Abstract

The transverse position of an optical vortex on propagation through atmospheric turbulence is studied. The probability density of the optical vortex position on a transverse plane in the atmosphere is formulated in weak turbulence by using the Born approximation. With these formulas, the effect of aperture averaging on topological charge detection is investigated. These results provide quantitative guidelines for the design of an optimal detector of topological charge, which has potential application in optical vortex communication systems.

© 2013 Optical Society of America

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References

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  2. G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 55, pp. 285–341.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  29. V. P. Lukin, V. A. Sennikov, and V. A. Tartakovski, “Optical vortices: creation, annihilation, and modeling,” Proc. SPIE 4724, 85–93 (2002).
    [CrossRef]

2012

2010

2009

G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009).
[CrossRef]

C. Chen, H. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett. 34, 419–421 (2009).
[CrossRef]

D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. 48, 036001 (2009).
[CrossRef]

A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E 80, 046609 (2009).
[CrossRef]

2008

2005

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

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

2004

2002

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

V. P. Lukin, V. A. Sennikov, and V. A. Tartakovski, “Optical vortices: creation, annihilation, and modeling,” Proc. SPIE 4724, 85–93 (2002).
[CrossRef]

1997

1992

D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
[CrossRef]

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

1989

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

1988

1987

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

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

Andrews, L. C.

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

Barnett, S. M.

Baykal, Y.

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98, 857–863 (2010).
[CrossRef]

Beijersbergen, M. W.

Bergé, L.

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

Borah, D. K.

Bouchal, Z.

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

Boyd, R. W.

Chen, C.

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Courtial, J.

Dainty, C.

Dainty, J. C.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Dipankar, A.

A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E 80, 046609 (2009).
[CrossRef]

Durnin, J.

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

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (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.

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98, 857–863 (2010).
[CrossRef]

Feng, X.

Flatté, S. M.

Franke-Arnold, S.

Fried, D. L.

Gbur, G.

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–230 (2008).
[CrossRef]

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 55, pp. 285–341.

Gibson, G.

Gil, L.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Glindemann, A.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Ji, X.

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98, 857–863 (2010).
[CrossRef]

Karman, G. P.

Kravtsov, Y. A.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer, 1989).

Lane, R. G.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Lavery, M. P. J.

Leach, J.

Lukin, V. P.

V. P. Lukin, V. A. Sennikov, and V. A. Tartakovski, “Optical vortices: creation, annihilation, and modeling,” Proc. SPIE 4724, 85–93 (2002).
[CrossRef]

Malik, M.

Marchiano, R.

A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E 80, 046609 (2009).
[CrossRef]

Martin, J. M.

Miceli, J. J.

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

Mirhosseini, M.

Murphy, K.

O’Sullivan, M.

O’Sullivan, M. N.

Padgett, M.

Padgett, M. J.

Pas’ko, V.

Paterson, C.

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

Phillips, R. L.

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

Robertson, D. J.

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Rodenburg, B.

Rytov, S. M.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer, 1989).

Sagaut, P.

A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E 80, 046609 (2009).
[CrossRef]

Schmidt, J. D.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

Sennikov, V. A.

V. P. Lukin, V. A. Sennikov, and V. A. Tartakovski, “Optical vortices: creation, annihilation, and modeling,” Proc. SPIE 4724, 85–93 (2002).
[CrossRef]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219–276.

Tartakovski, V. A.

V. P. Lukin, V. A. Sennikov, and V. A. Tartakovski, “Optical vortices: creation, annihilation, and modeling,” Proc. SPIE 4724, 85–93 (2002).
[CrossRef]

Tatarskii, V. I.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer, 1989).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Tyler, G. A.

Tyson, R. K.

van Duijl, A.

Vasnetsov, M.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219–276.

Vaughn, J. L.

Vinçotte, A.

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

Visser, T. D.

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 55, pp. 285–341.

Voelz, D. G.

D. K. Borah and D. G. Voelz, “Spatially partially coherent beam parameter optimization for free space optical communications,” Opt. Express 18, 20746–20758 (2010).
[CrossRef]

D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. 48, 036001 (2009).
[CrossRef]

Wang, H.

Woerdman, J. P.

Xiao, X.

D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. 48, 036001 (2009).
[CrossRef]

Yang, H.

Appl. Opt.

Appl. Phys. B

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98, 857–863 (2010).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

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

Opt. Eng.

D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. 48, 036001 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. E

A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E 80, 046609 (2009).
[CrossRef]

Phys. Rev. Lett.

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

A. Vinçotte and L. Bergé, “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]

Proc. SPIE

V. P. Lukin, V. A. Sennikov, and V. A. Tartakovski, “Optical vortices: creation, annihilation, and modeling,” Proc. SPIE 4724, 85–93 (2002).
[CrossRef]

Waves Random Media

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Other

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer, 1989).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 219–276.

L. Allen, S. M. Barnett, and M. J. Padgett, eds. Optical Angular Momentum (IOP, 2003).

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

G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 55, pp. 285–341.

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Figures (5)

Fig. 1.
Fig. 1.

(a), (b) Intensity pattern and phase map of an LG01 beam on the source plane z=0. (c), (d) Intensity pattern and phase map of an LG01 beam on the receiver plane z=5km. Here the wavelength is taken to be λ=1.55μm, the turbulence strength is taken to be Cn2=1015m2/3, and the width of the beam is taken to be w0=0.05m.

Fig. 2.
Fig. 2.

(a), (b) Histogram and probability density of xv. (c), (d) Histogram and probability density of yv. The histograms in (a), (c) and unfilled shapes (open circles) in (b), (d) are obtained by numerical simulation. The solid (theory) curves in (b) and (d) are the marginal probability densities of xv and yv calculated from Eqs. (16)–(18). The parameters are the same as in Fig. 1.

Fig. 3.
Fig. 3.

Illustration of xv2 as a function of initial width w0 of LG01. The parameters are the same as in Fig. 1.

Fig. 4.
Fig. 4.

Illustration of probability of successful topological charge detection as a function of the diameter of a circular detector or the side length of a square detector. The detector geometry is shown as the inset. The parameters are the same as in Fig. 1.

Fig. 5.
Fig. 5.

Illustration of the joint probability density of (xv,yv) for (a) κ=(1.5κx,κy,0) and (b) κ=(κx,3κy,0). The parameters are the same as in Fig. 1.

Tables (2)

Tables Icon

Table 1. First- and Second-Order Moments of xv and yv on the Receiver Planea

Tables Icon

Table 2. Standard Deviation and Correlation Coefficient of xv and yv on the Receiver Planea

Equations (49)

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

U(r)=A(r)eiϕ(r).
t=12πCϕ(r)·dl.
Unm(r,θ)=Cnm(2rw0)mLnm(2r2w02)exp(imθr2/w02),
U(r,z)=U0(r,z)exp[ψ1(r,z)+ψ2(r,z)+],
U(r,z)=U0(r,z)+U1(r,z)+U2(r,z)+,
U0(x,y,L)=C01exp(ikL)q2(L)(x+iy)exp[x2+y2q(L)w02],
U1(r,L)=k22π0Ldzexp[ik(Lz)+ik|sr|22(Lz)]×U0(r,L)n1(s,z)Lzd2s,
U2(r,L)=k22π0Ldzexp[ik(Lz)+ik|sr|22(Lz)]×U1(r,L)n1(s,z)Lzd2s.
n1(r,z)=exp(iκ·r)dv(κ,z),
U1(x,y,L)=C01exp(ikL)q2(L)exp[x2+y2q(L)w02][T11aiT11b+T12(x+iy)],
T11a=0L(Lz)dzκyexp[iγκ·riγκ2(Lz)2k]dv(κ,z),
T11b=0L(Lz)dzκxexp[iγκ·riγκ2(Lz)2k]dv(κ,z),
T12=ik0Ldzexp[iγκ·riγκ2(Lz)2k]dv(κ,z).
U2(x,y,L)=C01exp(ikL)q2(L)exp[x2+y2q(L)w02]×[T21a+iT21b+T22a+iT22b+T23(x+iy)],
T21a=0Ldz0zkγ(zz)dzκxexp[iγκ2(zz)2k]×exp[iγ(Lz)2k|κ+γκ|2]×exp[iγ(κ+γκ)·r]dv(κ,z)dv(κ,z),
T21b=0Ldz0zkγ(zz)dzκyexp[iγκ2(zz)2k]×exp[iγ(Lz)2k|κ+γκ|2]×exp[iγ(κ+γκ)·r]dv(κ,z)dv(κ,z),
T22a=k0Ldz0z(Lz)dz(κx+γκx)×exp[iγκ2(zz)2k]exp[iγ(Lz)2k|κ+γκ|2]×exp[iγ(κ+γκ)·r]dv(κ,z)dv(κ,z),
T22b=k0Ldz0z(Lz)dz(κy+γκy)×exp[iγκ2(zz)2k]exp[iγ(Lz)2k|κ+γκ|2]×exp[iγ(κ+γκ)·r]dv(κ,z)dv(κ,z),
T23=k20Ldz0zdzexp[iγκ2(zz)2k]×exp[iγ(Lz)2k|κ+γκ|2]×exp[iγ(κ+γκ)·r]dv(κ,z)dv(κ,z).
{Re(U0+U1+U2)=0Im(U0+U1+U2)=0,
{xv=ImT11bReT11ayv=ImT11a+ReT11b.
xv2=12ReT11bT11b*12ReT11bT11b+12ReT11aT11a+12ReT11aT11a*+ImT11aT11bImT11aT11a*,
yv2=12ReT11aT11a*12ReT11aT11a+12ReT11bT11b+12ReT11bT11b*ImT11aT11bImT11aT11b*,
xvyv=12ImT11aT11a12ImT11bT11bReT11aT11b,
T11aT11a=2π0L(Lη)2dηκy2exp[iγκ2(Lη)k]Φn(κ)d2κ,
T11aT11a*=2π0L(Lη)2dηκy2exp[i(γγ*)κ2(Lη)2k]Φn(κ)d2κ,
T11bT11b=2π0L(Lη)2dηκx2exp[iγκ2(Lη)k]Φn(κ)d2κ,
T11bT11b*=2π0L(Lη)2dηκx2exp[i(γγ*)κ2(Lη)2k]Φn(κ)d2κ,
T11aT11b=2π0L(Lη)2dηκxκyexp[iγκ2(Lη)k]Φn(κ)d2κ,
T11aT11b*=2π0L(Lη)2dηκxκyexp[i(γγ*)κ2(Lη)2k]Φn(κ)d2κ.
p(xv,yv)=12πσxσy1ρ2exp[12(1ρ2)(xv2σx2+yv2σy22ρxvyvσxσy)],
Φn(κ)=0.033Cn2exp(κ2/κm2)(κ2+κ02)11/6,
P(xv2+yv2D24)=1exp(D28σ2),
P(|xv|D2,|yv|D2)=erf(D22σ)2,
{(1+ReT12+ReT23)xv(ImT12+ImT23)yv=ReT11aImT11bReT21a+ImT21bReT22a+ImT22b(1+ReT12+ReT23)yv+(ImT12+ImT23)xv=ImT11a+ReT11bImT21aReT21bImT22aReT22b.
{xv=ImT11bReT11a1+2ReT12+T12T12*+2ReT23yv=ImT11a+ReT11b1+2ReT12+T12T12*+2ReT23.
{xv=ImT11bReT11a1+2ReT12+T12T12*yv=ImT11a+ReT11b1+2ReT12+T12T12*.
T12=ik0Ldzexp[iγκ2(Lz)2k]dv(κ,z),
T12T12*=2πk20Ldηexp[iκ22k(γγ*)(Lη)]Φn(κ)d2κ,
(ReT12)2=(T12T12*+ReT12T12)/2.
T12T12=2πk20Ldηexp[iκ2γk(Lη)]Φn(κ)d2κ.
T23=k20Ldz0zdzexp[iγκ2(zz)2k]×exp[iγ(Lz)2k|κ+γκ|2]dv(κ,z)dv(κ,z),
T23=πk20LdηΦn(κ)d2κ,
T11a=0L(Lz)dzκyexp[iγκ2(Lz)2k]dv(κ,z),
T11aT11a=0L(Lz)dz0L(Lz)dz×exp[iγκ2(Lz)2k]exp[iγκ2(Lz)2k]κyκydv(κ,z)dv(κ,z).
dv(κ,z)dv(κ,z)=Fn(κ,|zz|)δ(κ+κ)d2κd2κ,
Φn(κ)=12πFn(κ,μ)dμ,
T11aT11a=2π0L(Lη)2dηκy2exp[iγκ2(Lη)k]Φn(κ)d2κ,
dv(κ,z)dv*(κ,z)=Fn(κ,|zz|)δ(κκ)d2κd2κ.

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