Abstract

The propagation of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere has been investigated. The analytical expressions for the average intensity and the degree of the polarization of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system are derived in turbulent atmosphere, respectively. The average intensity distribution and the degree of the polarization of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere are numerically demonstrated. The influences of the beam parameters, the topological charge, the transverse coherent lengths, and the structure constant of the atmospheric turbulence on the propagation of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere are also examined in detail. This research is beneficial to the practical applications in free-space optical communications and the remote sensing of the dark hollow beams.

©2012 Optical Society of America

1. Introduction

Due to the important applications in atom optics, dark hollow beams have received considerable interest in the past decades [18]. Different theoretical beam models, e.g. the TEM01doughnut beam [1], the superposition of off-axis Gaussian beams [9], the higher order Bessel-Gaussian beam [10], the hollow Gaussian beam [11], and the controllable dark hollow beam [12] have been proposed to describe dark hollow laser beams, respectively. In experiment, dark hollow beams can be generated by using different methods. A dark hollow beam can be generated from a multimode fiber, and the dependence of the output beam profile on the incident angle of laser beam has been analyzed [13]. Hollow Gaussian beams can be created by spatial ðltering in the Fourier domain with spatial ðlters that consist of binomial combinations of even-order Hermite polynomials [14]. By coupling a partially coherent beam into a multimode ðber with a suitable incident angle, a high-quality partially coherent dark hollow beam can be experimentally generated [15]. The dark hollow beam can be produced via coherent combination based on adaptive optics [16]. Also, the dark hollow femtosecond pulsed beam has been generated by means of phase-only liquid crystal spatial light modulator [17]. By means of the different beam models, the properties of dark hollow beams propagating in free space and the turbulent atmosphere have been extensively theoretically investigated [1825]. However, the researches denote that the existing beam models have good propagation stability only in the region close to the source [11, 21]. With the increase of the propagation distance, the dark region decreases until it finally disappears. This defect seriously affects the atom manipulation with dark hollow beams. To overcome the above defect, a hollow vortex Gaussian beam is introduced, whose dark region always exists under arbitrary conditions. If the hollow Gaussian beam goes through a spiral phase plate, it becomes a hollow vortex Gaussian beam. The spiral phase plate can modulate the wave-front phase of the hollow Gaussian beam. As the initial vortex phase is introduced, the hollow vortex Gaussian beam can no longer be regarded as the superposition of a series of Laguerre-Gaussian beams. The propagation of a hollow vortex Gaussian beam in free space indicates that it has the remarkable propagation stability. In practical optical systems, however, laser beams are almost partially coherent [26], which denotes that fully coherent laser sources are only the ideal cases. Therefore, we now consider the partially coherent hollow vortex Gaussian beam. The research in the propagation of laser beams in a turbulent atmosphere is vital to the applications in free-space optical communications and the remote sensing. The propagation of various kinds of laser beams in a turbulent atmosphere has been extensively investigated [2736]. As to the practical applications of a laser beam in turbulent atmosphere, a series of optics systems is often used to direct or redirect a laser beam to a distant target plane. Accordingly, the analysis of propagation of a laser beam through an ABCD optical system in turbulent atmosphere is indispensable. In the remainder of this paper, therefore, the propagation of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere is investigated. Analytical expressions of the average intensity and the degree of the polarization are derived and illustrated by numerical examples.

2. Propagation of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere

In the Cartesian coordinate system, the z-axis is taken to be the propagation axis. The second order coherence and polarization properties of a partially coherent hollow vortex Gaussian beam in the source plane z = 0 is characterized by the following 2 × 2 cross spectral density matrix [37]

W(ρ01,ρ02,θ01,θ02,0)=[Wxx(ρ01,ρ02,θ01,θ02,0)Wxy(ρ01,ρ02,θ01,θ02,0)Wyx(ρ01,ρ02,θ01,θ02,0)Wyy(ρ01,ρ02,θ01,θ02,0)],
with Wxx(ρ01,ρ02,θ01,θ02,0)and Wyy(ρ01,ρ02,θ01,θ02,0)being given by
Wjj(ρ01,ρ02,θ01,θ02,0)=Ij0(ρ012ρ022w04)nexp(ρ012+ρ022w02)exp[im(θ01θ02)]×exp[ρ012+ρ0222ρ01ρ02cos(θ01θ02)δjj2],j=x,y,
where Ij0=|Ej0|2 and Ej0 is the characteristics amplitude. (ρ01, θ01) and (ρ02, θ02) are the radial and the azimuthal coordinates of two points in the source plane, respectively. w0 is the waist size of the Gaussian part. n denotes the beam order, and m represents the topological charge. δjj is the transverse coherent length in the j-direction. The complex degree of spatial coherence in Eq. (2) is generated by a Schell-model source. The off-diagonal elements of the cross spectral density matrix for a partially coherent hollow vortex Gaussian beam in the source plane Wxy(ρ01,ρ02,θ01,θ02,0)and Wyx(ρ01,ρ02,θ01,θ02,0) are set to be zero, which denotes that the two mutually orthogonal components Ex and Ey of the electric vector are uncorrelated at each point of the source [3840]. This class of sources is a special case, which results in the simple outcome. The average intensity and the degree of the polarization for the partially coherent hollow vortex Gaussian beam propagating through a paraxial ABCD optical system in turbulent atmosphere are given by [37]
I(ρ,θ,z)=TrW(ρ,ρ,θ,θ,z),
P(ρ,θ,z)={14detW(ρ,ρ,θ,θ,z)[TrW(ρ,ρ,θ,θ,z)]2}1/2,
with W(ρ,ρ,θ,θ,z)being given by
W(ρ,ρ,θ,θ,z)=[Wxx(ρ,ρ,θ,θ,z)Wxy(ρ,ρ,θ,θ,z)Wyx(ρ,ρ,θ,θ,z)Wyy(ρ,ρ,θ,θ,z)],
where (ρ, θ) are the radial and the azimuthal coordinates in the observation z-plane, respectively. Tr denotes the trace, and det stands for the determinant. W(ρ,ρ,θ,θ,z) is the cross spectral density matrix of the partially coherent hollow vortex Gaussian beam propagating through a paraxial ABCD optical system in turbulent atmosphere. A, B, C, and D are the transfer matrix elements of the paraxial optical system between the source and the observation planes. Apparently, Wxy(ρ, ρ, θ, θ, z) and Wyx(ρ, ρ, θ, θ, z) are equal to zero. Based on the extended Huygens-Fresnel diffraction integral, the cross spectral density of the partially coherent hollow vortex Gaussian beam propagating through a paraxial ABCD optical system in turbulent atmosphere can be obtained by [41]
Wjj(ρ,ρ,θ,θ,z)=1λ2BBexp(ikDρ22BikDρ22B)0002π02πρ01ρ02Wjj(ρ01,ρ02,θ01,θ02,0)×<exp[ψ(ρ01,ρ)+ψ(ρ02,ρ)]>exp{ik[Aρ0222BAρ0122B+ρρ01Bcos(θθ01)]}×exp[ikρρ02Bcos(θθ02)]dρ01dρ02dθ01dθ02,
where k = 2π/λ with λ being the optical wavelength. ψ(ρ01, ρ) and ψ(ρ02, ρ) are the solutions to the Rytov method that represents the random part of the complex phase. The angle brackets denote the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere. The asterisk means the complex conjugation. The ensemble average term is given by [42]
<exp[ψ(ρ01,ρ)+ψ(ρ02,ρ)]>=exp[ρ012+ρ0222ρ01ρ02cos(θ01θ02)σ02],
where σ0 is the spherical-wave lateral coherence radius due to the turbulence of the entire optical system and is defined as [42]
σ0=B[1.46k2Cn20Lb5/3(z)dz]3/5,
where Cn2 is the constant of refraction index structure and describes the turbulence level. b(z) corresponds to the approximate matrix element for a ray propagating backwards through the system. L is the axial distance between the source and the observation planes. Substituting Eqs. (2) and (7) into Eq. (6), the cross spectral density of the partially coherent hollow vortex Gaussian beam in the observation plane reads as
Wjj(ρ,ρ,θ,θ,z)=Ij0exp(iξρ2)λ2BBw04|n|0002π02π(ρ01ρ02)2n+1exp(α1ρ012α2ρ022)exp[im(θ02θ01)]×exp[ikρρ01Bcos(θθ01)ikρρ02Bcos(θθ02)+2α3ρ01ρ02cos(θ01θ02)]×dρ01dρ02dθ01dθ02,
with the auxiliary parameters ξ, α1, α2, and α3 being given by
ξ=kD2BkD2B,α1=1w02+1σ02+1δjj2+ikA2B,
α2=1w02+1σ02+1δjj2ikA2B,α3=1σ02+1δjj2.
We recall that the following formulae [43]
exp[ikρρ01Bcos(θθ01)]=l=ilJl(kρρ01B)exp[il(θθ01)],
02πexp[imθ01+2α3ρ01ρ02cos(θ01θ02)]dθ01=2πexp(imθ02)Im(2α3ρ01ρ02),
Jl(x)=(i)l2π02πexp(ixcosφ+ilφ)dφ,
where Jl(⋅)is the l-th order Bessel function of the first kind. Im(⋅)is the m-th order modified Bessel function of the first kind. The cross spectral density of the partially coherent hollow vortex Gaussian beam in the observation plane turns out to be
Wjj(ρ,ρ,θ,θ,z)=Ij0k2exp(iξρ2)BBl=00ρ01ρ02(ρ012ρ022w04)nexp(α1ρ012α2ρ022)×Jl(kρρ01B)Jl(kρρ02B)Il+m(2α3ρ01ρ02)dρ01dρ02.
We note that [43]
Il+m(2α3ρ01ρ02)=s=0(α3ρ01ρ02)2s+l+ms!Γ(s+l+m),
0tm1exp(αt2)Jl(βt)dt=αm/22l!Γ(l+m2)(β24α)l/2exp(β24α)F11(lm2+1;l+1;β24α),
where Γ(.) is a Gamma function and F11(;;) is a Kummer function. The cross spectral density of the partially coherent hollow vortex Gaussian beam in the observation plane finally can be analytically expressed as
Wjj(ρ,ρ,θ,θ,z)=Ij0k2exp(iξρ2)4BBw04nl=s=0[Γ(p1)]2(α1α2)p1α32s+l+ms!(l!)2Γ(s+l+m)(k2ρ24BB)l×exp(k2ρ24α1B2k2ρ24α2B2)F11(p2;l+1;k2ρ24α1B2)F11(p2;l+1;k2ρ24α2B2),
where the auxiliary parameters p1 and p2 are given by
p1=n+0.5m+s+l+1,p2=n0.5ms.
As a result, the average intensity of the partially coherent hollow vortex Gaussian beam in the observation plane is given by
I(ρ,θ,z)=Wxx(ρ,ρ,θ,θ,z)+Wyy(ρ,ρ,θ,θ,z).
The degree of the polarization of the partially coherent hollow vortex Gaussian beam in the observation plane yields
P(ρ,θ,z)=|Wxx(ρ,ρ,θ,θ,z)Wyy(ρ,ρ,θ,θ,z)|Wxx(ρ,ρ,θ,θ,z)+Wyy(ρ,ρ,θ,θ,z).
In the source plane, the degree of the polarization of the partially coherent hollow vortex Gaussian beam turns out to be

P(0)=|Ix0Iy0|Ix0+Iy0.

3. The numerical results and analyses

Now, the average intensity and the degree of the polarization of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere are calculated by using the formulae derived above. For simplicity, no other optical element is placed in turbulent atmosphere, which denotes that the matrix elements are A = 1, B = z, C = 0, and D = 1. In this case, σ0=(0.545Cn2k2z)3/5. λ is set to be 632.8nm. Figures 1 and 2 represent the normalized intensity distributions of the partially coherent hollow vortex Gaussian beams with different beam parameters in the reference planes of turbulent atmosphere z = 0, 0.5km, 1km, and 1.5 km. In Figs. 1 and 2, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3. With the increase of the propagation distance in turbulent atmosphere, the on-axis normalized intensity gradually increases from zero and finally becomes the maximum, which is caused by the isotropic influence of the atmosphere turbulence. When the propagation distance is large enough, the dark region disappears and the intensity distribution of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere tends to a Gaussian-like distribution. With increasing the waist size of the Gaussian part w0, the on-axis normalized intensity in the fixed reference plane decreases and the propagation distance within which the dark region will preserve increases. The effect of the increase of the beam order n is similar to that of the increase of the waist size of the Gaussian part w0. If one wants to keep the propagation distance within which the dark region will preserve longer, the increase of the beam order n is inferior to the increase of the waist size of the Gaussian part w0. The normalized average intensity distributions of a partially coherent hollow vortex Gaussian beam with different topological charge in the reference planes of the turbulent atmosphere are shown in Fig. 3 . The reference planes are same as those in Figs. 1 and 2. Besides the topological charge m, other parameters are kept unvaried. The sign of the topological charge m doesn’t affect the average intensity distribution, which is omitted to save space. The effect of the increase of the absolute value of the topological charge m is slight superior to that of the increase of the waist size of the Gaussian part w0. Figure 4 represents the normalized average intensity distributions of a partially coherent hollow vortex Gaussian beam with different transverse coherent lengths in the reference planes of the turbulent atmosphere. Except the transverse coherent lengths, the rest of the parameters are kept unvaried. With the increase of the transverse coherent lengths, the propagation distance within which the dark region will preserve also increases. However, the effect of the increase of the transverse coherent lengths is inferior to that of the increase of the beam order n. Figure 5 denotes the normalized average intensity distribution of a partially coherent hollow vortex Gaussian beam in the reference planes of the atmosphere with different turbulent level. Except Cn2, other parameters are unvaried. When propagating in the atmosphere of the weak turbulence, the propagation distance within which the dark region will preserve will be lengthened. The effect of the decrease of the structure constant of the atmospheric turbulence is approximately equivalent to that of the increase of the transverse coherent lengths. Moreover, the effect of the decrease of the structure constant of the atmospheric turbulence is also inferior to that of the increase of the beam order n. If only the degree of the polarization in the source plane is changed, the normalized intensity distributions of the partially coherent hollow vortex Gaussian beams in the fixed reference plane of turbulent atmosphere will not changed, the corresponding figures are omitted to save space.

 figure: Fig. 1

Fig. 1 Normalized average intensity distribution of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = m = 1, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3.

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 figure: Fig. 2

Fig. 2 Normalized average intensity distribution of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. m = 1, w0 = 0.02m, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3.

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 figure: Fig. 7

Fig. 7 The degree of the polarization of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. m = 1, w0 = 0.02m, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3.

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 figure: Fig. 8

Fig. 8 The degree of the polarization of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = 1, w0 = 0.02m, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3.

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 figure: Fig. 9

Fig. 9 The degree of the polarization of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = m = 1, w0 = 0.02m, P(0) = 0.5, and Cn2 = 10−14m-2/3.

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 figure: Fig. 3

Fig. 3 Normalized average intensity distribution of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = 1, w0 = 0.02m, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3.

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 figure: Fig. 4

Fig. 4 Normalized average intensity distribution of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = m = 1, w0 = 0.02m, P(0) = 0.5, and Cn2 = 10−14m-2/3.

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 figure: Fig. 5

Fig. 5 Normalized average intensity distribution of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = m = 1, w0 = 0.02m, δxx = 0.02m, δyy = 0.04m, and P(0) = 0.5.

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Figures 6 -10 represent the degree of the polarization of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. Figures 6-10 correspond to Figs. 1-5, respectively. Although the degree of the polarization in the everywhere of the source plane is identical, the degree of the polarization of a partially coherent hollow vortex Gaussian beam propagating in turbulent atmosphere is generally varied. Only under the condition of δxx = δyy, the degree of the polarization of a partially coherent hollow vortex Gaussian beam propagating in turbulent atmosphere is kept unvaried and is equal to the degree of the polarization in the source plane. Except the fluctuation in the edge, the degrees of the polarization in the different reference planes have the nearly same variational law. In a general way, the degrees of the polarization will first decrease, then increase, and finally fluctuate with increasing the value of the radial coordinateρ. However, there are also exceptions. When δxx = 0.03m and δyy = 0.04m, the degree of the polarization in the reference plane z = 1.5km gradually increases with increasing the value of ρ. This phenomenon may be interpreted as follow. Equations (10) and (11) denote that the smallest one of w0, σ0, δxx or δyy plays the dominant role on the intensity distribution. With the increase of the axial propagation distance z, the spherical-wave lateral coherence radius σ0 decreases. When z = 1.5km, σ0 = 0.018m. Therefore, w0 and σ0 both dominate the intensity distribution. Secondly, δxx is close to δyy. The combination of the above two situations results in the strange behavior of the degree of the polarization. When δxx = 0.02m and δyy = 0.01m, the degree of the polarization in the different reference planes mainly experiences first increases, then decreases, and finally increases with increasing the value of ρ, which is caused by δyy being far smaller than w0, σ0, and δxx. In this case, the degree of the polarization even reaches to zero, which means that Wxx(ρ, ρ, θ, θ, z) is equivalent to Wyy(ρ, ρ, θ, θ, z). The fluctuation of the degree of the polarization at larger value of the radial coordinate ρ decreases with increasing the propagation distance z. The degree of the polarization is related to the divergence of the beam. When the propagation distance z is small, the beam spot is also small. The local change of the degree of the polarization is relatively drastic in the edge of the small beam spot. The beam spot augments with increasing the propagation distance z. The local change of the degree of the polarization is relatively slow in the edge of the large beam spot.

 figure: Fig. 6

Fig. 6 The degree of the polarization of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = m = 1, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3.

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 figure: Fig. 10

Fig. 10 The degree of the polarization of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = m = 1, w0 = 0.02m, δxx = 0.02m, δyy = 0.04m, and P(0) = 0.5.

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Upon propagation in the turbulent atmosphere, the dark region of the partially coherent hollow vortex Gaussian beam will diminish until it disappears. Now, we consider the variation of the degree of the polarization in the axis upon propagation, which is shown in Fig. 11 . The degree of the polarization in the axis first fluctuates, then slowly decreases, and finally tends to P(0). The fluctuation of the degree of the polarization in the axis at small z values is also caused by the small beam spot in this case.

 figure: Fig. 11

Fig. 11 The degree of the polarization in the axis of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = 1, δxx = 0.02m, δyy = 0.04m, and P(0) = 0.5. (a) m = 1. (b) w0 = 0.02m.

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4. Conclusions

The propagation of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere is investigated. By means of the mathematical techniques, the analytical expressions of the average intensity and the degree of the polarization of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere are derived without any approximations. The average intensity distribution and the degree of the polarization of a partially coherent hollow vortex Gaussian beam are numerically examined. The influences of the beam parameters, the topological charge, the transverse coherent lengths, and the structure constant of the atmospheric turbulence on the propagation of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere are also discussed, respectively. With the increase of the propagation distance in turbulent atmosphere, the dark region gradually diminishes until it completely disappears. If one wants to keep the propagation distance within which the dark region will preserve longer, the optimal method is the increase of the topological charge m, and the second preferential method is the increase of the waist size of the Gaussian part w0, and the third preferential method is the increase of the beam order n. Of course, the increase of the transverse coherent lengths or the decrease of the structure constant of the atmospheric turbulence will also elongate the propagation distance within which the dark region will preserve. When δxxδyy, the degree of the polarization of a partially coherent hollow vortex Gaussian beam propagating in turbulent atmosphere is varied. In a general way, the degree of the polarization will first decreases, then increases, and finally fluctuates with increasing the value of the radial coordinate ρ. However, there are also exceptions. This research is beneficial to the practical applications in free-space optical communications and remote sensing involving in the dark hollow beams.

Acknowledgments

Guoquan Zhou acknowledges the support by the National Natural Science Foundation of China under Grant Nos. 10974179 and 61178016. Yangjian Cai acknowledges the support by the National Natural Science Foundation of China under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant No. 200928, the Natural Science of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009, and the Key Project of Chinese Ministry of Education under Grant No. 210081. The authors are indebted to the reviewers for valuable comments.

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40. D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).

41. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in turbulent atmosphere,” Waves Random Media 14, 513–523 (2004).

42. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).

43. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, CA, 1980).

References

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  1. V. I. Balykin and V. S. Letokhov, “The possibility of deep laser focusing of an atom beam into the A region,” Opt. Commun. 64, 151–156 (1987).
  2. X. Wang and M. G. Littman, “Laser cavity for generation of variable-radius rings of light,” Opt. Lett. 18(10), 767–768 (1993).
    [PubMed]
  3. H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
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  4. S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994).
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  5. C. Paterson and R. Smith, “High-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
  6. J. Yin, Y. Zhu, W. Wang, Y. Wang, and W. Jhe, “Optical potential for atom guidance in a dark hollow laser beam,” J. Opt. Soc. Am. B 15, 25–33 (1998).
  7. M. Yan, J. Yin, and Y. Zhu, “Dark-hollow-beam guiding and splitting of a low-velocity atomic beam,” J. Opt. Soc. Am. B 17, 1817–1820 (2000).
  8. Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A, Pure Appl. Opt. 7, 147–153 (2005).
  9. K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).
  10. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
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  13. H. Ma, H. Cheng, W. Zhang, L. Liu, and Y. Wang, “Generation of a hollow laser beam by a multimode fiber,” Chin. Opt. Lett. 5, 460–462 (2007).
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    [PubMed]
  15. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008).
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  16. Y. Zheng, X. Wang, F. Shen, and X. Li, “Generation of dark hollow beam via coherent combination based on adaptive optics,” Opt. Express 18(26), 26946–26958 (2010).
    [PubMed]
  17. Y. Nie, H. Ma, X. Li, W. Hu, and J. Yang, “Generation of dark hollow femtosecond pulsed beam by phase-only liquid crystal spatial light modulator,” Appl. Opt. 50(21), 4174–4179 (2011).
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  20. H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
  21. G. Zhou, X. Chu, and J. Zheng, “Investigation in hollow Gaussian beam from vectorial structure,” Opt. Commun. 281, 5653–5658 (2008).
  22. Z. Mei and D. Zhao, “Nonparaxial propagation of controllable dark-hollow beams,” J. Opt. Soc. Am. A 25(3), 537–542 (2008).
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  23. D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B 26, 2044–2049 (2009).
  24. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
    [PubMed]
  25. H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).
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  29. T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47, 036002 (2008).
  30. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25(1), 225–230 (2008).
    [PubMed]
  31. X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17(6), 4257–4262 (2009).
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    [PubMed]
  33. X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18(7), 6922–6928 (2010).
    [PubMed]
  34. P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).
  35. C. Li and J. Pu, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599–604 (2010).
  36. X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19(27), 26444–26450 (2011).
    [PubMed]
  37. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
  38. J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5 Pt 2), 056610 (2007).
    [PubMed]
  39. A. Al-Qasimi, O. Korotkova, D. James, and E. Wolf, “Definitions of the degree of polarization of a light beam,” Opt. Lett. 32(9), 1015–1016 (2007).
    [PubMed]
  40. D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).
  41. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in turbulent atmosphere,” Waves Random Media 14, 513–523 (2004).
  42. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
  43. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, San Diego, CA, 1980).

2011 (2)

2010 (5)

X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18(7), 6922–6928 (2010).
[PubMed]

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

C. Li and J. Pu, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599–604 (2010).

Y. Zheng, X. Wang, F. Shen, and X. Li, “Generation of dark hollow beam via coherent combination based on adaptive optics,” Opt. Express 18(26), 26946–26958 (2010).
[PubMed]

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).

2009 (4)

2008 (8)

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[PubMed]

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

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25(1), 225–230 (2008).
[PubMed]

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).

G. Zhou, X. Chu, and J. Zheng, “Investigation in hollow Gaussian beam from vectorial structure,” Opt. Commun. 281, 5653–5658 (2008).

Z. Mei and D. Zhao, “Nonparaxial propagation of controllable dark-hollow beams,” J. Opt. Soc. Am. A 25(3), 537–542 (2008).
[PubMed]

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008).
[PubMed]

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).

2007 (4)

2006 (1)

2005 (4)

2004 (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in turbulent atmosphere,” Waves Random Media 14, 513–523 (2004).

2003 (2)

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28(13), 1084–1086 (2003).
[PubMed]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).

2002 (1)

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).

2000 (1)

1998 (1)

1996 (2)

C. Paterson and R. Smith, “High-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

1994 (2)

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[PubMed]

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994).
[PubMed]

1993 (1)

1987 (2)

V. I. Balykin and V. S. Letokhov, “The possibility of deep laser focusing of an atom beam into the A region,” Opt. Commun. 64, 151–156 (1987).

H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).

Ahmad, M. A.

Al-Qasimi, A.

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Balykin, V. I.

V. I. Balykin and V. S. Letokhov, “The possibility of deep laser focusing of an atom beam into the A region,” Opt. Commun. 64, 151–156 (1987).

Baykal, Y.

Cai, Y.

Chen, Z.

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

Cheng, H.

Choi, K.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[PubMed]

Chu, X.

G. Zhou, X. Chu, and J. Zheng, “Investigation in hollow Gaussian beam from vectorial structure,” Opt. Commun. 281, 5653–5658 (2008).

Deng, D.

Dogariu, A.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in turbulent atmosphere,” Waves Random Media 14, 513–523 (2004).

Dong, Y.

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A, Pure Appl. Opt. 7, 147–153 (2005).

Du, X.

Eyyuboglu, H. T.

Fan, Z.

Fenichel, H.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[PubMed]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Fu, X.

Gbur, G.

Guo, Q.

Hanson, S. G.

He, S.

Hu, W.

James, D.

Jhe, W.

Ji, X.

Korotkova, O.

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[PubMed]

A. Al-Qasimi, O. Korotkova, D. James, and E. Wolf, “Definitions of the degree of polarization of a light beam,” Opt. Lett. 32(9), 1015–1016 (2007).
[PubMed]

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5 Pt 2), 056610 (2007).
[PubMed]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in turbulent atmosphere,” Waves Random Media 14, 513–523 (2004).

Lee, H. S.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[PubMed]

Letokhov, V. S.

V. I. Balykin and V. S. Letokhov, “The possibility of deep laser focusing of an atom beam into the A region,” Opt. Commun. 64, 151–156 (1987).

Li, C.

C. Li and J. Pu, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599–604 (2010).

Li, X.

Lin, J.

Lin, Q.

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A, Pure Appl. Opt. 7, 147–153 (2005).

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28(13), 1084–1086 (2003).
[PubMed]

Littman, M. G.

Liu, J.

Liu, L.

Liu, S.

Liu, T.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).

Liu, X.

Liu, Z.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32(15), 2076–2078 (2007).
[PubMed]

Lu, X.

Ma, H.

Ma, Y.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

Marksteiner, S.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994).
[PubMed]

Mei, Z.

Nie, Y.

Paterson, C.

C. Paterson and R. Smith, “High-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).

Pu, J.

X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19(27), 26444–26450 (2011).
[PubMed]

C. Li and J. Pu, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599–604 (2010).

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

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5 Pt 2), 056610 (2007).
[PubMed]

Qu, J.

Rolston, S. L.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994).
[PubMed]

Salem, M.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in turbulent atmosphere,” Waves Random Media 14, 513–523 (2004).

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Savage, C. M.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994).
[PubMed]

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Shao, J.

Shen, F.

Smith, R.

C. Paterson and R. Smith, “High-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).

Spagnolo, G. S.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Stewart, B. W.

H. S. Lee, B. W. Stewart, K. Choi, and H. Fenichel, “Holographic nondiverging hollow beam,” Phys. Rev. A 49(6), 4922–4927 (1994).
[PubMed]

Sun, X.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).

Tang, H.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).

Tyson, R. K.

Wang, F.

Wang, T.

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

Wang, W.

Wang, X.

Y. Zheng, X. Wang, F. Shen, and X. Li, “Generation of dark hollow beam via coherent combination based on adaptive optics,” Opt. Express 18(26), 26946–26958 (2010).
[PubMed]

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).

X. Wang and M. G. Littman, “Laser cavity for generation of variable-radius rings of light,” Opt. Lett. 18(10), 767–768 (1993).
[PubMed]

Wang, Y.

Wang, Z.

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A, Pure Appl. Opt. 7, 147–153 (2005).

Wei, C.

Wolf, E.

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5 Pt 2), 056610 (2007).
[PubMed]

A. Al-Qasimi, O. Korotkova, D. James, and E. Wolf, “Definitions of the degree of polarization of a light beam,” Opt. Lett. 32(9), 1015–1016 (2007).
[PubMed]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in turbulent atmosphere,” Waves Random Media 14, 513–523 (2004).

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).

Xu, X.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

Yan, M.

Yang, J.

Yin, J.

Yuan, Y.

Yura, H. T.

Zhang, W.

Zhao, C.

Zhao, D.

Zhao, H.

Zheng, J.

G. Zhou, X. Chu, and J. Zheng, “Investigation in hollow Gaussian beam from vectorial structure,” Opt. Commun. 281, 5653–5658 (2008).

Zheng, Y.

Zhou, G.

G. Zhou, X. Chu, and J. Zheng, “Investigation in hollow Gaussian beam from vectorial structure,” Opt. Commun. 281, 5653–5658 (2008).

Zhou, P.

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

Zhu, K.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).

Zhu, Y.

Zoller, P.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994).
[PubMed]

Appl. Opt. (2)

Appl. Phys. B (2)

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).

C. Li and J. Pu, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599–604 (2010).

Chin. Opt. Lett. (1)

J. Mod. Opt. (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

J. Opt. (1)

P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. 12, 01540–01549 (2010).

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

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A, Pure Appl. Opt. 7, 147–153 (2005).

J. Opt. Soc. Am. A (6)

J. Opt. Soc. Am. B (3)

Opt. Commun. (5)

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, “Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams,” Opt. Commun. 207, 29–34 (2002).

V. I. Balykin and V. S. Letokhov, “The possibility of deep laser focusing of an atom beam into the A region,” Opt. Commun. 64, 151–156 (1987).

C. Paterson and R. Smith, “High-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).

G. Zhou, X. Chu, and J. Zheng, “Investigation in hollow Gaussian beam from vectorial structure,” Opt. Commun. 281, 5653–5658 (2008).

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).

Opt. Eng. (1)

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Figures (11)

Fig. 1
Fig. 1 Normalized average intensity distribution of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = m = 1, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3.
Fig. 2
Fig. 2 Normalized average intensity distribution of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. m = 1, w0 = 0.02m, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3.
Fig. 7
Fig. 7 The degree of the polarization of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. m = 1, w0 = 0.02m, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3.
Fig. 8
Fig. 8 The degree of the polarization of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = 1, w0 = 0.02m, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3.
Fig. 9
Fig. 9 The degree of the polarization of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = m = 1, w0 = 0.02m, P(0) = 0.5, and Cn2 = 10−14m-2/3.
Fig. 3
Fig. 3 Normalized average intensity distribution of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = 1, w0 = 0.02m, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3.
Fig. 4
Fig. 4 Normalized average intensity distribution of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = m = 1, w0 = 0.02m, P(0) = 0.5, and Cn2 = 10−14m-2/3.
Fig. 5
Fig. 5 Normalized average intensity distribution of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = m = 1, w0 = 0.02m, δxx = 0.02m, δyy = 0.04m, and P(0) = 0.5.
Fig. 6
Fig. 6 The degree of the polarization of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = m = 1, δxx = 0.02m, δyy = 0.04m, P(0) = 0.5, and Cn2 = 10−14m-2/3.
Fig. 10
Fig. 10 The degree of the polarization of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = m = 1, w0 = 0.02m, δxx = 0.02m, δyy = 0.04m, and P(0) = 0.5.
Fig. 11
Fig. 11 The degree of the polarization in the axis of a partially coherent hollow vortex Gaussian beam in the different reference planes of the turbulent atmosphere. n = 1, δxx = 0.02m, δyy = 0.04m, and P(0) = 0.5. (a) m = 1. (b) w0 = 0.02m.

Equations (22)

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W ( ρ 01 , ρ 02 , θ 01 , θ 02 ,0)=[ W xx ( ρ 01 , ρ 02 , θ 01 , θ 02 ,0) W xy ( ρ 01 , ρ 02 , θ 01 , θ 02 ,0) W yx ( ρ 01 , ρ 02 , θ 01 , θ 02 ,0) W yy ( ρ 01 , ρ 02 , θ 01 , θ 02 ,0) ],
W jj ( ρ 01 , ρ 02 , θ 01 , θ 02 ,0)= I j0 ( ρ 01 2 ρ 02 2 w 0 4 ) n exp( ρ 01 2 + ρ 02 2 w 0 2 )exp[im( θ 01 θ 02 )] × exp[ ρ 01 2 + ρ 02 2 2 ρ 01 ρ 02 cos( θ 01 θ 02 ) δ jj 2 ],j=x,y,
I(ρ,θ,z)=Tr W (ρ,ρ,θ,θ,z),
P(ρ,θ,z)= { 1 4det W (ρ,ρ,θ,θ,z) [Tr W (ρ,ρ,θ,θ,z)] 2 } 1/2 ,
W (ρ,ρ,θ,θ,z)=[ W xx (ρ,ρ,θ,θ,z) W xy (ρ,ρ,θ,θ,z) W yx (ρ,ρ,θ,θ,z) W yy (ρ,ρ,θ,θ,z) ],
W jj (ρ,ρ,θ,θ,z)= 1 λ 2 B B exp( ik D ρ 2 2 B ikD ρ 2 2B ) 0 0 0 2π 0 2π ρ 01 ρ 02 W jj ( ρ 01 , ρ 02 , θ 01 , θ 02 ,0) ×<exp[ψ( ρ 01 ,ρ)+ ψ ( ρ 02 ,ρ)]>exp{ ik[ A ρ 02 2 2 B A ρ 01 2 2B + ρ ρ 01 B cos(θ θ 01 ) ] } ×exp[ ik ρ ρ 02 B cos(θ θ 02 ) ]d ρ 01 d ρ 02 d θ 01 d θ 02 ,
<exp[ψ( ρ 01 ,ρ)+ ψ ( ρ 02 ,ρ)]>=exp[ ρ 01 2 + ρ 02 2 2 ρ 01 ρ 02 cos( θ 01 θ 02 ) σ 0 2 ],
σ 0 =B [ 1.46 k 2 C n 2 0 L b 5/3 (z)dz ] 3/5 ,
W jj (ρ,ρ,θ,θ,z)= I j0 exp(iξ ρ 2 ) λ 2 B B w 0 4| n | 0 0 0 2π 0 2π ( ρ 01 ρ 02 ) 2n+1 exp( α 1 ρ 01 2 α 2 ρ 02 2 )exp[im( θ 02 θ 01 )] × exp[ ikρ ρ 01 B cos(θ θ 01 ) ikρ ρ 02 B cos(θ θ 02 )+2 α 3 ρ 01 ρ 02 cos( θ 01 θ 02 ) ] × d ρ 01 d ρ 02 d θ 01 d θ 02 ,
ξ= k D 2 B kD 2B , α 1 = 1 w 0 2 + 1 σ 0 2 + 1 δ jj 2 + ikA 2B ,
α 2 = 1 w 0 2 + 1 σ 0 2 + 1 δ jj 2 ik A 2 B , α 3 = 1 σ 0 2 + 1 δ jj 2 .
exp[ ikρ ρ 01 B cos(θ θ 01 ) ]= l= i l J l ( kρ ρ 01 B )exp[il(θ θ 01 )],
0 2π exp[im θ 01 +2 α 3 ρ 01 ρ 02 cos( θ 01 θ 02 )] d θ 01 =2πexp(im θ 02 ) I m (2 α 3 ρ 01 ρ 02 ),
J l (x)= (i) l 2π 0 2π exp(ixcosφ+ilφ) dφ,
W jj (ρ,ρ,θ,θ,z)= I j0 k 2 exp(iξ ρ 2 ) B B l= 0 0 ρ 01 ρ 02 ( ρ 01 2 ρ 02 2 w 0 4 ) n exp( α 1 ρ 01 2 α 2 ρ 02 2 ) × J l ( kρ ρ 01 B ) J l ( kρ ρ 02 B ) I l+m (2 α 3 ρ 01 ρ 02 )d ρ 01 d ρ 02 .
I l+m (2 α 3 ρ 01 ρ 02 )= s=0 ( α 3 ρ 01 ρ 02 ) 2s+l+m s!Γ(s+l+m) ,
0 t m1 exp(α t 2 ) J l (βt)dt = α m/2 2l! Γ( l+m 2 ) ( β 2 4α ) l/2 exp( β 2 4α ) F 1 1 ( lm 2 +1;l+1; β 2 4α ),
W jj (ρ,ρ,θ,θ,z)= I j0 k 2 exp(iξ ρ 2 ) 4B B w 0 4 n l= s=0 [ Γ( p 1 ) ] 2 ( α 1 α 2 ) p 1 α 3 2s+l+m s! (l!) 2 Γ(s+l+m) ( k 2 ρ 2 4B B ) l × exp( k 2 ρ 2 4 α 1 B 2 k 2 ρ 2 4 α 2 B 2 ) F 1 1 ( p 2 ;l+1; k 2 ρ 2 4 α 1 B 2 ) F 1 1 ( p 2 ;l+1; k 2 ρ 2 4 α 2 B 2 ),
p 1 =n+0.5m+s+l+1, p 2 =n0.5ms.
I(ρ,θ,z)= W xx (ρ,ρ,θ,θ,z)+ W yy (ρ,ρ,θ,θ,z).
P(ρ,θ,z)= | W xx (ρ,ρ,θ,θ,z) W yy (ρ,ρ,θ,θ,z) | W xx (ρ,ρ,θ,θ,z)+ W yy (ρ,ρ,θ,θ,z) .
P (0) = | I x0 I y0 | I x0 + I y0 .

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