## Influence of atmospheric turbulence on the propagation of quantum states of light using plane-wave encoding |

Optics Express, Vol. 19, Issue 19, pp. 18310-18317 (2011)

http://dx.doi.org/10.1364/OE.19.018310

Acrobat PDF (729 KB)

### Abstract

We consider the possibility of performing quantum key distribution (QKD) by encoding information onto individual photons using plane-wave basis states. We compare the results of this calculation to those obtained by earlier workers, who considered encoding using OAM-carrying vortex modes of the field. We find theoretically that plane-wave encoding is less strongly influenced by atmospheric turbulence than is OAM encoding, with potentially important implications for free-space quantum key distribution.

© 2011 OSA

1. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**, 313–316 (2001). [CrossRef] [PubMed]

2. N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics **1**, 165–171 (2007). [CrossRef]

3. N. J. 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. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A **35**, 10065–10076 (2002). [CrossRef]

5. 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]

1. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**, 313–316 (2001). [CrossRef] [PubMed]

6. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. **88**, 013601 (2002). [CrossRef] [PubMed]

7. 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**, A33–A41 (2008). [CrossRef]

8. J. Leach, M. R. Denis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. **7**, 55 (2005). [CrossRef]

9. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science **329**, 662–665 (2010). [CrossRef] [PubMed]

10. B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A **74**, 062104 (2006). [CrossRef]

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

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

19. Y. Wang, Y. Zhang, J. Wang, and J. Jia, “Degree of polarization for quantum light field propagating through non-Kolmogorov turbulence,” Opt. Laser Technol. **43**, 776–780 (2011). [CrossRef]

14. 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**, 142–144 (2009). To allow straightforward comparison between the OAM and plane-wave cases, in the present article we use notation similar to that of this earlier paper. [CrossRef] [PubMed]

*V*(

*r*,

*ϕ*) =

*A*

_{0}exp(

*iℓϕ*), where

*r*and

*ϕ*are radial and azimuthal coordinates, respectively, and

*ℓ*is the OAM quantum number. It was found that these beams showed essentially the same sensitivity to turbulence as do LG beams, thereby suggesting that the specific form of the transverse field distribution used to encode information is not important in determining the robustness of the communication protocol.

20. We note that a security analysis of such a plane-wave encoding scheme has been presented earlier for the case of a continuous-variable protocol by L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. **100**, 110504 (2008). In the present paper we discretize the propagation direction to better compare our results to those of earlier work based on an OAM basis. [CrossRef] [PubMed]

*L*. We represent each of the plane wave modes in the paraxial approximation as Here

*m*is any positive or negative integer that identifies a particular mode and

*q*is some characteristic transverse wave vector component. We choose the value of

*q*to ensure that the modes are orthogonal over the transmitting aperture. To do so, we consider the overlap integral which, through use of Eq. (1), becomes For

*m*=

*n*this integral is simply evaluated to give

*O*= |

_{nn}*A*|

^{2}

*L*, whereas more generally we obtain Note that if we choose

*q*to be equal to 2

*π*/

*L*, this equation becomes which vanishes for any non-zero value of the integer

*m*–

*n*. For this value of

*q*, the plane wave modes are orthogonal over the transmitting aperture. We note that this result has the simple interpretation that for

*q*= 2

*π*/

*L*the angle between adjacent modes

*α*=

*q*/

*k*is just equal to

*λ*/

*L*where

*λ*is the optical wavelength, which is the approximate angular spread of each such mode. Thus, these modes are just barely resolved in the far field.

*A*

_{0}is the (spatially uniform) field amplitude,

*W*(

*ξ*) is the aperture function defined so that

*W*(

*ξ*) = 1 for |

*η*| ≤ 1 and zero otherwise, and

*l*is the mode index of the launched field. We further assume that the transmitted beam remains sufficiently well collimated that the field at the receiver aperture can be well described by where

*ϕ*(

*x*,

*y*) represents the turbulence-induced distortion of the wavefront and where we have omitted the overall phase factor exp(

*ikz*). Here we are describing the influence of turbulence in the phase-screen approximation, which is valid when the turbulence is not too strong. We are also assuming that the apertures sizes and separations are chosen so that essentially all of the light leaving the transmitter is intercepted by the receiver. This condition is readily achieved. The necessary condition is that the Fresnel number (

*F*=

*L*

^{2}/

*λZ*

_{0}, where

*Z*

_{0}is the distance from the transmitter to the receiver) of the communication link be much greater than unity. For a wavelength

*λ*of 1

*μ*m and a separation of

*Z*

_{0}of 10 km, a telescope aperture of only

*L*= 0.1 m is needed to obtain a Fresnel number of unity.

*iϕ*(

*x*,

*y*)] in a Fourier series in the

*x*direction only as where the expansion coefficients

*g*(

_{m}*y*) are given by Of course, we could have expanded exp[

*iϕ*(

*x*,

*y*)] in Fourier series in both the

*x*and

*y*directions, but doing so is not needed and in fact would complicate the ensuing analysis. It is crucial to note that we are

*not*assuming that the atmospheric properties are uniform in the

*y*direction. Rather, we are arguing that

*y*-dependent inhomogeneities will not lead to beam deflections in the

*x*direction, the direction in which information is encoded. Similarly, we expand the received field

*V*(

*x*,

*y*) in a Fourier series as where each Fourier component

*V*(

_{n}*y*) is given by Eqs. (7) and (8) are now substituted into Eq. (11) which becomes

*L*for

*n*–

*l*–

*m*= 0 and vanishes otherwise. Using this result, the summation in Eq. (12) can be performed directly to give This result illustrate the manner in which the Fourier components

*g*

_{n–l}(

*r*) associated with atmospheric turbulence are coupled to the plane-wave states of the received field. Specifically, if mode

*l*is launched, mode

*n*will be received with a probability amplitude that is proportional to the quantity

*g*

_{n–l}(

*y*), which represents the amplitude of the spectrum of atmospheric fluctuations at spatial frequency (

*n – l*)

*q*.

*n*–

*l*.

*s*

_{Δ}=

*P*

_{Δ}/

*P*of the received power contained in each plane-wave mode. This quantity is given by For any statistical realization of the atmospheric turbulence,

*s*

_{Δ}gives the probability that the quantum number

*n*of the received photon departs from that

*l*of the transmitted photon by the amount Δ =

*n*–

*l*.

*y*can be performed directly. We also make a change of integration variables to

*ζ*= (

*x*

_{1}+

*x*

_{2})/2

*L*and

*η*= (

*x*

_{1}–

*x*

_{2})/2

*L*. Eq. (22) thus becomes which can be expressed as We note that 〈

*s*

_{Δ}〉 clearly depends only on

*L*/

*r*

_{0}.

14. 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**, 142–144 (2009). To allow straightforward comparison between the OAM and plane-wave cases, in the present article we use notation similar to that of this earlier paper. [CrossRef] [PubMed]

14. 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**, 142–144 (2009). To allow straightforward comparison between the OAM and plane-wave cases, in the present article we use notation similar to that of this earlier paper. [CrossRef] [PubMed]

*L*/2 to

*L*/2, whereas the integral for the OAM case extends over an angular range of 2

*π*radians or a distance range of

*π*times the diameter of the OAM mode. For an OAM mode that just fills the aperture of the transmitter telescope, this will lead to about a factor-of-

*π*increase in the robustness of the plane-wave modes as compared to the OAM modes, as is seen in the numerical results of Fig. 2. In fact, the two curves labeled Δ = 0 in this figure have essentially identical shapes, with the curves being offset horizontally from one another by a value of

*π*. Stated somewhat differently, these results indicate that plane-wave modes are corrupted by fluctuations that occur along the x direction in the range –

*L*/2 to

*L*/2, whereas OAM modes are corrupted by fluctuations that occur anywhere along the circumference of the OAM mode.

*L*/

*r*

_{0}→ 0) we can expand the exponent in Eq. (24) in a power series in (

*L*/

*r*

_{0})

^{5/3}and retain only the lowest-order correction term. For Δ = 0 we integrate the resulting equation directly to obtain For Δ ≠ 0, we find that to good approximation For the opposite limit of very large receiver aperture, we obtain We have also found a highly accurate (at most 1.6% error) approximate expression for 〈

*s*

_{0}〉 valid over the entire domain of

*L*/

*r*

_{0}, namely

## Acknowledgments

## References and links

1. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature |

2. | N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics |

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

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

5. | 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 |

6. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. |

7. | 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. |

8. | J. Leach, M. R. Denis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. |

9. | J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science |

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

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

12. | C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. |

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

14. | 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. |

15. | C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. |

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

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

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

19. | Y. Wang, Y. Zhang, J. Wang, and J. Jia, “Degree of polarization for quantum light field propagating through non-Kolmogorov turbulence,” Opt. Laser Technol. |

20. | We note that a security analysis of such a plane-wave encoding scheme has been presented earlier for the case of a continuous-variable protocol by L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. |

21. | W. A. Miller, “Efficient photon sorter in a high-dimensional state space,” Quantum Inf. Comput. |

22. | D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very short and very long exposures,” J. Opt. Soc. Am. |

23. | The propagation of light through atmospheric turbulence is reviewed by P. W. Milonni, “Adaptive optics for astronomy,” Am. J. Phys. |

**OCIS Codes**

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(270.5568) Quantum optics : Quantum cryptography

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: July 26, 2011

Revised Manuscript: August 18, 2011

Manuscript Accepted: August 19, 2011

Published: September 2, 2011

**Citation**

Robert W. Boyd, Brandon Rodenburg, Mohammad Mirhosseini, and Stephen M. Barnett, "Influence of atmospheric turbulence on the propagation of quantum states of light using plane-wave encoding," Opt. Express **19**, 18310-18317 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-18310

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### References

- A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412, 313–316 (2001). [CrossRef] [PubMed]
- N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics1, 165–171 (2007). [CrossRef]
- N. J. 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]
- M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A35, 10065–10076 (2002). [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]
- G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett.88, 013601 (2002). [CrossRef] [PubMed]
- 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, A33–A41 (2008). [CrossRef]
- J. Leach, M. R. Denis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys.7, 55 (2005). [CrossRef]
- J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662–665 (2010). [CrossRef] [PubMed]
- B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A74, 062104 (2006). [CrossRef]
- C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett.94, 153901 (2005). [CrossRef] [PubMed]
- C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys.9, 94 (2007). [CrossRef]
- G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A25, 225–260 (2008). [CrossRef]
- 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, 142–144 (2009). To allow straightforward comparison between the OAM and plane-wave cases, in the present article we use notation similar to that of this earlier paper. [CrossRef] [PubMed]
- C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys.11, 045017 (2009). [CrossRef]
- B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express19, 6671–6683 (2011). [CrossRef] [PubMed]
- F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A83, 053822 (2011). [CrossRef]
- Y.-X. Zhang, Y.-G. Wang, J.-C. Xu, J.-Y. Wang, and J.-J Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun.284, 1132–1138 (2011). [CrossRef]
- Y. Wang, Y. Zhang, J. Wang, and J. Jia, “Degree of polarization for quantum light field propagating through non-Kolmogorov turbulence,” Opt. Laser Technol.43, 776–780 (2011). [CrossRef]
- We note that a security analysis of such a plane-wave encoding scheme has been presented earlier for the case of a continuous-variable protocol by L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett.100, 110504 (2008). In the present paper we discretize the propagation direction to better compare our results to those of earlier work based on an OAM basis. [CrossRef] [PubMed]
- W. A. Miller, “Efficient photon sorter in a high-dimensional state space,” Quantum Inf. Comput.11, 0313–0325 (2011).
- D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very short and very long exposures,” J. Opt. Soc. Am.56, 1372–1379 (1966). [CrossRef]
- The propagation of light through atmospheric turbulence is reviewed by P. W. Milonni, “Adaptive optics for astronomy,” Am. J. Phys.67, 476–485 (1999). [CrossRef]

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