## Aggregate behavior of branch points - persistent pairs |

Optics Express, Vol. 20, Issue 2, pp. 1046-1059 (2012)

http://dx.doi.org/10.1364/OE.20.001046

Acrobat PDF (6249 KB)

### 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

## 1. Introduction

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]

*π*circulation, described by the complex function

*e*

^{±iθ}, where

*θ*describes the azimuthal angle.

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

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

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]

12. 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]

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]

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]

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]

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). [CrossRef] [PubMed]

20. 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]

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]

## 2. Problem formulation and assumptions

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]

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

*π*phase on a 256

*x*256 spatial grid. Our WFS gathers short exposure frames at a constant frequency. These measurements include both photon and read noise.

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

*π*a branch point is assigned to the intersection of the four pixels. An array of these locations of size 255

*x*255, is called the polarity array. It is composed of zeros or ±1 for those positions of branch points with positive or negative helicity respectively.

*k*

_{0}is the wave number,

*z*is the altitude with the telescope at

*z*= 0, and

*L*is the maximum height of the turbulence. We have observed experimentally that branch points begin to appear in the data at

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]

*π*discontinuity extending outward from the branch point. Figure 1(b) shows two branch points of opposite helicity sharing a common branch cut. In typical branch cut placing problems as those presented in [3], the work is of pairing branch points is being done on a single frame of phase data. Although the locations of branch points in the wrapped phase data are fixed by the turbulence, the arrangement of the branch cuts is not and in order to obtain a correct placement additional information is needed [3]. The suggestion of additional information usually takes the form of guide maps or other metrics to weigh how the branch cuts are placed. In previous work, no consideration is given to using a time series of phase data as is collected with a WFS in adaptive optics. Our additional constraint relies on identifying the arrangement of branch cuts through time. This additional constraint allows for proper pairing of branch points, as is shown in the next section.

## 3. Pairing algorithm

*mod*

_{2π}phase our pairing algorithm has four main stages; (1) branch point identification, (2) determining branch point velocities, (3) 2-D pairing, and (4) velocity filtering. The process is shown at a cartoon level in Fig. 2.

*x*256 pixels per frame.

### 3.1. Branch point identification

*x*255

*x*100 zeros. A few frames of this array are represented in Fig. 2(a). Branch points are identified, as described in Section 2. Then at the locations of the branch points the polarity array filled with ±1 appropriately. The locations of positive and negative circulations are shown as red and green dots in Fig. 2(b). The polarity array forms the basis of subsequent processing.

### 3.2. Estimation of branch point velocities

*x/*Δ

*tî*+ Δ

*y/*Δ

*tj*̂, is calculated with respect to every other point of like polarity throughout the 255

*x*255

*x*100 array, with

*î*and

*ĵ*representing unit vectors in the

*x*and

*y*directions. Our previous research has shown that a histogram of all these velocity components contains strong correlation peaks that signify the branch point group velocities [15

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]

### 3.3. 2-D pairing

*π*discontinuities in the phase data

#### 3.3.1. The difference of gradients

*π*discontinuities in a frame of WFS data, we compare the signs of the x and y gradients computed as with those computed in

*mod*

_{2π}space, We find that the signs of Eqs. (1) and (2) are reversed in the presence of a discontinuity. This leads to a simple process for reducing a frame of WFS data to a map of 2

*π*discontinuities. Combining the signs of Eqs. (1) and (2), creates two arrays composed only of delta functions at the positions of discontinuities, The arrays of Eq. (3), by the nature of the differences in Eqs. (1) and (2), are 256x255 and 255x256 for x and y respectively. Shifting and adding the absolute values of these arrays creates a 255x255 map of the 2

*π*discontinuities, This is shown pictorially in Fig. 3.

#### 3.3.2. Scanning piston

*π*discontinuities and those due to wrapping effects from the phase being represented in

*mod*

_{2}

*space, this approach can lead to incorrect or missed pairs. A sample of this is shown in Fig. 4. Here several branch point pairs have been highlighted in the insets to show how the discontinuities fell in the difference of gradient method. Sometimes the pair is clearly connected by a branch cut. However, wrapping discontinuities can interact with branch cuts in such a way as to obscure the correct branch cut placement.*

_{π}*π*discontinuities not associated with the branch points move, providing a different view of the branch cuts, see Fig. 5. Through varying the amount of added piston between 0 and 2

*π*while applying the walking algorithm at each step, provides a large set of possible branch point pairs for a given frame of WFS data. Sorting these pairings according to frequency greatly reduces the number of errors due to phase wrapping.

### 3.4. Velocity filtering

## 4. Two layer demonstration

### 4.1. The experimental apparatus

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]

*r*

_{0}), Rytov number (

*f*), are controllable and repeatable. All of the data presented here was collected with a 256x256 pixel resolution SRI WFS, scaled to a 1.5 meter aperture. The placement of the phase wheels selects the

_{G}*r*

_{0}and

### 4.2. Turbulence parameters

*r*

_{0}and

### 4.3. Close-up of Case 1

#### 4.3.1. Examination of the unpaired points

#### 4.3.2. 1 to 2 layer comparison

### 4.4. Other configuration results

## 5. Discussion

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]

## Acknowledgments

## References and links

1. | D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. |

2. | D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A |

3. | D. C. Ghiglia and M. D. Pritt, |

4. | J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A |

5. | G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. |

6. | I. Freund, “Optical vortices in gaussian random fields: Statistical probability densities,” J. Opt. Soc. Am. |

7. | F. S. Roux, “Coupling of noncanonical optical vortices,” J. Opt. Soc. Am. |

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 |

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 |

10. | J. E. Curtis and D. G. Grier, “Structure of optical vorticies,” Phys. Rev. |

11. | I. D. Maleev and G. A. Swartzlander Jr., “Composite optical vorticies,” J. Opt. Soc. Am. B |

12. | E. O. Le Bigot and W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. |

13. | T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express |

14. | 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 |

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 |

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 |

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 |

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 |

19. | 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. |

20. | K. Murphy, D. Burke, N. Devaney, and C. Dainty, “Experimental detection of optical vorticies with a shack-hartmann wavefront sensor,” Opt. Express |

21. | D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Opt. Express |

22. | D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Express |

23. | T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE |

24. | D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electron. Eng. |

25. | M. Schöck and E. J. Spillar, “Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor,” Proc. SPIE |

26. | M. Schöck and E. J. Spillar, “Method for a quantitative investiagation of the frozen flow hypothesis,” J. Opt. Soc. Am. A |

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 |

28. | R. J. Sasiela, |

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. |

30. | S. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(010.1285) Atmospheric and oceanic optics : Atmospheric correction

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: August 12, 2011

Revised Manuscript: November 10, 2011

Manuscript Accepted: December 1, 2011

Published: January 4, 2012

**Citation**

Denis W. Oesch, Darryl J. Sanchez, and Carolyn M. Tewksbury-Christle, "Aggregate behavior of branch points - persistent pairs," Opt. Express **20**, 1046-1059 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1046

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### References

- D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt.31, 2865–2882 (1992). [CrossRef] [PubMed]
- D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A15, 2759–2768 (1998). [CrossRef]
- D. C. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley and Sons, 1998).
- J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London A336, 165–190 (1974). [CrossRef]
- G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt.40, 73–87 (1993). [CrossRef]
- 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]
- 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. A27, 2138–2143 (2008). [CrossRef]
- 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. A45, 8185–8189 (1992). [CrossRef] [PubMed]
- 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. B20, 1169–1176 (2003). [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]
- T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express16, 6985–6998 (2008). [CrossRef] [PubMed]
- 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. SPIE7466, 0501–0512 (2009).
- 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. Express18, 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. SPIE7466, 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. SPIE7816, 0501–0513 (2010).
- 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. Express14, 11919–11924 (2006). [CrossRef] [PubMed]
- 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. Express18, 15448–15460 (2010). [CrossRef] [PubMed]
- D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Opt. Express19, 25388–25396 (2011). [CrossRef]
- D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Express19, 24596–24068 (2011). [CrossRef] [PubMed]
- T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE5553 (2004). [CrossRef]
- 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]
- M. Schöck and E. J. Spillar, “Analyzing atmospheric turbulence with a shack-hartmann wavefront sensor,” Proc. SPIE3353, 1092–1099 (1998). [CrossRef]
- M. Schöck and E. J. Spillar, “Method for a quantitative investiagation of the frozen flow hypothesis,” J. Opt. Soc. Am. A17, 1650–1658 (2000). [CrossRef]
- 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. A26, 833–846 (2009). [CrossRef]
- R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE Press, 2007).
- 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]
- S. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE5553, 290–300 (2004). [CrossRef]

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