Abstract

This is the second of two papers demonstrating that photons with orbital angular momentum can be created in optical waves propagating through distributed turbulence. In the companion paper, it is shown that propagation through atmospheric turbulence can create non-trivial angular momentum. Here, we extend the result and demonstrate that this momentum is, at least in part, orbital angular momentum. Specifically, we demonstrate that branch points (in the language of the adaptive optic community) indicate the presence of photons with non-zero OAM. Furthermore, the conditions required to create photons with non-zero orbital angular momentum are ubiquitous. The repercussions of this statement are wide ranging and these are cursorily enumerated.

© 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,” Applied Optics 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
  2. 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,” Physical Review A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
  3. J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Optics Express 14, 11919–11924 (2006).
    [CrossRef] [PubMed]
  4. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  5. 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,” Optics Express 18, 22377–22392 (2010).
    [CrossRef] [PubMed]
  6. D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Optics Express (2011). Accepted for publication.
    [PubMed]
  7. 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,” SPIE 7466, 0501–0512 (2009).
  8. D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - persistent pairs,” Optics Express (2011). Submitted for publication.
  9. D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - characterization in wave optical simulation,” Optics Express (2011). Submitted for publication.
  10. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (SPIE Press, Bellingham, Wa, USA, 2007), 2nd ed.
  11. J. W. Goodman, Statistical Optics (John Wiley & Sons, New York, New York, 2000), Wiley Classics Library ed.
  12. See for instance Ref. [11] page 394.
  13. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, USA, 1975), 2nd ed.
  14. R. A. Beth, “Mechanical detection of the angular momentum of light,” Physical Review 50, 115–125 (1936).
    [CrossRef]
  15. This analysis is true for any type of wavefront sensor. The Shack-Hartmann is presented here because it is well known. In our lab, we use a self-referencing interferometer.
  16. This is in keeping with the notation in Fried’s seminal paper, Ref. [4].
  17. First pointed out by Terry Brennan, the Optical Science Corporation, in a private conversation.
  18. D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - modeling parameters,” Optical Society of America - Frontiers in Optics proceedings (2011). Accepted for publication.
  19. D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “Branch points in deep turbulence and its relevance to adaptive optics – an overview of the ASALT laboratory’s deep turbulence research,” in “2010 DEPS Annual Conference,”, D. Herrick, ed. (Directed Energy Professional Society, 2010).
  20. D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - a proposal for an atmospheric turbulence layer sensor,” SPIE 7816, 0601–0616 (2010).
  21. 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,” SPIE 7816, 0501–0513 (2010).
  22. 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,” SPIE 7466, 0601–0610 (2009).
  23. This result holds in general to include diffraction, not merely for the these simplifying assumptions Specifically, if the assumption were relaxed to allow diffraction, each small region after propagation would have non-zero components that extend to infinity. But other than in the local region, these components are trivial and would at most cause the location of the null to move slightly.
  24. D. W. Oesch and D. J. Sanchez, “Studying the optical field in and through the failure of the Rytov approximation,” Optics Express (2011). In preparation.
  25. D. J. Sanchez and D. W. Oesch, “The effect of orbital angular momentum in the Rytov approximation,” In preparation.
  26. D. J. Sanchez, D. W. Oesch, and S. M. Gregory, “Orbital angular momentum in waves propagating through galactic clouds and dust,” In preparation.
  27. We have taken great care in our research to make a distinction between the persistant topological features of the propagating wave from transient phenonena. The persistent topological features--pairs of zeros in amplitude with opposite winding number--we call branch points; all other ciculations, we label as noise. Prior to our work, this distinction was vague. As a point in fact, in the earlest days of adaptive optics all 2π circulations were lumped into what was then called the “slope discrepancy” or “null” space, and even today in the phase reconstruction process, standard wavefront sensors lump all 2π circulations into a single group so that they may be summarily discarded, hence Fried’s [4] terminology “hidden phase”.

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,” Optics Express 18, 22377–22392 (2010).
[CrossRef] [PubMed]

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - a proposal for an atmospheric turbulence layer sensor,” SPIE 7816, 0601–0616 (2010).

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,” SPIE 7816, 0501–0513 (2010).

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,” 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,” SPIE 7466, 0501–0512 (2009).

2006

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Optics Express 14, 11919–11924 (2006).
[CrossRef] [PubMed]

1998

1992

D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Applied Optics 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,” Physical Review A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

1936

R. A. Beth, “Mechanical detection of the angular momentum of light,” Physical Review 50, 115–125 (1936).
[CrossRef]

Allen, L.

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,” Physical Review 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,” Physical Review A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Beth, R. A.

R. A. Beth, “Mechanical detection of the angular momentum of light,” Physical Review 50, 115–125 (1936).
[CrossRef]

Fried, D. L.

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

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

Goodman, J. W.

J. W. Goodman, Statistical Optics (John Wiley & Sons, New York, New York, 2000), Wiley Classics Library ed.

Gregory, S. M.

D. J. Sanchez, D. W. Oesch, and S. M. Gregory, “Orbital angular momentum in waves propagating through galactic clouds and dust,” In preparation.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, USA, 1975), 2nd ed.

Keen, S.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Optics Express 14, 11919–11924 (2006).
[CrossRef] [PubMed]

Kelly, P. R.

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - a proposal for an atmospheric turbulence layer sensor,” SPIE 7816, 0601–0616 (2010).

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,” 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,” 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,” 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 - persistent pairs,” Optics Express (2011). Submitted for publication.

D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - characterization in wave optical simulation,” Optics Express (2011). Submitted for publication.

D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - modeling parameters,” Optical Society of America - Frontiers in Optics proceedings (2011). Accepted for publication.

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “Branch points in deep turbulence and its relevance to adaptive optics – an overview of the ASALT laboratory’s deep turbulence research,” in “2010 DEPS Annual Conference,”, D. Herrick, ed. (Directed Energy Professional Society, 2010).

Leach, J.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Optics Express 14, 11919–11924 (2006).
[CrossRef] [PubMed]

Love, G. D.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Optics Express 14, 11919–11924 (2006).
[CrossRef] [PubMed]

Matson, C. L.

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,” Optics Express 18, 22377–22392 (2010).
[CrossRef] [PubMed]

Oesch, D. W.

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,” Optics Express 18, 22377–22392 (2010).
[CrossRef] [PubMed]

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - a proposal for an atmospheric turbulence layer sensor,” SPIE 7816, 0601–0616 (2010).

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,” 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,” 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,” 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 - persistent pairs,” Optics Express (2011). Submitted for publication.

D. J. Sanchez, D. W. Oesch, and S. M. Gregory, “Orbital angular momentum in waves propagating through galactic clouds and dust,” In preparation.

D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - characterization in wave optical simulation,” Optics Express (2011). Submitted for publication.

D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - modeling parameters,” Optical Society of America - Frontiers in Optics proceedings (2011). Accepted for publication.

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “Branch points in deep turbulence and its relevance to adaptive optics – an overview of the ASALT laboratory’s deep turbulence research,” in “2010 DEPS Annual Conference,”, D. Herrick, ed. (Directed Energy Professional Society, 2010).

D. J. Sanchez and D. W. Oesch, “The effect of orbital angular momentum in the Rytov approximation,” In preparation.

D. W. Oesch and D. J. Sanchez, “Studying the optical field in and through the failure of the Rytov approximation,” Optics Express (2011). In preparation.

D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Optics Express (2011). Accepted for publication.
[PubMed]

Padgett, M. J.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Optics Express 14, 11919–11924 (2006).
[CrossRef] [PubMed]

Sanchez, D. J.

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,” Optics Express 18, 22377–22392 (2010).
[CrossRef] [PubMed]

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - a proposal for an atmospheric turbulence layer sensor,” SPIE 7816, 0601–0616 (2010).

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,” 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,” 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,” 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 - persistent pairs,” Optics Express (2011). Submitted for publication.

D. J. Sanchez, D. W. Oesch, and S. M. Gregory, “Orbital angular momentum in waves propagating through galactic clouds and dust,” In preparation.

D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - characterization in wave optical simulation,” Optics Express (2011). Submitted for publication.

D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - modeling parameters,” Optical Society of America - Frontiers in Optics proceedings (2011). Accepted for publication.

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “Branch points in deep turbulence and its relevance to adaptive optics – an overview of the ASALT laboratory’s deep turbulence research,” in “2010 DEPS Annual Conference,”, D. Herrick, ed. (Directed Energy Professional Society, 2010).

D. J. Sanchez and D. W. Oesch, “The effect of orbital angular momentum in the Rytov approximation,” In preparation.

D. W. Oesch and D. J. Sanchez, “Studying the optical field in and through the failure of the Rytov approximation,” Optics Express (2011). In preparation.

D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Optics Express (2011). Accepted for publication.
[PubMed]

Sasiela, R. J.

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

Saunter, C.

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Optics Express 14, 11919–11924 (2006).
[CrossRef] [PubMed]

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,” Physical Review A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

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,” 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 - a proposal for an atmospheric turbulence layer sensor,” SPIE 7816, 0601–0616 (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,” 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,” 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 - persistent pairs,” Optics Express (2011). Submitted for publication.

D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - characterization in wave optical simulation,” Optics Express (2011). Submitted for publication.

D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - modeling parameters,” Optical Society of America - Frontiers in Optics proceedings (2011). Accepted for publication.

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “Branch points in deep turbulence and its relevance to adaptive optics – an overview of the ASALT laboratory’s deep turbulence research,” in “2010 DEPS Annual Conference,”, D. Herrick, ed. (Directed Energy Professional Society, 2010).

Vaughn, J. L.

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

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,” Physical Review A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Applied Optics

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

J. Opt. Soc. Am. A

Optics Express

J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Optics Express 14, 11919–11924 (2006).
[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,” Optics Express 18, 22377–22392 (2010).
[CrossRef] [PubMed]

Physical Review

R. A. Beth, “Mechanical detection of the angular momentum of light,” Physical Review 50, 115–125 (1936).
[CrossRef]

Physical Review A

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,” Physical Review A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

SPIE

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,” SPIE 7466, 0501–0512 (2009).

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - a proposal for an atmospheric turbulence layer sensor,” SPIE 7816, 0601–0616 (2010).

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,” 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,” SPIE 7466, 0601–0610 (2009).

Other

This result holds in general to include diffraction, not merely for the these simplifying assumptions Specifically, if the assumption were relaxed to allow diffraction, each small region after propagation would have non-zero components that extend to infinity. But other than in the local region, these components are trivial and would at most cause the location of the null to move slightly.

D. W. Oesch and D. J. Sanchez, “Studying the optical field in and through the failure of the Rytov approximation,” Optics Express (2011). In preparation.

D. J. Sanchez and D. W. Oesch, “The effect of orbital angular momentum in the Rytov approximation,” In preparation.

D. J. Sanchez, D. W. Oesch, and S. M. Gregory, “Orbital angular momentum in waves propagating through galactic clouds and dust,” In preparation.

We have taken great care in our research to make a distinction between the persistant topological features of the propagating wave from transient phenonena. The persistent topological features--pairs of zeros in amplitude with opposite winding number--we call branch points; all other ciculations, we label as noise. Prior to our work, this distinction was vague. As a point in fact, in the earlest days of adaptive optics all 2π circulations were lumped into what was then called the “slope discrepancy” or “null” space, and even today in the phase reconstruction process, standard wavefront sensors lump all 2π circulations into a single group so that they may be summarily discarded, hence Fried’s [4] terminology “hidden phase”.

D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - persistent pairs,” Optics Express (2011). Submitted for publication.

D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - characterization in wave optical simulation,” Optics Express (2011). Submitted for publication.

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

J. W. Goodman, Statistical Optics (John Wiley & Sons, New York, New York, 2000), Wiley Classics Library ed.

See for instance Ref. [11] page 394.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, USA, 1975), 2nd ed.

This analysis is true for any type of wavefront sensor. The Shack-Hartmann is presented here because it is well known. In our lab, we use a self-referencing interferometer.

This is in keeping with the notation in Fried’s seminal paper, Ref. [4].

First pointed out by Terry Brennan, the Optical Science Corporation, in a private conversation.

D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - modeling parameters,” Optical Society of America - Frontiers in Optics proceedings (2011). Accepted for publication.

D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “Branch points in deep turbulence and its relevance to adaptive optics – an overview of the ASALT laboratory’s deep turbulence research,” in “2010 DEPS Annual Conference,”, D. Herrick, ed. (Directed Energy Professional Society, 2010).

D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Optics Express (2011). Accepted for publication.
[PubMed]

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Figures (5)

Fig. 1
Fig. 1

A three dimensional view of measured branch point positions and helicities. In each of the plots, the two horizontal axes are spatial and the vertical is time. Fig. (a) plots the data after the circulations are identified in each frame but with all frames kept and “stacked” one atop the other. The red and green dots are the location of circulations with red positive helicity and green negative. The bottom row plots the data after creation pairs are identified and grouped with similar velocities. In this case, two distinct velocities are apparent. This is the number of atmospheric turbulence layers. Our later work [8,9] demonstrated that the separation, δ1, for all the creation pairs in group v1 are identical, and similarly there was a single δ2 for v2. (b) A plot of creation pairs with velocity v1 and separation δ1. (c) A plot of creation pairs with velocity v2 and separation δ2. After density was calculated for each of these two groups of creation pairs, it was shown that for each atmospheric turbulence layer from these branch point parameters, the layer’s altitude, strength and velocity can be estimated.

Fig. 2
Fig. 2

Left: Optical path difference for a sample atmospheric turbulence. Right: A first order model of the OPD on the left, keeping only the tilt terms in each coherence patch. The arrows indicate the direction of the beam based on the tilt in that patch. The green dot is the location of a null with the green arrows indication patches that contribute to that null. The blue arrows are patch that don’t contribute to the null at the green dot.

Fig. 3
Fig. 3

Branch point density versus normalized propagation distance.

Fig. 4
Fig. 4

A plot of theoretical Rytov (plotted along the horizontal axis) versus measured Rytov (along the vertical axis). The experimental data saturates as the Rytov parameters gets large.

Fig. 5
Fig. 5

Plotted is the distance between the branch points within a creation pair versus normalized distance from the turbulence layer.

Equations (7)

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2 E + k 2 n 2 E = 0
2 E + k 2 n 2 E + 2 ( E ) log n = 0 .
ϕ hid = Im { k = 1 J + K m k log [ ( x x k ) + i ( y y k ) ] }
Ω ρ 0 ϕ ( r )
j = j 0 j = j 0 + 3 ϕ ( r j )
0.5631 k 0 7 / 6 0 L C n 2 ( z ) z 5 / 6 d z > 0.1
S = E × B { S x = E z B y S y = E z B x S z = E x B y E y B x

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