Abstract

The effect of isotropic and homogeneous random media on propagation characteristics of recently introduced multi-Gaussian Schell-model (MGSM) vortex beams is investigated. The analytical formula for the cross-spectral density function of such a beam propagating in random turbulent media is derived and used to explore the evolution of the spectral density, the degree of coherence and the turbulence-induced spreading. An example illustrates the fact that, at sufficiently large distance from the source, the source correlations modulation of the spectral distribution in free space is shown to be suppressed by the uniformly correlated turbulence. The impacts, arising from the index M, the correlation width of the source and the properties of the medium on such characteristics are analyzed in depth.

© 2015 Optical Society of America

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References

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    [Crossref]
  27. Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
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    [Crossref]
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    [Crossref]
  30. F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005).
    [Crossref]
  31. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
    [Crossref]

2014 (6)

M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312, 89–93 (2014).
[Crossref]

Z. Mei, “Two types of sinc Schell-model beams and their propagation characteristics,” Opt. Lett. 39(14), 4188–4191 (2014).
[Crossref] [PubMed]

Z. Mei and Y. Mao, “Electromagnetic sinc Schell-model beams and their statistical properties,” Opt. Express 22(19), 22534–22546 (2014).
[Crossref] [PubMed]

Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
[Crossref] [PubMed]

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

2013 (1)

2012 (3)

2011 (2)

2010 (2)

2009 (1)

J. Li and B. Lü, “Evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
[Crossref]

2008 (3)

2007 (3)

2005 (1)

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005).
[Crossref]

2004 (1)

2003 (4)

2002 (1)

1990 (1)

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37(4), 671–684 (1990).
[Crossref]

Amarande, S.

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Baykal, Y.

Borghi, R.

Cai, Y.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Chan, C. T.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref] [PubMed]

Chen, Q.

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

Chen, Z.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[Crossref]

Dogariu, A.

Du, X.

Duan, Z.

Eyyuboglu, H.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Flossmann, F.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005).
[Crossref]

Gao, Z.

Gbur, G.

Gori, F.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

Guo, H.

He, X.

Hua, L.

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

Huang, Y.

Korotkova, O.

Li, J.

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
[Crossref]

J. Li and B. Lü, “Evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
[Crossref]

Lin, Z.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref] [PubMed]

Liu, L.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Lü, B.

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
[Crossref]

X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov atmospheric turbulence,” J. Opt. Soc. Am. A 28(9), 1941–1948 (2011).
[Crossref]

J. Li and B. Lü, “Evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
[Crossref]

Luo, B.

Luo, M.

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

Maier, M.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005).
[Crossref]

Mao, Y.

Mei, Z.

Ng, J.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref] [PubMed]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Pu, J.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[Crossref]

Ramírez-Sánchez, V.

Sahin, S.

Santarsiero, M.

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005).
[Crossref]

Shchepakina, E.

Shirai, T.

Tang, M.

M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312, 89–93 (2014).
[Crossref]

Tong, Z.

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Tyson, R. K.

Wang, T.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[Crossref]

Wolf, E.

Wu, G.

Wu, J.

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37(4), 671–684 (1990).
[Crossref]

Yu, S.

Zhang, B.

Zhang, Y.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Zhao, C.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Zhao, D.

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312, 89–93 (2014).
[Crossref]

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

Zhao, G.

J. Mod. Opt. (1)

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37(4), 671–684 (1990).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

J. Li and B. Lü, “Evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11(4), 045710 (2009).
[Crossref]

J. Opt. Soc. Am. A (7)

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

Opt. Commun. (3)

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005).
[Crossref]

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
[Crossref]

M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312, 89–93 (2014).
[Crossref]

Opt. Eng. (1)

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[Crossref]

Opt. Express (4)

Opt. Lett. (7)

Phys. Lett. A (2)

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Phys. Rev. Lett. (1)

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref] [PubMed]

Proc. SPIE (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Other (2)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (Bellingham, 1998)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995)

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Figures (7)

Fig. 1
Fig. 1 Evolution of normalized spectral intensity of a GSM vortex beam ( M=1 ) and a MGSM vortex beam with ( M=5 ) propagating in (a) free space and (b) non-Kolmogorov atmospheric turbulence.
Fig. 2
Fig. 2 (a) The normalized spectral distribution of the MGSM vortex beam with M=5 at propagation distance z=10km with different δ in (a) free space and (b) non-Kolmogorov atmospheric turbulence.
Fig. 3
Fig. 3 The normalized spectral distribution of a MGSM vortex beam propagation in atmospheric turbulence at propagation distance z=500m , (a) for different Mwith C ˜ n 2 = 10 14 m 3α ; (b) for different C ˜ n 2 with M=5 .
Fig. 4
Fig. 4 Modulus of the spectral degree of coherence of a MGSM vortex beam with several values of M as a function of separation distance x d at different propagation distances in free space, (a) z=0.5km ; (b) z=1km ; (c) z=2km ; (d) z=5km .
Fig. 5
Fig. 5 As in Fig. 4 but on propagation in non-Kolmogorov turbulence.
Fig. 6
Fig. 6 Modulus of the spectral degree of coherence of the MGSM vortex beam propagating in atmospheric turbulence as a function of separation distance x d at propagation distance z=2km , (a) for different C ˜ n 2 with δ=0.5cm ; (b) for different δwith C ˜ n 2 = 10 14 m 3α .
Fig. 7
Fig. 7 Relative width of the MGSM vortex beams propagating through atmospheric turbulence, (a) for different Mwith δ=0.5cm as a function of z; (b) for different δwith M=5 as a function of z; (c) at several propagation distances as a function of α.

Equations (30)

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U( ρ ,z=0)=u( ρ ) [ x +isgn(l) y ] l ,
W (0) ( ρ 1 , ρ 2 ,0)= 1 C 0 [ ( x 1 x 2 + y 1 y 2 )+isgn(l)( x 1 y 2 x 2 y 1 ) ] l ×exp( ρ 1 2 + ρ 2 2 w 0 2 ) m=1 M ( M m ) (1) m1 m exp[ ( ρ 1 ρ 2 ) 2 2m δ 2 ],
W( ρ 1 , ρ 2 ,z)= ( k 2πz ) 2 W (0) ( ρ 1 , ρ 2 ,0)exp[ ik ( ρ 1 ρ 1 ) 2 ( ρ 2 ρ 2 ) 2z ] × exp[ ψ * ( ρ 1 , ρ 1 ,z)+ψ( ρ 2 , ρ 2 ,z) ] R d 2 ρ 1 d 2 ρ 2 ,
exp[ ψ * ( ρ 1 , ρ 1 ' ,z)+ψ( ρ 2 , ρ 2 ' ,z) ] R = exp{ π 2 k 2 z 3 [ ( ρ 1 ρ 2 ) 2 +( ρ 1 ρ 2 )( ρ 1 ρ 2 )+ ( ρ 1 ρ 2 ) 2 ] 0 κ 3 Φ n (κ)dκ },
Φ n (κ)=A(α) C ˜ n 2 exp[ κ 2 / κ m 2 ] ( κ 2 + κ 0 2 ) α/2 , 0κ<,
κ 0 = 2π L 0 , κ m = c(α) l 0 ,
c(α)= [ 2π 3 Γ( 5 α 2 )A( α ) ] 1 α5 ,
A(α)= 1 4 π 2 Γ( α1 )cos( απ 2 ),
I= 0 κ 3 Φ n (κ)dκ = A(α) 2(α2) C ˜ n 2 κ m 2α βexp( κ 0 2 κ m 2 )Γ( 2 α 2 , κ 0 2 κ m 2 )2 κ 0 4α ,
W( ρ 1 , ρ 2 ,z)= 1 C 0 ( k 2πz ) 2 m=1 M ( M m ) (1) m1 m exp[ ik 2z ( ρ 1 2 ρ 2 2 ) ]exp[ π 2 k 2 zI 3 ( ρ 1 ρ 2 ) 2 ] × d 2 u d 2 v[ ( u 2 v 2 4 )i( u x v y u y v x ) ] exp( 2 w 0 2 u 2 )exp[ ik z ( ρ 1 ρ 2 )u ] ×exp( A v 2 ik z uv )exp{ v[ ik 2z ( ρ 1 + ρ 2 ) π 2 k 2 zI 3 ( ρ 1 ρ 2 ) ] },
A= 1 2 w 0 2 + 1 2m δ 2 + π 2 k 2 zI 3 ,
exp(p x 2 +2qx) dx=exp( q 2 p ) π p ,
xexp(p x 2 +2qx) dx=exp( q 2 p ) π p ( q p ),
x 2 exp(p x 2 +2qx) dx= 1 2p exp( q 2 p ) π p ( 1+ 2 q 2 p ),
W( ρ 1 , ρ 2 ,z)= 1 C 0 ( k 2πz ) 2 m=1 M ( M m ) (1) m1 m exp[ ik 2z ( ρ 1 2 ρ 2 2 ) ] ×exp{ π 2 k 2 zI 3 [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ] } ×[ ( N 1 N 2 )( N 3 N 4 ) ],
N 1 = π 2 AC B x B y exp( D x 2 + D y 2 C )( D x 2 + D y 2 C 2 + 1 C ),
N 2 = π 2 w 0 2 8F exp( k 2 w 0 2 8 z 2 | ρ 1 ρ 2 | 2 )exp( G x 2 + G y 2 F )( G x 2 + G y 2 F 2 + 1 C ),
N 3 = i π 2 w 0 2ACF B y D x G y CF exp[ k 2 w 0 2 8 z 2 ( y 1 y 2 ) 2 ]exp( D x 2 C + G y 2 F ),
N 4 = i π 2 w 0 2ACF B y D y G x CF exp[ k 2 w 0 2 8 z 2 ( x 1 x 2 ) 2 ]exp( D y 2 C + G x 2 F ),
B x =exp[ k 2 16A z 2 ( x 1 + x 2 ) 2 ]exp[ π 4 k 4 z 2 I 2 36A ( x 1 x 2 ) 2 ]exp[ i π 2 k 3 I 12A ( x 1 2 x 2 2 ) ],
D x = 1 2 [ ik z ( x 1 x 2 )+ k 2 4A z 2 ( x 1 + x 2 )+ i π 2 k 3 I 6A ( x 1 x 2 ) ],
G x = 1 2 [ ik z ( x 1 + x 2 ) π 2 k 2 zI 3 ( x 1 x 2 )+ k 2 w 0 2 4 z 2 ( x 1 x 2 ) ],
C= 2 w 0 2 + k 2 4A z 2 ,
F=A+ k 2 w 0 2 8 z 2 .
S( ρ,z )=W(ρ,ρ,z)= 1 C 0 ( k 2πz ) 2 m=1 M ( M m ) (1) m1 m { π 2 AC exp[ ( H 2 C H ) ρ 2 ] ×( H 2 ρ 2 C 2 + 1 C ) π 2 w 0 2 8F exp( J 2 ρ 2 F )( J 2 ρ 2 F 2 + 1 F ) },
H= k 2 4A z 2 , J= ik 2z .
μ( ρ 1 , ρ 2 ,z )= W( ρ 1 , ρ 2 ,z ) [ S( ρ 1 ,z )S( ρ 2 ,z ) ] 1 2 .
w( z )= ρ 2 S( ρ,z ) d 2 ρ S( ρ,z ) d 2 ρ .
x 2p exp( β x 2q )dx = Γ( 2p+1 2q ) / q β 2p+1 2q ,
w( z )= m=1 M ( M m ) (1) m1 m { π 2 AC ( H H 2 C ) 3 [ 2 H 2 C 2 + 1 C ( H H 2 C ) ]+ π 2 w 0 2 J 4 8 } m=1 M ( M m ) (1) m1 m { π 2 AC ( H H 2 C ) 2 [ H 2 C 2 + 1 C ( H H 2 C ) ] } .

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