Abstract

Evolution of branch points in the distorted optical field is studied when a laser beam propagates through turbulent atmosphere along an uplink path. Two categories of propagation events are mainly explored for the same propagation height: fixed wavelength with change of the turbulence strength and fixed turbulence strength with change of the wavelength. It is shown that, when the beam propagates to a certain height, the density of the branch-points reaches its maximum and such a height changes with the turbulence strength but nearly remains constant with different wavelengths. The relationship between the density of branch-points and the Rytov number is also given. A fitted formula describing the relationship between the density of branch-points and propagation height with different turbulence strength and wavelength is found out. Interestingly, this formula is very similar to the formula used for describing the Blackbody radiation in physics. The results obtained may be helpful for atmospheric optics, astronomy and optical communication.

© 2014 Optical Society of America

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    [CrossRef]
  3. N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73(5), 525–528 (1983).
    [CrossRef]
  4. V. A. Tartakovski, N. N. Mayer, “Phase dislocation and minimal phase representation of the wave function,” Atmos. Oceanic Opt. 8, 231–235 (1995).
  5. B. V. Fortes, V. Lukin, “The effects of wavefront dislocations on the atmospheric adaptive optical systems performance,” Proc. SPIE 2778, 1002–1003 (1996).
  6. I. Freund, N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
    [CrossRef] [PubMed]
  7. R. Rao, “Statistics of the fractal structure and phase singularity of a plane light wave propagation in atmospheric turbulence,” Appl. Opt. 47(2), 269–276 (2008).
    [CrossRef] [PubMed]
  8. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31(15), 2865–2882 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  27. D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, “Aggregate behavior of branch points--persistent pairs,” Opt. Express 20(2), 1046–1059 (2012).
    [CrossRef] [PubMed]
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  29. D. J. Sanchez, D. W. Oesch, “Localization of angular momentum in optical waves propagating through turbulence,” Opt. Express 19(25), 25388–25396 (2011).
    [CrossRef] [PubMed]
  30. D. J. Sanchez, D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed turbulence,” Opt. Express 19(24), 24596–24608 (2011).
    [CrossRef] [PubMed]
  31. X. Qian, W. Zhu, R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sin. 58, 6633–6638 (2009).
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2013 (1)

2012 (2)

D. J. Sanchez, D. W. Oesch, P. R. Kelly, “The aggregate behavior of branch points - theoretical calculation of branch point velocity,” Proc. SPIE 8380, 83800P (2012).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, “Aggregate behavior of branch points--persistent pairs,” Opt. Express 20(2), 1046–1059 (2012).
[CrossRef] [PubMed]

2011 (3)

2010 (2)

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. L. Matson, “The aggregate behavior of branch points--measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18(21), 22377–22392 (2010).
[CrossRef] [PubMed]

2009 (3)

D. J. Sanchez, D. W. Oesch, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 746605 (2009).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 746606 (2009).
[CrossRef]

X. Qian, W. Zhu, R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sin. 58, 6633–6638 (2009).

2008 (2)

2004 (2)

2003 (1)

C. Fan, Y. Wang, Z. Gong, “Effect of branch points on adaptive optics,” High Power Laser Particle Beams 15, 435–438 (2003).

2001 (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200(1–6), 43–72 (2001).
[CrossRef]

1999 (1)

B. Wang, A. C. Koivunen, M. C. Roggemann, “Comparison of branch point and least squares reconstructors for laser beam transmission through the atmosphere,” Proc. SPIE 3763, 41–49 (1999).
[CrossRef]

1998 (4)

E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Branch point reconstructors for discontinuous light phase functions,” Proc. SPIE 3381, 76–87 (1998).
[CrossRef]

V. V. Voitsekhovich, D. Kouznetsov, D. K. Morozov, “Density of turbulence-induced phase dislocations,” Appl. Opt. 37(21), 4525–4535 (1998).
[CrossRef] [PubMed]

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15(10), 2759–2768 (1998).
[CrossRef]

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

1996 (1)

B. V. Fortes, V. Lukin, “The effects of wavefront dislocations on the atmospheric adaptive optical systems performance,” Proc. SPIE 2778, 1002–1003 (1996).

1995 (2)

V. A. Tartakovski, N. N. Mayer, “Phase dislocation and minimal phase representation of the wave function,” Atmos. Oceanic Opt. 8, 231–235 (1995).

C. A. Primmerman, T. R. Price, R. A. Humphreys, B. G. Zollars, H. T. Barclay, J. H. Herrmann, “Atmospheric-compensation experiments in strong-scintillation conditions,” Appl. Opt. 34(12), 2081–2088 (1995).
[CrossRef] [PubMed]

1994 (1)

I. Freund, N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

1992 (1)

1983 (1)

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73(5), 525–528 (1983).
[CrossRef]

1981 (1)

J. F. Nye, “The motion and structure of dislocation in wavefronts,” Proc. R. Soc. London A Math. Phys. Sci. 378(1773), 219–239 (1981).
[CrossRef]

1974 (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

Baranova, N. B.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73(5), 525–528 (1983).
[CrossRef]

Barclay, H. T.

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

Chen, M.

Fan, C.

C. Fan, Y. Wang, Z. Gong, “Effects of different beacon wavelengths on atmospheric compensation in strong scintillation,” Appl. Opt. 43(22), 4334–4338 (2004).
[CrossRef] [PubMed]

C. Fan, Y. Wang, Z. Gong, “Effect of branch points on adaptive optics,” High Power Laser Particle Beams 15, 435–438 (2003).

Fortes, B. V.

B. V. Fortes, V. Lukin, “The effects of wavefront dislocations on the atmospheric adaptive optical systems performance,” Proc. SPIE 2778, 1002–1003 (1996).

Freund, I.

I. Freund, N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

Fried, D. L.

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200(1–6), 43–72 (2001).
[CrossRef]

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15(10), 2759–2768 (1998).
[CrossRef]

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31(15), 2865–2882 (1992).
[CrossRef] [PubMed]

Gong, Z.

C. Fan, Y. Wang, Z. Gong, “Effects of different beacon wavelengths on atmospheric compensation in strong scintillation,” Appl. Opt. 43(22), 4334–4338 (2004).
[CrossRef] [PubMed]

C. Fan, Y. Wang, Z. Gong, “Effect of branch points on adaptive optics,” High Power Laser Particle Beams 15, 435–438 (2003).

Herrmann, J. H.

Humphreys, R. A.

Kelly, P. R.

D. J. Sanchez, D. W. Oesch, P. R. Kelly, “The aggregate behavior of branch points - theoretical calculation of branch point velocity,” Proc. SPIE 8380, 83800P (2012).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 746606 (2009).
[CrossRef]

Kibblewhite, E. J.

E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Branch point reconstructors for discontinuous light phase functions,” Proc. SPIE 3381, 76–87 (1998).
[CrossRef]

Koivunen, A. C.

B. Wang, A. C. Koivunen, M. C. Roggemann, “Comparison of branch point and least squares reconstructors for laser beam transmission through the atmosphere,” Proc. SPIE 3763, 41–49 (1999).
[CrossRef]

Kouznetsov, D.

Le Bigot, E.-O.

E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Branch point reconstructors for discontinuous light phase functions,” Proc. SPIE 3381, 76–87 (1998).
[CrossRef]

Li, Y.

Y. Li, “Branch point effect on adaptive correction,” Proc. SPIE 5490, 1064–1070 (2004).
[CrossRef]

Lukin, V.

B. V. Fortes, V. Lukin, “The effects of wavefront dislocations on the atmospheric adaptive optical systems performance,” Proc. SPIE 2778, 1002–1003 (1996).

Mamaev, A. V.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73(5), 525–528 (1983).
[CrossRef]

Matson, C. L.

Mayer, N. N.

V. A. Tartakovski, N. N. Mayer, “Phase dislocation and minimal phase representation of the wave function,” Atmos. Oceanic Opt. 8, 231–235 (1995).

Morozov, D. K.

Nye, J. F.

J. F. Nye, “The motion and structure of dislocation in wavefronts,” Proc. R. Soc. London A Math. Phys. Sci. 378(1773), 219–239 (1981).
[CrossRef]

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

Oesch, D. W.

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, “Aggregate behavior of branch points--persistent pairs,” Opt. Express 20(2), 1046–1059 (2012).
[CrossRef] [PubMed]

D. J. Sanchez, D. W. Oesch, P. R. Kelly, “The aggregate behavior of branch points - theoretical calculation of branch point velocity,” Proc. SPIE 8380, 83800P (2012).
[CrossRef]

D. J. Sanchez, D. W. Oesch, “Localization of angular momentum in optical waves propagating through turbulence,” Opt. Express 19(25), 25388–25396 (2011).
[CrossRef] [PubMed]

D. J. Sanchez, D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed turbulence,” Opt. Express 19(24), 24596–24608 (2011).
[CrossRef] [PubMed]

D. W. Oesch, D. J. Sanchez, C. L. Matson, “The aggregate behavior of branch points--measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18(21), 22377–22392 (2010).
[CrossRef] [PubMed]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
[CrossRef]

D. J. Sanchez, D. W. Oesch, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 746605 (2009).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 746606 (2009).
[CrossRef]

Pilipetsky, N. F.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73(5), 525–528 (1983).
[CrossRef]

Price, T. R.

Primmerman, C. A.

Qian, X.

X. Qian, W. Zhu, R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sin. 58, 6633–6638 (2009).

Rao, R.

X. Qian, W. Zhu, R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sin. 58, 6633–6638 (2009).

R. Rao, “Statistics of the fractal structure and phase singularity of a plane light wave propagation in atmospheric turbulence,” Appl. Opt. 47(2), 269–276 (2008).
[CrossRef] [PubMed]

Roggemann, M. C.

B. Wang, A. C. Koivunen, M. C. Roggemann, “Comparison of branch point and least squares reconstructors for laser beam transmission through the atmosphere,” Proc. SPIE 3763, 41–49 (1999).
[CrossRef]

Roux, F. S.

Sanchez, D. J.

D. J. Sanchez, D. W. Oesch, P. R. Kelly, “The aggregate behavior of branch points - theoretical calculation of branch point velocity,” Proc. SPIE 8380, 83800P (2012).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, “Aggregate behavior of branch points--persistent pairs,” Opt. Express 20(2), 1046–1059 (2012).
[CrossRef] [PubMed]

D. J. Sanchez, D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed turbulence,” Opt. Express 19(24), 24596–24608 (2011).
[CrossRef] [PubMed]

D. J. Sanchez, D. W. Oesch, “Localization of angular momentum in optical waves propagating through turbulence,” Opt. Express 19(25), 25388–25396 (2011).
[CrossRef] [PubMed]

D. W. Oesch, D. J. Sanchez, C. L. Matson, “The aggregate behavior of branch points--measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18(21), 22377–22392 (2010).
[CrossRef] [PubMed]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
[CrossRef]

D. J. Sanchez, D. W. Oesch, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 746605 (2009).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 746606 (2009).
[CrossRef]

Shkunov, V. V.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73(5), 525–528 (1983).
[CrossRef]

Shvartsman, N.

I. Freund, N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

Tartakovski, V. A.

V. A. Tartakovski, N. N. Mayer, “Phase dislocation and minimal phase representation of the wave function,” Atmos. Oceanic Opt. 8, 231–235 (1995).

Tewksbury-Christle, C. M.

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, “Aggregate behavior of branch points--persistent pairs,” Opt. Express 20(2), 1046–1059 (2012).
[CrossRef] [PubMed]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 746606 (2009).
[CrossRef]

Vaughn, J. L.

Voitsekhovich, V. V.

Wang, B.

B. Wang, A. C. Koivunen, M. C. Roggemann, “Comparison of branch point and least squares reconstructors for laser beam transmission through the atmosphere,” Proc. SPIE 3763, 41–49 (1999).
[CrossRef]

Wang, Y.

C. Fan, Y. Wang, Z. Gong, “Effects of different beacon wavelengths on atmospheric compensation in strong scintillation,” Appl. Opt. 43(22), 4334–4338 (2004).
[CrossRef] [PubMed]

C. Fan, Y. Wang, Z. Gong, “Effect of branch points on adaptive optics,” High Power Laser Particle Beams 15, 435–438 (2003).

Wild, W. J.

E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Branch point reconstructors for discontinuous light phase functions,” Proc. SPIE 3381, 76–87 (1998).
[CrossRef]

Zel’dovich, B. Y.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73(5), 525–528 (1983).
[CrossRef]

Zhu, W.

X. Qian, W. Zhu, R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sin. 58, 6633–6638 (2009).

Zollars, B. G.

Acta Phys. Sin. (1)

X. Qian, W. Zhu, R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sin. 58, 6633–6638 (2009).

Appl. Opt. (5)

Atmos. Oceanic Opt. (2)

V. A. Tartakovski, N. N. Mayer, “Phase dislocation and minimal phase representation of the wave function,” Atmos. Oceanic Opt. 8, 231–235 (1995).

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

High Power Laser Particle Beams (1)

C. Fan, Y. Wang, Z. Gong, “Effect of branch points on adaptive optics,” High Power Laser Particle Beams 15, 435–438 (2003).

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200(1–6), 43–72 (2001).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. A (1)

I. Freund, N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[CrossRef] [PubMed]

Proc. R. Soc. London A Math. Phys. Sci. (2)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A Math. Phys. Sci. 336(1605), 165–190 (1974).
[CrossRef]

J. F. Nye, “The motion and structure of dislocation in wavefronts,” Proc. R. Soc. London A Math. Phys. Sci. 378(1773), 219–239 (1981).
[CrossRef]

Proc. SPIE (8)

B. V. Fortes, V. Lukin, “The effects of wavefront dislocations on the atmospheric adaptive optical systems performance,” Proc. SPIE 2778, 1002–1003 (1996).

E.-O. Le Bigot, W. J. Wild, E. J. Kibblewhite, “Branch point reconstructors for discontinuous light phase functions,” Proc. SPIE 3381, 76–87 (1998).
[CrossRef]

B. Wang, A. C. Koivunen, M. C. Roggemann, “Comparison of branch point and least squares reconstructors for laser beam transmission through the atmosphere,” Proc. SPIE 3763, 41–49 (1999).
[CrossRef]

Y. Li, “Branch point effect on adaptive correction,” Proc. SPIE 5490, 1064–1070 (2004).
[CrossRef]

D. J. Sanchez, D. W. Oesch, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 746605 (2009).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 746606 (2009).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 781605 (2010).
[CrossRef]

D. J. Sanchez, D. W. Oesch, P. R. Kelly, “The aggregate behavior of branch points - theoretical calculation of branch point velocity,” Proc. SPIE 8380, 83800P (2012).
[CrossRef]

Other (3)

D. W. Oesch, D. J. Sanchez, and P. R. Kelly, “Optical vortex density in Rytov saturated atmospheric turbulence,” in FiO (2012), FW3A. 3.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE, 2007).

D. C. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley, 1998).

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Figures (3)

Fig. 1
Fig. 1

The vertical profile of turbulence strength with H-V model

Fig. 2
Fig. 2

The density of branch-points versus (a) propagation height and (b) Rytov number with different turbulence strength (λ = 1.06μm).

Fig. 3
Fig. 3

The density of branch-points versus (a) propagation height and (b) Rytov number with different wavelength (Cn2(0) = 4.104 × 10−14m-2/3).

Tables (2)

Tables Icon

Table 1 The values of fitting parameters for different turbulence strength with the fixed wavelength λ = 1.06μm

Tables Icon

Table 2 The values of fitting parameters for different wavelength with the fixed surface level value of turbulence strength Cn2(0) = 4.104 × 10−14m-2/3

Equations (9)

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φ( r p,q )= tan 1 ( Im{U( r p,q )} Re{U( r p,q )} ),
g( r p,q )=φ( r p,q ) {φ( r p+1,q )φ( r p,q )} pv d l x + {φ( r p,q+1 )φ( r p,v )} pv d l y ,
g( r p,q )= d 1 [ tan 1 ( Im{U[(p+1)d,qd] U (pd,qd)} Re{U[(p+1)d,qd] U (pd,qd)} )] l x + d 1 [ tan 1 ( Im{U[pd,(q+1)d] U (pd,qd)} Re{U[pd,(q+1)d] U (pd,qd)} )] l y .
C g(r)dr ={ ±2π, if a branch point is enclosesd 0 , if no branch point is enclosesd ,
S p,q =g( r p,q ) l x d+g( r p+1,q ) l y dg( r p,q+1 ) l x dg( r p,q ) l y d.
2ik Φ(z,r) z + 2 Φ(z,r)+2 k 2 δn(z,r)Φ(z,r)=0,
C n 2 (h)=8.2× 10 26 w 2 h 10 e h +2.7× 10 16 e h/1.5 + C n 2 (0) e 10h ,
σ χ 2 =0.5631 k 7/6 0 L dh C n 2 (h) (Lh) 5/6 .
y= A x n ( e B/x 1) ,

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