Abstract

We derive the analytical formula of the energy weight of each orbital angular momentum (OAM) mode of twisted Gaussian Schell-model (TGSM) beams propagating in weak turbulent atmosphere. The evolution of its OAM spectrum is studied by numerical calculation. Our results show that the OAM spectrum of a TGSM beam changes with the beam propagating in turbulent atmosphere, which is completely different from that of the TGSM beam propagating in free space. Furthermore, influences of the source parameters and the turbulence parameters on the OAM spectrum of a TGSM beam in turbulent atmosphere are analyzed. It is found that the source parameters and turbulence parameters, such as twist factor, coherence length, beam waist size, and structure constant, have a significant influence on the OAM spectrum, but the value of the wavelength and inner scale have little influence. Increasing the beam waist size or decreasing the coherence length would lead to the OAM spectrum broadened in the source plane, but would be robust for the OAM modes of the TGSM beam in the turbulent atmosphere. It is clear that the bigger the value of the twist factor, the more asymmetric the OAM mode of the TGSM beam is, and the better mode distribution can be maintained when it propagates in turbulent atmosphere. Our results have potential applications in reducing the error rate of free-space optical communication and detecting the atmospheric parameters.

© 2019 Optical Society of America

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References

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2017 (1)

2016 (3)

2015 (2)

2014 (3)

2012 (2)

2010 (1)

2009 (2)

2007 (1)

2006 (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

2005 (1)

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

2003 (1)

2002 (2)

2001 (4)

P. Ostlund and A. T. Friberg, “Radiometry and radiation efficiency of twisted Gaussian Schell-model sources,” Opt. Rev. 8, 1–8 (2001).
[Crossref]

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150–156 (2001).
[Crossref]

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 (2001).
[Crossref]

J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26, 405–407 (2001).
[Crossref]

1998 (1)

1993 (1)

1979 (1)

1972 (1)

Baykal, Y.

Boyd, R. W.

Cai, Y.

Chen, M.

Chen, Y.

Cheng, M.

Dan, W.

Davidson, F. M.

Dogariu, A.

Eyyubôlu, H. T.

Friberg, A. T.

P. Ostlund and A. T. Friberg, “Radiometry and radiation efficiency of twisted Gaussian Schell-model sources,” Opt. Rev. 8, 1–8 (2001).
[Crossref]

Gao, J.

Gbur, G.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

Guo, L.

He, H.

J. Ou, Y. Zhang, J. Tang, H. He, and Y. Wang, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Hu, Z.

Huang, Q.

Huang, Y.

Korotkova, O.

Li, J.

Li, Y.

Lin, Q.

Liu, L.

Molina-Terriza, G.

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 (2001).
[Crossref]

Movilla, J. M.

Mukunda, N.

Nairat, M.

Ostlund, P.

P. Ostlund and A. T. Friberg, “Radiometry and radiation efficiency of twisted Gaussian Schell-model sources,” Opt. Rev. 8, 1–8 (2001).
[Crossref]

Ou, J.

J. Ou, Y. Zhang, J. Tang, H. He, and Y. Wang, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Paterson, C.

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

Plonus, M. A.

Ponomarenko, S. A.

Ricklin, J. C.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

Santarsiero, F. G. M.

Serna, J.

Shirai, T.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Simon, R.

Sun, R.

Tang, J.

J. Ou, Y. Zhang, J. Tang, H. He, and Y. Wang, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Tong, Z.

Torner, L.

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 (2001).
[Crossref]

Torres, J. P.

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 (2001).
[Crossref]

Tyler, G. A.

Voelz, D.

Wang, F.

Wang, S. C. H.

Wang, Y.

J. Ou, Y. Zhang, J. Tang, H. He, and Y. Wang, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Wolf, E.

Yan, X.

Yura, H. T.

Zhang, L.

Zhang, Y.

Zhao, F.

Zhu, Y.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Chin. Opt. Lett. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Opt. Commun. (1)

J. Ou, Y. Zhang, J. Tang, H. He, and Y. Wang, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Opt. Express (5)

Opt. Lett. (6)

Opt. Rev. (1)

P. Ostlund and A. T. Friberg, “Radiometry and radiation efficiency of twisted Gaussian Schell-model sources,” Opt. Rev. 8, 1–8 (2001).
[Crossref]

Phys. Rev. Lett. (2)

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

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 (2001).
[Crossref]

Other (2)

A. E. Siegman, Lasers (University Science Books, 1986).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

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Figures (9)

Fig. 1.
Fig. 1. Distribution of the OAM spectrum of a TGSM beam at different propagating distances $ z $ in atmospheric turbulence. Note that different vertical scales are used.
Fig. 2.
Fig. 2. Energy weight of each OAM mode varying with the propagating distance $z$. Lines with OAM mode of $ \pm m $ are coincident.
Fig. 3.
Fig. 3. Decrement of the energy weight of mode zero (red line) and the increment of the energy weight of the sum of other modes (green line). The blue line denotes the total change of the energy weight of all modes.
Fig. 4.
Fig. 4. Distribution of the OAM spectrum of a TGSM beam with different values of twist factor $ u $ at different propagating distances.
Fig. 5.
Fig. 5. Distribution of the OAM spectrum of a TGSM beam with different beam waist sizes $ {\omega _0} $ in atmospheric turbulence.
Fig. 6.
Fig. 6. Distribution of the OAM spectrum of a TGSM beam with different coherence lengths $ {\delta _0} $ in atmospheric turbulence.
Fig. 7.
Fig. 7. Distribution of the OAM spectrum of a TGSM beam with different wavelengths $ \lambda $ in atmospheric turbulence.
Fig. 8.
Fig. 8. Energy weight of each OAM mode varying with the inner scale $ {l_0} $ in atmospheric turbulence of $z = {2000}\,\,{\rm m}$.
Fig. 9.
Fig. 9. Distribution of the OAM spectrum of a TGSM beam for different generalized structure constants $ C_n^2 $ in atmospheric turbulence.

Tables (1)

Tables Icon

Table 1. Reduction Proportion of the Central OAM Mode with Different Propagation Distances

Equations (22)

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W ( ρ , φ 1 , φ 2 ) = 1 2 π m | β m ( ρ ) | 2 exp [ i m ( φ 1 φ 2 ) ] ,
| β m ( ρ ) | 2 = 1 2 π 0 2 π 0 2 π W ( ρ , φ 1 , φ 2 ) × exp [ i m ( φ 1 φ 2 ) ] d φ 1 d φ 2 .
C m = 0 | β m ( ρ ) | 2 ρ d ρ .
C m N = C m m = C m ,
W ( ρ , φ 1 , φ 2 ; z ) = W f r e e ( ρ , φ 1 , φ 2 ; z ) × exp [ ψ ( ρ , φ 1 , z ) + ψ ( ρ , φ 2 , z ) ] ,
exp [ ψ ( ρ , φ 1 , z ) + ψ ( ρ , φ 2 , z ) ] = exp [ 2 ρ 2 1 cos ( φ 1 φ 2 ) ρ 0 2 ] ,
ρ 0 = [ π 2 3 k 2 z 0 κ 3 ϕ n ( κ ) d κ ] 1 / 2 ,
ϕ n ( κ ) = 0.033 C n 2 κ 11 / 3 exp ( κ 2 / κ m 2 ) ,
ρ 0 = ( 0.5466 k 2 z C n 2 l 0 1 / 3 ) 1 / 2 .
W ( ρ 1 , ρ 2 ) = exp [ ρ 1 2 + ρ 2 2 ω 0 2 ( ρ 1 ρ 2 ) 2 2 δ 0 2 i u ( ρ 1 × ρ 2 ) ] ,
W ( ρ 1 , ρ 2 ) = m 0 = μ m 0 N m 0 I m 0 ( 2 α ρ 1 ρ 2 ) exp [ s ( ρ 1 2 + ρ 2 2 ) ] × exp [ i m 0 ( φ 1 φ 2 ) ] ,
μ m 0 = π t m 0 2 s 2 α 2 ( α s + s 2 α 2 ) | m 0 | ,
N m 0 = 2 π s 2 α 2 ( s + s 2 α 2 α ) | m 0 | ,
s = a + b ; α = b 2 u 2 4 ; t = b + u 2 b u 2 ,
a = 1 / ( ω 0 2 ) ; b = 1 / ( 2 δ 0 2 ) .
W f r e e ( ρ 1 , ρ 2 ; z ) = m 0 = ω 0 2 ω 2 ( z ) μ m 0 N m 0 I m 0 [ ω 0 2 ω 2 ( z ) 2 α ρ 1 ρ 2 ] × exp [ ω 0 2 ω 2 ( z ) s ( ρ 1 2 + ρ 2 2 ) ] × exp [ i m 0 ( φ 1 φ 2 ) ] ,
ω 2 ( z ) = ω 0 2 + 4 z 2 / ( k ω 0 ) 2 .
0 2 π exp [ i m φ 1 + η cos ( φ 1 φ 2 ) ] d φ 1 = 2 π exp [ i m φ 2 ] I m ( η ) ,
| β m ( ρ , z ) | 2 = m 0 = 2 π ω 0 2 ω 2 ( z ) μ m 0 N m 0 × exp [ ω 0 2 ω 2 ( z ) 2 s ρ 2 2 ρ 2 ρ 0 2 ] × I | m 0 | [ ω 0 2 ω 2 ( z ) 2 α ρ 2 ] I | m 0 m | [ 2 ρ 2 ρ 0 2 ] ,
0 x λ 1 exp ( α x ) J μ ( β x ) J ν ( γ x ) d x = β μ γ ν Γ ( ν + 1 ) 2 ν μ α λ μ ν l = 0 Γ ( λ + μ + ν + 2 l ) l ! Γ ( μ + l + 1 ) × F ( l , μ l ; ν + 1 ; γ 2 β 2 ) ( β 2 4 α 2 ) l ; R e ( λ + μ + ν ) > 0 , R e ( α ± i β ± i γ ) > 1 ,
C m = m 0 = π ω 0 2 ω 2 ( z ) μ m 0 N m 0 i | m 0 | | m 0 m | × ( 2 i α ) | m 0 | ( 2 i / ρ 0 2 ) | m 0 m | Γ ( | m 0 m | + 1 ) 2 | m 0 | | m 0 m | × ( 2 s ) 1 | m 0 | | m 0 m | l = 0 Γ ( 1 + | m 0 | + | m 0 m | + 2 l ) l ! Γ ( | m 0 | + l + 1 ) × F [ l , | m 0 | l ; | m 0 m | + 1 ; ( 2 i / ρ 0 2 ) 2 ( 2 i α ) 2 ] × [ ( 2 i α ) 2 4 ( 2 s ) 2 ] l ,
s = ω 0 2 ω 2 ( z ) s + 1 ρ 0 2 , α = ω 0 2 ω 2 ( z ) α .

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