## Evolution of branch points for a laser beam propagating through an uplink turbulent atmosphere |

Optics Express, Vol. 22, Issue 6, pp. 6569-6576 (2014)

http://dx.doi.org/10.1364/OE.22.006569

Acrobat PDF (1091 KB)

### 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

## 1. Introduction

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

6. I. Freund and 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]

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

2. 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]

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

3. N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and 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]

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

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

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

17. 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]

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

*2π*jump. The branch point is at the origin of the

*2π*discontinuity that causes these wave dislocations. Fried and Vaughn [8

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

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

19. M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam,” J. Opt. Soc. Am. A **25**(6), 1279–1286 (2008). [CrossRef] [PubMed]

21. F. S. Roux, “How to distinguish between the annihilation and the creation of optical vortices,” Opt. Lett. **38**(19), 3895–3898 (2013). [CrossRef] [PubMed]

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

23. 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**, 746606 (2009). [CrossRef]

^{’}altitude and strength [24

24. 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**, 781605 (2010). [CrossRef]

25. 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**(21), 22377–22392 (2010). [CrossRef] [PubMed]

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

## 2. Theory of branch points detection

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

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

*U(*

*r**, in which*

_{p,q})

*r**is the position vector of the sampling point*

_{p,q}*(p, q)*(

*p, q*are the sequence number of the points in x and y direction, respectively). Its phase can be determined simply bywhere

*Re{ U(*

*r**and*

_{p,q})}*Im{ U(*

*r**represent the real and the imaginary parts of the complex amplitude, respectively. The spatial gradient of the phase distribution can be defined aswhere the notation*

_{p,q})}*{…}*denotes taking a principal-value,

_{pv}*d*is the sampling interval,

*l**and*

_{x}

*l**are unit vectors parallel to x- and y-axis, respectively. Equation (2) can be also expressed as the function of the complex amplitude by substituting Eq. (1) into Eq. (2), that is [9*

_{y}9. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A **15**(10), 2759–2768 (1998). [CrossRef]

*C*denotes the closed loop contour and

*d*

**is a vector parallel to the tangent of the contour**

*r**C*. Using the discrete form of the phase gradient described above, the discrete form of Eq. (4) can be written asHere the contour

*C*is set to be a square whose four corners are at

*r*

_{p,q}_{,}

*r*

_{p + 1,q}_{,}

*r*

_{p + 1,q + 1}_{,}and

*r**. Because of the opposite direction between the integral orientation and vectors*

_{p,q + 1}

*l**,*

_{x}

*l**, two minuses appear in Eq. (5). If*

_{y}*S*in Eq. (5) equals to

_{p,q}*+ 2π*, it means that a positive branch point exists in the contour, while a negative branch point will appears in the contour if

*S*in Eq. (5) is equal to

_{p,q}*-2π.*If the result of Eq. (5) equals to

*0*, it means that no branch point is enclosed in the contour. Thus, the location of branch points and their number in distorted optical field can be detected by Eq. (5). Because branch points are created and annihilated in pairs with opposite rotation or sign, we could calculate the density of positive branch points instead of calculating the density of all the branch-points.

## 3. Simulation results

*h*is the height above ground level in kilometers,

*w*is an upper level wind speed, given as the Root Mean Square wind speed [m/s] average over the 5~20 km above ground level range. C

_{n}

^{2}(0) is the refractive index structure parameter at the ground level, which could be used to represent the atmospheric turbulence strength on the surface level. The vertical profile for the H-V model is shown in Fig. 1 with C

_{n}

^{2}(0) = 1.7 × 10

^{−14}m

^{-2/3}and w = 21 m/s .

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

_{n}

^{2}(0) is set to be 2.736 × 10

^{−14}m

^{-2/3}, 3.420 × 10

^{−14}m

^{-2/3}, 4.104 × 10

^{−14}m

^{-2/3}, 4.788 × 10

^{−14}m

^{-2/3}, and 5.472 × 10

^{−14}m

^{-2/3}, respectively. From Fig. 2(a) it can be seen that the density of branch-points changes with the turbulence and propagation height, which can be divided into four regions with different behaviors of the branch-points density. The boundaries of the regions are indicated by thin dotted lines (shown in Figs. 1, 2(a) and 3(a)). The first region is near the ground (0≤h≤1km) with rapidly dropped turbulence strength, in which the density of branch-points starts at zero and increases nonlinearly with the increasing of the height. The second one is a region (1km≤h≤5km) with the declining turbulence strength, in which the density of branch-points begins to grow rapidly, almost linearly. We know that branch points are created in distorted optical field under the combined influences of turbulence effect and diffraction, and the evolution of the branch-points density is mainly determined by the rates of branch-points pair creation and annihilation. The probability of annihilation would increase as the separation distance between the pair of branch-points decreases [21

21. F. S. Roux, “How to distinguish between the annihilation and the creation of optical vortices,” Opt. Lett. **38**(19), 3895–3898 (2013). [CrossRef] [PubMed]

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

*A*,

*B*and

*n*are parameters changed with propagation conditions. It is interesting that this formula is very similar to the formula used to describe Blackbody radiation in physics.

*A*decreases with the increase of C

_{n}

^{2}(0) and λ, while the parameter

*B*will decrease with the increase of C

_{n}

^{2}(0) and increase with the increase of λ. Further analysis to the fitted dada indicates that

*B*is nearly proportional to (λ/C

_{n}

^{2}(0))

^{2/3}.

^{2}are all larger than 0.99. For example, the correlation coefficient of the fitting is equal to 0.99716 for the situation of C

_{n}

^{2}(0) = 2.736 × 10

^{−14}m

^{-2/3}.

^{−14}m

^{-2/3}. From the fitted curves we see that the fitted result will be better with the increase of the wavelength. For example, when the wavelength is equal to 1.319μm, the correlation coefficient is equal to 0.9978, while the correlation coefficient is just 0.96762 when the wavelength is reduced to 0.6μm.

## 4. Conclusions

## Acknowledgments

## References and links

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

2. | J. F. Nye, “The motion and structure of dislocation in wavefronts,” Proc. R. Soc. London A Math. Phys. Sci. |

3. | N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A |

4. | V. A. Tartakovski and N. N. Mayer, “Phase dislocation and minimal phase representation of the wave function,” Atmos. Oceanic Opt. |

5. | B. V. Fortes and V. Lukin, “The effects of wavefront dislocations on the atmospheric adaptive optical systems performance,” Proc. SPIE |

6. | I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A |

7. | R. Rao, “Statistics of the fractal structure and phase singularity of a plane light wave propagation in atmospheric turbulence,” Appl. Opt. |

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

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

10. | C. Fan, Y. Wang, and Z. Gong, “Effect of branch points on adaptive optics,” High Power Laser Particle Beams |

11. | C. Fan, Y. Wang, and Z. Gong, “Effects of different beacon wavelengths on atmospheric compensation in strong scintillation,” Appl. Opt. |

12. | Y. Li, “Branch point effect on adaptive correction,” Proc. SPIE |

13. | C. A. Primmerman, T. R. Price, R. A. Humphreys, B. G. Zollars, H. T. Barclay, and J. H. Herrmann, “Atmospheric-compensation experiments in strong-scintillation conditions,” Appl. Opt. |

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

15. | E.-O. Le Bigot, W. J. Wild, and E. J. Kibblewhite, “Branch point reconstructors for discontinuous light phase functions,” Proc. SPIE |

16. | B. Wang, A. C. Koivunen, and M. C. Roggemann, “Comparison of branch point and least squares reconstructors for laser beam transmission through the atmosphere,” Proc. SPIE |

17. | D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. |

18. | V. V. Voitsekhovich, D. Kouznetsov, and D. K. Morozov, “Density of turbulence-induced phase dislocations,” Appl. Opt. |

19. | M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam,” J. Opt. Soc. Am. A |

20. | F. S. Roux, “Anomalous transient behavior from an inhomogeneous initial optical vortex density,” J. Opt. Soc. Am. A |

21. | F. S. Roux, “How to distinguish between the annihilation and the creation of optical vortices,” Opt. Lett. |

22. | D. J. Sanchez and D. W. Oesch, “The aggregate behavior of branch points - the creation and evolution of branch points,” Proc. SPIE |

23. | 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 |

24. | 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 |

25. | 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 |

26. | D. J. Sanchez, D. W. Oesch, and P. R. Kelly, “The aggregate behavior of branch points - theoretical calculation of branch point velocity,” Proc. SPIE |

27. | D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “Aggregate behavior of branch points--persistent pairs,” Opt. Express |

28. | D. W. Oesch, D. J. Sanchez, and P. R. Kelly, “Optical vortex density in Rytov saturated atmospheric turbulence,” in |

29. | D. J. Sanchez and D. W. Oesch, “Localization of angular momentum in optical waves propagating through turbulence,” Opt. Express |

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

31. | X. Qian, W. Zhu, and R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sin. |

32. | D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. |

33. | R. J. Sasiela, |

**OCIS Codes**

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(010.7060) Atmospheric and oceanic optics : Turbulence

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: January 7, 2014

Revised Manuscript: February 13, 2014

Manuscript Accepted: February 27, 2014

Published: March 13, 2014

**Citation**

Xiao-Lu Ge, Xuan Liu, and Cheng-Shan Guo, "Evolution of branch points for a laser beam propagating through an uplink turbulent atmosphere," Opt. Express **22**, 6569-6576 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6569

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### References

- 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]
- 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]
- V. A. Tartakovski, N. N. Mayer, “Phase dislocation and minimal phase representation of the wave function,” Atmos. Oceanic Opt. 8, 231–235 (1995).
- B. V. Fortes, V. Lukin, “The effects of wavefront dislocations on the atmospheric adaptive optical systems performance,” Proc. SPIE 2778, 1002–1003 (1996).
- I. Freund, N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994). [CrossRef] [PubMed]
- 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]
- D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31(15), 2865–2882 (1992). [CrossRef] [PubMed]
- D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15(10), 2759–2768 (1998). [CrossRef]
- C. Fan, Y. Wang, Z. Gong, “Effect of branch points on adaptive optics,” High Power Laser Particle Beams 15, 435–438 (2003).
- 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]
- Y. Li, “Branch point effect on adaptive correction,” Proc. SPIE 5490, 1064–1070 (2004). [CrossRef]
- 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]
- D. C. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley, 1998).
- 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]
- 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]
- V. V. Voitsekhovich, D. Kouznetsov, D. K. Morozov, “Density of turbulence-induced phase dislocations,” Appl. Opt. 37(21), 4525–4535 (1998). [CrossRef] [PubMed]
- M. Chen, F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam,” J. Opt. Soc. Am. A 25(6), 1279–1286 (2008). [CrossRef] [PubMed]
- F. S. Roux, “Anomalous transient behavior from an inhomogeneous initial optical vortex density,” J. Opt. Soc. Am. A 28(4), 621–626 (2011). [CrossRef] [PubMed]
- F. S. Roux, “How to distinguish between the annihilation and the creation of optical vortices,” Opt. Lett. 38(19), 3895–3898 (2013). [CrossRef] [PubMed]
- 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. 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. 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. W. Oesch, D. J. Sanchez, and P. R. Kelly, “Optical vortex density in Rytov saturated atmospheric turbulence,” in FiO (2012), FW3A. 3.
- 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]
- 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).
- D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).
- R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE, 2007).

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