OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 19 — Sep. 12, 2011
  • pp: 18310–18317
« Show journal navigation

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

Robert W. Boyd, Brandon Rodenburg, Mohammad Mirhosseini, and Stephen M. Barnett  »View Author Affiliations


Optics Express, Vol. 19, Issue 19, pp. 18310-18317 (2011)
http://dx.doi.org/10.1364/OE.19.018310


View Full Text Article

Acrobat PDF (729 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

There has recently been great interest in encoding information onto the transverse degree of freedom of individual photons [1

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]

] for applications such as optical communication and especially for quantum key distribution (QKD) [2

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

]. The transverse degree of freedom of the photon is associated with an infinite-dimensional Hilbert space, and thus there is in principle no limit to how much information can be encoded in this manner onto an individual photon. One motivation for impressing many bits of information onto each photon is to increase the data transmission rate of a QKD system. Another more subtle motivation is to exploit the increased security of a QKD system that is afforded by the use of a larger state space [3

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

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]

].

Much of the prior work on exploiting the transverse degree of freedom of the photon for optical communication has made use of states that carry orbital angular momentum (OAM) [5

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]

], such as the Laguerre-Gauss (LG) states [1

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

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

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]

]. Single-photon states that carry OAM are readily synthesized in the laboratory [8

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

], and the quantum optical properties of these states have been studied by many workers [9

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]

].

Fig. 1 Schematic diagram of a free-space quantum communication link.

For practical reasons, it is important to know if some protocols for encoding information onto the transverse degree of freedom of the photon are more robust than others in the presence of atmospheric turbulence. If one could identify such states, one could use this knowledge to develop free-space QKD protocols with increased reliability.

In our earlier study [14

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]

], an analysis was presented of an encoding scheme based on the use of pure vortex beams, that is, beams having the form V (r,ϕ) = A 0 exp(iℓϕ), where r and ϕ are radial and azimuthal coordinates, respectively, and 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.

In the present article, we explore the possibility of encoding information through the use of plane-wave states of light [20

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]

, 21

21. W. A. Miller, “Efficient photon sorter in a high-dimensional state space,” Quantum Inf. Comput. 11, 0313–0325 (2011).

]. We choose to explore this form of encoding for two different reasons. One is that a system based on such an encoding scheme should prove easier to implement in the laboratory than one based on the use of OAM states. The other reason is that we want to examine the conceptual question of whether some states are more robust than others, and one might well expect that encoding by impressing phase information onto a cartesian coordinate would be significantly different from encoding onto an azimuthal coordinate.

Specifically, we consider the situation in which the propagation direction of each of the plane-wave modes lies in the x–z plane, and these modes are launched through a square aperture of side L. We represent each of the plane wave modes in the paraxial approximation as
En=Aexp(ikz+imqx).
(1)
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
Omn=L/2L/2EmEn*dx,
(2)
which, through use of Eq. (1), becomes
Omn=|A|2L/2L/2eiq(mn)xdx.
(3)
For m = n this integral is simply evaluated to give Onn = |A|2 L, whereas more generally we obtain
Omn=|A|22sin[q(mn)L/2]q(mn).
(4)
Note that if we choose q to be equal to 2π/L, this equation becomes
Omn=|A|2sin[π(mn)]π(mn)/L,
(5)
which vanishes for any non-zero value of the integer mn. 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 = 2π/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.

As established above in connection with Eq. (5), the integral in this expression is equal to L for nlm = 0 and vanishes otherwise. Using this result, the summation in Eq. (12) can be performed directly to give
Vn(y)=A0W(y/L)gnl(y).
(13)
This result illustrate the manner in which the Fourier components g nl (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 nl (y), which represents the amplitude of the spectrum of atmospheric fluctuations at spatial frequency (n – l)q.

Under many practical situations, one is interested primarily in determining the power contained in each plane-wave component of the received field. The total power collected by the receiver is given by
P=12ε0cdxdyV*(r)V(r)=12ε0c|A0|2L2.
(14)
where in obtaining the last form we have used the field given by Eq. (7). This power is distributed among the various (orthogonal) plane-wave modes of the received field according to
P=Δ=PΔ
(15)
where
PΔ=12ε0c|A0|2LL/2L/2dygΔ*(y)gΔ(y)
(16)
and where we have made use of Eq. (13) and have defined Δ as Δ = nl.

It is also useful to define the fraction s Δ = P Δ/P of the received power contained in each plane-wave mode. This quantity is given by
sΔ=1LL/2L/2dygΔ*(y)gΔ(y).
(17)
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 = nl.

The result presented in Eq. (17) is valid for any realization of atmospheric turbulence. Usually we are interested in the ensemble average of this quantity, which is given by
sΔ=1LL/2L/2dygΔ*(y)gΔ(y)
(18)
where the angle brackets 〈. . .〉, represent an ensemble average over the turbulence statistics.

To proceed, Eq. (9) is used to express Eq. (18) in terms of the random phase associated with atmospheric turbulence. One obtains
sΔ=1LL/2L/2dy1LL/2L/2dx1eiϕ(x1,y)eiΔqx11LL/2L/2dx2eiϕ(x2,y)eiΔqx2=1L3L/2L/2dyL/2L/2dx1L/2L/2dx2eiϕ(x1,y)eiϕ(x2,y)eiΔq(x1x2)
(19)

The analysis proceeds using standard methods [14

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]

, 22

22. 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]

, 23

23. 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]

]. Since the random aberrations introduced by atmospheric turbulence are normal random variables, the ensemble average present in Eq. (19) can be expressed as
ei[ϕ(x1,y)ϕ(x2,y)]=e12[ϕ(x1,y)ϕ(x2,y)]2.
(20)
The quantity 〈[ϕ(x 1,y) – ϕ(x 2,y)]2〉 is known as the phase structure function. It can be evaluated [22

22. 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]

] by means of Kolmogorov turbulence theory to give the result
[ϕ(x1,y)ϕ(x2,y)]2=6.88|x1x2r0|5/3,
(21)
where r 0 is Fried’s coherence diameter, which is a measure of the transverse distance scale over which refractive index correlations remain correlated. When Eqs. (20) and (21) are introduced into Eq. (19), the resulting integral simplifies dramatically. The result becomes
sΔ=1L3dydx1dx2eiΔq(x1x2)e3.44(|x2x1|/r0)5/3
(22)

We evaluate this expression as follows. The integral over y can be performed directly. We also make a change of integration variables to ζ = (x 1 + x 2)/2L and η = (x 1x 2)/2L. Eq. (22) thus becomes
sΔ=21212dη(12+|η|12|η|dζ)e3.44(L/r0)5/3|η|5/3e2iΔLqη
(23)
which can be expressed as
sΔ=8012dη(12η)e3.44(ηL/r0)5/3cos(4πΔη).
(24)
We note that 〈s Δ〉 clearly depends only on L/r 0.

This expression can be evaluated numerically under general conditions. Some of the results of this numerical evaluation are shown in Fig. 2. The solid curve labeled Δ = 0 represents the fraction of the power that remains in the the launched mode after propagation through atmospheric turbulence. The other solid curves indicate the modes into which this power is transferred. For comparison, the dashed curve represents the results for the OAM case studied in reference [14

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]

]. We see that plane-wave encoding is more robust to the influence of turbulence that is OAM encoding. The origin of the difference between the two situation can be traced to the form of Eq. (24) and the analogous equation (Eq. 14) of reference [14

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]

] for the OAM case. Specifically, the integrands of these two expressions are essentially the same, but the integral in the plane-wave case (expressed in physical units) extends from –L/2 to L/2, whereas the integral for the OAM case extends over an angular range of 2π 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.

Fig. 2 The quantity 〈s Δ〉 plotted against the strength of the atmospheric turbulence as quantified by the ratio of the linear size L of the telescope aperture to the Fried parameter r 0 for several values of Δ. 〈s Δ〉 is the ensemble average of the fraction of the received power that is found to be in plane-wave mode n = + Δ, assuming that the transmitted beam was in plane-wave mode . Solid lines give the predictions based on a numerical evaluation of the integral in Eq. (24). The dashed curve refers to the OAM case treated in reference [14]. The quantity L represents the diameter of the circular aperture for the OAM case and the length of each side of the square aperture for the plane wave case. Note that the the plane wave encoding is more robust (by about as much as a factor of three) than the OAM encoding.

In addition to the numerical results presented in Fig. 2, we have found analytic expressions for the integral of Eq. (24) valid in the limits of very small and very large receiver aperture. For small aperture (i.e. for 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
s0=10.22(L/r0)5/3.
(25)
For Δ ≠ 0, we find that to good approximation
sΔ=0.0674Δ2(Lr0)5/3.
(26)
For the opposite limit of very large receiver aperture, we obtain
sΔ=1.70(r0/L)
(27)
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
s0=[1+(0.659L/r0)2]1/2.
(28)

In any real implementation of a QKD system, it is necessary to be able to transmit in at least two different mutually orthogonal bases. One of the bases can be the plane-wave states considered in this paper. The other can be some linear combination of these states. To assess the performance of the entire system, a calculation of the sort presented here would need to be performed for each one of the additional bases. Such an analysis lies outside of the scope of this article. However, we can make some general comments. Certain linear combination of plane waves are highly localized in space. We expect these states to be less affected by turbulence than the more extended plane-wave states.

Acknowledgments

We gratefully acknowledge valuable discussions with A. Jha, J. Leach, M. Padgett, D. Gauthier, G. Tyler, M. Gruneisen, and W. Miller. We gratefully acknowledge support from the DARPA/DSO InPho program. SMB also thanks the Royal Society and the Wolfson Foundation for support

References and links

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]

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]

12.

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94 (2007). [CrossRef]

13.

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–260 (2008). [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]

15.

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]

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 19, 6671–6683 (2011). [CrossRef] [PubMed]

17.

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

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]

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]

21.

W. A. Miller, “Efficient photon sorter in a high-dimensional state space,” Quantum Inf. Comput. 11, 0313–0325 (2011).

22.

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]

23.

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]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412, 313–316 (2001). [CrossRef] [PubMed]
  2. N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics1, 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. A35, 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. A45, 8185–8189 (1992). [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,” Science329, 662–665 (2010). [CrossRef] [PubMed]
  10. B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A74, 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]
  12. C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys.9, 94 (2007). [CrossRef]
  13. 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]
  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]
  15. 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]
  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. Express19, 6671–6683 (2011). [CrossRef] [PubMed]
  17. 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]
  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]
  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]
  21. W. A. Miller, “Efficient photon sorter in a high-dimensional state space,” Quantum Inf. Comput.11, 0313–0325 (2011).
  22. 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]
  23. 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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited