## Creating well-defined orbital angular momentum states with a random turbulent medium |

Optics Express, Vol. 20, Issue 11, pp. 12292-12302 (2012)

http://dx.doi.org/10.1364/OE.20.012292

Acrobat PDF (930 KB)

### Abstract

Previous work by Allen, demonstrated that optical beams possess orbital angular momentum. Other work has shown that a random, phase-only disturbance can impart ±1 orbital angular momentum states to propagating waves. However, the field preceding the formation of these ±1 states was unknown. In this paper, we identify the unique field that leads to the formation of a pair of branch points, indicators of orbital angular momentum. This field is then verified in a bench-top optical experiment.

© 2012 OSA

## 1. Introduction

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]

4. A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. **12**, 083053 (2010). [CrossRef]

5. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetso, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**, 5448–5456 (2004). [CrossRef] [PubMed]

7. G. A. Tyler and R. W. Boyd, “Influenece of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. **34**, 142–144 (2009). [CrossRef] [PubMed]

8. T. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. **14**, 033001 (2012). [CrossRef]

9. S. J. van Enk, “Entanglement of electromagnetic fields,” Phys. Rev. A **67**, 022303 (2003). [CrossRef]

10. B. J. Pors, C. H. Monken, E. R. Eliel, and J. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express **19**, 6671–6683 (2011). [CrossRef] [PubMed]

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

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

13. W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of *H*_{2} pure rotational line emission from the GG Tauri binary system,” Astrophys. J. Lett. **521**, L63–L66 (1999). [CrossRef]

15. N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. **492**, 883–922 (2008). [CrossRef]

16. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. **121**, 36–40 (1995). [CrossRef]

17. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. **184**, 67–71 (2000). [CrossRef]

18. S. S. R. Oemrawsingh, J. A. W. van Houweilingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosteboer, and G. W. ‘t Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. **43**, 688–694 (2004). [CrossRef] [PubMed]

5. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetso, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**, 5448–5456 (2004). [CrossRef] [PubMed]

19. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with patial light modulators: application to quantum key distribution,” Appl. Opt. **47**, A32–A42 (2008). [CrossRef] [PubMed]

20. 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]

21. 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]

22. 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]

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

5. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetso, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**, 5448–5456 (2004). [CrossRef] [PubMed]

18. S. S. R. Oemrawsingh, J. A. W. van Houweilingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosteboer, and G. W. ‘t Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. **43**, 688–694 (2004). [CrossRef] [PubMed]

19. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with patial light modulators: application to quantum key distribution,” Appl. Opt. **47**, A32–A42 (2008). [CrossRef] [PubMed]

## 2. Background

21. 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]

*z*

_{0}[23], following a turbulence layer before branch points will form.

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

*z*

_{0}the phase gradients can be separated into two orthogonal components. Now consider the possibility that there exists a component of the phase prior to

*z*

_{0}that goes on to form a branch point pair, and hence OAM. Here, we place that supposition on firm grounds and identify the part of the phase that goes on to create a branch point pair.

### 2.1. Geometry

*z*>

*z*

_{0}, after the disturbance.

*z*=

*ℓ*

_{1}and Plane 2 at

*z*=

*ℓ*

_{2}, orthogonal to the direction of propagation, such that

*ℓ*

_{1}<

*z*

_{0}<

*ℓ*

_{2}and

*ℓ*

_{2}–

*ℓ*

_{1}=

*L*. The planes are defined by the transverse coordinates,

*x*and

*y*, see Fig. 1. The positions of the positive and negative branch points in the pair are identified by (

*x*,

_{p}*y*) and (

_{p}*x*,

_{n}*y*) respectively. The branch point pair separation,

_{n}*δ*, is given by

### 2.2. Unique mapping

*λ*, and wave number,

*k*= 2

*π*/

*λ*. For clarity, we describe the field at Plane 1 as the precursor field. Then the precursor field,

*A*

_{1}

*e*

^{iϕ1}, and the field associated with a pair of branch points,

*A*

_{2}

*e*

^{iϕ2}, form a unique Fresnel transform pair [27].

## 3. The Field at Plane 1

### 3.1. Functional behavior

### 3.2. The Precursor phase

*x*

_{0},

*y*

_{0}) and an angular dependence,

*θ*about the midpoint for orienting the branch point pair. Substituting Eq. (5) into

*ϕ*=

_{pre}*P*+

_{s}*P*and making the addition of these two new parameters gives the general form, with

_{a}### 3.3. Characteristics of the precursor phase

*c*

_{1},

*c*

_{0}and

*c*of Eq. (5), there is a linear relationship between the precursor phase and the branch point pair separation.

_{a}*P*, forms the rings of the precursor phase, seen in Fig. 2(a). These are a form of Fresnel diffraction [26] resulting from the zero amplitude of a branch point acting as an obscuration to the back-propagated field. As Fresnel rings this underlying distribution of the precursor phase is dependent on the propagation distance between the planes. This is reflected in the dependence of the arguments of the Bessel functions,

_{s}*ω*and

_{s}*ω*, on the Fresnel number,

_{a}*N*, in Eq. (5).

_{F}*P*, the branch point separation affects the steepness of the precursor phase between it’s extremum. The location of these extremum in Plane 1, seen in Fig. 2(a), lie on a line perpendicular to the one connecting the branch points in Plane 2 on Fig. 2(b). The slope through the origin along the line connecting the extremum of the precursor phase becomes more severe with increasing separation.

_{a}*δ*is too large for the distance between the planes used in the transform. In other words, a residual discontinuity in Plane 1, indicates that

*ℓ*

_{1}>

*z*

_{0}and the system does not follow our assumed geometry, leading to an incorrect solution.

## 4. Experimental demonstration

*N*= 31. At Plane 1 sits the DM with a resolution of 31x31 actuators, and a temporal, self-referencing interferometer (SRI) [31

_{F}31. T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE **5553**, 112–126 (2004). [CrossRef]

*θ*=

*π*/3 rotation in our WFS plane. This places the positive and negative branch points at approximately (23.00,15.19) and (17.00,4.80) respectively. Next we calculated the precursor phase in the DM plane as was done in Sec. 3. A plot of the result is shown in Fig. 3(a). This precursor phase is then binned down to the 31x31 actuator density of our DM, Fig. 3(b). The binned phase is shown in numbers of waves before being scaled into DM commands. Interestingly, the range of the binned phase is on the order of a half of a wave, peak-to-valley.

*x*,

_{p}*y*) ≃ (32,15) and (

_{p}*x*,

_{n}*y*) ≃ (23,2) for the positive and negative circulations respectively. This gives the created pair a measured separation of

_{n}*δ*= 15.8 pixels. The position of the midpoint of the branch point pair rests at (

*x*

_{0},

*y*

_{0}) = (27.5,8.5) with a rotation

*θ*≃

*π*/3.25.

*ϕ*described by Eq. (6), is linear with respect to

_{pre}*δ*, so an increase to the scale of the phase is reflected as an increase in the separation of the points at a fixed distance. Branch point pairs, once formed, separate with additional propagation [23]. Therefore, if the Fresnel number for our experimental planes is less than

*N*= 31, the pair will also appear with a larger separation.

_{F}## 5. Discussion and summary

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

6. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. **94**, 153901 (2005). [CrossRef] [PubMed]

7. G. A. Tyler and R. W. Boyd, “Influenece of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. **34**, 142–144 (2009). [CrossRef] [PubMed]

21. 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]

## A. Numerical solution for the precursor phase

29. Y. M. Engelberg and S. Ruschin, “Fast method for physical optics propagation of high-numerical-aperutre beams,” J. Opt. Soc. Am. A **21**, 2135–2145 (2004). [CrossRef]

*H*(

*z*) is the Fourier transform of the propagation kernel,

*h*(

*x*,

*y*) (Eq. (2)).

*A*

_{2}(

*x*,

*y*). The Fresnel rings phase can be reproduced by applying the inverse transform to only

*A*

_{2}(

*x*,

*y*), a top-hat function with uniform, non-zero amplitude and constant phase, Fig. 4(b). The precursor phase then is the difference between the phases in Figs. 4(a) and (b), shown in Fig. 4(c).

## B. Fitting precursor phases

*N*= 26, in the left column. In the middle and right columns the symmetric and asymmetric cross sections of the precursor phases are shown. Red and blue curves represent the precursor phase estimate from angular spectrum propagation, Eq. (7), and the identified general form, Eq. (6), respectively. As can be seen from comparing the images of the 2-D precursor phase in Fig. 5 column(b), the only change to the symmetric portion of the phase is an increase in the magnitude of the pattern.

_{F}*δ*. This deviation is almost entirely limited to the left side of the asymmetric curves, those values where the phase is greater than zero. This deviation is small however compared to the magnitude of the current term,

*P*.

_{a}*δ*= 3. In this case, the general form matches well across all Fresnel numbers for both the symmetric and asymmetric fits to some radius. At larger Fresnel numbers the obscuration rings become suppressed in the numerical solution beyond some radius. This suppression can be matched by adding a weak Gaussian envelope function to the symmetric term. However, doing so increases the complexity of the functional dependencies in Eq. (5) without significant improvement in the resulting fits and so wasn’t included in this paper.

## Acknowledgments

## References and links

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 |

2. | H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Bull. Am. Phys. Soc. |

3. | S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre-Gaussian beams,” J. Opt. Soc. Am. A |

4. | A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. |

5. | G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetso, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express |

6. | C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. |

7. | G. A. Tyler and R. W. Boyd, “Influenece of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. |

8. | T. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. |

9. | S. J. van Enk, “Entanglement of electromagnetic fields,” Phys. Rev. A |

10. | B. J. Pors, C. H. Monken, E. R. Eliel, and J. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express |

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

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

13. | W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of |

14. | J. S. Bary, D. A. Weintraub, and J. H. Kastner, “Detection of molecular hydrogen orbiting a ‘naked’ T Tauri star,” Astrophys. J. Lett. |

15. | N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. |

16. | M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. |

17. | L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. |

18. | S. S. R. Oemrawsingh, J. A. W. van Houweilingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosteboer, and G. W. ‘t Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. |

19. | M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with patial light modulators: application to quantum key distribution,” Appl. Opt. |

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

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

22. | 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 |

23. | 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 |

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

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

26. | J. W. Goodman, |

27. | J. D. Gaskill, |

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

29. | Y. M. Engelberg and S. Ruschin, “Fast method for physical optics propagation of high-numerical-aperutre beams,” J. Opt. Soc. Am. A |

30. | G. N. Watson, |

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

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

**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: March 30, 2012

Revised Manuscript: April 19, 2012

Manuscript Accepted: April 19, 2012

Published: May 16, 2012

**Citation**

Denis W. Oesch and Darryl J. Sanchez, "Creating well-defined orbital angular momentum states with a random turbulent medium," Opt. Express **20**, 12292-12302 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-12292

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### References

- 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]
- H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Bull. Am. Phys. Soc.75, 826–829 (1995).
- S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre-Gaussian beams,” J. Opt. Soc. Am. A26, 625–638 (2009). [CrossRef]
- A. Picon, A. Benseny, J. Mompart, J. R. Vazquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12, 083053 (2010). [CrossRef]
- G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetso, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express12, 5448–5456 (2004). [CrossRef] [PubMed]
- C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett.94, 153901 (2005). [CrossRef] [PubMed]
- G. A. Tyler and R. W. Boyd, “Influenece of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett.34, 142–144 (2009). [CrossRef] [PubMed]
- F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys.14, 033001 (2012). [CrossRef]
- S. J. van Enk, “Entanglement of electromagnetic fields,” Phys. Rev. A67, 022303 (2003). [CrossRef]
- B. J. Pors, C. H. Monken, E. R. Eliel, and J. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express19, 6671–6683 (2011). [CrossRef] [PubMed]
- G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt.40, 73–87 (1993). [CrossRef]
- I. D. Maleev and G. A. Swartzlander, “Composite optical vorticies,” J. Opt. Soc. Am. B20, 1169–1176 (2003). [CrossRef]
- W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of H2 pure rotational line emission from the GG Tauri binary system,” Astrophys. J. Lett.521, L63–L66 (1999). [CrossRef]
- J. S. Bary, D. A. Weintraub, and J. H. Kastner, “Detection of molecular hydrogen orbiting a ‘naked’ T Tauri star,” Astrophys. J. Lett.576, L73–L76 (2002). [CrossRef]
- N. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys.492, 883–922 (2008). [CrossRef]
- M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun.121, 36–40 (1995). [CrossRef]
- L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun.184, 67–71 (2000). [CrossRef]
- S. S. R. Oemrawsingh, J. A. W. van Houweilingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosteboer, and G. W. ‘t Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt.43, 688–694 (2004). [CrossRef] [PubMed]
- M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with patial light modulators: application to quantum key distribution,” Appl. Opt.47, A32–A42 (2008). [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]
- 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 - altitude and strength of atmospheric turbulence layers,” Proc. SPIE7816, 0501–0513 (2010).
- D. C. Johnston and B. M. Welsh, “Estimating the contribution of different parts of the atmosphere to optical wavefront aberration,” Comput. Electr. Eng.18, 467–483 (1992). [CrossRef]
- D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt.31, 2865–2882 (1992). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
- J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, 1st ed (John Wiley & Sons, 1978).
- D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A15, 2759–2768 (1998). [CrossRef]
- Y. M. Engelberg and S. Ruschin, “Fast method for physical optics propagation of high-numerical-aperutre beams,” J. Opt. Soc. Am. A21, 2135–2145 (2004). [CrossRef]
- G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed (Cambridge Univesity Press, 1962).
- T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE5553, 112–126 (2004). [CrossRef]
- D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “The aggregate behavior of branch points - persistent pairs,” Opt. Express2, 1046–1059 (2012). [CrossRef]

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