Abstract

We analyze the influence of atmospheric turbulence on the propagation of an optical vortex beam having the form V(r,θ)=A0eimθ. The probability that a detected photon after propagating through the atmosphere has the same value of the orbital angular momentum as the launched photon is found to be given by s0=[1+(1.845Dr0)2]12, where D is the aperture diameter and r0 is the Fried coherence diameter. These vortex beams behave very similarly to Laguerre–Gauss beams under the influence of atmospheric turbulence. These results have important implications for atmospheric laser communication systems that employ quantum encryption.

© 2009 Optical Society of America

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References

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  1. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
    [CrossRef] [PubMed]
  2. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
    [CrossRef] [PubMed]
  3. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
    [CrossRef] [PubMed]
  4. G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
    [CrossRef] [PubMed]
  5. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, Appl. Opt. 47, A33 (2008).
    [CrossRef]
  6. N. Gisin and R. Thew, Nat. Photonics 1, 165 (2007).
    [CrossRef]
  7. C. Paterson, Phys. Rev. Lett. 94, 153901 (2005).
    [CrossRef] [PubMed]
  8. C. Gopaul and R. Andrews, New J. Phys. 9, 94 (2007).
    [CrossRef]
  9. G. Gbur and R. K. Tyson, J. Opt. Soc. Am. A 25, 255 (2008).
    [CrossRef]

2008 (2)

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, Appl. Opt. 47, A33 (2008).
[CrossRef]

G. Gbur and R. K. Tyson, J. Opt. Soc. Am. A 25, 255 (2008).
[CrossRef]

2007 (2)

C. Gopaul and R. Andrews, New J. Phys. 9, 94 (2007).
[CrossRef]

N. Gisin and R. Thew, Nat. Photonics 1, 165 (2007).
[CrossRef]

2005 (1)

C. Paterson, Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

2004 (2)

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef] [PubMed]

2002 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Andrews, R.

C. Gopaul and R. Andrews, New J. Phys. 9, 94 (2007).
[CrossRef]

Barnett, S. M.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef] [PubMed]

Courtial, J.

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Dymale, R. C.

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, Appl. Opt. 47, A33 (2008).
[CrossRef]

Franke-Arnold, S.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef] [PubMed]

Gbur, G.

G. Gbur and R. K. Tyson, J. Opt. Soc. Am. A 25, 255 (2008).
[CrossRef]

Gibson, G.

Gisin, N.

N. Gisin and R. Thew, Nat. Photonics 1, 165 (2007).
[CrossRef]

Gopaul, C.

C. Gopaul and R. Andrews, New J. Phys. 9, 94 (2007).
[CrossRef]

Gruneisen, M. T.

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, Appl. Opt. 47, A33 (2008).
[CrossRef]

Leach, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Miller, W. A.

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, Appl. Opt. 47, A33 (2008).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

Padgett, M. J.

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Pas'ko, V.

Paterson, C.

C. Paterson, Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Sweiti, A. M.

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, Appl. Opt. 47, A33 (2008).
[CrossRef]

Thew, R.

N. Gisin and R. Thew, Nat. Photonics 1, 165 (2007).
[CrossRef]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

Tyson, R. K.

G. Gbur and R. K. Tyson, J. Opt. Soc. Am. A 25, 255 (2008).
[CrossRef]

Vasnetsov, M.

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Appl. Opt. (1)

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, Appl. Opt. 47, A33 (2008).
[CrossRef]

J. Opt. Soc. Am. A (1)

G. Gbur and R. K. Tyson, J. Opt. Soc. Am. A 25, 255 (2008).
[CrossRef]

Nat. Photonics (1)

N. Gisin and R. Thew, Nat. Photonics 1, 165 (2007).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

New J. Phys. (1)

C. Gopaul and R. Andrews, New J. Phys. 9, 94 (2007).
[CrossRef]

Opt. Express (1)

Phys. Rev. Lett. (3)

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

C. Paterson, Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1
Fig. 1

Schematic of a free-space quantum communication link.

Fig. 2
Fig. 2

Quantity s Δ plotted against the strength of the atmospheric turbulence as quantified by the ratio of the telescope diameter D 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 OAM mode n = m + Δ , assuming that the transmitted beam was in OAM mode m. Solid curves give the predictions based on a numerical evaluation of the integral in Eq. (14). The dashed curves, shown only for Δ = 0 , give the predictions of the asymptotic expressions of Eqs. (15, 16).

Equations (17)

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A ( r ) = A 0 W ( r R ) e i m θ ,
V ( r ) = A 0 W ( r R ) e i m θ e i ϕ ( r ) ,
e i ϕ ( r , θ ) = l = g l ( r ) e i l θ ,
g l ( r ) = 1 2 π 0 2 π d i ϕ ( r , θ ) e i l θ .
V n ( r ) = 1 2 π 0 2 π d θ V ( r , θ ) e i n θ .
V n ( r ) = A 0 2 π W ( r R ) l = g l ( r ) 0 2 π d θ e i ( n l m ) θ .
V n ( r ) = A 0 W ( r R ) g Δ ( r ) ,
P = 1 2 ϵ 0 c d r W ( r R ) V * ( r ) V ( r ) = 1 2 ϵ 0 c A 0 2 π R 2 ,
P = Δ = P Δ , where P Δ = 2 π A 0 2 0 R d r r g Δ * ( r ) g Δ ( r ) .
s Δ = 2 R 2 0 R d r r g Δ * ( r ) g Δ ( r ) .
s Δ = K 0 R d r r 0 2 π d θ 1 0 2 π d θ 2 e i [ ϕ ( r , θ 1 ) ϕ ( r , θ 2 ) ] e i Δ ( θ 1 θ 2 ) ,
e i [ ϕ ( r , θ 1 ) ϕ ( r , θ 2 ) ] = e 1 2 [ ϕ ( r , θ 1 ) ϕ ( r , θ 2 ) ] 2 .
[ ϕ ( r 1 ) ϕ ( r 2 ) ] 2 = 6.88 r 1 r 2 r 0 5 3 ,
s Δ = 1 π 0 1 d ρ ρ 0 2 π d θ e 3.44 ( D r 0 ) 5 3 ( ρ sin θ 2 ) 5 3 cos Δ θ ,
s Δ = { 1 1.01 ( D r 0 ) 5 3 for Δ = 0 0.142 Γ ( Δ 5 6 ) Γ ( Δ + 11 6 ) ( D r 0 ) 5 3 otherwise } ,
s Δ = 12 Γ ( 3 5 ) 5 π ( 3.44 ) 3 5 ( D r 0 ) 1 = 0.542 ( D r 0 ) 1 .
s 0 = [ 1 + ( 1.845 D r 0 ) 2 ] 1 2 .

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