Abstract

Based on partially coherent Bessel–Gaussian beams (BGBs), the coherence evolution of the partially coherent beams carrying optical vortices in non-Kolmogorov turbulence is investigated in detail. The analytical formulas for the spatial coherence length of partially coherent BGBs with optical vortices in non-Kolmogorov turbulence have been derived by using the combination of a coherence superposition approximation of decentered Gaussian beams and the extended Huygens–Fresnel principle. The influences of beam and turbulence parameters on spatial coherence are investigated by numerical examples. Numerical results reveal that the coherence of the partially coherent laser beam with vortices is independent of the optical vortices, and the spatial correlation length of the beams does not decrease monotonically during propagation in non-Kolmogorov turbulence. Within a certain propagation distance, the coherence of the partially coherent beam will improve, and the improvement of the coherence of the partially coherent beams is closely related to the beam and turbulence parameters.

© 2013 Optical Society of America

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References

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2014

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

2012

2011

2010

2009

X. Li, X. Chen, and X. Ji, “Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite–Gaussian beams,” Opt. Commun. 282, 7–13 (2009).
[CrossRef]

2008

2007

2006

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A 8, 537–545 (2006).
[CrossRef]

2005

A. Vinotte and L. Berg, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef]

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

2004

2003

2002

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[CrossRef]

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[CrossRef]

2000

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” J. Exp. Theor. Phys. Lett. 71, 130–133 (2000).
[CrossRef]

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre–Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[CrossRef]

1999

1997

1994

1990

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

1987

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gaussian beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

1979

1978

1974

Allen, L.

Amarande, S.

Andrews, L. C.

Arlt, J.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre–Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[CrossRef]

Baykal, Y.

Berg, L.

A. Vinotte and L. Berg, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef]

Bouchal, Z.

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[CrossRef]

Cai, Y.

Cang, J.

Chen, B.

B. Chen, Z. Chen, and J. Pu, “Propagation of partially coherent Bessel–Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008).
[CrossRef]

Chen, S.

Chen, X.

X. Li, X. Chen, and X. Ji, “Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite–Gaussian beams,” Opt. Commun. 282, 7–13 (2009).
[CrossRef]

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554–3563 (2007).
[CrossRef]

Chen, Z.

B. Chen, Z. Chen, and J. Pu, “Propagation of partially coherent Bessel–Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008).
[CrossRef]

Cheong, W. C.

Chu, X.

Dang, A.

Dholakia, K.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre–Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997).
[CrossRef]

Dogariu, A.

Du, X.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Eyyuboglu, H. T.

Fante, R. L.

Gbur, G.

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gaussian beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gaussian beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Guo, H.

Hitomi, T.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre–Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[CrossRef]

Ji, X.

X. Li, X. Chen, and X. Ji, “Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite–Gaussian beams,” Opt. Commun. 282, 7–13 (2009).
[CrossRef]

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554–3563 (2007).
[CrossRef]

Korotkova, O.

Leader, J. C.

Lee, W. M.

Li, S.

Li, X.

Liu, X.

Liu, Z.

Lü, B.

Luo, B.

Ma, Y.

Marienko, I. G.

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” J. Exp. Theor. Phys. Lett. 71, 130–133 (2000).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Miller, W. B.

Okayama, H.

Ou, B.

Padgett, M. J.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gaussian beams,” Opt. Commun. 64, 491–495 (1987).
[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.

Pu, J.

X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19, 26444–26450 (2011).
[CrossRef]

B. Chen, Z. Chen, and J. Pu, “Propagation of partially coherent Bessel–Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008).
[CrossRef]

Qin, Z.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

Ricklin, J. C.

Schmidt, J. D.

Shchepakina, E.

Shirai, T.

Simpson, N. B.

Soskin, M. S.

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” J. Exp. Theor. Phys. Lett. 71, 130–133 (2000).
[CrossRef]

Tang, H.

Tang, Y.

Tao, R.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

Vasnetsov, M. V.

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” J. Exp. Theor. Phys. Lett. 71, 130–133 (2000).
[CrossRef]

Vinotte, A.

A. Vinotte and L. Berg, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef]

Wang, L. Z.

Wang, S. C. H.

Wang, X.

Wheeler, D. J.

Wolf, E.

Wu, G.

Wu, J.

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

Wu, Y.

Xu, X.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

Yu, S.

Yu, Y.

Yuan, X.-C.

Zhao, D.

Zhao, H.

Zheng, X.

Zhou, G.

Zhou, P.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35, 1043–1045 (2010).
[CrossRef]

Zhu, K.

Zhu, Y.

Appl. Opt.

Appl. Phys. B

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre–Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[CrossRef]

J. Exp. Theor. Phys. Lett.

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” J. Exp. Theor. Phys. Lett. 71, 130–133 (2000).
[CrossRef]

J. Mod. Opt.

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

J. Opt. A

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A 8, 537–545 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
[CrossRef]

H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial Gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28, 1016–1021 (2011).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891–2901 (2007).
[CrossRef]

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554–3563 (2007).
[CrossRef]

X. Chu, Z. Liu, and Y. Wu, “Propagation of a general multi-Gaussian beam in turbulent atmosphere in a slant path,” J. Opt. Soc. Am. A 25, 74–79 (2008).
[CrossRef]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[CrossRef]

L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11, 1653–1660 (1994).
[CrossRef]

K. Zhu, S. Li, Y. Tang, Y. Yu, and H. Tang, “Study on the propagation parameters of Bessel–Gaussian beams carrying optical vortices through atmospheric turbulence,” J. Opt. Soc. Am. A 29, 251–257 (2012).
[CrossRef]

Opt. Commun.

X. Li, X. Chen, and X. Ji, “Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite–Gaussian beams,” Opt. Commun. 282, 7–13 (2009).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gaussian beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[CrossRef]

Opt. Express

Opt. Laser Technol.

Z. Qin, R. Tao, P. Zhou, X. Xu, and Z. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
[CrossRef]

B. Chen, Z. Chen, and J. Pu, “Propagation of partially coherent Bessel–Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

A. Vinotte and L. Berg, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

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

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Figures (4)

Fig. 1.
Fig. 1.

Spatial correlation length of the beam in non-Kolmogorov turbulence as a function of propagation distance for different values of initial spatial correlation length.

Fig. 2.
Fig. 2.

Spatial correlation length of the beam in non-Kolmogorov turbulence along z for different values of beam and turbulence parameters. (a) σ0=0.009m; l0=0.001m; L0=5m; α=3.5; and C˜n2=1×1014m3α. (b) σ0=0.009m; w0=0.05m; l0=0.001m; L0=5m; and C˜n2=1×1014m3α. (c) σ0=0.009m; w0=0.05m; l0=0.001m; L0=5m; and α=3.5. (d) σ0=0.009m; w0=0.05m; L0=5m; α=3.5; and C˜n2=1×1014m3α. (e) σ0=0.009m; w0=0.05m; l0=0.001m; α=3.5; and C˜n2=1×1014m3α.

Fig. 3.
Fig. 3.

Values of propagation distance and spatial correlation length of the prime turning point for different values of beam and turbulence parameters. (a) l0=0.001m; L0=5m; α=3.5; and C˜n2=1×1014m3α. (b) σ0=0.009m; w0=0.02m; l0=0.001m; and L0=5m. (c) σ0=0.009m; w0=0.02m; α=3.5; and C˜n2=1×1014m3α.

Fig. 4.
Fig. 4.

Variation of the modulus of the complex DOC against the propagation length: σ0=0.01m; w0=0.02m; l0=0.001m; L0=5m; and C˜n2=1×1014m3α.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

E(x,y,0)1Minexp(R24w02)m=0M1exp[(xi2Rcosθm)2+(yi2Rsinθm)2w02+iφm],
W(x1,y1;x2,y2;0)=1M2p=0M1q=0M1exp[in(pq)α0]exp((x12+x22)+(y12+y22)w02)×exp[iR(x1cospα0x2cosqα0)+iR(y1sinpα0y2sinqα0)w02]×exp[(x1x2)2+(y1y2)22σ02]exp{ik[(x12+y12)(x22+y22)]2F}.
W(r⃗1,r⃗2,z)=(k2πz)2×dr⃗1dr⃗2W(r⃗1,r⃗2,0)exp{ik2z[(r⃗1r⃗1)2(r⃗2r⃗2)2]}×exp[ψ(r⃗1,r⃗1)+ψ*(r⃗2,r⃗2)],
exp[ψ(r⃗1,r⃗1)+ψ*(r⃗2,r⃗2)]=exp{4π2k2z010dκdξΦn(κ,α)[1J0(κ|(1ξ)(r⃗2r⃗1)+ξ(r⃗2r⃗1)|)]},
Φn(κ,α)=H(α)C˜n2exp(κ2/κm2)(κ2+κ02)α/2,0κ<,3<α<4,
W(r⃗1,r⃗2,z)=1M2p=0M1q=0M1exp[in(pq)α0]Wx(x1,x2,z)Wy(y1,y2,z)
Wx(x1,x2,z)=1τexp(A1)exp{1w02τ2[(x1G1)2+(x2G2)2]}×exp{(x1x2)22σ2+ik2z(x12x22)+z2ikτ2b1[(x1b22b1)2(x2b32b1)2]},
Wy(y1,y2,z)=1τexp(A2)exp{1w02τ2[(y1G3)2+(y2G4)2]}×exp{(y1y2)22σ2+ik2z(y12y22)+z2ikτ2b1[(y1b42b1)2(y2b52b1)2]},
σ={2T(α,z)[1+1τ2(1+τ1τ22w02T(α,z)2)]+1τ2σ02}1/2,
A1=z2R2(cospα0cosqα0)2k2τ2w02×[z22(3T(α,z)zT(α,z)F+1zσ02)2+zw02(3T(α,z)zT(α,z)F+1zσ02)1w02(T(α,z)+12σ02)],
A2=z2R2(sinpα0sinqα0)2k2τ2w02×[z22(3T(α,z)zT(α,z)F+1zσ02)2+zw02(3T(α,z)zT(α,z)F+1zσ02)1w02(T(α,z)+12σ02)],
G1=z2R2k[2cospα0zw02+(3T(α,z)zT(α,z)F+1zσ02)(cospα0cosqα0)],
G2=z2R2k[2cosqα0zw02(3T(α,z)zT(α,z)F+1zσ02)(cospα0cosqα0)],
G3=z2R2k[2sinpα0zw02+(3T(α,z)zT(α,z)F+1zσ02)(sinpα0sinqα0)],
G4=z2R2k[2sinqα0zw02(3T(α,z)zT(α,z)F+1zσ02)(sinpα0sinqα0)],
b1=2T(α,z)w02zk2τ12z3,
b2=kRw02z2[τ1cospα0+τ424(cospα0+cosqα0)],
b3=kRw02z2[τ1cosqα0+τ424(cospα0+cosqα0)],
b4=kRw02z2[τ1sinpα0+τ424(sinpα0+sinqα0)],
b5=kRw02z2[τ1sinqα0+τ424(sinpα0+sinqα0)],
τ1=1zFτ2=2zkw02τ3=2zkw0σ0τ4=8z2T(α,z)k2w02,
τ2=τ12+τ22+τ32+τ42,
T(α,z)=13π2k2z[A(α)C˜n22κm2αβexp(κ02κm2)Γ(2α2,κ02κm2)2κ04αα2],
μ(x1,y1;x2,y2;z)=W(x1,y1,x2,y2,z)W(x1,y1,x1,y1,z)W(x2,y2,x2,y2,z)=p=0M1q=0M1exp[in(pq)α0]Wx(x1,x2,z)Wy(y1,y2,z)p=0M1q=0M1exp[in(pq)α0]Wx(x1,x1,z)Wy(y1,y1,z)p=0M1q=0M1exp[in(pq)α0]Wx(x2,x2,z)Wy(y2,y2,z).
|μ(x1,y1;x2,y2;z)|=exp{[(x1x2)22σ2+(y1y2)22σ2]}.

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