Abstract

The wave structure function (WSF) for a plane wave, calculated from the basic Rytov theory, is usually expressed as 6.88(r/r0)5/3, but this does not include the effect of a finite outer scale (or of a nonzero inner scale) of turbulence. When separation distance r is only 5% of the outer scale, this expression overpredicts the WSF by a factor of approximately 2. Accurate evaluations of the Rytov formulas are given for the WSFs of plane and spherical waves in Kolmogorov and von Karman turbulence and for the structure function of the atmosphere's index of refraction. Simple formulas make the results easy to use.

© 2007 Optical Society of America

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  1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 1998), especially Subsection 6.4 and Appendix III.
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
  3. J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 8.
  4. R. Beland, "Propagation through atmospheric optical turbulence," in Atmospheric Propagation of Radiation, Vol. 2 of The Infrared and Electro-Optical Systems Handbook, J. S. Accetta and D. L. Shumaker, eds. (SPIE, 1993), pp. 157-232.
  5. S. F. Clifford, "The classical theory of wave propagation in a turbulent medium," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, 1978), pp. 9-43.
  6. D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967).
    [CrossRef]
  7. V. P. Lukin, "Estimation of behavior of outer scale of turbulence from optical measurements," Proc. SPIE 4538, 74-84 (2002).
  8. D. I. Cooper, W. E. Eichinger, J. Archuleta, L. Hipps, J. Kao, and J. Prueger, "An advanced method for deriving latent energy flux from a scanning Raman lidar," Agron. J. (to be published).
  9. V. I. Tatarskii, Wave Propagation in a Turbulent Atmosphere (Wiley, 1985).
  10. C. E. Coulman, J. Vernin, Y. Coqueuniot, and J. L. Caccia, "Outer scale of turbulence appropriate to modeling refractive-index structure profiles," Appl. Opt. 27, 155-160 (1988).
    [CrossRef] [PubMed]
  11. A. Ziad, M. Schock, G. A. Chanan, M. Troy, R. Dekany, B. F. Lane, J. Borgnino, and F. Martin, "Comparison of measurements of the outer scale of turbulence by three different techniques," Appl. Opt. 43, 2316-2324 (2004).
    [CrossRef] [PubMed]
  12. G. Rousset and J.-L. Beuzit, "The COME-ON/ADONIS systems," in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, 1999), pp. 171-203.
  13. F. Rigaut, G. Rousset, P. Kern, J.-C. Fontanella, J.-P. Gaffard, and F. Merkle, "Adaptive optics on a 3.6-m telescope: results and performance," Astron. Astrophys. 250, 280-290 (1991).
  14. G. Rousset, P.-Y. Madec, and F. Rigaut, "Temporal analysis of turbulent wavefronts sensed by adaptive optics," in Atmospheric, Volume and Surface Scattering and Propagation, ICO Topical Meeting, A. Consortini, ed. (ICO, 1991), pp. 77-80.
  15. D. L. Fried, DLFried@cruzio.com (personal communication, 8 July 2004).
  16. A. Abahamid, J. Vernin, Z. Benkhaldoun, A. Jabiri, M. Azouit, and A. Agabi, "Seeing, outer scale of optical turbulence, and coherence outer scale at different astronomical sites using instruments on meteorological balloons," Astron. Astrophys. 422, 1123-1127 (2004).
    [CrossRef]
  17. R. L. Lucke, "Synthetic aperture ladar simulations with phase screens and Fourier propagation," in IEEE Aerospace Conference Proceedings (IEEE, 2004), Vol. 3, p. 1788.
  18. R. F. Lutomirski and H. T. Yura, "Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere," J. Opt. Soc. Am. 61, 482-487 (1971).
    [CrossRef]
  19. L. C. Andrews, S. Vester, and C. E. Richardson, "Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations," J. Mod. Opt. 40, 931-938 (1993).
    [CrossRef]
  20. V. V. Voitsekhovich, "Outer scale of turbulence: comparison of different models," J. Opt. Soc. Am. A 12, 1346-1353 (1995).
    [CrossRef]
  21. C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, "The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory," Waves Random Media 14, 75-96 (2004).
    [CrossRef]
  22. R. L. Fante, "Turbulence-induced distortion of synthetic aperture radar images," IEEE Trans. Geosci. Remote Sens. 32, 958-961 (1994).
    [CrossRef]
  23. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994).
  24. R. Conan, A. Ziad, J. Borgnino, F. Martin, and A. Tokovinin, "Measurements of the wavefront outer scale at Paranal: influence of this parameter in interferometry," Proc. SPIE 4006, 963-973 (2000).
    [CrossRef]
  25. R. J. Sasiela, MIT Lincoln Laboratory, Cambridge, Mass. (personal communication, 15 June 2005).
  26. V. P. Lukin, "Optical measurements of the outer scale of the atmospheric turbulence," Proc. SPIE 1968, 327-336 (1993).
  27. R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
    [CrossRef]
  28. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).
  29. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).

2004 (3)

A. Abahamid, J. Vernin, Z. Benkhaldoun, A. Jabiri, M. Azouit, and A. Agabi, "Seeing, outer scale of optical turbulence, and coherence outer scale at different astronomical sites using instruments on meteorological balloons," Astron. Astrophys. 422, 1123-1127 (2004).
[CrossRef]

C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, "The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory," Waves Random Media 14, 75-96 (2004).
[CrossRef]

A. Ziad, M. Schock, G. A. Chanan, M. Troy, R. Dekany, B. F. Lane, J. Borgnino, and F. Martin, "Comparison of measurements of the outer scale of turbulence by three different techniques," Appl. Opt. 43, 2316-2324 (2004).
[CrossRef] [PubMed]

2002 (1)

V. P. Lukin, "Estimation of behavior of outer scale of turbulence from optical measurements," Proc. SPIE 4538, 74-84 (2002).

2000 (1)

R. Conan, A. Ziad, J. Borgnino, F. Martin, and A. Tokovinin, "Measurements of the wavefront outer scale at Paranal: influence of this parameter in interferometry," Proc. SPIE 4006, 963-973 (2000).
[CrossRef]

1995 (1)

1994 (1)

R. L. Fante, "Turbulence-induced distortion of synthetic aperture radar images," IEEE Trans. Geosci. Remote Sens. 32, 958-961 (1994).
[CrossRef]

1993 (2)

V. P. Lukin, "Optical measurements of the outer scale of the atmospheric turbulence," Proc. SPIE 1968, 327-336 (1993).

L. C. Andrews, S. Vester, and C. E. Richardson, "Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations," J. Mod. Opt. 40, 931-938 (1993).
[CrossRef]

1992 (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

1991 (1)

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

1988 (1)

1971 (1)

1967 (1)

D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967).
[CrossRef]

Abahamid, A.

A. Abahamid, J. Vernin, Z. Benkhaldoun, A. Jabiri, M. Azouit, and A. Agabi, "Seeing, outer scale of optical turbulence, and coherence outer scale at different astronomical sites using instruments on meteorological balloons," Astron. Astrophys. 422, 1123-1127 (2004).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).

Agabi, A.

A. Abahamid, J. Vernin, Z. Benkhaldoun, A. Jabiri, M. Azouit, and A. Agabi, "Seeing, outer scale of optical turbulence, and coherence outer scale at different astronomical sites using instruments on meteorological balloons," Astron. Astrophys. 422, 1123-1127 (2004).
[CrossRef]

Andrews, L. C.

L. C. Andrews, S. Vester, and C. E. Richardson, "Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations," J. Mod. Opt. 40, 931-938 (1993).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 1998), especially Subsection 6.4 and Appendix III.

Archuleta, J.

D. I. Cooper, W. E. Eichinger, J. Archuleta, L. Hipps, J. Kao, and J. Prueger, "An advanced method for deriving latent energy flux from a scanning Raman lidar," Agron. J. (to be published).

Azouit, M.

A. Abahamid, J. Vernin, Z. Benkhaldoun, A. Jabiri, M. Azouit, and A. Agabi, "Seeing, outer scale of optical turbulence, and coherence outer scale at different astronomical sites using instruments on meteorological balloons," Astron. Astrophys. 422, 1123-1127 (2004).
[CrossRef]

Beland, R.

R. Beland, "Propagation through atmospheric optical turbulence," in Atmospheric Propagation of Radiation, Vol. 2 of The Infrared and Electro-Optical Systems Handbook, J. S. Accetta and D. L. Shumaker, eds. (SPIE, 1993), pp. 157-232.

Benkhaldoun, Z.

A. Abahamid, J. Vernin, Z. Benkhaldoun, A. Jabiri, M. Azouit, and A. Agabi, "Seeing, outer scale of optical turbulence, and coherence outer scale at different astronomical sites using instruments on meteorological balloons," Astron. Astrophys. 422, 1123-1127 (2004).
[CrossRef]

Beuzit, J.-L.

G. Rousset and J.-L. Beuzit, "The COME-ON/ADONIS systems," in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, 1999), pp. 171-203.

Borgnino, J.

A. Ziad, M. Schock, G. A. Chanan, M. Troy, R. Dekany, B. F. Lane, J. Borgnino, and F. Martin, "Comparison of measurements of the outer scale of turbulence by three different techniques," Appl. Opt. 43, 2316-2324 (2004).
[CrossRef] [PubMed]

R. Conan, A. Ziad, J. Borgnino, F. Martin, and A. Tokovinin, "Measurements of the wavefront outer scale at Paranal: influence of this parameter in interferometry," Proc. SPIE 4006, 963-973 (2000).
[CrossRef]

Caccia, J. L.

Chanan, G. A.

Clifford, S. F.

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

Conan, R.

R. Conan, A. Ziad, J. Borgnino, F. Martin, and A. Tokovinin, "Measurements of the wavefront outer scale at Paranal: influence of this parameter in interferometry," Proc. SPIE 4006, 963-973 (2000).
[CrossRef]

Cooper, D. I.

D. I. Cooper, W. E. Eichinger, J. Archuleta, L. Hipps, J. Kao, and J. Prueger, "An advanced method for deriving latent energy flux from a scanning Raman lidar," Agron. J. (to be published).

Coqueuniot, Y.

Coulman, C. E.

Dainty, J. C.

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Dekany, R.

Eichinger, W. E.

D. I. Cooper, W. E. Eichinger, J. Archuleta, L. Hipps, J. Kao, and J. Prueger, "An advanced method for deriving latent energy flux from a scanning Raman lidar," Agron. J. (to be published).

Fante, R. L.

R. L. Fante, "Turbulence-induced distortion of synthetic aperture radar images," IEEE Trans. Geosci. Remote Sens. 32, 958-961 (1994).
[CrossRef]

Fontanella, J.-C.

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

Fried, D. L.

D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967).
[CrossRef]

D. L. Fried, DLFried@cruzio.com (personal communication, 8 July 2004).

Gaffard, J.-P.

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

Glindemann, A.

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 8.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Hipps, L.

D. I. Cooper, W. E. Eichinger, J. Archuleta, L. Hipps, J. Kao, and J. Prueger, "An advanced method for deriving latent energy flux from a scanning Raman lidar," Agron. J. (to be published).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Jabiri, A.

A. Abahamid, J. Vernin, Z. Benkhaldoun, A. Jabiri, M. Azouit, and A. Agabi, "Seeing, outer scale of optical turbulence, and coherence outer scale at different astronomical sites using instruments on meteorological balloons," Astron. Astrophys. 422, 1123-1127 (2004).
[CrossRef]

Kao, J.

D. I. Cooper, W. E. Eichinger, J. Archuleta, L. Hipps, J. Kao, and J. Prueger, "An advanced method for deriving latent energy flux from a scanning Raman lidar," Agron. J. (to be published).

Kern, P.

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

Lane, B. F.

Lane, R. G.

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Lucke, R. L.

R. L. Lucke, "Synthetic aperture ladar simulations with phase screens and Fourier propagation," in IEEE Aerospace Conference Proceedings (IEEE, 2004), Vol. 3, p. 1788.

Lukin, V. P.

V. P. Lukin, "Estimation of behavior of outer scale of turbulence from optical measurements," Proc. SPIE 4538, 74-84 (2002).

V. P. Lukin, "Optical measurements of the outer scale of the atmospheric turbulence," Proc. SPIE 1968, 327-336 (1993).

Lutomirski, R. F.

Madec, P.-Y.

G. Rousset, P.-Y. Madec, and F. Rigaut, "Temporal analysis of turbulent wavefronts sensed by adaptive optics," in Atmospheric, Volume and Surface Scattering and Propagation, ICO Topical Meeting, A. Consortini, ed. (ICO, 1991), pp. 77-80.

Martin, F.

A. Ziad, M. Schock, G. A. Chanan, M. Troy, R. Dekany, B. F. Lane, J. Borgnino, and F. Martin, "Comparison of measurements of the outer scale of turbulence by three different techniques," Appl. Opt. 43, 2316-2324 (2004).
[CrossRef] [PubMed]

R. Conan, A. Ziad, J. Borgnino, F. Martin, and A. Tokovinin, "Measurements of the wavefront outer scale at Paranal: influence of this parameter in interferometry," Proc. SPIE 4006, 963-973 (2000).
[CrossRef]

Masino, A. J.

C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, "The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory," Waves Random Media 14, 75-96 (2004).
[CrossRef]

Merkle, F.

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

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 1998), especially Subsection 6.4 and Appendix III.

Prueger, J.

D. I. Cooper, W. E. Eichinger, J. Archuleta, L. Hipps, J. Kao, and J. Prueger, "An advanced method for deriving latent energy flux from a scanning Raman lidar," Agron. J. (to be published).

Richardson, C. E.

L. C. Andrews, S. Vester, and C. E. Richardson, "Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations," J. Mod. Opt. 40, 931-938 (1993).
[CrossRef]

Rigaut, F.

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

G. Rousset, P.-Y. Madec, and F. Rigaut, "Temporal analysis of turbulent wavefronts sensed by adaptive optics," in Atmospheric, Volume and Surface Scattering and Propagation, ICO Topical Meeting, A. Consortini, ed. (ICO, 1991), pp. 77-80.

Rousset, G.

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

G. Rousset, P.-Y. Madec, and F. Rigaut, "Temporal analysis of turbulent wavefronts sensed by adaptive optics," in Atmospheric, Volume and Surface Scattering and Propagation, ICO Topical Meeting, A. Consortini, ed. (ICO, 1991), pp. 77-80.

G. Rousset and J.-L. Beuzit, "The COME-ON/ADONIS systems," in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, 1999), pp. 171-203.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994).

R. J. Sasiela, MIT Lincoln Laboratory, Cambridge, Mass. (personal communication, 15 June 2005).

Schock, M.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).

Subich, C. J.

C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, "The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory," Waves Random Media 14, 75-96 (2004).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Atmosphere (Wiley, 1985).

Thomas, F. E.

C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, "The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory," Waves Random Media 14, 75-96 (2004).
[CrossRef]

Tokovinin, A.

R. Conan, A. Ziad, J. Borgnino, F. Martin, and A. Tokovinin, "Measurements of the wavefront outer scale at Paranal: influence of this parameter in interferometry," Proc. SPIE 4006, 963-973 (2000).
[CrossRef]

Troy, M.

Vernin, J.

A. Abahamid, J. Vernin, Z. Benkhaldoun, A. Jabiri, M. Azouit, and A. Agabi, "Seeing, outer scale of optical turbulence, and coherence outer scale at different astronomical sites using instruments on meteorological balloons," Astron. Astrophys. 422, 1123-1127 (2004).
[CrossRef]

C. E. Coulman, J. Vernin, Y. Coqueuniot, and J. L. Caccia, "Outer scale of turbulence appropriate to modeling refractive-index structure profiles," Appl. Opt. 27, 155-160 (1988).
[CrossRef] [PubMed]

Vester, S.

L. C. Andrews, S. Vester, and C. E. Richardson, "Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations," J. Mod. Opt. 40, 931-938 (1993).
[CrossRef]

Voitsekhovich, V. V.

Young, C. Y.

C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, "The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory," Waves Random Media 14, 75-96 (2004).
[CrossRef]

Yura, H. T.

Ziad, A.

A. Ziad, M. Schock, G. A. Chanan, M. Troy, R. Dekany, B. F. Lane, J. Borgnino, and F. Martin, "Comparison of measurements of the outer scale of turbulence by three different techniques," Appl. Opt. 43, 2316-2324 (2004).
[CrossRef] [PubMed]

R. Conan, A. Ziad, J. Borgnino, F. Martin, and A. Tokovinin, "Measurements of the wavefront outer scale at Paranal: influence of this parameter in interferometry," Proc. SPIE 4006, 963-973 (2000).
[CrossRef]

Agron. J. (1)

D. I. Cooper, W. E. Eichinger, J. Archuleta, L. Hipps, J. Kao, and J. Prueger, "An advanced method for deriving latent energy flux from a scanning Raman lidar," Agron. J. (to be published).

Appl. Opt. (2)

Astron. Astrophys. (2)

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

A. Abahamid, J. Vernin, Z. Benkhaldoun, A. Jabiri, M. Azouit, and A. Agabi, "Seeing, outer scale of optical turbulence, and coherence outer scale at different astronomical sites using instruments on meteorological balloons," Astron. Astrophys. 422, 1123-1127 (2004).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

R. L. Fante, "Turbulence-induced distortion of synthetic aperture radar images," IEEE Trans. Geosci. Remote Sens. 32, 958-961 (1994).
[CrossRef]

J. Mod. Opt. (1)

L. C. Andrews, S. Vester, and C. E. Richardson, "Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations," J. Mod. Opt. 40, 931-938 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Proc. IEEE (1)

D. L. Fried, "Optical heterodyne detection of an atmospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967).
[CrossRef]

Proc. SPIE (1)

R. Conan, A. Ziad, J. Borgnino, F. Martin, and A. Tokovinin, "Measurements of the wavefront outer scale at Paranal: influence of this parameter in interferometry," Proc. SPIE 4006, 963-973 (2000).
[CrossRef]

Waves Random Media (2)

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, "The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory," Waves Random Media 14, 75-96 (2004).
[CrossRef]

Other (16)

R. L. Lucke, "Synthetic aperture ladar simulations with phase screens and Fourier propagation," in IEEE Aerospace Conference Proceedings (IEEE, 2004), Vol. 3, p. 1788.

G. Rousset, P.-Y. Madec, and F. Rigaut, "Temporal analysis of turbulent wavefronts sensed by adaptive optics," in Atmospheric, Volume and Surface Scattering and Propagation, ICO Topical Meeting, A. Consortini, ed. (ICO, 1991), pp. 77-80.

D. L. Fried, DLFried@cruzio.com (personal communication, 8 July 2004).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).

G. Rousset and J.-L. Beuzit, "The COME-ON/ADONIS systems," in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, 1999), pp. 171-203.

R. J. Sasiela, MIT Lincoln Laboratory, Cambridge, Mass. (personal communication, 15 June 2005).

V. P. Lukin, "Optical measurements of the outer scale of the atmospheric turbulence," Proc. SPIE 1968, 327-336 (1993).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994).

V. P. Lukin, "Estimation of behavior of outer scale of turbulence from optical measurements," Proc. SPIE 4538, 74-84 (2002).

V. I. Tatarskii, Wave Propagation in a Turbulent Atmosphere (Wiley, 1985).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 1998), especially Subsection 6.4 and Appendix III.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 8.

R. Beland, "Propagation through atmospheric optical turbulence," in Atmospheric Propagation of Radiation, Vol. 2 of The Infrared and Electro-Optical Systems Handbook, J. S. Accetta and D. L. Shumaker, eds. (SPIE, 1993), pp. 157-232.

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

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Figures (6)

Fig. 1
Fig. 1

Case 1. Functions plotted are taken from Eqs. (6), (7), (9), (10), and (11). The solid curve is the numerically calculated normalized WSF, the other curves are analytic approximations (see text). From Eq. (9), D 1 * is the normalized form of the usual approximation to the WSF, that is, D 1 * ( r / L 0 ) r 5 / 3 .

Fig. 2
Fig. 2

Case 2. Functions given by Eqs. (16)–(18). D 1 * is the same as in Fig. 1. Parameter a is the ratio of the cutoff frequency to the roll-off frequency (see text).

Fig. 3
Fig. 3

Case 3. Functions given by Eqs. (21)–(23).

Fig. 4
Fig. 4

Case 4. Functions given by Eqs. (24) and (26). D 1 * is the same as in Fig. 3.

Fig. 5
Fig. 5

Normalized index-of-refraction structure function for Kolmogorov turbulence, from Eqs. (28) and (29).

Fig. 6
Fig. 6

Similar to Fig. 5, with functions given by Eqs. (31)–(33). D n * 1 is the same as in Fig. 5.

Equations (48)

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D ( r ) = 8 π 2 k 2 Z 0 Φ n ( K ) [ 1 J 0 ( K r ) ] K d K ,
D ( r ) = 8 π 2 k 2 Z 0 1 0 Φ n ( K ) [ 1 J 0 ( K r ξ ) ] K d K d ξ .
Φ n ( K ) = 0.033 C n 2 K 11 / 3 , K 0 K K 1 ,
Φ n ( K ) = 0.033 C n 2   exp ( K 2 / K m 2 ) ( K 2 + K 0 2 ) 11 / 6 = 0.033 C n 2 ( K 2 + K 0 2 ) 11 / 6 , a K 0 K ,
D ( r ) = 8 π 2 k 2 Z × 0.033 C n 2 K 0 1 J 0 ( K r ) K 11 / 3 K d K = 24 π 2 0.033 C n 2 k 2 Z 5 K 0 5 / 3 [ 5 3 ( K 0 r ) 5 / 3 K 0 r 1 J 0 ( u ) u 8 / 3  d u ] = 0.073 C n 2 k 2 Z L 0 5 / 3 D * ( r L 0 ) = 0.173 ( L 0 r 0 ) 5 / 3 D * ( r L 0 ) ,
D * ( r L 0 ) 5 3 ( K 0 r ) 5 / 3 K 0 r 1 J 0 ( u ) u 8 / 3 d u = 5 3 ( K 0 r ) 5 / 3 [ 0 1 J 0 ( u ) u 8 / 3 d u 0 K 0 r 1 J 0 ( u ) u 8 / 3 d u ] .
D * ( r L 0 ) = 5 3 1 1 J 0 ( K 0 r v ) v 8 / 3 d v
  5 3 1 d v v 8 / 3 = 1 , as r L 0 ,
D * ( r L 0 ) 5 3 ( K 0 r ) 5 / 3 [ 1.1183 3 4 ( K 0 r ) 1 / 3 + 3 448 ( K 0 r ) 7 / 3 ] = 35.66 ( r L 0 ) 5 / 3 [ 1.1183 1.384 ( r L 0 ) 1 / 3 + 0.488 ( r L 0 ) 7 / 3 ] = 39.88 ( r L 0 ) 5 / 3 [ 1 1.238 ( r L 0 ) 1 / 3 + 0.436 ( r L 0 ) 7 / 3 ] .
D 1 * ( r L 0 ) = 39.88 ( r L 0 ) 5 / 3 ,
D 2 * ( r L 0 ) = 39.88 ( r L 0 ) 5 / 3 [ 1 1.238 ( r L 0 ) 1 / 3 ] ,
D 3 * ( r L 0 ) = 39.88 ( r L 0 ) 5 / 3 [ 1 1.238 ( r L 0 ) 1 / 3 + 0.436 ( r L 0 ) 7 / 3 ] .
D ( r ) = 0.073 C n 2 k 2 Z L 0 5 / 3 D * ( r L 0 ) 0.073 C n 2 k 2 Z L 0 5 / 3 for   r L 0 > 0.28 2.91 C n 2 k 2 Z r 5 / 3 [ 1 1.238 ( r L 0 ) 1 / 3 + 0.436 ( r L 0 ) 7 / 3 ] for   r L 0 0.28 = 6.88 ( r L 0 ) 5 / 3 [ 1 1.238 ( r L 0 ) 1 / 3 + 0.436 ( r L 0 ) 7 / 3 ] .
D 1 ( r ) = 6.88 ( r r 0 ) 5 / 3
D 2 ( r ) = 6.88 ( r r 0 ) 5 / 3 [ 1 1.24 ( r L 0 ) 1 / 3 ] ,
D ( r ) = 8 π 2 0.033 C n 2 k 2 Z a K 0 1 J 0 ( K r ) ( K 2 + K 0 2 ) 11 / 6 K d K = 24 π 2 0.033 C n 2 k 2 Z 5 K 0 5 / 3 ( 5 3 ( K 0 r ) 5 / 3 × a K 0 r 1 J 0 ( u ) ( u 2 + K 0 2 r 2 ) 11 / 6 u d u ) .
D * ( r L 0 ) = 5 3 ( K 0 r ) 5 / 3 [ 0 1 J 0 ( u ) ( u 2 + K 0 2 r 2 ) 11 / 6 u d u 0 a K 0 r 1 J 0 ( u ) ( u 2 + K 0 2 r 2 ) 11 / 6 u d u ] .
D 3 * ( r L 0 ) = 39.88 ( r L 0 ) 5 / 3 [ 1 1.485 ( r L 0 ) 1 / 3 + 0.64 ( r L 0 ) 4 / 3 ] .
D * ( ) = 5 3 a v d v ( v 2 + 1 ) 11 / 6 = 5 6 [ 6 5 ( v 2 + 1 ) 5 / 6 ] a = ( 1 ,  0.561 ) .
D ( r ) = 0.073 C n 2 k 2 Z L 0 5 / 3 D * ( r L 0 ) ( 0.073 ,  0.041 ) C n 2 k 2 Z L 0 5 / 3 for   r L 0 > ( 0.53 ,  0.20 ) 6.88 ( r r 0 ) 5 / 3 [ 1 1.485 ( r L 0 ) 1 / 3 + 0.64 ( r L 0 ) 4 / 3 ] for   r L 0 ( 0.53 ,  0.20 ) .
D ( r ) = 8 π 2 0.033 C n 2 k 2 Z 0 1 K 0 1 J 0 ( K r ξ ) K 11 / 3 K d K d ξ = 24 π 2 0.033 C n 2 k 2 Z 5 K 0 5 / 3 { 5 3 0 1 ( K 0 r ξ ) 5 / 3 ×[ 0 1 J 0 ( u ) u 8 / 3 d u 0 K 0 r ξ 1 J 0 ( u ) u 8 / 3 d u ] d ξ } .
  D * ( r L 0 ) 5 3 0 1 [ 1.118 ( K 0 r ξ ) 5 / 3 3 4 ( K 0 r ξ ) 2 + 3 448 ( K 0 r ξ ) 4 ] d ξ = 5 3 [ 1.118 3 8 ( K 0 r ) 5 / 3 1 4 ( K 0 r ) 2 + 3 2240 ( K 0 r ) 4 ] = 14.95 ( r L 0 ) 5 / 3 [ 1 1.100 ( r L 0 ) 1 / 3 + 0.233 ( r L 0 ) 7 / 3 ] .
  D * ( r L 0 ) 5 3 0 1 ( K 0 r ξ ) 5 / 3 K 0 ξ r 1 J 0 ( u ) u 8 / 3 d u d ξ = 5 3 0 1 1 1 J 0 ( K 0 r ξ v ) v 8 / 3 d v d ξ 1 , as r L 0 .
D ( r ) = 0.073 C n 2 k 2 Z L 0 5 / 3 D * ( r L 0 ) 0.073 C n 2 k 2 Z L 0 5 / 3 for   r L 0 > 0.68 1.09 C n 2 k 2 Z r 5 / 3 [ 1 1.100 ( r L 0 ) 1 / 3 + 0.233 ( r L 0 ) 7 / 3 ] for   r L 0 0.68 = 2.58 ( r r 0 ) 5 / 3 [ 1 1.100 ( r L 0 ) 1 / 3 + 0.233 ( r L 0 ) 7 / 3 ] .
D ( r ) = 24 π 2 0.033 C n 2 k 2 Z 5 K 0 5 / 3 [ 5 3 0 1 ( K 0 r ξ ) 5 / 3 × a K 0 r ξ 1 J 0 ( u ) ( u 2 + K 0 2 r 2 ξ 2 ) 11 / 6 u d u d ξ ] ,
D * ( r L 0 ) = 5 3 0 1 ( K 0 r ξ ) 5 / 3 [ 0 1 J 0 ( u ) ( u 2 + K 0 2 r 2 ) 11 / 6 u d u 0 a K 0 r 1 J 0 ( u ) ( u 2 + K 0 2 r 2 ) 11 / 6 u d u ] d ξ .
D 3 * ( r L 0 ) = 14.95 ( r L 0 ) 5 / 3 [ 1 1.32 ( r L 0 ) 1 / 3 + 0.39 ( r L 0 ) 4 / 3 ] ,
D ( r ) = 0.073 C n 2 k 2 Z L 0 5 / 3 D s p h , v K * ( r L 0 ) ( 0.073 ,  0.041 ) C n 2 k 2 Z L 0 5 / 3 for   r L 0 > ( 0.97 ,  0.63 ) 2.58 ( r r 0 ) 5 / 3 [ 1 1.32 ( r L 0 ) 1 / 3 + 0.39 ( r L 0 ) 4 / 3 ] for   r L 0 ( 0.97 ,  0.63 ) .
D n ( r ) = 8 π × 0.033 C n 2 K 0 1 sin ( K r ) / ( K r ) K 11 / 3 K 2 d K
= 12 π × 0.033 C n 2 K 0 2 / 3 [ 2 3 ( K 0 r ) 2 / 3 K 0 r 1 sin   u / u u 5 / 3 d u ]
= 0.365 C n 2 L 0 2 / 3 D n * ( r L 0 ) = 0.86 ( L 0 r 0 ) 2 / 3 D n * ( r L 0 ) .
D n * ( r L 0 ) 2 3 ( K 0 r ) 2 / 3 [ 0 1 sin u / u u 5 / 3 d u 0 K 0 r ( u 1 / 3 6 u 7 / 3 120 ) d u ] = 2 3 ( K 0 r ) 2 / 3 [ 1.21 ( K 0 r ) 4 / 3 8 + ( K 0 r ) 10 / 3 400 ] = 2.75 ( r L 0 ) 2 / 3 [ 1 1.20 ( r L 0 ) 4 / 3 + 0.95 ( r L 0 ) 10 / 3 ] ,
D n ( r ) = 0.365 C n 2 L 0 2 / 3 D n * ( r L 0 ) 0.365 C n 2 L 0 2 / 3   for   r L 0 > 0.35 C n 2 r 2 / 3 [ 1 1.20 ( r L 0 ) 4 / 3 + 0.95 ( r L 0 ) 10 / 3 ] for   r L 0 0.35.
D n ( r ) = 12 π × 0.033 C n 2 K 0 2 / 3 ×[ 2 3 ( K 0 r ) 2 / 3 a K 0 r 1 sin   u / u ( u 2 + K 0 2 r 2 ) 11 / 6 u 2 d u ] = 12 π × 0.033 C n 2 K 0 2 / 3 { 2 3 ( K 0 r ) 2 / 3 ×[ 0 1 sin   u / u ( u 2 + K 0 2 r 2 ) 11 / 6 u 2 d u 0 a K 0 r 1 sin   u / u ( u 2 + K 0 2 r 2 ) 11 / 6 u 2 d u ] } .
D n * 3 ( r L 0 ) = 2.75 ( r L 0 ) 2 / 3 [ 1 4.55 ( r L 0 ) 4 / 3 + 5.9 ( r L 0 ) 2 ] ,
D n * ( ) = 2 3 a v 2 d v ( v 2 + 1 ) 11 / 6 = 0 ( a 2 + 1 ) 1 / 3 1 w 3 d w 0 ( a 2 + 1 ) 1 / 3 ( 1 w 3 2 ) d w = ( 0.875 ,  0.744 ) .
D n ( r ) = 0.365 C n * L 0 2 / 3 D n , v K * ( r L 0 ) ( 0.31 ,  0.27 ) C n 2 L 0 2 / 3 for   r L 0 > ( 0.36 ,  0.28 ) C n 2 r 2 / 3 [ 1 4.55 ( r L 0 ) 4 / 3 + 5.9 ( r L 0 ) 2 ] for   r L 0 ( 0.36 ,  0.28 ) .
0 u p [ 1 J 0 ( u ) ] d u = [ u p + 1 p + 1 [ 1 J 0 ( u ) ] ] 0 + 1 p 1 0 u p + 1 J 1 ( u ) d u
= 0 + 1 ( p 1 ) Γ ( 1 ( p 1 ) / 2 ) 2 p 1 Γ ( 1 + ( p 1 ) / 2 )
= Γ ( 1 / 6 ) ( 5 / 3 ) 2 5 / 3 Γ ( 11 / 6 ) = 1.1183 ,
0 K 0 r u 2 / 4 u 4 / 64 u 8 / 3 d u = [ 3 4 u 1 / 3 3 7 × 64 u 7 / 3 ] u = 0 u = K 0 r = 3 4 ( K 0 r ) 1 / 3 3 448 ( K 0 r ) 7 / 3 .
  K ν ( x ) = 1 2 n = 0 ( 1 ) n n ! [ Γ ( n ν ) ( x 2 ) 2 n + ν + Γ ( n + ν ) ( x 2 ) 2 n ν ] 1 2 [ Γ ( 5 / 6 ) ( x 2 ) 5 / 6 + Γ ( 5 / 6 ) ( x 2 ) 5 / 6 Γ ( 11 / 6 ) ( x 2 ) 17 / 6 Γ ( 1 / 6 ) ( x 2 ) 7 / 6 ] .
0 1 J 0 ( u ) ( u 2 + K 0 2 r 2 ) 11 / 6 u d u = [ 3 5 ( u 2 + K 0 2 r 2 ) 5 / 6 ] 0 0 J 0 ( u ) ( u 2 + K 0 2 r 2 ) 11 / 6 u d u = 3 5 ( K 0 r ) 5 / 3 K 5 / 6 ( K 0 r ) ( 2 K 0 r ) 5 / 6 Γ ( 11 / 6 ) 1.118 1.660 ( r L 0 ) 1 / 3 + 6.020 ( r L 0 ) 2 = 1.118 [ 1 1.485 ( r L 0 ) 1 / 3 + 5.385 ( r L 0 ) 2 ] ,
0 a K 0 r u 2 / 4 ( u 2 + K 0 2 r 2 ) 11 / 6 u d u = 1 8 0 ( a K 0 r ) 2 u 2 ( u 2 + K 0 2 r 2 ) 11 / 6 d ( u 2 ) = 1 8 [ 6 ( u 2 + K 0 2 r 2 ) 1 / 6 + 6 5 K 0 2 r 2 ( u 2 + K 0 2 r 2 ) 5 / 6 ] u 2 = 0 u 2 = ( a K 0 r ) 2 = ( K 0 r ) 1 / 3 1 8 [ 6 × ( a 2 + 1 ) 1 / 6 + 6 5 × ( a 2 + 1 ) 5 / 6 6 6 5 ] = ( 0 ,  0.048 ) ( r L 0 ) 1 / 3 ,
D * ( r L 0 ) = 1 5 3 Γ ( 11 / 6 ) ( K 0 r 2 ) 5 / 6 K 5 / 6 ( K 0 r ) .
0 u 8 / 3 ( u sin   u ) d u = [ 3 5 u 5 / 3 ( u sin   u ) ] 0 + 3 5 [ 3 2 u 2 / 3 ( 1 cos   u ) ] 0 + 3 5 3 2 0 u 2 / 3   sin   u d u = 0 + 0 + 0.9 Γ ( 1 / 3 ) sin ( π / 6 ) 1.21.
0 1 sin   u / u ( u 2 + K 0 2 r 2 ) 11 / 6 u 2 d u = 0 u 2 ( u 2 + K 0 2 r 2 ) 11 / 6 d u 0 u   sin   u ( u 2 + K 0 2 r 2 ) 11 / 6 d u = Γ ( 3 / 2 ) Γ ( 1 / 3 ) 2 Γ ( 11 / 6 ) ( K 0 r ) 2 / 3 cos ( 4 π / 3 ) Γ ( 5 / 6 ) 2 4 / 3 π × ( K 0 r ) 1 / 3 K 1 / 3 ( K 0 r ) 1.2055 0.4733 ( K 0 r ) 4 / 3 + 0.2260 ( K 0 r ) 2 = 1.21 [ 1 4.55 ( r L 0 ) 4 / 3 + 7.40 ( r L 0 ) 2 ] ,
D n ( r L 0 ) = C n 2 K 0 2 / 3 [ 1.0468 0.6203 ( K 0 r ) 1 / 3 K 1 / 3 ( K 0 r ) ] .

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