Abstract

Light propagating through atmospheric turbulence acquires spatial and temporal phase variations. For strong enough turbulence conditions, interference from these phase variations within the optical wave can produce branch points; positions of zero amplitude. Under the assumption of a layered turbulence model, our previous work has shown that these branch points can be used to estimate the number and velocities of atmospheric layers. Key to this previous demonstration was the property of branch point persistence. Branch points from a single turbulence layer persist in time and through additional layers. In this paper we extend persistence to include branch point pairs. We develop an algorithm for isolating persistent pairs and show that through experimental data that they exist through time and through additional turbulence.

© 2011 OSA

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References

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  1. D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
  2. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  3. D. C. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley and Sons, 1998).
  4. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A 336, 165–190 (1974).
    [CrossRef]
  5. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  6. I. Freund, “Optical vortices in gaussian random fields: Statistical probability densities,” J. Opt. Soc. Am. 11, 1644–1652 (1994).
    [CrossRef]
  7. F. S. Roux, “Coupling of noncanonical optical vortices,” J. Opt. Soc. Am. 21, 664–670 (2004).
    [CrossRef]
  8. M. Chen and L. Roux, “Evolution of the scintillation index and the optical vortex density in speckle fields after removal of the least-squares phase,” J. Opt. Soc. Am. A 27, 2138–2143 (2008).
    [CrossRef]
  9. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
  10. J. E. Curtis and D. G. Grier, “Structure of optical vorticies,” Phys. Rev. 90, 133901 (2003).
  11. I. D. Maleev and G. A. Swartzlander, “Composite optical vorticies,” J. Opt. Soc. Am. B 20, 1169–1176 (2003).
    [CrossRef]
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    [CrossRef]
  13. T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express 16, 6985–6998 (2008).
    [CrossRef] [PubMed]
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  15. D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points - measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18, 22377–22392 (2010).
    [CrossRef] [PubMed]
  16. D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 0601–0610 (2009).
  17. D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).
  18. J. Leach, S. Keen, M. J. Padget, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the poynting vector in a helically phased beam,” Opt. Express 14, 11919–11924 (2006).
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    [CrossRef] [PubMed]
  21. D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Opt. Express 19, 25388–25396 (2011).
    [CrossRef]
  22. D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Express 19, 24596–24068 (2011).
    [CrossRef] [PubMed]
  23. T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553 (2004).
    [CrossRef]
  24. D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electron. Eng. 18, 467–483 (1992).
    [CrossRef]
  25. M. Schöck and E. J. Spillar, “Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor,” Proc. SPIE 3353, 1092–1099 (1998).
    [CrossRef]
  26. M. Schöck and E. J. Spillar, “Method for a quantitative investiagation of the frozen flow hypothesis,” J. Opt. Soc. Am. A 17, 1650–1658 (2000).
    [CrossRef]
  27. L. Poyneer, M. van Dam, and J. P. Véran, “Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemtry,” J. Opt. Soc. Am. A 26, 833–846 (2009).
    [CrossRef]
  28. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE Press, 2007).
  29. M. C. Roggeman and A. C. Koivunen, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction,” J. Opt. Soc. Am. 17, 53–62 (2000).
    [CrossRef]
  30. S. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004).
    [CrossRef]

2011 (2)

2010 (4)

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22 (2010).

K. Murphy, D. Burke, N. Devaney, and C. Dainty, “Experimental detection of optical vorticies with a shack-hartmann wavefront sensor,” Opt. Express 18, 15448–15460 (2010).
[CrossRef] [PubMed]

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

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).

2009 (3)

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 0601–0610 (2009).

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 0501–0512 (2009).

L. Poyneer, M. van Dam, and J. P. Véran, “Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemtry,” J. Opt. Soc. Am. A 26, 833–846 (2009).
[CrossRef]

2008 (2)

2006 (1)

2004 (3)

F. S. Roux, “Coupling of noncanonical optical vortices,” J. Opt. Soc. Am. 21, 664–670 (2004).
[CrossRef]

S. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004).
[CrossRef]

T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553 (2004).
[CrossRef]

2003 (2)

J. E. Curtis and D. G. Grier, “Structure of optical vorticies,” Phys. Rev. 90, 133901 (2003).

I. D. Maleev and G. A. Swartzlander, “Composite optical vorticies,” J. Opt. Soc. Am. B 20, 1169–1176 (2003).
[CrossRef]

2000 (2)

M. Schöck and E. J. Spillar, “Method for a quantitative investiagation of the frozen flow hypothesis,” J. Opt. Soc. Am. A 17, 1650–1658 (2000).
[CrossRef]

M. C. Roggeman and A. C. Koivunen, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction,” J. Opt. Soc. Am. 17, 53–62 (2000).
[CrossRef]

1999 (1)

E. O. Le Bigot and W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. 16, 1724–1729 (1999).
[CrossRef]

1998 (2)

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

M. Schöck and E. J. Spillar, “Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor,” Proc. SPIE 3353, 1092–1099 (1998).
[CrossRef]

1994 (1)

I. Freund, “Optical vortices in gaussian random fields: Statistical probability densities,” J. Opt. Soc. Am. 11, 1644–1652 (1994).
[CrossRef]

1993 (1)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

1992 (3)

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

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electron. Eng. 18, 467–483 (1992).
[CrossRef]

1974 (1)

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

Allen, L.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22 (2010).

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Berry, M. V.

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

Burke, D.

Chen, M.

Curtis, J. E.

J. E. Curtis and D. G. Grier, “Structure of optical vorticies,” Phys. Rev. 90, 133901 (2003).

Dainty, C.

Devaney, N.

Dholakia, K.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22 (2010).

Freund, I.

I. Freund, “Optical vortices in gaussian random fields: Statistical probability densities,” J. Opt. Soc. Am. 11, 1644–1652 (1994).
[CrossRef]

Fried, D. L.

Ghiglia, D. C.

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

Glas, R. S.

S. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004).
[CrossRef]

Grier, D. G.

J. E. Curtis and D. G. Grier, “Structure of optical vorticies,” Phys. Rev. 90, 133901 (2003).

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Johnston, D. C.

D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electron. Eng. 18, 467–483 (1992).
[CrossRef]

Keen, S.

Kelly, P. R.

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 0501–0512 (2009).

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 0601–0610 (2009).

Koivunen, A. C.

M. C. Roggeman and A. C. Koivunen, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction,” J. Opt. Soc. Am. 17, 53–62 (2000).
[CrossRef]

Le Bigot, E. O.

E. O. Le Bigot and W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. 16, 1724–1729 (1999).
[CrossRef]

Leach, J.

Love, G. D.

Maleev, I. D.

Mantravadi, S.

S. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004).
[CrossRef]

Matson, C. L.

Murphy, K.

Nye, J. F.

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

Oesch, D. W.

D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Express 19, 24596–24068 (2011).
[CrossRef] [PubMed]

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

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).

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

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 0601–0610 (2009).

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 0501–0512 (2009).

Padget, M. J.

Padgett, M. J.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22 (2010).

Poyneer, L.

Pritt, M. D.

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

Rhoadarmer, T. A.

T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553 (2004).
[CrossRef]

S. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004).
[CrossRef]

Roggeman, M. C.

M. C. Roggeman and A. C. Koivunen, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction,” J. Opt. Soc. Am. 17, 53–62 (2000).
[CrossRef]

Roux, F. S.

F. S. Roux, “Coupling of noncanonical optical vortices,” J. Opt. Soc. Am. 21, 664–670 (2004).
[CrossRef]

Roux, L.

Sanchez, D. J.

D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Express 19, 24596–24068 (2011).
[CrossRef] [PubMed]

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

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

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 0601–0610 (2009).

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 0501–0512 (2009).

Sasiela, R. J.

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

Saunter, C.

Schmidt, J. D.

Schöck, M.

M. Schöck and E. J. Spillar, “Method for a quantitative investiagation of the frozen flow hypothesis,” J. Opt. Soc. Am. A 17, 1650–1658 (2000).
[CrossRef]

M. Schöck and E. J. Spillar, “Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor,” Proc. SPIE 3353, 1092–1099 (1998).
[CrossRef]

Simpson, N. B.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22 (2010).

Spillar, E. J.

M. Schöck and E. J. Spillar, “Method for a quantitative investiagation of the frozen flow hypothesis,” J. Opt. Soc. Am. A 17, 1650–1658 (2000).
[CrossRef]

M. Schöck and E. J. Spillar, “Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor,” Proc. SPIE 3353, 1092–1099 (1998).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Swartzlander, G. A.

Tewksbury-Christle, C. M.

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 0601–0610 (2009).

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 0501–0512 (2009).

van Dam, M.

Vaughn, J. L.

Venema, T. M.

Véran, J. P.

Welsh, B. M.

D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electron. Eng. 18, 467–483 (1992).
[CrossRef]

Wild, W. J.

E. O. Le Bigot and W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. 16, 1724–1729 (1999).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Appl. Opt. (1)

Comput. Electron. Eng. (1)

D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electron. Eng. 18, 467–483 (1992).
[CrossRef]

J. Mod. Opt. (1)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

J. Opt. Soc. Am. (4)

I. Freund, “Optical vortices in gaussian random fields: Statistical probability densities,” J. Opt. Soc. Am. 11, 1644–1652 (1994).
[CrossRef]

F. S. Roux, “Coupling of noncanonical optical vortices,” J. Opt. Soc. Am. 21, 664–670 (2004).
[CrossRef]

E. O. Le Bigot and W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. 16, 1724–1729 (1999).
[CrossRef]

M. C. Roggeman and A. C. Koivunen, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction,” J. Opt. Soc. Am. 17, 53–62 (2000).
[CrossRef]

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

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

Opt. Express (6)

Opt. Lett. (1)

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22 (2010).

Phys. Rev. (1)

J. E. Curtis and D. G. Grier, “Structure of optical vorticies,” Phys. Rev. 90, 133901 (2003).

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Proc. R. Soc. London A (1)

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

Proc. SPIE (6)

T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553 (2004).
[CrossRef]

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” Proc. SPIE 7466, 0601–0610 (2009).

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - altitude and strength of atmospheric turbulence layers,” Proc. SPIE 7816, 0501–0513 (2010).

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE 7466, 0501–0512 (2009).

M. Schöck and E. J. Spillar, “Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor,” Proc. SPIE 3353, 1092–1099 (1998).
[CrossRef]

S. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE 5553, 290–300 (2004).
[CrossRef]

Other (2)

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

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

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Figures (10)

Fig. 1
Fig. 1

The phase associated with branch points. (a) phase of a single branch point. (b) the phase of a branch point pair. Note, the phase for a single branch point extends to infinity, i.e. it occupies the extent of the propagating beam. The phase for a pair of branch points also extends to infinity, but goes to zero as the distance from the pair increases.

Fig. 2
Fig. 2

Cartoon of the steps in the identification of persistent pupil plane branch point pairs. Red and green dots indicate the positive and negative circulations respectively. (a) The empty polarity array. (b) The polarity array with ±1’s at the locations of positive and negative circulations respectively. (c)Diagram showing the estimated velocities from the positive branch point locations. (d)Blue lines indicate the initial pairings following frame by frame use of a walking algorithm. (e)Pairings following the use of piston shifting technique. (f)Overlay of measured velocity to incorporate 3rd dimension of WFS data.

Fig. 3
Fig. 3

(a) Sample WFS phase and (b) a map of the discontinuities from difference of gradients method, as produced by Eq. (4).

Fig. 4
Fig. 4

A sample map of discontinuities with enlarged sections showing some of the branch point pairs and how the discontinuities are distributed around them.

Fig. 5
Fig. 5

Sample WFS phase (left), (a) original and (b) with a piston shift of π added. Maps of discontinuities from the difference of gradients method (right), (c) from the original phase and (d) from the phase with the added piston.

Fig. 6
Fig. 6

Single layer example of pairings using only single frame techniques. (a) The polarity array shown in three dimensions and (b) a close-up of the array showing numerous 2-D only pairings.

Fig. 7
Fig. 7

Single layer example of pairing now incorporating velocity filtering. (a) The polarity array shown in three dimensions and (b) a close-up of the array showing numerous 2-D only pairings.

Fig. 8
Fig. 8

3D examination of Case 1. (a)Identified circulations for two layer configuration. (b) Persistent pairs identified for the low altitude layer. (c) Persistent pairs identified for the high altitude layer. (d) Circulations not found to follow one of the measured velocities, noise circulations.

Fig. 9
Fig. 9

One and Two layer comparison. Top row: Persistent pairs identified for (a) the lower layer and (b) the upper layer from the combined two layer atmosphere. Bottom row: Persistent pairs identified for (c) the lower layer alone and (d) the upper layer alone (right). Pairs in the isolated layers appear in the combined layer results - indicating that the persistent pair method finds pairs through additional turbulence.

Fig. 10
Fig. 10

One and Two layer comparisons for all five configurations, on Table 1. Each row is proceeded by the Case number for the presented results. Identified persistent pairs for (a) the isolated low altitude layer, (b) the isolated high altitude layer, (c) the full polarity array for the combined two layer experiment and then (c,1) and (c,2) the low and high altitude layer pairs identified from our technique respectively.

Tables (1)

Tables Icon

Table 1 Turbulence conditions for one and two layer simulations. Dashes indicate when a phase wheel wasn’t included in the optical path.

Equations (5)

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σ χ 2 = 0.5631 k 0 7 / 6 0 L C n 2 ( z ) z 5 / 6 d z ,
δ x i , j = ϕ i , j ϕ i + 1 , j δ y i , j = ϕ i , j ϕ i , j + 1 ,
δ e x i , j = mod 2 π ( ϕ i , j ϕ i + 1 , j ) δ e y i , j = mod 2 π ( ϕ i , j ϕ i , j + 1 ) .
Δ x i , j = ( sign ( δ x i , j ) sign ( δ e x i , j ) ) / 2 Δ y i , j = ( sign ( δ y i . j ) sign ( δ e y i , j ) ) / 2 .
B C i , j = | Δ x i , j | + | Δ x i + 1 , j | + | Δ y i , j | + | Δ y i , j + 1 | .

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