## Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence |

Optics Express, Vol. 22, Issue 7, pp. 7765-7772 (2014)

http://dx.doi.org/10.1364/OE.22.007765

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### Abstract

We develop a novel model of the probability density of the orbital angular momentum (OAM) modes for Hankel-Bessel beams in paraxial turbulence channel based on the Rytov approximation. The results show that there are multi-peaks of the mode probability density along the radial direction. The peak position of the mode probability density moves to beam center with the increasing of non-Kolmogorov turbulence-parameters and the generalized refractive-index structure parameters and with the decreasing of OAM quantum number, propagation distance and wavelength of the beams. Additionally, larger OAM quantum number and smaller non-Kolmogorov turbulence-parameter can be selected in order to obtain larger mode probability density. The probability density of the OAM mode crosstalk is increasing with the decreasing of the quantum number deviation and the wavelength. Because of the focusing properties of Hankel-Bessel beams in turbulence channel, compared with the Laguerre-Gaussian beams, Hankel-Bessel beams are a good light source for weakening turbulence spreading of the beams and mitigating the effects of turbulence on the probability density of the OAM mode.

© 2014 Optical Society of America

## 1. Introduction

1. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. **94**(15), 153901 (2005). [CrossRef] [PubMed]

10. S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express **20**(1), 452–461 (2012). [CrossRef] [PubMed]

1. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. **94**(15), 153901 (2005). [CrossRef] [PubMed]

2. F. E. S. Vetelino and R. J. Morgana, “Model validation of turbulence effects on orbital angular momentum of single photons for optical communication,” Proc. SPIE **7685**, 76850R (2010). [CrossRef]

3. Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. **303**, 38–41 (2013). [CrossRef]

4. X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. **43**(6–10), 121–127 (2012). [CrossRef]

5. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. **47**(13), 2414–2429 (2008). [CrossRef] [PubMed]

6. G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. **34**(2), 142–144 (2009). [CrossRef] [PubMed]

7. B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. **37**(17), 3735–3737 (2012). [CrossRef] [PubMed]

8. Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. **284**(5), 1132–1138 (2011). [CrossRef]

10. S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express **20**(1), 452–461 (2012). [CrossRef] [PubMed]

## 2. Mode probability density of OAM

*z*

^{2}+

*r*

^{2})

^{1/2}≈

*z*+

*r*

^{2}/2

*z*, traveling scalar wave

11. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Hankel-Bessel laser beams,” J. Opt. Soc. Am. A **29**(5), 741–747 (2012). [CrossRef] [PubMed]

3. Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. **303**, 38–41 (2013). [CrossRef]

*β*(

_{l}*r*,

*z*) is given by the integral

*I*

_{n}(

*η*) is the Bessel function of second kind with

*r*where

*ρ*

_{0}is the spatial coherence radius of a spherical wave propagating in the non-Kolmogorov turbulence [13

13. X. Sheng, Y. Zhang, F. Zhao, L. Zhang, and Y. Zhu, “Effects of low-order atmosphere-turbulence aberrations on the entangled orbital angular momentum states,” Opt. Lett. **37**(13), 2607–2609 (2012). [CrossRef] [PubMed]

*α*is the non-Kolmogorov turbulence-parameter, Г(

*α*) denotes the Gamma function and

^{3-α}.

## 3. Numerical results

*r*for the wavelength

*λ*= 1550nm, propagation distance

*z*= 1km,

*α*= 11/3 and the refractive index structure parameter

^{3-}

*is shown in Fig. 1(a), where the effects of different initial mode indices*

^{α}*l*

_{0}= 1,2,3 are considered. It is clear that the maximum of the

*r*= 0 as the value

*l*

_{0}increases, and the

*l*

_{0}. The increasing of the OAM quantum number

*l*

_{0}gives rise to the broadening of the first ring of the OAM mode probability. Larger

*l*

_{0}can be the better choice for HB beams to propagate in the turbulent channel. The probability density

*l*

_{0}and

*l*is shown in Fig. 1(b). Clearly, the

*l*

_{0}and

*l*

_{0}± 1 are larger than that of between the OAM states

*l*

_{0}and

*l*

_{0}+ 2, and the maximum crosstalk of the mode probability density moves away from

*r*= 0 while the values Δ

*l*=

*l*±

*l*

_{0}increase. Compared with LG beams in the same parameters in Figs. 1(c) and 1(d), the spreading of LG beams is more seriously for different initial mode indices and LG beams have only one peak. Additionally, the mode probability density of the HB beams is much larger than that of LG beams, but the crosstalk probability density of the HB beams is smaller.

*α*= 3.07, 3.37, 3.67 and 3.97 with the same wavelength

*λ*= 1550nm,

*z*= 1km, mode indice

*l*

_{0}= 1 and the refractive index structure parameter

^{3-}

*is presented in Figs. 2(a)-2(c). Clearly, the peak position of the mode probability densitiy moves to the beam center as the non-Kolmogorov turbulence-parameter*

^{α}*α*increases. Note, the mode probability densities are the same at the values

*α*= 3.37 and

*α*= 3.67. It can be seen in Figs. 2(a)-2(c) that the ring breadth of the probability density of the mode or mode crosstalk is independent on the non-Kolmogorov property of the turbulence, while the mode probability density at the non-Kolmogorov turbulence-parameter

*α*= 3.07 is larger than that of

*α*= 3.97, that is to say the probability density of the crosstalk mode at

*α*= 3.97 is larger than that of

*α*= 3.07. We should choose smaller

*α*to hold more energy in the turbulent atmosphere. Moreover, we draw comparisons with the situation in LG beams (Figs. 2(d) and 2(e)). Their mode probability density and crosstalk probability density are almost the same.

*r*with the mode indice

*l*

_{0}= 1,

*λ*= 1550nm, propagation distance

*z*= 1km and

*α*= 11/3 for the structure parameters

^{3-}

*. From Fig. 3(a), it is observed that increasing*

^{α}*r*of the mode probability density moves to beam center. The effects of the refractive index structure parameter

*l*= 2, and

*l*

_{0}= 1 are depicted in Fig. 3(b).Note the ring breadth of the probability density of the mode or mode crosstalk is independent on the

*r*in Fig. 4(a) for propagation distance

*z*= 0.5 and 1km, assuming the mode indice

*l*

_{0}= 1, wavelength

*λ*= 1550nm,

*α*= 11/3 and the refractive index structure parameter

^{3-}

*. As is indicated in Fig. 4(a), the mode probability density*

^{α}*z*, and increasing

*z*, the broader breadth of the first ring of the mode probability density is brought about. Moreover, the maximum value position of the mode probability density moves away from

*r*= 0 as the propagation distance

*z*increases. As is shown in Fig. 4(b), the longer propagation distance, the larger crosstalk of the mode probability densities of the OAM modes. Clearly, increasing the propagation distance, also results in broader breadth of the first ring of the probability density of the mode crosstalk. By comparison to LG beams in Figs. 4(c) and 4(d), the mode probability density of LG beams is larger, but the spreading of the main peak of HB beams is much smaller.

*r*for wavelengths

*λ*= 690, 785, 850 and 1550nm, assuming the mode indice

*l*

_{0}= 1, propagation distance

*z*= 0.5km,

*α*= 11/3 and the refractive index structure parameter

^{3-}

*. It can be seen from Fig. 5(a) that the mode probability density decreases as the wavelength decreases, while in Fig. 5(b) the probability density of the mode crosstalk increases as the wavelength decreases. When the wavelength increases, the width of the first ring of the mode probability and the probability density of the mode crosstalk all become broader. The energy which the larger*

^{α}*λ*holds is larger, but its noise is larger too. It is the opposite way round-the mode probability density of LG beams raises as

*λ*decreases(Figs. 5(c) and 5(d)). The probability density of LG beams is much larger than HB beams, and in similar matters, the spreading of the main light spot is smaller.

## 4. Conclusions

*r*axis. The peak position of the mode probability density moves away from

*r*= 0 with the increasing of the OAM quantum number

*l*

_{0}, the propagation distance

*z*and the wavelength

*λ*of HB beams and the decreasing of the non-Kolmogorov turbulence-parameters

*α*and the generalized refractive-index structure parameters

*l*

_{0}or increasing

*z*result in decreasing of the mode probability density, but the mode probability densities are the same at the values of

*α*= 3.37 and

*α*= 3.67. We should choose larger

*l*

_{0}, smaller

*λ*and the deviation Δ

*l*of OAM quantum number. In addition, the ring breadth of the probability density of the mode or mode crosstalk is independent with

*α*and

*z*,

*λ*and

*l*

_{0}. Smaller

*λ*and

*l*

_{0}are the better choice to mitigate beam spreading in the turbulence atmospheric. Compared with LG beams in the same parameters, we can conclude that although HB beams have multi-peak mode probability densities, most of the energy is located between

*r*= 0 and

*r*= 5cm, which is really a small region. The spreading of LG beams is more seriously than HB beams. Similarly to the case of no turbulence [11

11. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Hankel-Bessel laser beams,” J. Opt. Soc. Am. A **29**(5), 741–747 (2012). [CrossRef] [PubMed]

## References and links

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

2. | F. E. S. Vetelino and R. J. Morgana, “Model validation of turbulence effects on orbital angular momentum of single photons for optical communication,” Proc. SPIE |

3. | Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. |

4. | X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. |

5. | J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. |

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

7. | B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. |

8. | Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. |

9. | J. Wang, J. Jia, J. Xu, Y. Wang, and Y. Zhang, “The probability of orbital angular momentum states of single photons with Z-tilt corrected residual aberration in a slant path turbulent atmosphere,” Optik |

10. | S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express |

11. | V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Hankel-Bessel laser beams,” J. Opt. Soc. Am. A |

12. | I. S. Gradshteyn and I. M. Ryzhik, |

13. | X. Sheng, Y. Zhang, F. Zhao, L. Zhang, and Y. Zhu, “Effects of low-order atmosphere-turbulence aberrations on the entangled orbital angular momentum states,” Opt. Lett. |

14. | F. Li, H. Tang, Y. Jiang, and J. Ou, “Spiral spectrum of Laguerre-Gaussian beams propagating in turbulent atmosphere,” Acta Phys. Sin. |

**OCIS Codes**

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(010.3310) Atmospheric and oceanic optics : Laser beam transmission

(060.2605) Fiber optics and optical communications : Free-space optical communication

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: October 29, 2013

Manuscript Accepted: December 2, 2013

Published: March 27, 2014

**Citation**

Yu Zhu, Xiaojun Liu, Jie Gao, Yixin Zhang, and Fengsheng Zhao, "Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence," Opt. Express **22**, 7765-7772 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-7765

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### References

- C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005). [CrossRef] [PubMed]
- F. E. S. Vetelino, R. J. Morgana, “Model validation of turbulence effects on orbital angular momentum of single photons for optical communication,” Proc. SPIE 7685, 76850R (2010). [CrossRef]
- Y. Jiang, S. Wang, J. Zhang, J. Ou, H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013). [CrossRef]
- X. Sheng, Y. Zhang, X. Wang, Z. Wang, Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43(6–10), 121–127 (2012). [CrossRef]
- J. A. Anguita, M. A. Neifeld, B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008). [CrossRef] [PubMed]
- G. A. Tyler, R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34(2), 142–144 (2009). [CrossRef] [PubMed]
- B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012). [CrossRef] [PubMed]
- Y. Zhang, Y. Wang, J. Xu, J. Wang, J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011). [CrossRef]
- J. Wang, J. Jia, J. Xu, Y. Wang, Y. Zhang, “The probability of orbital angular momentum states of single photons with Z-tilt corrected residual aberration in a slant path turbulent atmosphere,” Optik 122(11), 996–999 (2011). [CrossRef]
- S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express 20(1), 452–461 (2012). [CrossRef] [PubMed]
- V. V. Kotlyar, A. A. Kovalev, V. A. Soifer, “Hankel-Bessel laser beams,” J. Opt. Soc. Am. A 29(5), 741–747 (2012). [CrossRef] [PubMed]
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).
- X. Sheng, Y. Zhang, F. Zhao, L. Zhang, Y. Zhu, “Effects of low-order atmosphere-turbulence aberrations on the entangled orbital angular momentum states,” Opt. Lett. 37(13), 2607–2609 (2012). [CrossRef] [PubMed]
- F. Li, H. Tang, Y. Jiang, J. Ou, “Spiral spectrum of Laguerre-Gaussian beams propagating in turbulent atmosphere,” Acta Phys. Sin. 60(1), 014204 (2011).

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