Abstract

It is generally true that the orbital angular momentum (OAM) mode persistently degenerate when a vortex beam propagates in the atmospheric turbulence. Here, however, we unveil an interesting self-recovery effect of OAM mode of the circular beam (CiB) in weak non-Kolmogorov turbulence. We show that the CiB displays the self-focusing effect and has clear focus in the weak non-Kolmogorov turbulence if we choose proper complex parameters, and the detection probability of the original OAM mode reaches the maximum at the focus. Our study proposes a method to alleviate the turbulent effects on OAM-based communication.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]

2015 (3)

2014 (2)

2012 (1)

2011 (1)

2010 (1)

2009 (1)

2008 (3)

2007 (1)

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

2005 (2)

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

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
[Crossref] [PubMed]

2004 (1)

2000 (1)

C. H. Rao, W. H. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

1992 (1)

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Ahmed, N.

Allen, L.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
[Crossref]

Andrews, R.

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

Anguita, J. A.

Ashrafi, N.

Ashrafi, S.

Bandres, M. A.

Bao, C.

Barnett, S.

Beijersbergen, M.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Boyd, R. W.

Cao, Y.

Carrasco, S.

Chen, Z.

Cheng, M. J.

Christodoulides, D. N.

Courtial, J.

Dan, W. Y.

Efremidis, N. K.

Franke-Arnold, S.

Gao, C. Q.

Y.-D. Liu, C. Q. Gao, M. W. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Gao, J.

Gao, M. W.

Y.-D. Liu, C. Q. Gao, M. W. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Gibson, G.

Gopaul, C.

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

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

Gutierrez-Vega, J. C.

Hu, Z. D.

Huang, H.

Jiang, W. H.

C. H. Rao, W. H. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

Lavery, M.

Lavery, M. P. J.

Li, F.

Y.-D. Liu, C. Q. Gao, M. W. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Li, L.

Ling, N.

C. H. Rao, W. H. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

Liu, X. J.

Liu, Y.-D.

Y.-D. Liu, C. Q. Gao, M. W. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Malik, M.

Mills, M. S.

Mirhosseini, M.

Molisch, A.

Neifeld, M.

O’Sullivan, M. N.

Padgett, M.

Pas’ko, V.

Paterson, C.

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

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
[Crossref]

Prakash, J.

Ramachandran, S.

Rao, C. H.

C. H. Rao, W. H. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

Ren, Y.

Robertson, D. J.

Rodenburg, B.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

Spreeuw, R.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Torner, L.

Torres, J. P.

Tur, M.

Tyler, G. A.

Vallone, G.

Vasic, B.

Vasnetsov, M.

Wang, J.

Willner, A.

Woerdman, J.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Xie, G.

Yan, Y.

Zhang, L. C.

Zhang, P.

Zhang, Y. X.

Zhang, Z.

Zhao, F. S.

Zhao, Z.

Zhu, Y.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

J. Mod. Opt. (1)

C. H. Rao, W. H. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

New J. Phys. (1)

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

Opt. Commun. (1)

Y.-D. Liu, C. Q. Gao, M. W. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Opt. Express (5)

Opt. Lett. (6)

Phys. Rev. A (1)

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

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

Other (2)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
[Crossref]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic, 2007).

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Figures (4)
Fig. 1
Fig. 1 The waist position ze against (a) Re(q0) and Re(q1) with p=2+20i; (b) p with Re(q0)=Re(q1)=0m; (c) Re(q0) and Re(q1) with p=2+10i; (d) p with Re(q0)=Re(q1)=1000m. Other parameters: l0 = 1, C n 2 = 10 − 15 m 3 − α and α = 3.67, respectively.

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Fig. 2
Fig. 2 (a) beam width W (z) of the CiB against propagation distance z in non-Kolmogorov turbulence. Phase patterns of the CiB on x − y plane of (b) z=0m, (c) z=2264m and (d) z=5000m in non-Kolmogorov turbulence. Other parameters: C n 2 = 10 − 15 m 3 − α , α=3.67, q1=1200i m, l0=1 and p=2+20i, respectively.

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Fig. 3
Fig. 3 The received power weight C l 0 for the CiB against propagation distance z (a) with different Re(p) where C n 2 = 10 − 15 m 3 − α , α=3.97, Im(p) = 20 and l0=1; (b) with different α where C n 2 = 10 − 15 m 3 − α , l0=1, p=2+20i; (c) with different C n 2 where α=3.67, l0=1, p=2+20i; (d) with different l0 where C n 2 = 10 − 15 m 3 − α , α=3.67, p=2+20i. Other parameter: q1=1200i m.

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Fig. 4
Fig. 4 The crosstalk power weight Cl for the CiB against propagation distance z with Δl=|l − l0|=1,2,3,4. Other parameters: l0=1, p=2+20i, q1=1200i m, α=3.67 and C n 2 = 10 − 15 m 3 − α .

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Equations (11)

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CiB p , l 0 ( q 0 , q 1 ) ( r , φ , z ) = ( i 2 z 0 W 0 ) | l 0 | + 1 [ π | l 0 | ! Ψ p , l 0 ( ξ ) ] − 1 2 1 q ( z ) exp [ − i k r 2 2 q ( z ) ] × [ ( 1 + ξ ) q ˜ ( z ) q ( z ) ] p 2 [ r q ( z ) ] | l 0 | F 1 1 ( − p 2 , | l 0 | + 1 ; r 2 χ 2 ( z ) ) . × exp ( i l 0 φ )
Ψ p , q 0 ( q 0 , q 1 ) ( r , φ , z ) = CiB p , l 0 ( q 0 , q 1 ) ( r , φ , z ) ⋅ exp [ i ψ ( r , φ , z ) ] ,
Ψ p , l 0 ( q 0 , q 1 ) ( r , φ , z ) = 1 2 π ∑ l = − ∞ ∞ β p , l 0 , l ( q 0 , q 1 ) ( r , z ) exp ( i l φ )
β p , l 0 , l ( q 0 , q 1 ) ( r , z ) = 1 2 π ∫ 0 2 π CiB p , l 0 ( q 0 , q 1 ) ( r , φ , z ) exp [ i ψ ( r , φ , z ) ] exp ( − i l φ ) d φ .
〈 | β p , l 0 , l ( q 0 , q 1 ) ( r , z ) | 2 〉 = 1 2 π ∫ 0 2 π ∫ 0 2 π CiB p , l 0 ( q 0 , q 1 ) ( r , φ 1 , z ) CiB p , l 0 ( q 0 , q 1 ) * ( r , φ 2 , z ) × exp [ − i l ( φ 1 − φ 2 ) ] × 〈 exp { i [ ψ ( r , φ 1 , z ) − ψ ( r , φ 2 , z ) ] } 〉 d φ 1 d φ 2
〈 exp { i [ ψ ( r , φ 1 , z ) − ψ ( r , φ 2 , z ) ] } 〉 = exp [ − 2 r 2 − 2 r 2 cos ( φ 1 − φ 2 ) ρ 0 2 ] ,
ρ 0 = [ 8 α − 2 Γ ( 2 α − 2 ) ] 1 2 ⋅ [ 2 ( α − 1 ) Γ ( 3 − α 2 ) π Γ ( 2 − α 2 ) k 2 C n 2 z ] 1 α − 2                   ( 3 < α < 4 ) ,
〈 | β p , l 0 , l ( q 0 , q 1 ) ( r , z ) | 2 〉 = 1 2 | l 0 | | l 0 | ! Ψ p , l 0 ( ξ ) ( k W 0 | q ( z ) | ) 2 | l 0 | + 2 | [ ( 1 + ξ ) q ˜ ( z ) q ( z ) ] p | × exp ( − k 2 r 2 W 0 2 2 | q ( z ) | 2 ) | F 1 1 ( − p 2 , | l 0 | + 1 ; r 2 χ 2 ( z ) ) | 2 , × r 2 | l 0 | exp ( − 2 r 2 ρ 0 2 ) I l − l 0 ( 2 r 2 ρ 0 2 )
C l = P l ∑ m = − ∞ ∞ P m .
C l = i ( − 1 ) | l 0 | + 1 k W 0 ρ 0 2 π | q ( z ) | ⋅ | l 0 | ! Ψ p , l 0 ( ξ ) ( η − 1 η + 1 ) | l 0 | 2 + 1 4 | [ ( 1 + ξ ) q ˜ ( z ) q ( z ) ] p | × ∑ m = 0 ∞ ∑ n = 0 ∞ ( − p * 2 ) m ( − p 2 ) n ( | l 0 | + 1 ) m ( | l 0 | + 1 ) n n ! m ! × [ − ρ 0 2 2 ( χ 2 ) * ( η 2 − 1 ) 1 2 ] m [ − ρ 0 2 2 χ 2 ( η 2 − 1 ) 1 2 ] n × Q l − l 0 − 1 2 | l 0 | + n + m + 1 2 ( η ) ,
W 2 ( z ) = 4 〈 r 2 〉 = 4 ∬ r 2 | CiB p , l 0 ( q 0 , q 1 ) ( r → , z ) | 2 d 2 r →                 ( z ≥ 0 ) ,

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