Abstract

We present an improved technique for imaging through turbulence at visible wavelengths using a rotation shearing pupil-plane interferometer, intended for astronomical and terrestrial imaging applications. While previous astronomical rotation shearing interferometers have made only visibility modulus measurements, this interferometer makes four simultaneous measurements on each interferometric baseline, with phase differences of π/2 between each measurement, allowing complex visibility measurements (modulus and phase) across the entire input pupil in a single exposure. This technique offers excellent wavefront resolution, allowing operation at visible wavelengths on large apertures, is potentially immune to amplitude fluctuations (scintillation), and may offer superior calibration capabilities to other imaging techniques. The interferometer has been tested in the laboratory under weakly aberrating conditions and at Palomar Observatory under ordinary astronomical observing conditions. This research is based partly on observations obtained at the Hale Telescope.

© 2005 Optical Society of America

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    [CrossRef]

2002

L. C. Roberts, C. R. Neyman, “Characterization of the AEOS adaptive optics system,” Pub. Astron. Soc. Pacific 114, 1260–1266 (2002).
[CrossRef]

2000

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

K. Wallace, G. Hardy, E. Serabyn. “Deep and stable interferometric nulling of broadband light with implications for observing planets around nearby stars,” Nature 406, 700–702 (2000).
[CrossRef] [PubMed]

B. D. Mason, W. I. Hartkopf, E. R. Holdenried, T. J. Rafferty, G. L. Wycoff, G. S. Hennessy, D. M. Hall, S. E. Urban, T. E. Corbin, “Speckle interferometery at the US Naval Observatory VI,” Astron. J. 120, 1120–1132 (2000).
[CrossRef]

1996

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

1995

N. Ageorges, J.-L. Monin, L. Desbat, E. Tessier, “Phase and image reconstruction from interferometric imaging: Integration of the phasors,” Astron. Astrophys., Suppl. Ser. 112, 163–171 (1995).

1994

C. A. Hummel, J. T. Armstrong, A. Quirrenbach, D. F. Buscher, D. Mozurkewich, N. M. Elias, R. E. Wilson, “Very high precision orbit of Capella by long baseline interferometry,” Astron. J. 107, 1859–1867 (1994).
[CrossRef]

1992

J. M. Mariotti, J. L. Monin, P. Ghez, C. Perrier, A. Zadrozny, “Pupil plane interferometry in the near infrared I. Methodology of observation and first results,” Astron. Astrophys. 255, 462–476 (1992).

1988

1987

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

1986

F. Roddier, C. Roddier, R. Petrov, R. Martin, G. Ricort, C. Aime. “New observations of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 305, L77–L80 (1986).
[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]

1974

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

J. B. Breckinridge, “Two-dimensional white light coherence interferometer,” Appl. Opt. 13, 2760–2762 (1974).
[CrossRef] [PubMed]

1972

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).

1965

Acton, D. S.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Ageorges, N.

N. Ageorges, J.-L. Monin, L. Desbat, E. Tessier, “Phase and image reconstruction from interferometric imaging: Integration of the phasors,” Astron. Astrophys., Suppl. Ser. 112, 163–171 (1995).

Aime, C.

F. Roddier, C. Roddier, R. Petrov, R. Martin, G. Ricort, C. Aime. “New observations of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 305, L77–L80 (1986).
[CrossRef]

An, J.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Armstrong, J. T.

C. A. Hummel, J. T. Armstrong, A. Quirrenbach, D. F. Buscher, D. Mozurkewich, N. M. Elias, R. E. Wilson, “Very high precision orbit of Capella by long baseline interferometry,” Astron. J. 107, 1859–1867 (1994).
[CrossRef]

Avicola, K.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Ayers, G. R.

Baldwin, J. E.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Beckett, M. G.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge University Press, 1999).
[CrossRef]

Boysen, R. C.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

Brase, J.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Breckinridge, J. B.

Bregman, J. D.

C. M. de Vos, J. D. Bregman, U. J. Schwarz, “Pupil plane interferometry: Some conclusions from SCASIS,” in Very High Angular Resolution Imaging, J. G. Robertson, W. J. Tango, eds. (Kluwer, 1994), pp. 419–420.
[CrossRef]

Burns, D.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

Buscher, D. F.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

C. A. Hummel, J. T. Armstrong, A. Quirrenbach, D. F. Buscher, D. Mozurkewich, N. M. Elias, R. E. Wilson, “Very high precision orbit of Capella by long baseline interferometry,” Astron. J. 107, 1859–1867 (1994).
[CrossRef]

Corbin, T. E.

B. D. Mason, W. I. Hartkopf, E. R. Holdenried, T. J. Rafferty, G. L. Wycoff, G. S. Hennessy, D. M. Hall, S. E. Urban, T. E. Corbin, “Speckle interferometery at the US Naval Observatory VI,” Astron. J. 120, 1120–1132 (2000).
[CrossRef]

Cox, G. C.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

Dainty, J. C.

de Vos, C. M.

C. M. de Vos, J. D. Bregman, U. J. Schwarz, “Pupil plane interferometry: Some conclusions from SCASIS,” in Very High Angular Resolution Imaging, J. G. Robertson, W. J. Tango, eds. (Kluwer, 1994), pp. 419–420.
[CrossRef]

Desbat, L.

N. Ageorges, J.-L. Monin, L. Desbat, E. Tessier, “Phase and image reconstruction from interferometric imaging: Integration of the phasors,” Astron. Astrophys., Suppl. Ser. 112, 163–171 (1995).

Elias, N. M.

C. A. Hummel, J. T. Armstrong, A. Quirrenbach, D. F. Buscher, D. Mozurkewich, N. M. Elias, R. E. Wilson, “Very high precision orbit of Capella by long baseline interferometry,” Astron. J. 107, 1859–1867 (1994).
[CrossRef]

Fried, D. L.

Gath-right, J.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Gavel, D.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Ghez, A.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Ghez, P.

J. M. Mariotti, J. L. Monin, P. Ghez, C. Perrier, A. Zadrozny, “Pupil plane interferometry in the near infrared I. Methodology of observation and first results,” Astron. Astrophys. 255, 462–476 (1992).

Hall, D. M.

B. D. Mason, W. I. Hartkopf, E. R. Holdenried, T. J. Rafferty, G. L. Wycoff, G. S. Hennessy, D. M. Hall, S. E. Urban, T. E. Corbin, “Speckle interferometery at the US Naval Observatory VI,” Astron. J. 120, 1120–1132 (2000).
[CrossRef]

Haniff, C. A.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Hardy, G.

K. Wallace, G. Hardy, E. Serabyn. “Deep and stable interferometric nulling of broadband light with implications for observing planets around nearby stars,” Nature 406, 700–702 (2000).
[CrossRef] [PubMed]

Hartkopf, W. I.

B. D. Mason, W. I. Hartkopf, E. R. Holdenried, T. J. Rafferty, G. L. Wycoff, G. S. Hennessy, D. M. Hall, S. E. Urban, T. E. Corbin, “Speckle interferometery at the US Naval Observatory VI,” Astron. J. 120, 1120–1132 (2000).
[CrossRef]

Hennessy, G. S.

B. D. Mason, W. I. Hartkopf, E. R. Holdenried, T. J. Rafferty, G. L. Wycoff, G. S. Hennessy, D. M. Hall, S. E. Urban, T. E. Corbin, “Speckle interferometery at the US Naval Observatory VI,” Astron. J. 120, 1120–1132 (2000).
[CrossRef]

Ho, K.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Holdenried, E. R.

B. D. Mason, W. I. Hartkopf, E. R. Holdenried, T. J. Rafferty, G. L. Wycoff, G. S. Hennessy, D. M. Hall, S. E. Urban, T. E. Corbin, “Speckle interferometery at the US Naval Observatory VI,” Astron. J. 120, 1120–1132 (2000).
[CrossRef]

Hummel, C. A.

C. A. Hummel, J. T. Armstrong, A. Quirrenbach, D. F. Buscher, D. Mozurkewich, N. M. Elias, R. E. Wilson, “Very high precision orbit of Capella by long baseline interferometry,” Astron. J. 107, 1859–1867 (1994).
[CrossRef]

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Labeyrie, A.

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

Lai, O.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Larkin, J.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Lupton, W.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Macintosh, B.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Mackay, C. D.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Mariotti, J. M.

J. M. Mariotti, J. L. Monin, P. Ghez, C. Perrier, A. Zadrozny, “Pupil plane interferometry in the near infrared I. Methodology of observation and first results,” Astron. Astrophys. 255, 462–476 (1992).

Martin, R.

F. Roddier, C. Roddier, R. Petrov, R. Martin, G. Ricort, C. Aime. “New observations of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 305, L77–L80 (1986).
[CrossRef]

Mason, B. D.

B. D. Mason, W. I. Hartkopf, E. R. Holdenried, T. J. Rafferty, G. L. Wycoff, G. S. Hennessy, D. M. Hall, S. E. Urban, T. E. Corbin, “Speckle interferometery at the US Naval Observatory VI,” Astron. J. 120, 1120–1132 (2000).
[CrossRef]

Max, C.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Monin, J. L.

J. M. Mariotti, J. L. Monin, P. Ghez, C. Perrier, A. Zadrozny, “Pupil plane interferometry in the near infrared I. Methodology of observation and first results,” Astron. Astrophys. 255, 462–476 (1992).

Monin, J.-L.

N. Ageorges, J.-L. Monin, L. Desbat, E. Tessier, “Phase and image reconstruction from interferometric imaging: Integration of the phasors,” Astron. Astrophys., Suppl. Ser. 112, 163–171 (1995).

Mozurkewich, D.

C. A. Hummel, J. T. Armstrong, A. Quirrenbach, D. F. Buscher, D. Mozurkewich, N. M. Elias, R. E. Wilson, “Very high precision orbit of Capella by long baseline interferometry,” Astron. J. 107, 1859–1867 (1994).
[CrossRef]

Neyman, C. R.

L. C. Roberts, C. R. Neyman, “Characterization of the AEOS adaptive optics system,” Pub. Astron. Soc. Pacific 114, 1260–1266 (2002).
[CrossRef]

Nightingale, N. S.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

Olivier, S.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Perrier, C.

J. M. Mariotti, J. L. Monin, P. Ghez, C. Perrier, A. Zadrozny, “Pupil plane interferometry in the near infrared I. Methodology of observation and first results,” Astron. Astrophys. 255, 462–476 (1992).

Petrov, R.

F. Roddier, C. Roddier, R. Petrov, R. Martin, G. Ricort, C. Aime. “New observations of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 305, L77–L80 (1986).
[CrossRef]

Quirrenbach, A.

C. A. Hummel, J. T. Armstrong, A. Quirrenbach, D. F. Buscher, D. Mozurkewich, N. M. Elias, R. E. Wilson, “Very high precision orbit of Capella by long baseline interferometry,” Astron. J. 107, 1859–1867 (1994).
[CrossRef]

Rafferty, T. J.

B. D. Mason, W. I. Hartkopf, E. R. Holdenried, T. J. Rafferty, G. L. Wycoff, G. S. Hennessy, D. M. Hall, S. E. Urban, T. E. Corbin, “Speckle interferometery at the US Naval Observatory VI,” Astron. J. 120, 1120–1132 (2000).
[CrossRef]

Ribak, E.

Ricort, G.

F. Roddier, C. Roddier, R. Petrov, R. Martin, G. Ricort, C. Aime. “New observations of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 305, L77–L80 (1986).
[CrossRef]

Roberts, L. C.

L. C. Roberts, C. R. Neyman, “Characterization of the AEOS adaptive optics system,” Pub. Astron. Soc. Pacific 114, 1260–1266 (2002).
[CrossRef]

Roddier, C.

E. Ribak, C. Roddier, F. Roddier, J. B. Breckinridge, “Signal to noise limitations in white light holography,” Appl. Opt. 27, 1183–1186 (1988).
[CrossRef] [PubMed]

F. Roddier, C. Roddier, R. Petrov, R. Martin, G. Ricort, C. Aime. “New observations of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 305, L77–L80 (1986).
[CrossRef]

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

Roddier, F.

E. Ribak, C. Roddier, F. Roddier, J. B. Breckinridge, “Signal to noise limitations in white light holography,” Appl. Opt. 27, 1183–1186 (1988).
[CrossRef] [PubMed]

F. Roddier, “Interferometric imaging in optical astronomy.”Phy. Rep. 170, 99–166 (1988).
[CrossRef]

F. Roddier, C. Roddier, R. Petrov, R. Martin, G. Ricort, C. Aime. “New observations of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 305, L77–L80 (1986).
[CrossRef]

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

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

F. Roddier, Adaptive Optics in Astronomy (Cambridge University Press, 1999).
[CrossRef]

Rogers, J.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

Scheuer, P. A. G.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

Schwarz, U. J.

C. M. de Vos, J. D. Bregman, U. J. Schwarz, “Pupil plane interferometry: Some conclusions from SCASIS,” in Very High Angular Resolution Imaging, J. G. Robertson, W. J. Tango, eds. (Kluwer, 1994), pp. 419–420.
[CrossRef]

Scott, T. R.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

Serabyn, E.

K. Wallace, G. Hardy, E. Serabyn. “Deep and stable interferometric nulling of broadband light with implications for observing planets around nearby stars,” Nature 406, 700–702 (2000).
[CrossRef] [PubMed]

Shelton, C.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Sivia, D.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Stomski, P.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Tessier, E.

N. Ageorges, J.-L. Monin, L. Desbat, E. Tessier, “Phase and image reconstruction from interferometric imaging: Integration of the phasors,” Astron. Astrophys., Suppl. Ser. 112, 163–171 (1995).

Thompson, A. R.

A. R. Thompson, Interferometry and Synthesis in Radio Astronomy (Wiley, 1986).

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Titterington, D. J.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Tsubota, K.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Tuthill, P. G.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

Urban, S. E.

B. D. Mason, W. I. Hartkopf, E. R. Holdenried, T. J. Rafferty, G. L. Wycoff, G. S. Hennessy, D. M. Hall, S. E. Urban, T. E. Corbin, “Speckle interferometery at the US Naval Observatory VI,” Astron. J. 120, 1120–1132 (2000).
[CrossRef]

Wallace, K.

K. Wallace, G. Hardy, E. Serabyn. “Deep and stable interferometric nulling of broadband light with implications for observing planets around nearby stars,” Nature 406, 700–702 (2000).
[CrossRef] [PubMed]

Warner, P. J.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

Wilson, D. M. A.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

Wilson, R. E.

C. A. Hummel, J. T. Armstrong, A. Quirrenbach, D. F. Buscher, D. Mozurkewich, N. M. Elias, R. E. Wilson, “Very high precision orbit of Capella by long baseline interferometry,” Astron. J. 107, 1859–1867 (1994).
[CrossRef]

Wilson, R. W.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

Wizinowich, P.

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge University Press, 1999).
[CrossRef]

Wycoff, G. L.

B. D. Mason, W. I. Hartkopf, E. R. Holdenried, T. J. Rafferty, G. L. Wycoff, G. S. Hennessy, D. M. Hall, S. E. Urban, T. E. Corbin, “Speckle interferometery at the US Naval Observatory VI,” Astron. J. 120, 1120–1132 (2000).
[CrossRef]

Zadrozny, A.

J. M. Mariotti, J. L. Monin, P. Ghez, C. Perrier, A. Zadrozny, “Pupil plane interferometry in the near infrared I. Methodology of observation and first results,” Astron. Astrophys. 255, 462–476 (1992).

Appl. Opt.

Astron. Astrophys.

J. E. Baldwin, M. G. Beckett, R. C. Boysen, D. Burns, D. F. Buscher, G. C. Cox, C. A. Haniff, C. D. Mackay, N. S. Nightingale, J. Rogers, P. A. G. Scheuer, T. R. Scott, P. G. Tuthill, P. J. Warner, D. M. A. Wilson, R. W. Wilson, “The first images from an optical aperture synthesis array: Mapping of Capella with COAST at two epochs,” Astron. Astrophys. 306, L13–L16 (1996).

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

J. M. Mariotti, J. L. Monin, P. Ghez, C. Perrier, A. Zadrozny, “Pupil plane interferometry in the near infrared I. Methodology of observation and first results,” Astron. Astrophys. 255, 462–476 (1992).

Astron. Astrophys., Suppl. Ser.

N. Ageorges, J.-L. Monin, L. Desbat, E. Tessier, “Phase and image reconstruction from interferometric imaging: Integration of the phasors,” Astron. Astrophys., Suppl. Ser. 112, 163–171 (1995).

Astron. J.

C. A. Hummel, J. T. Armstrong, A. Quirrenbach, D. F. Buscher, D. Mozurkewich, N. M. Elias, R. E. Wilson, “Very high precision orbit of Capella by long baseline interferometry,” Astron. J. 107, 1859–1867 (1994).
[CrossRef]

B. D. Mason, W. I. Hartkopf, E. R. Holdenried, T. J. Rafferty, G. L. Wycoff, G. S. Hennessy, D. M. Hall, S. E. Urban, T. E. Corbin, “Speckle interferometery at the US Naval Observatory VI,” Astron. J. 120, 1120–1132 (2000).
[CrossRef]

Astrophys. J.

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

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

F. Roddier, C. Roddier, R. Petrov, R. Martin, G. Ricort, C. Aime. “New observations of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 305, L77–L80 (1986).
[CrossRef]

J. Opt. Soc. Am.

Nature

K. Wallace, G. Hardy, E. Serabyn. “Deep and stable interferometric nulling of broadband light with implications for observing planets around nearby stars,” Nature 406, 700–702 (2000).
[CrossRef] [PubMed]

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Opt. Lett.

Phy. Rep.

F. Roddier, “Interferometric imaging in optical astronomy.”Phy. Rep. 170, 99–166 (1988).
[CrossRef]

Pub. Astron. Soc. Pacific

P. Wizinowich, D. S. Acton, C. Shelton, P. Stomski, J. Gath-right, K. Ho, W. Lupton, K. Tsubota, O. Lai, C. Max, J. Brase, J. An, K. Avicola, S. Olivier, D. Gavel, B. Macintosh, A. Ghez, J. Larkin, “First light adaptive optics images from the Keck II telescope: a new era of high angular resolution imagery,” Pub. Astron. Soc. Pacific 112, 315–319 (2000).
[CrossRef]

L. C. Roberts, C. R. Neyman, “Characterization of the AEOS adaptive optics system,” Pub. Astron. Soc. Pacific 114, 1260–1266 (2002).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Cambridge University Press, 1999).
[CrossRef]

A. R. Thompson, Interferometry and Synthesis in Radio Astronomy (Wiley, 1986).

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

F. Roddier, Adaptive Optics in Astronomy (Cambridge University Press, 1999).
[CrossRef]

C. M. de Vos, J. D. Bregman, U. J. Schwarz, “Pupil plane interferometry: Some conclusions from SCASIS,” in Very High Angular Resolution Imaging, J. G. Robertson, W. J. Tango, eds. (Kluwer, 1994), pp. 419–420.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Schematic description of interference between two points in the input pupil. Light from two subapertures of the input pupil is superposed (with the superposition shown as ⊕). The output intensity is modulated by the complex visibility, evaluated at a frequency defined by the separation between the two subapertures. In the case of a rotation shearing interferometer, the subapertures are defined by individual pixels on the detector, and all points in the input pupil are simultaneously interfered pairwise to form a two-dimensional interferogram.

Fig. 2
Fig. 2

Schematic rotation shearing geometry and coordinate systems. Light is incident on the input pupil at the top of the figure. Arms A and B each receive a copy of the input pupil, and rotate the copies 90 deg in opposite directions. When the two copies of the input pupil are recombined (through a beam splitter) they form two interferograms. Instrumental path-length differences, ΔL, shown here as a constant with respect to (ξ, η), effectively retard or advance the wavefronts in one arm. Vertical displacements in the figure represent path-length differences. The two output interferograms differ by π rad (180°) in phase, by virtue of conservation of energy, shown here as a λ/2 path-length difference in Interferogram 2. While two detectors are shown in this figure, the QPI is currently configured so that the two interferograms land side by side on the same detector.

Fig. 3
Fig. 3

Mirror arrangement giving rotation shear. Each arm of the interferometer contains three mirrors, arranged to fold the light path by 90 deg, and rotate the field of view by ±90 deg about the propagation direction. This figure depicts a fan of light rays, initially oriented vertically as it exits the first beam splitter (propagating in the z direction in the displayed coordinate frame), being folded to propagate in the −x direction and rotated to a horizontal orientation before entering the second beam splitter. The other arm of the interferometer performs the same rotation, in the opposite sense (−90 deg instead of +90 deg). The angles of incidence of the chief ray on each of the three mirrors are the same.

Fig. 4
Fig. 4

Sample laboratory interferograms of a pinhole. Each point in the interferogram shows the interference of two points in the input pupil (see Fig. 1). The pinhole has a diameter smaller than the diffraction limit of the interferometer, giving a near-uniform visibility modulus. Corresponding points in the two interferograms differ in interferometric phase by π rad. The constant phase gradient (manifested as fringes) shows that the object is off axis. No turbulent aberrations are present.

Fig. 5
Fig. 5

Sample laboratory interferograms of a pinhole, divided into four interferograms. The upper panels are identical to the upper halves of the interferograms in Fig. 4, while the bottom panels have been rotated 180 deg to emphasize the correlation between diametrically opposed points (x, y) in the upper panels and (−x, −y) in the lower panels. Four points that compose a quadrature-phase measurement are identified by crosses. The interferometric phases of the corresponding points in the left and right panels differ by π rad, while those in the upper and lower panels differ by π/2 rad.

Fig. 6
Fig. 6

Architecture of the KFS–CCD. The device is divided into 32 equivalent segments, each with a high-bandwidth floating diffusion amplifier.

Fig. 7
Fig. 7

Optical layout for the laboratory test. A Hg lamp illuminates a mask, and the light is collimated and stopped down. Turbulent aberrations are introduced into the collimated light by a He–air jet. A beam splitter breaks off 10% of the light as a direct image, shown here shaded as the DIRECT component, while the rest of the light passes through the interferometer, shown here as the PUPIL component. The CCD records the direct image and the two interferograms simultaneously (near the CCD, the direct image and one interferogram overlap as drawn, but are separated vertically). The mirrors in the interferometer, shown in Fig. 3, are not drawn here.

Fig. 8
Fig. 8

Direct and interferometric laboratory images. The left panels are direct images, and the right panels are interferometric images. Corresponding direct and interferometric images are acquired simultaneously through identical conditions. The top panels include no turbulent aberrations, and the bottom panels are imaged through a turbulent He–air jet that introduces phase and amplitude aberrations. Each of the bottom panels is an average of the same ten exposures. The interferometric image quality is only slightly affected by the aberrations, while the aberrated direct images are blurred beyond recognition. The diffraction limit of the system is approximately half the width of the line segments making up the letters.

Fig. 9
Fig. 9

Sample differenced interferogram of Vega. This figure is the difference between Interferogram 1 and 2 for a single exposure, removing scintillation effects. The visibility modulus is essentially uniform across the exposure. The input pupil was not correctly aligned with the interferometer center of rotation, leading to an oversized central obscuration and noncircular outer boundary.

Fig. 10
Fig. 10

Direct, raw interferometric, and differenced interferometric images of Vega. The left panel is the average of 10 s (400 frames) of direct images of Vega. The middle panel is a raw interferometric image, from the same 10 s of data as the left panel, formed without estimating the instrumental phase. The right panel is formed by calibrating one half of the 10 s dataset using the other half. The FWHM of these images is 1.0, 0.5, and 0.2 arc sec.

Fig. 11
Fig. 11

Binary separation of Capella. The diamonds mark the separation measured by the QPI, with two diamonds shown because the separation is uncertain by 180 deg. The crosses mark the origin and the expected binary separation at the time of the observation, with the dotted curve tracing out an entire orbit. The separation is measured by fitting the measured visibility modulus from 10 s of data to a sinusoid, appropriate to a binary pair. The disagreement between the measured and expected binary separations is 0.0012 arc sec.

Fig. 12
Fig. 12

Comparison of imaging performance under two QPI configurations. The left panel is the image produced by a split mirror QPI configuration, and the right panel is the image produced by the current QPI configuration. The split mirror configuration results in a loss of low-frequency information, producing a high-pass filtered image.

Equations (18)

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

[ B ( α , β ) ] = γ ( u , υ ) .
γ ( u , υ ) = V ( u , υ ) exp { i ϕ ( u , υ ) } ,
I ( u , υ ) / I 0 = 1 + { γ meas ( u , υ ) } , = 1 + V meas ( u , υ ) cos [ ϕ meas ( u , υ ) ] .
( u , υ ) = [ ( ξ B ξ A ) / λ , ( η B η A ) / λ ] ,
γ ( u , υ ) = γ * ( u , υ ) , V ( u , υ ) = V ( u , υ ) , ϕ ( u , υ ) = ϕ ( u , υ ) .
( u , υ ) = ( 2 y / λ , 2 x / λ ) .
ϕ meas = ϕ obj + ϕ inst + ϕ turb .
ϕ inst , 1 ( x , y ) = 2 π Δ L ( x , y ) / λ ϕ inst , 2 ( x , y ) = 2 π Δ L ( x , y ) / λ + π ,
ϕ 0 ( x , y ) ϕ obj ( x , y ) + ϕ turb ( x , y ) + π / 4 .
ϕ meas , 1 ( x , y ) = ϕ obj ( x , y ) + ϕ turb ( x , y ) + π / 4 = ϕ 0 ( x , y ) , ϕ meas , 1 ( x , y ) = ϕ obj ( x , y ) ϕ turb ( x , y ) + π / 4 = [ ϕ 0 ( x , y ) π / 2 ] , ϕ meas , 2 ( x , y ) = ϕ obj ( x , y ) + ϕ turb ( x , y ) + 5 π / 4 = ϕ 0 ( x , y ) + π , ϕ meas , 2 ( x , y ) = ϕ obj ( x , y ) ϕ turb ( x , y ) + 5 π / 4 = [ ϕ 0 ( x , y ) 3 π / 2 ] ,
I 1 ( x , y ) = I 0 ( x , y ) { 1 + V meas ( x , y ) cos [ ϕ 0 ( x , y ) ] } , I 1 ( x , y ) = I 0 ( x , y ) { 1 + V meas ( x , y ) sin [ ϕ 0 ( x , y ) ] } , I 2 ( x , y ) = I 0 ( x , y ) { 1 V meas ( x , y ) cos [ ϕ 0 ( x , y ) ] } , I 2 ( x , y ) = I 0 ( x , y ) { 1 V meas ( x , y ) sin [ ϕ 0 ( x , y ) ] } ,
I 0 ( x , y ) = [ I 1 ( x , y ) + I 2 ( x , y ) ] / 2 = [ I 1 ( x , y ) + I 2 ( x , y ) ] / 2 , V meas ( x , y ) = { [ I 1 ( x , y ) I 2 ( x , y ) ] 2 + [ I 1 ( x , y ) + I 2 ( x , y ) ] 2 } 1 / 2 2 I 0 ( x , y ) , ϕ obj ( x , y ) + ϕ turb ( x , y ) = tan 1 I 1 ( x , y ) I 2 ( x , y ) I 1 ( x , y ) I 2 ( x , y ) π / 4 .
V meas = V band V temp V spat V amp V obj ,
D φ ( ρ ) = [ φ turb ( ξ , η ) φ turb ( ξ + Δ ξ , η + Δ η ) ] 2 = 6.88 ( ρ / r 0 ) 5 / 3 rad 2 ,
ϕ turb 2 ( x , y ) = D φ [ ( x 2 + y 2 ) 1 / 2 ] ,
[ x ϕ turb ( x , y ) ] 2 2 D φ ( Δ x ) / Δ x 2 ,
Δ x < r 0 / 5 .
cos ϕ off ( x , y ) = 4 [ I 1 ( x , y ) I 2 ( x , y ) ] [ I 1 ( x , y ) I 2 ( x , y ) ] [ I 1 ( x , y ) + I 2 ( x , y ) ] [ I 1 ( x , y ) + I 2 ( x , y ) ] [ I 1 ( x , y ) I 2 ( x , y ) ] 2 [ I 1 ( x , y ) + I 2 ( x , y ) ] 2 + [ I 1 ( x , y ) I 2 ( x , y ) ] 2 [ I 1 ( x , y ) + I 2 ( x , y ) ] 2 .

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