atmospheric phase fluctuation measurements: interferometric resultsfrom the WHT and COAST telescopes,” in Amplitude and Intensity Spatial InterferometryII, J. B. Breckinridge, ed., Proc. SPIE2200, 407–417 (1994).
[CrossRef]

Barletti, R.

Beland, R. R.

R. R. Beland, J. H. Brown, “A deterministic temperature model for stratospheric optical turbulence,” Phys. Scr. 37, 419–423 (1988).
[CrossRef]

Berio, Ph.

Bester, M.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projectedperformances of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Booth, A. J.

J. Davis, P. R. Lawson, A. J. Booth, W. J. Tango, E. D. Thorvaldson, “Atmospheric path variations for baselines up to 80 m measured withthe Sydney University Stellar Interferometer,” Mon. Not. R. Astron. Soc. 277, L53–L58 (1995).

Boreman, G.

Borgnino, J.

Ph. Berio, D. Mourard, F. Vakili, J. Borgnino, A. Ziad, “Effects of atmospheric spectral decorrelation on visibility measurementsin Michelson interferometry,” J. Opt. Soc. Am. A 14, 114–121 (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).

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

A. Ziad, J. Borgnino, A. Agabi, F. Martin, “Optimized spectral bandwidth in high angular resolution imaging, effectof a finite spatial-coherence outer scale,” Exp. Astron. 5, 247–268 (1994).
[CrossRef]

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 outer scale on the covariances of angle-of-arrivalfluctuations,” Opt. Commun. 91, 267–269 (1992).
[CrossRef]

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

C. Aime, J. Borgnino, F. Martin, R. Petrov, G. Ricort, “Contribution to the space–time study of stellar speckle patterns,” J. Opt. Soc. Am. A 3, 1001–1009 (1986).
[CrossRef]

Brown, J. H.

R. R. Beland, J. H. Brown, “A deterministic temperature model for stratospheric optical turbulence,” Phys. Scr. 37, 419–423 (1988).
[CrossRef]

Bufton, J. L.

Buscher, D. F.

D. F. Buscher, J. T. Armstrong, C. A. Hummel, A. Quirrenbach, D. Mozurkewich, K. J. Johnston, C. S. Denison, M. M. Colavita, M. Shao, “Interferometric seeing measurements on Mt. Wilson: power spectra andouter scales,” Appl. Opt. 34, 1081–1096 (1995).
[CrossRef] [PubMed]

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

D. F. Buscher, J. T. Armstrong, D. Mozurkewich, C. S. Denison, “Atmospheric fluctuation measurements with the MKIII interferometer and the power spectra of everything,” in High Resolution Imaging by Interferometry II, Proceedings of European Southern Observatory Conference 39, J. M. Beckers, J. M. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1029–1038.

Caccia, J. L.

Ceppatelli, G.

Chanan, G.

F. Dekens, D. Kirkman, G. Chanan, T. Mast, J. Nelson, G. Illingworth, P. Wizinowich, “High speed seeing measurements at Keck telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 310–313 (1994).
[CrossRef]

Colavita, M. M.

Conan, J.-M.

J.-M. Conan, G. Rousset, P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
[CrossRef]

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

Coqueugniot, Y.

Coulman, C. E.

Crochet, M.

F. Delaudier, D. Sidi, M. Crochet, J. Vernin, “Direct evidence of `sheets' in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Cuevas, S.

Dainty, C.

Danchi, W. C.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projectedperformances of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Davis, J.

J. Davis, P. R. Lawson, A. J. Booth, W. J. Tango, E. D. Thorvaldson, “Atmospheric path variations for baselines up to 80 m measured withthe Sydney University Stellar Interferometer,” Mon. Not. R. Astron. Soc. 277, L53–L58 (1995).

Degiacomi, C. G.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projectedperformances of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Dekens, F.

F. Dekens, D. Kirkman, G. Chanan, T. Mast, J. Nelson, G. Illingworth, P. Wizinowich, “High speed seeing measurements at Keck telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 310–313 (1994).
[CrossRef]

Delaudier, F.

F. Delaudier, D. Sidi, M. Crochet, J. Vernin, “Direct evidence of `sheets' in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Denison, C. S.

D. F. Buscher, J. T. Armstrong, C. A. Hummel, A. Quirrenbach, D. Mozurkewich, K. J. Johnston, C. S. Denison, M. M. Colavita, M. Shao, “Interferometric seeing measurements on Mt. Wilson: power spectra andouter scales,” Appl. Opt. 34, 1081–1096 (1995).
[CrossRef] [PubMed]

D. F. Buscher, J. T. Armstrong, D. Mozurkewich, C. S. Denison, “Atmospheric fluctuation measurements with the MKIII interferometer and the power spectra of everything,” in High Resolution Imaging by Interferometry II, Proceedings of European Southern Observatory Conference 39, J. M. Beckers, J. M. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1029–1038.

Di Benedetto, G. P.

J. M. Mariotti, G. P. Di Benedetto, “Pathlength stability of synthetic aperture telescopes: the case of the 25 cm CERGA interferometer,” in Very Large Telescopes, Their Instrumentation and Programs, Proceedings of International Astronomical Union Colloquium 79, M. H. Ulrich, K. Jear, eds. (European Southern Observatory, Garching, Germany, 1984), pp. 257–265.

Fontanella, J. C.

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

Fuchs, A.

C. E. Coulman, J. Vernin, A. Fuchs, “Optical seeing-mechanism of formation of thin turbulent laminae inthe atmosphere,” Appl. Opt. 34, 5461–5474 (1995).
[CrossRef] [PubMed]

A. Fuchs, “Contribution à l'étude de l'apparition de la turbulence optique dans les couches minces. Concept du SCIDAR généralisé,” Thèse de doctorat (Université de Nice, Nice, France, 1995).

Gaffard, J. P.

F. Rigaut, G. Rousset, P. Kern, J. C. Fontanella, J. P. 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 opticsobservations,” Astrophys. Space Sci. 239, 221–228 (1996).
[CrossRef]

Gilli, J. M.

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

Graves, J. E.

Greenhill, L. J.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projectedperformances of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Haniff, C. A.

C. A. Haniff, J. E. Baldwin, P. J. Warner, T. R. Scott, “Atmospheric phase fluctuation measurements: interferometric resultsfrom the WHT and COAST telescopes,” in Amplitude and Intensity Spatial InterferometryII, J. B. Breckinridge, ed., Proc. SPIE2200, 407–417 (1994).
[CrossRef]

Hummel, C. A.

Illingworth, G.

F. Dekens, D. Kirkman, G. Chanan, T. Mast, J. Nelson, G. Illingworth, P. Wizinowich, “High speed seeing measurements at Keck telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 310–313 (1994).
[CrossRef]

Johnston, K. J.

Kern, P.

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

Kirkman, D.

F. Dekens, D. Kirkman, G. Chanan, T. Mast, J. Nelson, G. Illingworth, P. Wizinowich, “High speed seeing measurements at Keck telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 310–313 (1994).
[CrossRef]

Lawson, P. R.

J. Davis, P. R. Lawson, A. J. Booth, W. J. Tango, E. D. Thorvaldson, “Atmospheric path variations for baselines up to 80 m measured withthe Sydney University Stellar Interferometer,” Mon. Not. R. Astron. Soc. 277, L53–L58 (1995).

Lena, P.

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

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

F. Roddier, P. Lena, “Long-baseline Michelson interferometry with ground-based telescopesoperating at optical wavelengths,” J. Opt. (Paris) 15, 171–182 (1984).
[CrossRef]

Lopez, B.

B. Lopez, “How to monitor optimum exposure times for high resolution imaging modes?” Astron. Astrophys. 253, 635–640 (1992).

Lund, G.

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

Madec, P.-Y.

J.-M. Conan, G. Rousset, P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
[CrossRef]

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

G. Rousset, P.-Y. Madec, D. Rabaud, “Adaptive optics partial correction simulation for two telescope interferometry,” in High Resolution Imaging by Interferometry II, Proceedings of European Southern Observatory Conference 39, J. M. Beckers, J. M. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104.

Mariotti, J. M.

J. M. Mariotti, G. P. Di Benedetto, “Pathlength stability of synthetic aperture telescopes: the case of the 25 cm CERGA interferometer,” in Very Large Telescopes, Their Instrumentation and Programs, Proceedings of International Astronomical Union Colloquium 79, M. H. Ulrich, K. Jear, eds. (European Southern Observatory, Garching, Germany, 1984), pp. 257–265.

Martin, F.

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).

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

A. Ziad, J. Borgnino, A. Agabi, F. Martin, “Optimized spectral bandwidth in high angular resolution imaging, effectof a finite spatial-coherence outer scale,” Exp. Astron. 5, 247–268 (1994).
[CrossRef]

J. Borgnino, F. Martin, A. Ziad, “Effect of a finite outer scale on the covariances of angle-of-arrivalfluctuations,” Opt. Commun. 91, 267–269 (1992).
[CrossRef]

C. Aime, J. Borgnino, F. Martin, R. Petrov, G. Ricort, “Contribution to the space–time study of stellar speckle patterns,” J. Opt. Soc. Am. A 3, 1001–1009 (1986).
[CrossRef]

Martin, H. M.

H. M. Martin, “Image motion as a measure of seeing quality,” Publ. Astron. Soc. Pac. 99, 1360–1370 (1987).
[CrossRef]

Mast, T.

F. Dekens, D. Kirkman, G. Chanan, T. Mast, J. Nelson, G. Illingworth, P. Wizinowich, “High speed seeing measurements at Keck telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 310–313 (1994).
[CrossRef]

McKenna, D. L.

Merkle, F.

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

Mourard, D.

Mozurkewich, D.

D. F. Buscher, J. T. Armstrong, C. A. Hummel, A. Quirrenbach, D. Mozurkewich, K. J. Johnston, C. S. Denison, M. M. Colavita, M. Shao, “Interferometric seeing measurements on Mt. Wilson: power spectra andouter scales,” Appl. Opt. 34, 1081–1096 (1995).
[CrossRef] [PubMed]

D. F. Buscher, J. T. Armstrong, D. Mozurkewich, C. S. Denison, “Atmospheric fluctuation measurements with the MKIII interferometer and the power spectra of everything,” in High Resolution Imaging by Interferometry II, Proceedings of European Southern Observatory Conference 39, J. M. Beckers, J. M. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1029–1038.

Nelson, J.

F. Dekens, D. Kirkman, G. Chanan, T. Mast, J. Nelson, G. Illingworth, P. Wizinowich, “High speed seeing measurements at Keck telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 310–313 (1994).
[CrossRef]

Nightingale, N. S.

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

Northcott, M. J.

Paternò, L.

Petrov, R.

Quirrenbach, A.

Rabaud, D.

G. Rousset, P.-Y. Madec, D. Rabaud, “Adaptive optics partial correction simulation for two telescope interferometry,” in High Resolution Imaging by Interferometry II, Proceedings of European Southern Observatory Conference 39, J. M. Beckers, J. M. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104.

Ricort, G.

Rigaut, F.

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

Righini, A.

Roddier, F.

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, “One-dimensional spectra of turbulence-induced Zernike aberrations:time-delay and isoplanicity error in partial adaptive compensation,” J. Opt. Soc. Am. A 10, 957–965 (1993).
[CrossRef]

F. Roddier, P. Lena, “Long-baseline Michelson interferometry with ground-based telescopesoperating at optical wavelengths,” J. Opt. (Paris) 15, 171–182 (1984).
[CrossRef]

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

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.

Rousset, G.

J.-M. Conan, G. Rousset, P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
[CrossRef]

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

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

G. Rousset, P.-Y. Madec, D. Rabaud, “Adaptive optics partial correction simulation for two telescope interferometry,” in High Resolution Imaging by Interferometry II, Proceedings of European Southern Observatory Conference 39, J. M. Beckers, J. M. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104.

Scott, T. R.

C. A. Haniff, J. E. Baldwin, P. J. Warner, T. R. Scott, “Atmospheric phase fluctuation measurements: interferometric resultsfrom the WHT and COAST telescopes,” in Amplitude and Intensity Spatial InterferometryII, J. B. Breckinridge, ed., Proc. SPIE2200, 407–417 (1994).
[CrossRef]

Shao, M.

Sidi, D.

F. Delaudier, D. Sidi, M. Crochet, J. Vernin, “Direct evidence of `sheets' in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Speroni, N.

Staelin, D. H.

Takato, N.

N. Takato, I. Yamaguchi, “Spatial correlation of Zernike phase-expansion coefficients for atmosphericturbulence with finite outer scale,” J. Opt. Soc. Am. A 12, 957–965 (1995).
[CrossRef]

Tallon, M.

M. Tallon, “Contributions à l'imagerie à Haute Résolution Angulaire: analyse de la surface d'onde, source laser de référence, optique adaptative,” Thèse de doctorat (Université de Nice, Nice, France, 1989).

Tango, W. J.

J. Davis, P. R. Lawson, A. J. Booth, W. J. Tango, E. D. Thorvaldson, “Atmospheric path variations for baselines up to 80 m measured withthe Sydney University Stellar Interferometer,” Mon. Not. R. Astron. Soc. 277, L53–L58 (1995).

Thorvaldson, E. D.

J. Davis, P. R. Lawson, A. J. Booth, W. J. Tango, E. D. Thorvaldson, “Atmospheric path variations for baselines up to 80 m measured withthe Sydney University Stellar Interferometer,” Mon. Not. R. Astron. Soc. 277, L53–L58 (1995).

Tokovinin, A.

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).

Townes, C. H.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projectedperformances of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Vakili, F.

Vernin, J.

Voitsekhovich, V. V.

Warner, P. J.

C. A. Haniff, J. E. Baldwin, P. J. Warner, T. R. Scott, “Atmospheric phase fluctuation measurements: interferometric resultsfrom the WHT and COAST telescopes,” in Amplitude and Intensity Spatial InterferometryII, J. B. Breckinridge, ed., Proc. SPIE2200, 407–417 (1994).
[CrossRef]

Winker, D. M.

Wizinowich, P.

F. Dekens, D. Kirkman, G. Chanan, T. Mast, J. Nelson, G. Illingworth, P. Wizinowich, “High speed seeing measurements at Keck telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 310–313 (1994).
[CrossRef]

Yamaguchi, I.

N. Takato, I. Yamaguchi, “Spatial correlation of Zernike phase-expansion coefficients for atmosphericturbulence with finite outer scale,” J. Opt. Soc. Am. A 12, 957–965 (1995).
[CrossRef]

Ziad, A.

Ph. Berio, D. Mourard, F. Vakili, J. Borgnino, A. Ziad, “Effects of atmospheric spectral decorrelation on visibility measurementsin Michelson interferometry,” J. Opt. Soc. Am. A 14, 114–121 (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).

A. Ziad, J. Borgnino, A. Agabi, F. Martin, “Optimized spectral bandwidth in high angular resolution imaging, effectof a finite spatial-coherence outer scale,” Exp. Astron. 5, 247–268 (1994).
[CrossRef]

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

J. Borgnino, F. Martin, A. Ziad, “Effect of a finite outer scale on the covariances of angle-of-arrivalfluctuations,” Opt. Commun. 91, 267–269 (1992).
[CrossRef]

A. Ziad, “Estimation des échelles limites de cohérence spatiale des fronts d'onde et optimisation des observations à Haute Résolution Angulaire en Astronomie,” Thèse de doctorat (Université de Nice, Nice, France, 1993).

Appl. Opt. (7)

Astron. Astrophys. (3)

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

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

B. Lopez, “How to monitor optimum exposure times for high resolution imaging modes?” Astron. Astrophys. 253, 635–640 (1992).

Astron. Astrophys. Suppl. Ser. (2)

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, 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).

Astrophys. J. (1)

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projectedperformances of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Astrophys. Space Sci. (1)

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

Boundary-Layer Meteorol. (1)

C. E. Coulman, “Vertical profiles of small-scale temperature structure in the atmosphere,” Boundary-Layer Meteorol. 4, 169–177 (1973).
[CrossRef]

Exp. Astron. (1)

A. Ziad, J. Borgnino, A. Agabi, F. Martin, “Optimized spectral bandwidth in high angular resolution imaging, effectof a finite spatial-coherence outer scale,” Exp. Astron. 5, 247–268 (1994).
[CrossRef]

J. Atmos. Sci. (1)

F. Delaudier, D. Sidi, M. Crochet, J. Vernin, “Direct evidence of `sheets' in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

J. Opt. (Paris) (2)

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

F. Roddier, P. Lena, “Long-baseline Michelson interferometry with ground-based telescopesoperating at optical wavelengths,” J. Opt. (Paris) 15, 171–182 (1984).
[CrossRef]

J. Opt. Soc. Am. A (8)

Mon. Not. R. Astron. Soc. (2)

J. Davis, P. R. Lawson, A. J. Booth, W. J. Tango, E. D. Thorvaldson, “Atmospheric path variations for baselines up to 80 m measured withthe Sydney University Stellar Interferometer,” Mon. Not. R. Astron. Soc. 277, L53–L58 (1995).

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

Opt. Commun. (1)

J. Borgnino, F. Martin, A. Ziad, “Effect of a finite outer scale on the covariances of angle-of-arrivalfluctuations,” Opt. Commun. 91, 267–269 (1992).
[CrossRef]

Phys. Scr. (1)

R. R. Beland, J. H. Brown, “A deterministic temperature model for stratospheric optical turbulence,” Phys. Scr. 37, 419–423 (1988).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

H. M. Martin, “Image motion as a measure of seeing quality,” Publ. Astron. Soc. Pac. 99, 1360–1370 (1987).
[CrossRef]

Other (10)

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

G. Rousset, P.-Y. Madec, D. Rabaud, “Adaptive optics partial correction simulation for two telescope interferometry,” in High Resolution Imaging by Interferometry II, Proceedings of European Southern Observatory Conference 39, J. M. Beckers, J. M. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1095–1104.

F. Dekens, D. Kirkman, G. Chanan, T. Mast, J. Nelson, G. Illingworth, P. Wizinowich, “High speed seeing measurements at Keck telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 310–313 (1994).
[CrossRef]

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.

C. A. Haniff, J. E. Baldwin, P. J. Warner, T. R. Scott, “Atmospheric phase fluctuation measurements: interferometric resultsfrom the WHT and COAST telescopes,” in Amplitude and Intensity Spatial InterferometryII, J. B. Breckinridge, ed., Proc. SPIE2200, 407–417 (1994).
[CrossRef]

A. Fuchs, “Contribution à l'étude de l'apparition de la turbulence optique dans les couches minces. Concept du SCIDAR généralisé,” Thèse de doctorat (Université de Nice, Nice, France, 1995).

J. M. Mariotti, G. P. Di Benedetto, “Pathlength stability of synthetic aperture telescopes: the case of the 25 cm CERGA interferometer,” in Very Large Telescopes, Their Instrumentation and Programs, Proceedings of International Astronomical Union Colloquium 79, M. H. Ulrich, K. Jear, eds. (European Southern Observatory, Garching, Germany, 1984), pp. 257–265.

M. Tallon, “Contributions à l'imagerie à Haute Résolution Angulaire: analyse de la surface d'onde, source laser de référence, optique adaptative,” Thèse de doctorat (Université de Nice, Nice, France, 1989).

D. F. Buscher, J. T. Armstrong, D. Mozurkewich, C. S. Denison, “Atmospheric fluctuation measurements with the MKIII interferometer and the power spectra of everything,” in High Resolution Imaging by Interferometry II, Proceedings of European Southern Observatory Conference 39, J. M. Beckers, J. M. Merkle, eds. (European Southern Observatory, Garching, Germany, 1991), pp. 1029–1038.

A. Ziad, “Estimation des échelles limites de cohérence spatiale des fronts d'onde et optimisation des observations à Haute Résolution Angulaire en Astronomie,” Thèse de doctorat (Université de Nice, Nice, France, 1993).

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Figures (11)

Fig. 1
Fig. 1

General configuration for the calculation of the angle-of-arrival (AA) two-dimensional (2-D) covariance and temporal cross spectrum (TCS). α 1 and α 2 represent the AA fluctuations along the x direction, measured at pupils P 1 and P 2 , respectively. The baseline between both pupils has coordinates ( x 0 ,   y 0 ) in the Cartesian system ( x ,   y ) and polar coordinates ( B ,   γ ) . The mean wind velocity with polar coordinates ( v ,   η ) is also represented.

Fig. 2
Fig. 2

2-D plot of the AA covariance as a function of the baseline vector ( B ,   γ ) , obtained from numerical integration of Eq. (8) for D = 10   cm , L 0 = 10   m , λ = 0.5   μ m , and r 0 = 10   cm . The missing half of the surface is symmetrical to the one shown, with respect to the plane of equation γ = 0 ° . Covariance values higher than 0.01   arcsec 2 are not represented in order to reduce the dynamic range in the covariance axis, making visible the smooth transition from longitudinal ( γ = 0 ° ) to transverse ( γ = 90 ° ) covariances.

Fig. 3
Fig. 3

Distortion function H VK , obtained from numerical integration of Eq. (13) for D / B = 0.1 and for different values of the baseline angle γ.

Fig. 4
Fig. 4

AA longitudinal covariance ( γ = 0 ° ) obtained from numerical integration of Eq. (10) for D = 10   cm and r 0 = 10   cm at λ = 0.5   μ m for three values of the wave-front outer scale. The case of Kolmogorov's law is also presented.

Fig. 5
Fig. 5

AA longitudinal covariance normalized by the AA variance (solid curves) and by the AA structure function (dashed curves), calculated by using Eqs. (16) and (17) (with B 2 = 0.2   m and γ 2 = 0 °), respectively, for three values of the wave-front outer scale and D = 10   cm .

Fig. 6
Fig. 6

Baseline components in the direction of the wind velocity and perpendicular to it. α 1 and α 2 represent the AA fluctuations along the x direction, measured at pupils P 1 and P 2 , respectively. The baseline has polar coordinates ( B ,   γ ) , and those of the mean wind velocity are ( v ,   η ) . Dashed lines represent the perturbed wave front reaching P 1 at time t 1 and P 2 at time t 2 .

Fig. 7
Fig. 7

Influence of the baseline separation on the AA TCS. The TCS modulus multiplied by the wind speed (a) and the TCS phase (b) is represented as functions of ν / v for three values of B , other parameters being fixed to the values shown. The scheme in (b) represents the orientations of the mean wind velocity, the baseline vector (between two circular pupils), and the AA fluctuations (arrows on pupils). Wind velocity and baseline are parallel to each other, in which case the TCS modulus in (a) equals the AA temporal power spectrum for all B . Power laws and high-frequency cutoffs are shown. In (b) the total phase and the linear phase term [Eq. (28)] are plotted, but they are exactly superimposed for all B . The unit in the vertical axis of (a) is radian 2 meter.

Fig. 8
Fig. 8

Similar to Fig. 7, but with wind velocity perpendicular to the baseline, in which case the TCS phase is equal to 0 for all B and thus is not represented.

Fig. 9
Fig. 9

Influence of the baseline angle of the AA TCS. This figure is similar to Fig. 7, but the TCS (a) modulus and (b) phase are shown for three values of the baseline angle γ. Dashed lines in (b) represent the linear phase terms [Eq. (28)], which are almost superimposed on the total phase (solid lines) for γ = 10 ° and γ = 30 ° and are exactly superimposed for γ = 80 ° .

Fig. 10
Fig. 10

Influence of the mean wind velocity angle on the AA TCS. This figure is similar to Fig. 7, but the TCS (a) modulus and (b) phase are shown for three values of the mean wind velocity angle η. The dashed line in (b) represents the phase contribution φ(ν) [Eq. (28)] for η = 45 ° . For the two other values of η, this contribution appears superimposed on the total phase, represented as solid curves.  

Fig. 11
Fig. 11

Influence of the outer scale on the AA TCS. This figure is similar to Fig. 7, but the TCS (a) modulus and (b) phase are shown for three values of the outer scale L 0 . Low-frequency cutoffs are shown. No significant influence on the TCS phase is noticed.

Tables (1)

Tables Icon

Table 1 Reported Measurements of the Wave-Front Outer Scale L 0

Equations (32)

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

C α ( x 0 ,   y 0 ) = λ 2 - + - + d f x d f y   f x 2 W ϕ ( f x ,   f y ) × exp [ 2 i π ( x 0 f x + y 0 f y ) ] ,
C α ( x 0 ,   y 0 ) = λ 2 0 + d f   f 3 W ϕ ( f   ) I ( x 0 ,   y 0 ,   f   ) ,
I ( x 0 ,   y 0 ,   f   ) = 0 2 π d θ   cos 2   θ × exp [ 2 i π f ( x 0   cos   θ + y 0   sin θ ) ] ,
I ( x 0 ,   y 0 ,   f   ) = I ( B ,   γ ,   f   ) = 0 2 π d θ   cos 2   θ   exp [ 2 i π fB   cos ( θ - γ ) ] .
I ( B ,   γ ,   f   ) = π [ J 0 ( 2 π fB ) - cos ( 2 γ ) J 2 ( 2 π fB ) ] ,
C α ( x 0 ,   y 0 ) = C α ( B ,   γ ) = π λ 2 0 + d f   f 3 W ϕ ( f   ) × [ J 0 ( 2 π fB ) - cos ( 2 γ ) J 2 ( 2 π fB ) ] .
W ϕ ( f   ) = 0.0228 r 0 - 5 / 3 ( f 2 + 1 / L 0 2 ) - 11 / 6 ,
C α ( B ,   γ ,   D ,   L 0 ) = 0.0716 λ 2 r 0 - 5 / 3 × 0 + d f   f 3 f 2 + 1 L 0 2 - 11 / 6 × [ J 0 ( 2 π fB ) - cos ( 2 γ ) J 2 ( 2 π fB ) ] × 2   J 1 ( π Df   ) π Df 2 .
C α ( B ,   γ ,   D ,   L 0 ) = 0.0388 λ 2 r 0 - 5 / 3 B - 1 / 3 × 0 + d X   X 3 X 2 + 2 π B L 0 2 - 11 / 6 × [ J 0 ( X ) - cos ( 2 γ ) J 2 ( X ) ] × 2   J 1 DX 2 B DX 2 B 2 .
C α   π / 2   B ,   ,   D ,   L 0   0  
= 0.0388 λ 2 r 0 - 5 / 3 B - 1 / 3 0 + d X   X 3 X 2 + 2 π B L 0 2 - 11 / 6 × [ J 0 ( X ) ± J 2 ( X ) ] 2   J 1 DX 2 B DX 2 B 2 ,
C K α ( x 0 ,   y 0 ) = 0.145 λ 2 r 0 - 5 / 3 ( x 0 2 + y 0 2 ) - 1 / 6 - 1 3   x 0 2 ( x 0 2 + y 0 2 ) - 7 / 6 .
C K α ( B ,   γ ) = 0.145 λ 2 r 0 - 5 / 3 B - 1 / 3 [ 1 - ( cos 2   γ ) / 3 ] .
H VK ( B ,   γ ,   D ,   L 0 ) = 0.0388 0.145 [ 1 - ( cos 2   γ ) / 3 ] × 0 + d X   X 3 X 2 + 2 π B L 0 2 - 11 / 6 × [ J 0 ( X ) - cos ( 2 γ ) J 2 ( X ) ] × 2   J 1 DX 2 B DX 2 B 2 .
C α ( B ,   γ ,   D ,   L 0 ) = C K α ( B ,   γ ) H VK ( B ,   γ ,   D ,   L 0 ) .
D α ( B ,   γ ,   D ,   L 0 ) = 2 [ σ α 2 ( D ,   L 0 ) - C α ( B ,   γ ,   D ,   L 0 ) ] .
Γ α = C α ( B ,   γ ,   D ,   L 0 ) σ α 2 ( D ,   L 0 ) ,
Γ α = C α ( B ,   γ ,   D ,   L 0 ) D α ( B 2 ,   γ 2 ,   D ,   L 0 ) ,
W α , B ( ν ) = α ^ 1 ( ν ) α ^ 2 * ( ν ) ,
C α , B ( τ ) = α ( r ,   t ) α ( r + B ,   t + τ ) .
α ( r + B ,   t + τ ) = α ( r + B - v τ ,   t ) .
C α , B ( τ ) = α ( r ,   t ) α ( r + B - v τ ,   t ) = C α ( v τ - B ) .
W α , B ( ν ) = - + d τ   C α ( v τ - B ) exp ( 2 i π ν τ ) .
W α , B ( ν ) = - + d τ - + d f   W α ( f ) × exp [ - 2 i π f ( v τ - B ) ] exp ( 2 i π ν τ ) .
W α , B ( ν ) = - + d f   W α ( f ) δ ( ν - v f ) exp ( 2 i π f B ) .
W α , B ( ν ) = 1 v   exp 2 i π B   ν v   cos ( η - γ ) × - + d q   W α ν v   cos   η - q   sin   η , ν v   sin   η + q   cos   η exp [ 2 i π qB   sin ( γ - η ) ] ,
W α , B ( ν ) = 0.0228 λ 2 r 0 - 5 / 3   1 v   exp 2 i π B   ν v   cos   ( η - γ ) × - + d q ν v   cos   η - q   sin   η 2 × ν v 2 + q 2 + 1 L 0 2 - 11 / 6 × 2   J 1 ( π D [ ( ν / v ) 2 + q 2 ] 1 / 2 ) π D [ ( ν / v ) 2 + q 2 ] 1 / 2 2 × exp [ 2 i π qB   sin ( γ - η ) ] .
φ ( ν ) = 2 π ( t 2 - t 1 ) ν .
φ ( ν ) = 2 π   B   cos ( η - γ ) v   ν .
ν L 0 v / L 0 .
ν B v | CP 2 | = v B   sin ( | η - γ | ) .
ν D 0.5   v D ,

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