Abstract

In this paper, orbital angular momentum (OAM) modes transmission in the presence of atmosphere turbulence is studied via a coupled mode theory. The Laguerre-Gauss (LG) beams with OAM topological charges are emitted into free space and undergo interactions due to the random index variations in the atmosphere. The coupling between the LG beams can be characterized by a set of coupled average power equation, which resembles the Marcuse' coupled power equation (CPE) originally proposed for the optical waveguides. The coupling coefficients and the modal radiation losses for the equation can be evaluated analytically. The accurate solution and the first order approximate solution to the CPE match the published data and the Mont-Carlos simulation results with good accuracy. The CPE and its approximate analytical solution can work as powerful tools for the analysis of the OAM beam evolution with the presence of the atmosphere turbulence.

© 2015 Optical Society of America

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References

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2014 (2)

2012 (2)

B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012).
[Crossref] [PubMed]

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

2011 (1)

N. Baddour, “Two-Dimensional Fourier Transforms in Polar Coordinates,” Adv. Imaging Electron Phys. 165, 1–45 (2011).
[Crossref]

2009 (1)

2008 (1)

2007 (1)

2005 (1)

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

2004 (1)

2000 (1)

1972 (2)

D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51(1), 229–237 (1972).
[Crossref]

D. Marcuse, “Power distribution and radiation losses in multimode dielectric waveguides,” Bell Syst. Tech. J. 51(2), 429–454 (1972).
[Crossref]

1969 (1)

D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48(10), 3187–3215 (1969).
[Crossref]

Ahmed, N.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Anguita, J. A.

Bächtold, W.

Baddour, N.

N. Baddour, “Two-Dimensional Fourier Transforms in Polar Coordinates,” Adv. Imaging Electron Phys. 165, 1–45 (2011).
[Crossref]

Belmonte, A.

Boyd, R. W.

Cappuzzo, M.

Cheng, M.

Dan, W.

Dolinar, S.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Earnshaw, M.

Erni, D.

Fazal, I. M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Ferrari, C.

Fontaine, N. K.

Gallion, P.

J. Zhou and P. Gallion, “A comprehensive analytical model to characterize randomness in optical waveguides,” J. Lightwave Technol. (submitted to).

Gao, J.

Guan, B.

Hu, Z.

Huang, H.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Keller, B.

Klemens, F.

Lavery, M. P. J.

Lenz, D.

Malik, M.

Marcuse, D.

D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51(1), 229–237 (1972).
[Crossref]

D. Marcuse, “Power distribution and radiation losses in multimode dielectric waveguides,” Bell Syst. Tech. J. 51(2), 429–454 (1972).
[Crossref]

D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48(10), 3187–3215 (1969).
[Crossref]

Mirhosseini, M.

Neifeld, M. A.

O’Sullivan, M. N.

Padgett, M.

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]

Qin, C.

Ren, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Robertson, D. J.

Rodenburg, B.

Scott, R. P.

Su, T.

Tur, M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Tyler, G. A.

Vasic, B. V.

Wang, J.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Willner, A. E.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yan, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, J.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yoo, S. J. B.

Yue, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Zhang, Y.

Zhao, F.

Zhou, J.

J. Zhou and P. Gallion, “A comprehensive analytical model to characterize randomness in optical waveguides,” J. Lightwave Technol. (submitted to).

Zhu, Y.

Adv. Imaging Electron Phys. (1)

N. Baddour, “Two-Dimensional Fourier Transforms in Polar Coordinates,” Adv. Imaging Electron Phys. 165, 1–45 (2011).
[Crossref]

Appl. Opt. (3)

Bell Syst. Tech. J. (3)

D. Marcuse, “Mode conversion caused by surface imperfections of a dielectric slab waveguide,” Bell Syst. Tech. J. 48(10), 3187–3215 (1969).
[Crossref]

D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51(1), 229–237 (1972).
[Crossref]

D. Marcuse, “Power distribution and radiation losses in multimode dielectric waveguides,” Bell Syst. Tech. J. 51(2), 429–454 (1972).
[Crossref]

Nat. Photonics (1)

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

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 (5)

N. Bozinovic, P. Kristensen, and S. Ramachandran, “Long-range fiber-transmission of photons with orbital angular momentum,” in Proceedings of the Conference on Lasers and Electro-Optics (Optical Society of America, 2011), paper CTuB1.
[Crossref]

H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. Tur, B. Erkmen, K. Birnbaum, S. Dolinar, M. Lavery, M. Padgett, and A. E. Willner, “100 Tbit/s Free-Space Data Link using Orbital Angular Momentum Mode Division Multiplexing Combined with Wavelength Division Multiplexing,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper OTh4G.5.
[Crossref]

A. V. Oppenheim, A. S. Willsky, and S. Hamid, Signal and Systems, 2nd edition (Prentice Hall, 1997).

J. Zhou and P. Gallion, “A comprehensive analytical model to characterize randomness in optical waveguides,” J. Lightwave Technol. (submitted to).

V. P. Aksenov, “Scintillation index of a laser beam having the initial orbital angular momentum in the turbulent atmosphere,” in Imaging and Applied Optics 2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper PM2C.5.

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Figures (10)

Fig. 1
Fig. 1 Intensity (a) and phase (b) distribution of the output field when the input is OAM state + 5.
Fig. 2
Fig. 2 scintillation index versus propagation distance.
Fig. 3
Fig. 3 Correlation between OAM state 0 and other OAM states when propagation distance z = 1000.
Fig. 4
Fig. 4 OAM state conversion efficiency when the refractive index structure parameter Cn 2 is 10−14 m-2/3 and the initial beam waist is 1.6cm.
Fig. 5
Fig. 5 OAM state crosstalk from OAM state m to OAM state m-1 when the refractive index structure parameter Cn 2 is 10−14 m-2/3 and the initial beam waist is 1.6cm.
Fig. 6
Fig. 6 OAM state crosstalk from OAM state m to OAM state m-2 when the refractive index structure parameter Cn 2 is 10−14 m-2/3 and the initial beam waist is 1.6cm.
Fig. 7
Fig. 7 OAM state conversion efficiency when the refractive index structure parameter Cn 2 is 10−15 m-2/3, the initial beam waist is 1.6cm.
Fig. 8
Fig. 8 OAM state conversion efficiency when the refractive index structure parameter Cn 2 is 10−16 m-2/3, the initial beam waist is 1.6cm.
Fig. 9
Fig. 9 OAM state conversion efficiency when the refractive index structure parameter Cn 2 is 10−14 m-2/3 and the initial beam waist is 3cm.
Fig. 10
Fig. 10 OAM state 0 conversion efficiency deviation between the Mont Carlo simulation and the theory versus propagation distance, the refractive index structure parameter Cn 2 is 10−14 m-2/3 and the initial beam waist is 1.6cm. The simulation area is 70cmX70cm.

Equations (27)

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2 φ + k 2 φ = 0
( 1 + δ n ) k 0
Δ n = δ n δ n
k = ( n + Δ n ) k 0 n = 1 + δ n
2 φ + k 0 2 ( 1 + 2 Δ n ) φ = 0
φ ( x , y , z ) = m a m u m ( x , y , z ) e j k 0 z
2 ( u m ( x , y , z ) e j k 0 z ) + k 0 2 ( u m ( x , y , z ) e j k 0 z ) = 0
u m ( r , ϕ , z ) = 2 p ! π ( p + | m | ) ! 1 w ( z ) ( r 2 w ( z ) ) | m | L p m ( 2 r 2 w 2 ( z ) ) exp ( r 2 w 2 ( z ) ) exp ( j k 0 r 2 z 2 ( z 2 + z R 2 ) ) exp ( j ( 2 p + | m | + 1 ) tan 1 ( z z R ) ) exp ( j m ϕ )
w ( z ) = w 0 1 + ( z / z R ) 2 z R = π w 0 2 λ
u m * u n d S = δ m n
2 a m ( z ) z ( u m z + j k 0 u m ) e j k 0 z + 2 Δ n k 0 2 a m ( z ) u m e j k 0 z = 0
d a m ( z ) d z = m ' C m m ' ( z ) a m ' ( z ) C m m ' ( z ) = j k 0 x y u m * Δ n u m ' d S
R ( x x ' , y y ' , z z ' ) = Δ n ( x , y , z ) Δ n ( x ' , y ' , z ' )
K ( k x , k y , k z ) = F T ( R ( x , y , z ) ) = ( 2 π ) 3 0.033 C n 2 ( k 2 + 1 L 0 2 ) 11 / 6 f ( k , k l ) f ( k , k l ) = exp ( k 2 k l 2 ) ( 1 + 1.802 ( k k l ) 0.254 ( k k l ) 7 6 ) k 2 = k x 2 + k y 2 + k z 2 k l = 3.3 / l 0
R ( x , y , z ) R x y ( x , y ) f ( z ) R x y ( x , y ) = F T 1 ( K ( k x , k y , 0 ) ) f ( z ) f ( z ' ) = δ ( z z ' )
P m ( z ) = a m ( z ) a m * ( z )
d P m ( z ) d z = m ' κ m m ' ( z ) ( P m ' ( z ) P m ( z ) )
κ m m ' ( z ) = k 0 2 x y x ' y ' u m * ( x , y , z ) u m ( x ' , y ' , z ) u m ' ( x , y , z ) u m ' * ( x ' , y ' , z ) R x y ( x x ' , y y ' ) d S d S '
d P m ( z ) d z = m ' κ m m ' ( z ) ( P m ' ( z ) P m ( z ) ) α m ( z ) P m ( z )
κ m m ' ( z ) = k 0 2 x y ( u m * ( x , y , z ) u m ' ( x , y , z ) ) ( R x y ( x , y ) ( u m ( x , y , z ) u m ' * ( x , y , z ) ) ) d S
κ m m ' ( z ) = 1 4 π 2 k 0 2 k x k y | F T ( u m * ( x , y , z ) u m ' ( x , y , z ) ) | 2 K ( k x , k y , 0 ) d k x d k y
α m ( z ) = 1 4 π 2 k 0 2 k x k y x y x ' y ' u m * ( x , y , z ) u m ( x ' , y ' , z ) e j k x ( x x ' ) + j k y ( y y ' ) R x y ( x x ' , y y ' ) d S d S ' d k x d k y = k 0 2 x y x ' y ' u m * ( x , y , z ) u m ( x ' , y ' , z ) δ ( x x ' , y y ' ) R x y ( x x ' , y y ' ) d S d S ' = k 0 2 x y | u m ( x , y , z ) | 2 R x y ( 0 , 0 ) d S = 1 4 π 2 k 0 2 R x y ( 0 , 0 ) = 1 4 π 2 k 0 2 k x k y K ( k x , k y , 0 ) d k x d k y
d P ( z ) d z = ( κ ( z ) α ( z ) ) P ( z )
d P ( z ) d z = ( lim N Δ z 0 i = 0 N exp ( ( κ ( z i ) α ( z i ) ) Δ z ) ) P ( 0 )
κ m m ' ( z ) κ m m ' ( 0 ) α m ( z ) α m ( 0 )
P ( z ) exp ( ( κ ( 0 ) α ( 0 ) ) z ) P ( 0 )
P ( z ) = M P ( 0 )

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