## Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding |

Optics Express, Vol. 20, Issue 12, pp. 13195-13200 (2012)

http://dx.doi.org/10.1364/OE.20.013195

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### Abstract

We describe an experimental implementation of a free-space 11-dimensional communication system using orbital angular momentum (OAM) modes. This system has a maximum measured OAM channel capacity of 2.12 bits/photon. The effects of Kolmogorov thin-phase turbulence on the OAM channel capacity are quantified. We find that increasing the turbulence leads to a degradation of the channel capacity. We are able to mitigate the effects of turbulence by increasing the spacing between detected OAM modes. This study has implications for high-dimensional quantum key distribution (QKD) systems. We describe the sort of QKD system that could be built using our current technology.

© 2012 OSA

## 1. Introduction

2. P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys. **8**, 193–193 (2006). [CrossRef]

3. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. **3**, 481–486 (2007). [CrossRef]

2. P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys. **8**, 193–193 (2006). [CrossRef]

3. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. **3**, 481–486 (2007). [CrossRef]

4. M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A-Math. Gen. **35**, 10065–10076 (2002). [CrossRef]

5. S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. **8**, 75 (2006). [CrossRef]

6. N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-Level systems,” Phys. Rev. Lett. **88**, 127902 (2002). [CrossRef] [PubMed]

4. M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A-Math. Gen. **35**, 10065–10076 (2002). [CrossRef]

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

11. F. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A **83**, 053822 (2011). [CrossRef]

## 2. Quantum key distribution system

*N*-dimensional QKD system is equal to

*N*+ 1, for when

*N*is a prime number [12

12. W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys.-New York **191**, 363–381 (1989). [CrossRef]

*N*is equal to 2 and there are three available MUBs. Here we describe a proposed OAM-based QKD system with a dimensionality

*d*= 11, and 12 possible mutually unbiased bases. Out of these, we choose to use only two in order to maximize our key transmission rate, which is inversely proportional to the number of MUBs that are used. The first basis is made up of 11 OAM modes and is called the OAM basis. The OAM modes have the form where

*A*

_{0}is the spatially uniform field amplitude,

*W*(

*x*) is an aperture function such that

*W*(

*x*) = 1 for |

*x*| ≤ 1 and zero otherwise,

*r*and

*θ*are the radial and azimuthal coordinates, and

*l*is the OAM quantum number. The second basis is called the ANG basis and is made up of angular modes, which are a linear combination of the 11 OAM modes and have the form

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

*f*system [13

13. V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A **24**, 3500–3507 (2007). [CrossRef]

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

15. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. **105**, 153601 (2010). [CrossRef]

16. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express **20**, 2110–2115 (2012). [CrossRef] [PubMed]

## 3. Effects of turbulence on channel capacity

17. A. T. Young, “Seeing: its cause and cure,” Astrophys. J. **189**, 587–604 (1974). [CrossRef]

*N*= 11 is much lower than the theoretical maximum of log

_{2}(11) = 3.46 bits/photon. This is because the OAM sorter is not perfect and introduces some inherent crosstalk [15

15. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. **105**, 153601 (2010). [CrossRef]

*N*-dimensional system crosses the dotted line for a polarization-based system. At a turbulence strength of

*D/r*

_{0}= 10, the channel capacity vanishes for all systems.

_{2}(3) = 1.585. In addition, the channel capacity for this mode-spacing starts decreasing at a value of

*D/r*

_{0}that is almost an order of magnitude greater than that for a mode-spacing of one.

## 4. Conclusions

## References and links

1. | C. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” |

2. | P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys. |

3. | R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. |

4. | M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A-Math. Gen. |

5. | S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. |

6. | N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-Level systems,” Phys. Rev. Lett. |

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

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

9. | B. Smith and M. Raymer, “Two-photon wave mechanics,” Phys. Rev. A |

10. | G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A |

11. | F. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A |

12. | W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys.-New York |

13. | V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A |

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

15. | G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. |

16. | M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express |

17. | A. T. Young, “Seeing: its cause and cure,” Astrophys. J. |

18. | D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. |

19. | C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. |

**OCIS Codes**

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(200.2605) Optics in computing : Free-space optical communication

(270.5568) Quantum optics : Quantum cryptography

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: April 25, 2012

Revised Manuscript: May 17, 2012

Manuscript Accepted: May 21, 2012

Published: May 25, 2012

**Citation**

Mehul Malik, Malcolm O’Sullivan, Brandon Rodenburg, Mohammad Mirhosseini, Jonathan Leach, Martin P. J. Lavery, Miles J. Padgett, and Robert W. Boyd, "Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding," Opt. Express **20**, 13195-13200 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13195

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### References

- C. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int. Conf., 175–179 (Bangalore, 1984).
- P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys.8, 193–193 (2006). [CrossRef]
- R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys.3, 481–486 (2007). [CrossRef]
- M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A-Math. Gen.35, 10065–10076 (2002). [CrossRef]
- S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys.8, 75 (2006). [CrossRef]
- N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-Level systems,” Phys. Rev. Lett.88, 127902 (2002). [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. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett.34, 142–144 (2009). [CrossRef] [PubMed]
- B. Smith and M. Raymer, “Two-photon wave mechanics,” Phys. Rev. A74, 062104 (2006). [CrossRef]
- G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A25, 225–230 (2008). [CrossRef]
- F. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A83, 053822 (2011). [CrossRef]
- W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys.-New York191, 363–381 (1989). [CrossRef]
- V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A24, 3500–3507 (2007). [CrossRef]
- M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt.47, A32–A42 (2008). [CrossRef] [PubMed]
- G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett.105, 153601 (2010). [CrossRef]
- M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express20, 2110–2115 (2012). [CrossRef] [PubMed]
- A. T. Young, “Seeing: its cause and cure,” Astrophys. J.189, 587–604 (1974). [CrossRef]
- D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am.55, 1427–1431 (1965). [CrossRef]
- C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt.38, 2161–2170 (1999). [CrossRef]

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