Abstract

Under the Geometrics Optics approximation is possible to estimate the covariance between the displacements of two thin beams after they have propagated through a turbulent medium. Previous works have concentrated in long propagation distances to provide models for the wandering statistics. These models are useful when the separation between beams is smaller than the propagation path—regardless of the characteristics scales of the turbulence. In this work we give a complete model for these covariances, behavior introducing absolute limits to the validity of former approximations. Moreover, these generalizations are established for non-Kolmogorov atmospheric models.

© 2012 OSA

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  1. J. Zhang and Z. Zeng, “Statistical properties of optical turbulence in a convective tank: experimental results,” J. Opt. A: Pure Appl. Opt. 3, 236–241 (2001).
    [Crossref]
  2. C. Innocenti and A. Consortini, “Refractive index gradient of the atmosphere at near ground levels,” J. Mod. Optics 52, 671–689 (2005).
    [Crossref]
  3. J. H. Churnside and R. J. Lataitis, “Angle-of-arrival fluctuations of a reflected beam in atmospheric turbulence,” J. Opt. Soc. Am. A 4, 1264–1272 (1987).
    [Crossref]
  4. E. Masciadri and J. Vernin, “Optical technique for inner-scale measurement: possible astronomical applications,” Appl. Opt. 36, 1320–1327 (1997).
    [Crossref] [PubMed]
  5. M. S. Andreeva, A. V. Koryabin, V. A. Kulikov, and V. I. Shmalhausen, “Diagnostics of the scale of turbulence using a divergent laser beam,” Moscow Univ. Phys. Bull. 66, 627–630 (2011).
    [Crossref]
  6. M. S. Andreeva, N. G. Iroshnikov, A. B. Koryabin, A. V. Larichev, and V. I. Shmalgauzen, “Usage of wavefront sensor for estimation of atmospheric turbulence parameters,” Optoelectronics, Instrumentation and Data Processing 48, 197–204 (2012).
    [Crossref]
  7. A. Consortini and K. O’Donnell, “Beam wandering of thin parallel beams through atmospheric turbulence,” Wave. Random Media 3, S11–S28 (1991).
    [Crossref]
  8. A. Consortini and K. O’Donnell, “Measuring the inner-scale of atmospheric turbulence by correlation of lateral displacements of thin parallel laser beams,” Wave. Random Media 3, S11–S28 (1991).
    [Crossref]
  9. A. Consortini, C. Innocenti, and G. Paoli, “Estimate method for outer scale of atmospheric turbulence,” Opt. Comm. 214, 9–14 (2002).
    [Crossref]
  10. Y. Y. Sun, A. Consortini, and Z. P. Li, “A new method for measuring the outer scale of atmospheric turbulence,” Wave. Random Complex 17, 1–8 (2007).
    [Crossref]
  11. E. Golbraikh and N. S. Kopeika, “Behavior of structure function of refraction coefficients in different turbulent field,” Appl. Opt. 43(33), 6151–6156 (2004).
    [Crossref] [PubMed]
  12. E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Processes Geophys. 13, 297–301 (2006).
    [Crossref]
  13. G. D. Boreman and C. Dainty, “Zernike expansions for non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 13, 517–522 (1996).
    [Crossref]
  14. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
    [Crossref]
  15. C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wavefront sensors for non-Kolmogorov turbulence,” Opt. Eng. 41(2), 534–541 (2002).
    [Crossref]
  16. P. F. Lazorenko, “Differential image motion at non-Kolmogorov distortions of the turbulent wave-front,” Astron. Astrophys. 382, 1125–1137 (2002).
    [Crossref]
  17. P. F. Lazorenko, “Non-Kolmogorov features of differential image motion restored from the Multichannel Astrometric Photometer data,” Astron. Astrophys. 396, 353–360 (2002).
    [Crossref]
  18. E. Golbraikh and N. S. Kopeika, “Turbulence strength parameter in laboratory and natural optical experiments in non-Kolmogorov cases,” Opt. Commun. 242, 333–338 (2004).
    [Crossref]
  19. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283, 1229–1235 (2010).
    [Crossref]
  20. L. Zunino, D. G. Pérez, O. A. Rosso, and M. Garavaglia, “Characterization of Laser Propagation Through Turbulent Media by Quantifiers Based on the Wavelet Transform,” Fractals 12, 223–233 (2004).
    [Crossref]
  21. D. G. Pérez, L. Zunino, M. Garavaglia, and D. G. Pérez, “A fractional Brownian motion model for the turbulent refractive index in lightwave propagation,” Opt. Commun. 242, 57–63 (2004).
    [Crossref]
  22. D. G. Pérez, L. Zunino, and M. Garavaglia, “Modeling the turbulent wave-front phase as a fractional Brownian motion: a new approach,” J. Opt. Soc. Am. A 21, 1962–1969 (2004).
    [Crossref]
  23. D. G. Pérez and L. Zunino, “Generalized wave-front phase for non-Kolmogorov turbulence,” Opt. Lett. 33, 572–574 (2008).
    [Crossref] [PubMed]
  24. L. Zunino, D. G. Pérez, M. Garavaglia, O. A. Rosso, and D. G. Pérez, “Characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform: dynamic study,” Physica A 364, 79–86 (2006).
    [Crossref]
  25. G. Funes, D. D. Gulich, L. Zunino, D. G. Pérez, M. Garavaglia, and D. G. Pérez, “Behavior of the laser beam wandering variance with the turbulent path length,” Opt. Commun. 272, 476–479 (2007).
    [Crossref]
  26. D. D. Gulich, G. Funes, L. Zunino, D. G. Pérez, and M. Garavaglia, “Angle-of-arrival variance’s dependence on the aperture size for indoor convective turbulence,” Opt. Commun. 277, 241–246 (2007).
    [Crossref]
  27. P. Beckman, “Signal Degeneration in Laser Beams Propagated Through a Turbulent atmosphere,” Radio Sci. J. Res. (NBS/USNC-URSI) 69D, 629–640 (1965).
  28. Equations (1) and (2) are obtained in [7] from approximating the beam displacements through Geometric Optics. Since there is a linear relationship between these displacements and the refractive index perturbation, through integrals and derivatives, their covariances are functionals of it. This is true regardless of the model employed to evaluate the covariance of the turbulent refractive index.
  29. V. I. Tatarskĭ, Wave Propagation in a Turbulent Atmosphere (Nauka Press, Moscow, 1967).
  30. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  31. C. Innocenti and A. Consortini, “Estimate of characteristic scales of atmospheric turbulence by thin beams: Comparison between the von Karman and Hill-Andrews models,” J. Mod. Optics 51(3), 333–342 (2004).
    [Crossref]
  32. D. G. Pérez, A. Férnandez, G. Funes, D. Gulich, and L. Zunino, “Retrieving atmospheric turbulence features from differential laser tracking motion data,” Proc. SPIE 8535, 853508 (2012).
    [Crossref]

2012 (2)

M. S. Andreeva, N. G. Iroshnikov, A. B. Koryabin, A. V. Larichev, and V. I. Shmalgauzen, “Usage of wavefront sensor for estimation of atmospheric turbulence parameters,” Optoelectronics, Instrumentation and Data Processing 48, 197–204 (2012).
[Crossref]

D. G. Pérez, A. Férnandez, G. Funes, D. Gulich, and L. Zunino, “Retrieving atmospheric turbulence features from differential laser tracking motion data,” Proc. SPIE 8535, 853508 (2012).
[Crossref]

2011 (1)

M. S. Andreeva, A. V. Koryabin, V. A. Kulikov, and V. I. Shmalhausen, “Diagnostics of the scale of turbulence using a divergent laser beam,” Moscow Univ. Phys. Bull. 66, 627–630 (2011).
[Crossref]

2010 (1)

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283, 1229–1235 (2010).
[Crossref]

2008 (1)

2007 (4)

G. Funes, D. D. Gulich, L. Zunino, D. G. Pérez, M. Garavaglia, and D. G. Pérez, “Behavior of the laser beam wandering variance with the turbulent path length,” Opt. Commun. 272, 476–479 (2007).
[Crossref]

D. D. Gulich, G. Funes, L. Zunino, D. G. Pérez, and M. Garavaglia, “Angle-of-arrival variance’s dependence on the aperture size for indoor convective turbulence,” Opt. Commun. 277, 241–246 (2007).
[Crossref]

Y. Y. Sun, A. Consortini, and Z. P. Li, “A new method for measuring the outer scale of atmospheric turbulence,” Wave. Random Complex 17, 1–8 (2007).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

2006 (2)

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Processes Geophys. 13, 297–301 (2006).
[Crossref]

L. Zunino, D. G. Pérez, M. Garavaglia, O. A. Rosso, and D. G. Pérez, “Characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform: dynamic study,” Physica A 364, 79–86 (2006).
[Crossref]

2005 (1)

C. Innocenti and A. Consortini, “Refractive index gradient of the atmosphere at near ground levels,” J. Mod. Optics 52, 671–689 (2005).
[Crossref]

2004 (6)

E. Golbraikh and N. S. Kopeika, “Behavior of structure function of refraction coefficients in different turbulent field,” Appl. Opt. 43(33), 6151–6156 (2004).
[Crossref] [PubMed]

L. Zunino, D. G. Pérez, O. A. Rosso, and M. Garavaglia, “Characterization of Laser Propagation Through Turbulent Media by Quantifiers Based on the Wavelet Transform,” Fractals 12, 223–233 (2004).
[Crossref]

D. G. Pérez, L. Zunino, M. Garavaglia, and D. G. Pérez, “A fractional Brownian motion model for the turbulent refractive index in lightwave propagation,” Opt. Commun. 242, 57–63 (2004).
[Crossref]

D. G. Pérez, L. Zunino, and M. Garavaglia, “Modeling the turbulent wave-front phase as a fractional Brownian motion: a new approach,” J. Opt. Soc. Am. A 21, 1962–1969 (2004).
[Crossref]

E. Golbraikh and N. S. Kopeika, “Turbulence strength parameter in laboratory and natural optical experiments in non-Kolmogorov cases,” Opt. Commun. 242, 333–338 (2004).
[Crossref]

C. Innocenti and A. Consortini, “Estimate of characteristic scales of atmospheric turbulence by thin beams: Comparison between the von Karman and Hill-Andrews models,” J. Mod. Optics 51(3), 333–342 (2004).
[Crossref]

2002 (4)

C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wavefront sensors for non-Kolmogorov turbulence,” Opt. Eng. 41(2), 534–541 (2002).
[Crossref]

P. F. Lazorenko, “Differential image motion at non-Kolmogorov distortions of the turbulent wave-front,” Astron. Astrophys. 382, 1125–1137 (2002).
[Crossref]

P. F. Lazorenko, “Non-Kolmogorov features of differential image motion restored from the Multichannel Astrometric Photometer data,” Astron. Astrophys. 396, 353–360 (2002).
[Crossref]

A. Consortini, C. Innocenti, and G. Paoli, “Estimate method for outer scale of atmospheric turbulence,” Opt. Comm. 214, 9–14 (2002).
[Crossref]

2001 (1)

J. Zhang and Z. Zeng, “Statistical properties of optical turbulence in a convective tank: experimental results,” J. Opt. A: Pure Appl. Opt. 3, 236–241 (2001).
[Crossref]

1997 (1)

1996 (1)

1991 (2)

A. Consortini and K. O’Donnell, “Beam wandering of thin parallel beams through atmospheric turbulence,” Wave. Random Media 3, S11–S28 (1991).
[Crossref]

A. Consortini and K. O’Donnell, “Measuring the inner-scale of atmospheric turbulence by correlation of lateral displacements of thin parallel laser beams,” Wave. Random Media 3, S11–S28 (1991).
[Crossref]

1987 (1)

1965 (1)

P. Beckman, “Signal Degeneration in Laser Beams Propagated Through a Turbulent atmosphere,” Radio Sci. J. Res. (NBS/USNC-URSI) 69D, 629–640 (1965).

Andreeva, M. S.

M. S. Andreeva, N. G. Iroshnikov, A. B. Koryabin, A. V. Larichev, and V. I. Shmalgauzen, “Usage of wavefront sensor for estimation of atmospheric turbulence parameters,” Optoelectronics, Instrumentation and Data Processing 48, 197–204 (2012).
[Crossref]

M. S. Andreeva, A. V. Koryabin, V. A. Kulikov, and V. I. Shmalhausen, “Diagnostics of the scale of turbulence using a divergent laser beam,” Moscow Univ. Phys. Bull. 66, 627–630 (2011).
[Crossref]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Beckman, P.

P. Beckman, “Signal Degeneration in Laser Beams Propagated Through a Turbulent atmosphere,” Radio Sci. J. Res. (NBS/USNC-URSI) 69D, 629–640 (1965).

Boreman, G. D.

Branover, H.

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Processes Geophys. 13, 297–301 (2006).
[Crossref]

Churnside, J. H.

Consortini, A.

Y. Y. Sun, A. Consortini, and Z. P. Li, “A new method for measuring the outer scale of atmospheric turbulence,” Wave. Random Complex 17, 1–8 (2007).
[Crossref]

C. Innocenti and A. Consortini, “Refractive index gradient of the atmosphere at near ground levels,” J. Mod. Optics 52, 671–689 (2005).
[Crossref]

C. Innocenti and A. Consortini, “Estimate of characteristic scales of atmospheric turbulence by thin beams: Comparison between the von Karman and Hill-Andrews models,” J. Mod. Optics 51(3), 333–342 (2004).
[Crossref]

A. Consortini, C. Innocenti, and G. Paoli, “Estimate method for outer scale of atmospheric turbulence,” Opt. Comm. 214, 9–14 (2002).
[Crossref]

A. Consortini and K. O’Donnell, “Beam wandering of thin parallel beams through atmospheric turbulence,” Wave. Random Media 3, S11–S28 (1991).
[Crossref]

A. Consortini and K. O’Donnell, “Measuring the inner-scale of atmospheric turbulence by correlation of lateral displacements of thin parallel laser beams,” Wave. Random Media 3, S11–S28 (1991).
[Crossref]

Dainty, C.

Férnandez, A.

D. G. Pérez, A. Férnandez, G. Funes, D. Gulich, and L. Zunino, “Retrieving atmospheric turbulence features from differential laser tracking motion data,” Proc. SPIE 8535, 853508 (2012).
[Crossref]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Funes, G.

D. G. Pérez, A. Férnandez, G. Funes, D. Gulich, and L. Zunino, “Retrieving atmospheric turbulence features from differential laser tracking motion data,” Proc. SPIE 8535, 853508 (2012).
[Crossref]

G. Funes, D. D. Gulich, L. Zunino, D. G. Pérez, M. Garavaglia, and D. G. Pérez, “Behavior of the laser beam wandering variance with the turbulent path length,” Opt. Commun. 272, 476–479 (2007).
[Crossref]

D. D. Gulich, G. Funes, L. Zunino, D. G. Pérez, and M. Garavaglia, “Angle-of-arrival variance’s dependence on the aperture size for indoor convective turbulence,” Opt. Commun. 277, 241–246 (2007).
[Crossref]

Garavaglia, M.

G. Funes, D. D. Gulich, L. Zunino, D. G. Pérez, M. Garavaglia, and D. G. Pérez, “Behavior of the laser beam wandering variance with the turbulent path length,” Opt. Commun. 272, 476–479 (2007).
[Crossref]

D. D. Gulich, G. Funes, L. Zunino, D. G. Pérez, and M. Garavaglia, “Angle-of-arrival variance’s dependence on the aperture size for indoor convective turbulence,” Opt. Commun. 277, 241–246 (2007).
[Crossref]

L. Zunino, D. G. Pérez, M. Garavaglia, O. A. Rosso, and D. G. Pérez, “Characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform: dynamic study,” Physica A 364, 79–86 (2006).
[Crossref]

L. Zunino, D. G. Pérez, O. A. Rosso, and M. Garavaglia, “Characterization of Laser Propagation Through Turbulent Media by Quantifiers Based on the Wavelet Transform,” Fractals 12, 223–233 (2004).
[Crossref]

D. G. Pérez, L. Zunino, M. Garavaglia, and D. G. Pérez, “A fractional Brownian motion model for the turbulent refractive index in lightwave propagation,” Opt. Commun. 242, 57–63 (2004).
[Crossref]

D. G. Pérez, L. Zunino, and M. Garavaglia, “Modeling the turbulent wave-front phase as a fractional Brownian motion: a new approach,” J. Opt. Soc. Am. A 21, 1962–1969 (2004).
[Crossref]

Golbraikh, E.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283, 1229–1235 (2010).
[Crossref]

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Processes Geophys. 13, 297–301 (2006).
[Crossref]

E. Golbraikh and N. S. Kopeika, “Behavior of structure function of refraction coefficients in different turbulent field,” Appl. Opt. 43(33), 6151–6156 (2004).
[Crossref] [PubMed]

E. Golbraikh and N. S. Kopeika, “Turbulence strength parameter in laboratory and natural optical experiments in non-Kolmogorov cases,” Opt. Commun. 242, 333–338 (2004).
[Crossref]

Gulich, D.

D. G. Pérez, A. Férnandez, G. Funes, D. Gulich, and L. Zunino, “Retrieving atmospheric turbulence features from differential laser tracking motion data,” Proc. SPIE 8535, 853508 (2012).
[Crossref]

Gulich, D. D.

G. Funes, D. D. Gulich, L. Zunino, D. G. Pérez, M. Garavaglia, and D. G. Pérez, “Behavior of the laser beam wandering variance with the turbulent path length,” Opt. Commun. 272, 476–479 (2007).
[Crossref]

D. D. Gulich, G. Funes, L. Zunino, D. G. Pérez, and M. Garavaglia, “Angle-of-arrival variance’s dependence on the aperture size for indoor convective turbulence,” Opt. Commun. 277, 241–246 (2007).
[Crossref]

Innocenti, C.

C. Innocenti and A. Consortini, “Refractive index gradient of the atmosphere at near ground levels,” J. Mod. Optics 52, 671–689 (2005).
[Crossref]

C. Innocenti and A. Consortini, “Estimate of characteristic scales of atmospheric turbulence by thin beams: Comparison between the von Karman and Hill-Andrews models,” J. Mod. Optics 51(3), 333–342 (2004).
[Crossref]

A. Consortini, C. Innocenti, and G. Paoli, “Estimate method for outer scale of atmospheric turbulence,” Opt. Comm. 214, 9–14 (2002).
[Crossref]

Iroshnikov, N. G.

M. S. Andreeva, N. G. Iroshnikov, A. B. Koryabin, A. V. Larichev, and V. I. Shmalgauzen, “Usage of wavefront sensor for estimation of atmospheric turbulence parameters,” Optoelectronics, Instrumentation and Data Processing 48, 197–204 (2012).
[Crossref]

Jiang, W.

C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wavefront sensors for non-Kolmogorov turbulence,” Opt. Eng. 41(2), 534–541 (2002).
[Crossref]

Kopeika, N. S.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283, 1229–1235 (2010).
[Crossref]

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Processes Geophys. 13, 297–301 (2006).
[Crossref]

E. Golbraikh and N. S. Kopeika, “Behavior of structure function of refraction coefficients in different turbulent field,” Appl. Opt. 43(33), 6151–6156 (2004).
[Crossref] [PubMed]

E. Golbraikh and N. S. Kopeika, “Turbulence strength parameter in laboratory and natural optical experiments in non-Kolmogorov cases,” Opt. Commun. 242, 333–338 (2004).
[Crossref]

Koryabin, A. B.

M. S. Andreeva, N. G. Iroshnikov, A. B. Koryabin, A. V. Larichev, and V. I. Shmalgauzen, “Usage of wavefront sensor for estimation of atmospheric turbulence parameters,” Optoelectronics, Instrumentation and Data Processing 48, 197–204 (2012).
[Crossref]

Koryabin, A. V.

M. S. Andreeva, A. V. Koryabin, V. A. Kulikov, and V. I. Shmalhausen, “Diagnostics of the scale of turbulence using a divergent laser beam,” Moscow Univ. Phys. Bull. 66, 627–630 (2011).
[Crossref]

Kulikov, V. A.

M. S. Andreeva, A. V. Koryabin, V. A. Kulikov, and V. I. Shmalhausen, “Diagnostics of the scale of turbulence using a divergent laser beam,” Moscow Univ. Phys. Bull. 66, 627–630 (2011).
[Crossref]

Larichev, A. V.

M. S. Andreeva, N. G. Iroshnikov, A. B. Koryabin, A. V. Larichev, and V. I. Shmalgauzen, “Usage of wavefront sensor for estimation of atmospheric turbulence parameters,” Optoelectronics, Instrumentation and Data Processing 48, 197–204 (2012).
[Crossref]

Lataitis, R. J.

Lazorenko, P. F.

P. F. Lazorenko, “Differential image motion at non-Kolmogorov distortions of the turbulent wave-front,” Astron. Astrophys. 382, 1125–1137 (2002).
[Crossref]

P. F. Lazorenko, “Non-Kolmogorov features of differential image motion restored from the Multichannel Astrometric Photometer data,” Astron. Astrophys. 396, 353–360 (2002).
[Crossref]

Li, Z. P.

Y. Y. Sun, A. Consortini, and Z. P. Li, “A new method for measuring the outer scale of atmospheric turbulence,” Wave. Random Complex 17, 1–8 (2007).
[Crossref]

Ling, N.

C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wavefront sensors for non-Kolmogorov turbulence,” Opt. Eng. 41(2), 534–541 (2002).
[Crossref]

Masciadri, E.

O’Donnell, K.

A. Consortini and K. O’Donnell, “Measuring the inner-scale of atmospheric turbulence by correlation of lateral displacements of thin parallel laser beams,” Wave. Random Media 3, S11–S28 (1991).
[Crossref]

A. Consortini and K. O’Donnell, “Beam wandering of thin parallel beams through atmospheric turbulence,” Wave. Random Media 3, S11–S28 (1991).
[Crossref]

Paoli, G.

A. Consortini, C. Innocenti, and G. Paoli, “Estimate method for outer scale of atmospheric turbulence,” Opt. Comm. 214, 9–14 (2002).
[Crossref]

Pérez, D. G.

D. G. Pérez, A. Férnandez, G. Funes, D. Gulich, and L. Zunino, “Retrieving atmospheric turbulence features from differential laser tracking motion data,” Proc. SPIE 8535, 853508 (2012).
[Crossref]

D. G. Pérez and L. Zunino, “Generalized wave-front phase for non-Kolmogorov turbulence,” Opt. Lett. 33, 572–574 (2008).
[Crossref] [PubMed]

D. D. Gulich, G. Funes, L. Zunino, D. G. Pérez, and M. Garavaglia, “Angle-of-arrival variance’s dependence on the aperture size for indoor convective turbulence,” Opt. Commun. 277, 241–246 (2007).
[Crossref]

G. Funes, D. D. Gulich, L. Zunino, D. G. Pérez, M. Garavaglia, and D. G. Pérez, “Behavior of the laser beam wandering variance with the turbulent path length,” Opt. Commun. 272, 476–479 (2007).
[Crossref]

G. Funes, D. D. Gulich, L. Zunino, D. G. Pérez, M. Garavaglia, and D. G. Pérez, “Behavior of the laser beam wandering variance with the turbulent path length,” Opt. Commun. 272, 476–479 (2007).
[Crossref]

L. Zunino, D. G. Pérez, M. Garavaglia, O. A. Rosso, and D. G. Pérez, “Characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform: dynamic study,” Physica A 364, 79–86 (2006).
[Crossref]

L. Zunino, D. G. Pérez, M. Garavaglia, O. A. Rosso, and D. G. Pérez, “Characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform: dynamic study,” Physica A 364, 79–86 (2006).
[Crossref]

D. G. Pérez, L. Zunino, M. Garavaglia, and D. G. Pérez, “A fractional Brownian motion model for the turbulent refractive index in lightwave propagation,” Opt. Commun. 242, 57–63 (2004).
[Crossref]

D. G. Pérez, L. Zunino, M. Garavaglia, and D. G. Pérez, “A fractional Brownian motion model for the turbulent refractive index in lightwave propagation,” Opt. Commun. 242, 57–63 (2004).
[Crossref]

L. Zunino, D. G. Pérez, O. A. Rosso, and M. Garavaglia, “Characterization of Laser Propagation Through Turbulent Media by Quantifiers Based on the Wavelet Transform,” Fractals 12, 223–233 (2004).
[Crossref]

D. G. Pérez, L. Zunino, and M. Garavaglia, “Modeling the turbulent wave-front phase as a fractional Brownian motion: a new approach,” J. Opt. Soc. Am. A 21, 1962–1969 (2004).
[Crossref]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Rao, C.

C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wavefront sensors for non-Kolmogorov turbulence,” Opt. Eng. 41(2), 534–541 (2002).
[Crossref]

Rosso, O. A.

L. Zunino, D. G. Pérez, M. Garavaglia, O. A. Rosso, and D. G. Pérez, “Characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform: dynamic study,” Physica A 364, 79–86 (2006).
[Crossref]

L. Zunino, D. G. Pérez, O. A. Rosso, and M. Garavaglia, “Characterization of Laser Propagation Through Turbulent Media by Quantifiers Based on the Wavelet Transform,” Fractals 12, 223–233 (2004).
[Crossref]

Shmalgauzen, V. I.

M. S. Andreeva, N. G. Iroshnikov, A. B. Koryabin, A. V. Larichev, and V. I. Shmalgauzen, “Usage of wavefront sensor for estimation of atmospheric turbulence parameters,” Optoelectronics, Instrumentation and Data Processing 48, 197–204 (2012).
[Crossref]

Shmalhausen, V. I.

M. S. Andreeva, A. V. Koryabin, V. A. Kulikov, and V. I. Shmalhausen, “Diagnostics of the scale of turbulence using a divergent laser beam,” Moscow Univ. Phys. Bull. 66, 627–630 (2011).
[Crossref]

Sun, Y. Y.

Y. Y. Sun, A. Consortini, and Z. P. Li, “A new method for measuring the outer scale of atmospheric turbulence,” Wave. Random Complex 17, 1–8 (2007).
[Crossref]

Tatarski, V. I.

V. I. Tatarskĭ, Wave Propagation in a Turbulent Atmosphere (Nauka Press, Moscow, 1967).

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Vernin, J.

Zeng, Z.

J. Zhang and Z. Zeng, “Statistical properties of optical turbulence in a convective tank: experimental results,” J. Opt. A: Pure Appl. Opt. 3, 236–241 (2001).
[Crossref]

Zhang, J.

J. Zhang and Z. Zeng, “Statistical properties of optical turbulence in a convective tank: experimental results,” J. Opt. A: Pure Appl. Opt. 3, 236–241 (2001).
[Crossref]

Zilberman, A.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283, 1229–1235 (2010).
[Crossref]

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Processes Geophys. 13, 297–301 (2006).
[Crossref]

Zunino, L.

D. G. Pérez, A. Férnandez, G. Funes, D. Gulich, and L. Zunino, “Retrieving atmospheric turbulence features from differential laser tracking motion data,” Proc. SPIE 8535, 853508 (2012).
[Crossref]

D. G. Pérez and L. Zunino, “Generalized wave-front phase for non-Kolmogorov turbulence,” Opt. Lett. 33, 572–574 (2008).
[Crossref] [PubMed]

G. Funes, D. D. Gulich, L. Zunino, D. G. Pérez, M. Garavaglia, and D. G. Pérez, “Behavior of the laser beam wandering variance with the turbulent path length,” Opt. Commun. 272, 476–479 (2007).
[Crossref]

D. D. Gulich, G. Funes, L. Zunino, D. G. Pérez, and M. Garavaglia, “Angle-of-arrival variance’s dependence on the aperture size for indoor convective turbulence,” Opt. Commun. 277, 241–246 (2007).
[Crossref]

L. Zunino, D. G. Pérez, M. Garavaglia, O. A. Rosso, and D. G. Pérez, “Characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform: dynamic study,” Physica A 364, 79–86 (2006).
[Crossref]

D. G. Pérez, L. Zunino, M. Garavaglia, and D. G. Pérez, “A fractional Brownian motion model for the turbulent refractive index in lightwave propagation,” Opt. Commun. 242, 57–63 (2004).
[Crossref]

L. Zunino, D. G. Pérez, O. A. Rosso, and M. Garavaglia, “Characterization of Laser Propagation Through Turbulent Media by Quantifiers Based on the Wavelet Transform,” Fractals 12, 223–233 (2004).
[Crossref]

D. G. Pérez, L. Zunino, and M. Garavaglia, “Modeling the turbulent wave-front phase as a fractional Brownian motion: a new approach,” J. Opt. Soc. Am. A 21, 1962–1969 (2004).
[Crossref]

Appl. Opt. (2)

Astron. Astrophys. (2)

P. F. Lazorenko, “Differential image motion at non-Kolmogorov distortions of the turbulent wave-front,” Astron. Astrophys. 382, 1125–1137 (2002).
[Crossref]

P. F. Lazorenko, “Non-Kolmogorov features of differential image motion restored from the Multichannel Astrometric Photometer data,” Astron. Astrophys. 396, 353–360 (2002).
[Crossref]

Fractals (1)

L. Zunino, D. G. Pérez, O. A. Rosso, and M. Garavaglia, “Characterization of Laser Propagation Through Turbulent Media by Quantifiers Based on the Wavelet Transform,” Fractals 12, 223–233 (2004).
[Crossref]

J. Mod. Optics (2)

C. Innocenti and A. Consortini, “Estimate of characteristic scales of atmospheric turbulence by thin beams: Comparison between the von Karman and Hill-Andrews models,” J. Mod. Optics 51(3), 333–342 (2004).
[Crossref]

C. Innocenti and A. Consortini, “Refractive index gradient of the atmosphere at near ground levels,” J. Mod. Optics 52, 671–689 (2005).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

J. Zhang and Z. Zeng, “Statistical properties of optical turbulence in a convective tank: experimental results,” J. Opt. A: Pure Appl. Opt. 3, 236–241 (2001).
[Crossref]

J. Opt. Soc. Am. A (3)

Moscow Univ. Phys. Bull. (1)

M. S. Andreeva, A. V. Koryabin, V. A. Kulikov, and V. I. Shmalhausen, “Diagnostics of the scale of turbulence using a divergent laser beam,” Moscow Univ. Phys. Bull. 66, 627–630 (2011).
[Crossref]

Nonlin. Processes Geophys. (1)

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlin. Processes Geophys. 13, 297–301 (2006).
[Crossref]

Opt. Comm. (1)

A. Consortini, C. Innocenti, and G. Paoli, “Estimate method for outer scale of atmospheric turbulence,” Opt. Comm. 214, 9–14 (2002).
[Crossref]

Opt. Commun. (5)

E. Golbraikh and N. S. Kopeika, “Turbulence strength parameter in laboratory and natural optical experiments in non-Kolmogorov cases,” Opt. Commun. 242, 333–338 (2004).
[Crossref]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283, 1229–1235 (2010).
[Crossref]

G. Funes, D. D. Gulich, L. Zunino, D. G. Pérez, M. Garavaglia, and D. G. Pérez, “Behavior of the laser beam wandering variance with the turbulent path length,” Opt. Commun. 272, 476–479 (2007).
[Crossref]

D. D. Gulich, G. Funes, L. Zunino, D. G. Pérez, and M. Garavaglia, “Angle-of-arrival variance’s dependence on the aperture size for indoor convective turbulence,” Opt. Commun. 277, 241–246 (2007).
[Crossref]

D. G. Pérez, L. Zunino, M. Garavaglia, and D. G. Pérez, “A fractional Brownian motion model for the turbulent refractive index in lightwave propagation,” Opt. Commun. 242, 57–63 (2004).
[Crossref]

Opt. Eng. (1)

C. Rao, W. Jiang, and N. Ling, “Atmospheric characterization with Shack-Hartmann wavefront sensors for non-Kolmogorov turbulence,” Opt. Eng. 41(2), 534–541 (2002).
[Crossref]

Opt. Lett. (1)

Optoelectronics, Instrumentation and Data Processing (1)

M. S. Andreeva, N. G. Iroshnikov, A. B. Koryabin, A. V. Larichev, and V. I. Shmalgauzen, “Usage of wavefront sensor for estimation of atmospheric turbulence parameters,” Optoelectronics, Instrumentation and Data Processing 48, 197–204 (2012).
[Crossref]

Physica A (1)

L. Zunino, D. G. Pérez, M. Garavaglia, O. A. Rosso, and D. G. Pérez, “Characterization of laser propagation through turbulent media by quantifiers based on the wavelet transform: dynamic study,” Physica A 364, 79–86 (2006).
[Crossref]

Proc. SPIE (2)

D. G. Pérez, A. Férnandez, G. Funes, D. Gulich, and L. Zunino, “Retrieving atmospheric turbulence features from differential laser tracking motion data,” Proc. SPIE 8535, 853508 (2012).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Radio Sci. J. Res. (NBS/USNC-URSI) (1)

P. Beckman, “Signal Degeneration in Laser Beams Propagated Through a Turbulent atmosphere,” Radio Sci. J. Res. (NBS/USNC-URSI) 69D, 629–640 (1965).

Wave. Random Complex (1)

Y. Y. Sun, A. Consortini, and Z. P. Li, “A new method for measuring the outer scale of atmospheric turbulence,” Wave. Random Complex 17, 1–8 (2007).
[Crossref]

Wave. Random Media (2)

A. Consortini and K. O’Donnell, “Beam wandering of thin parallel beams through atmospheric turbulence,” Wave. Random Media 3, S11–S28 (1991).
[Crossref]

A. Consortini and K. O’Donnell, “Measuring the inner-scale of atmospheric turbulence by correlation of lateral displacements of thin parallel laser beams,” Wave. Random Media 3, S11–S28 (1991).
[Crossref]

Other (3)

Equations (1) and (2) are obtained in [7] from approximating the beam displacements through Geometric Optics. Since there is a linear relationship between these displacements and the refractive index perturbation, through integrals and derivatives, their covariances are functionals of it. This is true regardless of the model employed to evaluate the covariance of the turbulent refractive index.

V. I. Tatarskĭ, Wave Propagation in a Turbulent Atmosphere (Nauka Press, Moscow, 1967).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Supplementary Material (4)

» Media 1: PDF (779 KB)     
» Media 2: PDF (426 KB)     
» Media 3: PDF (152 KB)     
» Media 4: PDF (183 KB)     

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Figures (11)

Fig. 1
Fig. 1

Two parallels thin beams propagating in a turbulent media. The turbulent region is confined to the zone of length S, after that the beams propagate in a region of length P without being deflected. Originally, the beams are separated a distance d, and the arrival position with respect to the original trajectory is given by (η0, ζ0) and (ηd, ζd).

Fig. 2
Fig. 2

(a) The off-plane covariance, By, is depicted for different values of H. (b) The on-plane covariance, Bx, is shown to cross the δ-axis at different points depending on the value of the Hurst exponent. (c) The covariance difference, Δ, also diverges. (d) The axis cut for the on-plane covariance as a function of the Hurt exponent.

Fig. 3
Fig. 3

(a) The normalized off-plane covariance, By, is depicted for different values of Lm. (b) The normalized on-plane covariance, Bx, is shown to cross the δ-axis at the same point, δ ≃ 0.367, regardless of the value of Lm. (c) and (d) The normalized covariance difference, Δ, approaches the origin as the adimensional propagation distance increases.

Fig. 4
Fig. 4

Fixed the adimensional distance Lm = 250: (a) the normalized off-plane covariance for different Hurst exponents becomes rapidly decreasing with lower H; (b) the normalized on-plane covariance axis cut is sensitive to H; (c) the shape of the normalized covariance difference, and its maximum, changes with decreasing H.

Fig. 5
Fig. 5

(top left) 2π/Lm versus the position of the maximum δ for the difference function; a linear fit is found for most of the range (see zoomed area). The fits giving higher slopes belong to smaller Hurst exponents. (left bottom) Fits coefficients as functions of H; p(H) = −1.4260(±0.2969)H3 −0.4147(±0.1333)H2 −1.2890(±0.0157)H + 1.8200(±0.0005) and b(H) = −0.010103(±0.004187)H3 + 0.000677(±0.001832)H2 −0.000442(±0.000209)H − 0.000133(±0.000006). (right top) The roots for the on-plane covariance Bx as a function of H. The points corresponds to a numerical evaluation of the covariance’s crossing point, then a fit is performed using a quadratic polynomial; δ0(Hi) + δ1(Hi)(2π/Lm) + δ2(Hi)(2π/Lm)2. (right bottom) Origin ordinate δ0 as a function of H obtained from the right top fit, Eq (20).

Fig. 6
Fig. 6

Fixed the adimensional distance Lm = 250, for varying scale ratios compared against the modified Tatarskĭ spectrum case: (a) the normalized off-plane covariance; (b) the normalized on-plane covariance; (c) the normalized covariance difference.

Fig. 7
Fig. 7

m = 100: (a) normalized on-plane covariance,(b) normalized off-plane covariance, and (c) normalized difference. m = 10: (d) normalized on-plane covariance, (e) normalized off-plane covariance, (f) normalized difference. m = 0.1: (g) normalized on-plane covariance, (h) normalized off-plane covariance, (i) normalized difference.

Fig. 8
Fig. 8

Fixed the adimensional distance Lm = 250 and the scale ratio q = 0.01: (a) the normalized off-plane covariance for different Hurst exponents becomes rapidly decreasing with lower H; (b) the normalized on-plane covariance axis cut is sensitive to H; (c) the shape of the normalized covariance difference, and its maximum, changes with decreasing H.

Fig. 9
Fig. 9

The plotted points represents the changes in the maximum location with respect to the inverse of the adimensional scale, 2π/Lm, and the scale ratio, q, for different values of H. The surface that fits these points follow Eq. (24) (extended figure in Media 1): p(H) = 7.645(±3.664)H2 + 1.907(±0.921)H + 3.470(±0.045), c(H) = −0.5989(±0.599)H2 −0.3902(±0.376)H +0.6082(±0.029), and m(H) = 1.39(±1.000)H2 +0.765(±0.361)H + 1.023(±0.026).

Fig. 10
Fig. 10

The plotted points represents the changes in the axis-cut location with respect to the inverse of the adimensional scale, 2π/Lm, and the scale ratio, q, for different values of H. Interpolant surface fits are used to exemplify its behaviour, a true interpolant function was not found (extended figure in Media 2).

Fig. 11
Fig. 11

Comparison between Consortini’s asymptotic approximation and our numerical approach: (a) r.m.s. error for relative separation in 0 ≤ δ ≤ 0.02 (filled markers) and 0 ≤ δ ≤ 0.4 (hollow markers) in the Tatarskĭ case ( Media 3); (b) r.m.s. error for relative separation in 0 ≤ δ ≤ 0.02 (filled markers) and 0 ≤ δ ≤ 0.45 (hollow markers) in the von Kármán case ( Media 4). The L2 norm is used as a measure of the uniform convergence (r.m.s error) in both cases.

Equations (24)

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B y ( d ) = ζ d ζ 0 = 0 L F ( z , S , P ) g n ( z , d ) d z
B x ( d ) = η d η 0 = 0 L F ( z , S , P ) f n ( z , d ) d z ,
F ( z , S , P ) = 2 ( S 3 3 + P S 2 + S P 2 ) z ( S 2 + 2 P S + 2 P 2 ) + z 3 3
f n ( z , d ) = ( 2 π ) 3 / 2 0 κ Φ n ( κ ) [ κ 3 / 2 J 3 / 2 ( κ r ) r 3 / 2 d 2 κ 5 / 2 J 5 / 2 ( κ r ) r 5 / 2 ] d κ
g n ( z , d ) = ( 2 π ) 3 / 2 0 κ Φ n ( κ ) [ κ 3 / 2 J 3 / 2 ( κ r ) r 3 / 2 ] d κ ,
B y ( d ) = L m 5 / 2 [ 0 1 g ˜ n ( u , δ ) ( 2 3 u + u 3 3 ) d u p 2 0 1 g ˜ n ( u , δ ) ( u + 2 3 p ) d u ] ,
g ˜ n ( u , δ ) = ( 2 π ) 3 / 2 0 [ κ m k Φ n ( κ m k ) ] [ k 3 / 2 J 3 / 2 ( k L m u 2 + δ 2 ) ( u 2 + δ 2 ) 3 / 4 ] d k
Δ ( d ) = B y ( d ) B x ( d ) = = δ 2 L m 7 / 2 [ 0 1 h ˜ n ( u , δ ) ( 2 3 u + u 3 3 ) d u p 2 0 1 h ˜ n ( u , δ ) ( u + 2 3 p ) d u ] ,
h ˜ n ( u , δ ) = ( 2 π ) 3 / 2 0 [ κ m k Φ n ( κ m k ) ] [ k 5 / 2 J 5 / 2 ( k L m u 2 + δ 2 ) ( u 2 + δ 2 ) 5 / 4 ] d k .
Φ n ( κ ) = sin ( π H ) Γ ( 2 H + 2 ) 4 π 2 C n 2 1 κ 2 H + 3 , with 0 < H 1 / 3 .
g n ( u , δ ) = H C n 2 L 2 H 2 ( δ 2 + u 2 ) 1 H , and
h n ( u , δ ) = 2 H ( 1 H ) C n 2 L 2 H 4 ( δ 2 + u 2 ) 2 H .
B y ( d ) = H C n 2 L 2 H + 2 0 1 ( 2 / 3 u + u 3 / 3 ) ( δ 2 + u 2 ) 1 H d u = H C n 2 L 2 H + 2 δ 2 H 2 [ 2 3 F 1 2 ( 1 H , 1 2 ; 3 2 ; δ 2 ) + 1 2 F 1 2 ( 1 H , 1 ; 2 ; δ 2 ) + 1 12 F 1 2 ( 1 H , 2 ; 3 ; δ 2 ) ]
Δ ( d ) = 2 H ( 1 H ) C n 2 L 2 H + 2 δ 2 0 1 ( 2 / 3 u + u 3 / 3 ) ( δ 2 + u 2 ) 2 H d u = 2 H ( 1 H ) C n 2 L 2 H + 2 δ 2 H 2 [ 2 3 F 1 2 ( 2 H , 1 2 ; 3 2 ; δ 2 ) + 1 2 F 1 2 ( 2 H , 1 ; 2 ; δ 2 ) + 1 12 F 1 2 ( 2 H , 2 ; 3 ; δ 2 ) ] .
δ 0 = 1.945 ( ± 0.3735 ) H 2 + 0.3995 ( ± 0.1291 ) H + 0.0009689 ( ± 0.0092 ) .
Φ n ( κ ) = sin ( π H ) Γ ( 2 H + 2 ) 4 π 2 C n 2 exp ( κ 2 / κ m 2 ) κ 2 H + 3 , with 0 < H 1 / 3 .
g ˜ n ( u , δ ) = sin ( π H ) Γ ( 2 H + 2 ) Γ ( 1 H ) 6 π κ m 2 H + 2 C n 2 L m 3 / 2 F 1 1 [ 1 H , 5 / 2 , L m 2 4 ( δ 2 + u 2 ) ] , and
h ˜ n ( u , δ ) = sin ( π H ) Γ ( 2 H + 2 ) Γ ( 2 H ) 30 π κ m 2 H + 2 C n 2 L m 5 / 2 F 1 1 [ 2 H , 7 / 2 , L m 2 4 ( δ 2 + u 2 ) ] .
0 2 π = p ( H ) d + b ( H ) L , for d < 1.5 × 10 3 L ,
δ 0 = 1.962 ( ± 0.3840 ) H 2 + 0.3915 ( ± 0.1328 ) H + 0.0018 ( ± 0.0095 ) .
Φ n ( κ ) = sin ( π H ) Γ ( 2 H + 2 ) 4 π 2 C n 2 exp ( κ 2 / κ m 2 ) ( κ 0 2 + κ 2 ) H + 3 / 2 , with 0 < H 1 / 3 ,
g ˜ n ( u , δ ) = sin ( π H ) Γ ( 2 H + 2 ) π 1 / 2 2 3 κ m 2 + 2 H C n 2 L m 3 / 2 n = 0 ( 1 ) n n ! 2 2 n U [ 3 2 + H , H n ; q 2 ] L m 2 n ( δ 2 + u 2 ) n
h ˜ n ( u , δ ) = sin ( π H ) Γ ( 2 H + 2 ) π 1 / 2 2 4 κ m 2 + 2 H C n 2 L m 5 / 2 n = 0 ( 1 ) n n ! 2 2 n U [ 3 2 + H , H 1 n ; q 2 ] L m 2 n ( δ 2 + u 2 ) n ,
d = p ( H ) 0 cosh [ m ( H ) ( 0 L 0 ) c ( H ) ] ,

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