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Dynamic evolution of circular edge dislocations in free space and atmospheric turbulence

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Abstract

Based on the extended Huygens-Fresnel principle, the analytical expressions for the cross-spectral density function of circular edge dislocation beams propagating through atmospheric turbulence and free space have been derived, and used to study the dynamic evolution of circular edge dislocations. It is shown that the number of circular edge dislocations on propagation equals that at the source plane when propagating through free space. The radius of circular edge dislocations increases with increasing propagation distance z. n-circular edge dislocations vanish and transform to n pairs of optical vortices with the opposite topological charge when propagating through atmospheric turbulence, and the position of each pair of optical vortices are symmetric about the slanted axis y = x. All the optical vortices will annihilate as soon as the propagation distance becomes large enough. The smaller radius of circular edge dislocation corresponds with the sooner annihilation of optical vortices. The structure constant affects the annihilation distance of the pairs of optical vortices, and the annihilation distance of the pairs of optical vortices will increase with the decrement of the structure constant.

© 2017 Optical Society of America

1. Introduction

Optical fields may contain wavefront dislocations (phase singularities), that is to say, points or lines or rings on the wavefront surface where the phase of the optical field is undetermined (singular) and its amplitude vanishes [1–3]. It is well known that there exist three typical phase dislocations, i.e. screw dislocation, edge dislocation and mixed dislocations [1, 4]. A screw dislocation is spiral phase ramps around a dark spot, where the phase of the field is undefined, and thus the field amplitude vanishes. An edge dislocation is the π phase jump in the wave phase located along a line or ring in the transverse plane. According to the shape introduced by Soskin and Vasnetsov [3], there are mainly two types of edge dislocation: linear edge dislocation and circular edge dislocations. The mixed dislocations are linear edge-screw dislocations or circular edge-screw dislocations. Optical beams carrying phase singularities have recently attracted much interests because of their importance from a theoretical aspect [3, 5] and potential applications in optical communications [6–8], quantum information and entanglement [9–11], optical manipulation and trapping of small particles [12–17], high-resolution fluorescence microscopy [18, 19], stellar coronagraph [20, 21], optical parametric oscillator [22, 23], biological tissues [24, 25] and so on.

Nowadays, the screw dislocation is often termed the optical vortices or coherent vortices, optical vortices can be generated by various methods such as mode conversions [26, 27], computer generated holograms [28, 29], spiral phase plates [30, 31], multi-level spiral phase plate [32], spiral-like phase plate [33], inhomogeneous birefringent elements [34], subwavelength gratings [35], optical wedges [36, 37], nanoantennas [38] multilevel vortex-producing lens [39], Bragg reflector waveguide [40] and optical mirror [41, 42]. Verbeeck et al. have described the production of optical vortices by using versatile holographic reconstruction technique which is a reproducible method of producing optical vortices in the conventional electron microscope [14]. Vyas and Senthilkumaran have proposed the creation of optical vortex arrays using the modified Michelson interferometer and the modified Mach–Zehnder interferometer [43, 44]. The evolution of optical vortex and propagation dynamics of optical vortex beams have been studied extensively. Most of the previous researches have been focused on the motion, creation, annihilation, shift, transformation and trajectories of optical vortices, coherent vortices or composite optical vortices in free space, optical system, linear media, atmospheric turbulence, and so on [45–57]. The vortex evolution and decay of high-order optical vortices in media with an anisotropic nonlocal nonlinearity were analyzed in [45, 46]. The dynamic inversion of the topological charge of an optical vortices in the free space propagation were discovered by Molina-Terriza et al [47]. Singh and Chowdhury have conducted an experimental study on the trajectory and shape of the noncanonical optical vortices propagating in free space [48]. Roux has described a method for computing the trajectories of optical vortices, and analytically computed the positions of the vortices as a function of the propagation distance [49, 50]. Dipankar et al. have investigated the trajectory of an optical vortices in atmospheric turbulence using numerical simulations in 2009 [51]. We have studied the evolution of coherent vortices and propagation properties of vortex beams through atmospheric turbulence, and found that the beams quality of the vortex beams are less influenced by atmospheric turbulence than that of non-vortex beams [52–55]. In our recent paper we have analyzed the classification of coherent vortices creation and distance of topological charge conservation, which showed that the coherent vortices are grouped into three classes according to the creation [56, 57]. The propagation dynamics of optical vortex dipole (i.e. pair of optical vortices of opposite topological charge) in free space and astigmatic optical system have been researched in [58–61].

An edge dislocation can be produced experimentally by using a phase plate to embed a π-phase jump in the host beam. Vasnetsov et al. have analyzed how a circular edge dislocation of a wavefront can be created in an interference of two uniaxial Gaussian beams [62]. Phase singularities and spectral changes of higher-order Bessel–Gauss pulsed beams in free space have been studied in the literature [63], it is shown that some circular edge dislocations appear and the spectrum changes on propagation. The dynamic evolution of a linear edge dislocation nested in a general elliptical Gaussian beams through aligned and misaligned paraxial optical ABCD systems was analyzed by Yan and Lü [64]. We also have studied the transformation of the linear edge dislocation in atmospheric turbulence, and found that the linear edge dislocation disappears and transforms to a noncanonical coherent vortex [65]. It is interesting to ask: what will happen when the circular edge dislocations propagating through free space and atmospheric turbulence?

This paper is devoted to studying the dynamic evolution of circular edge dislocations and to expose some richer behavior of singularities in atmospheric turbulence. In Sec. 2, the cross-spectral density function of circular edge dislocations beams in atmospheric turbulence is derived. The evolution behavior of circular edge dislocations in free space is described in Sec. 3. The transforms of circular edge dislocations and 3D trajectory of the pairs of the optical vortices propagating through atmospheric turbulence are analyzed and illustrated by numerical examples in Sec. 4. Finally, Sec. 5 provides the concluding remarks.

2. Theoretical model

The initial field distribution of Laguerre–Gaussian (LG) beam at the source plane z = 0 can be expressed as [66, 67]

E(s,θ,0)=(2sw0)mLnm(2s2w02)exp(s2w02)exp(imθ),
where s and θ are the radial coordinate and azimuthal (angle) coordinate, respectively. w0 denotes the waist width, Lnm denotes the Laguerre polynomial with mode orders n and m. For n = 0 and m = 0, Eq. (1) degenerates to the initial field of fundamental Gaussian beam.

Using the relations between Laguerre polynomial and Hermite polynomial [68]

exp(imθ)s2Lnm(s2)=(1)n22n+mn!t=0nr=0mir(tn)(rm)H2t+mr(sx)H2n2t+r(sy),
with Hn being the Hermite polynomial of order n, (tn)(rm)being binomial coefficients, Eq. (1) can be expressed in following alternative form in Cartesian coordinates

E(s,0)=(1)n22n+mn!t=0nr=0mir(tn)(rm)H2t+mr(2sxw0)H2n2t+r(2syw0)exp(s2w02).

LG beam is a typical mixed circular edge-screw dislocations beam. For m≠0 and n = 0, Eq. (3) reduces to the initial field distribution of screw dislocation beam (vortex beam) which can be written as

E(s,0)=12mr=0mir(rm)Hmr(2sxw0)Hr(2syw0)exp(s2w02),
m is topological charge of vortex beam. For m = 0 and n≠0, Eq. (3) degenerates to the initial field distribution of circular edge dislocations beam which can be written as
E(s,0)=(1)n22nn!t=0n(tn)H2t(2sxw0)H2n2t(2syw0)exp(s2w02).
n is the number of circular edge dislocations.

Figure 1 gives the intensity and phase distribution of circular edge dislocation beams and vortex beams at the source plane. The calculation parameters are w0 = 3cm n = 1 in Figs. 1(a) and 1(c), m = 1 in Figs. 1(b) and 1(d). Figs. 1(a) and 1(c) show that there exists a circular edge dislocation at the source plane, whereas Figs. 1(b) and 1(d) suggest that there exists an optical vortex with topological charge + 1 at the source plane.

 figure: Fig. 1

Fig. 1 (a, b) The intensity and (c, d) phase distribution of (a, c) circular edge dislocation beam and (b, d) vortex beam at the source plane.

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Now we study the propagation of circular edge dislocations beams in atmospheric turbulence, the cross-spectral density function of circular edge dislocations beams at the source plane z = 0 is expressed as

W0(s1,s2,0)=124n(n!)2t1=0nt2=0n(t1n)(t2n)H2t1(2s1xw0)H2t2(2s2xw0)×H2n2t1(2s1yw0)H2n2t2(2s2yw0)exp(s12+s22w02).

Based on the extended Huygens-Fresnel principle [69], the cross-spectral density function of circular edge dislocations beams propagating through atmospheric turbulence is given by

W(ρ1,ρ2,z)=(k2πz)2W0(s1,s2,0)exp{ik2z[(s1ρ1)2(s2ρ2)2]}×exp[ψ(s1,ρ1)+ψ*(s2,ρ2)]ds1ds2,
where ρ1 = (ρ1x, ρ1y) and ρ2 = (ρ2x, ρ2y) denote the position vector at the z plane, k is the wave number related to the wavelength λ by k = 2π/λ, * denotes the complex conjugate. ψ(s,ρ) stands for the random part of the complex phase of a spherical wave due to the turbulence, <·>m means the average over the ensemble of the turbulent medium, and [70]
exp[ψ(s1,ρ1)+ψ*(s2,ρ2)]=exp[(s1s2)2+(ρ1ρ2)2+(s1s2)(ρ1ρ2)ρ02],
where ρ0=(0.545Cn2k2z)-3/5 denotes the spatial coherence radius of a spherical wave propagation through atmospheric turbulence and Cn2 specifies the refraction index structure constant. The larger Cn2 is, the stronger the atmospheric turbulence is. That Cn2 = 0 means ρ0→∞, namely, exp[ψ(s1,ρ1)+ψ*(s2,ρ2)]=1, thus Eq. (7) reduces to the cross-spectral density function of circular edge dislocations beams in free space.

Substituting Eqs. (6) and (8) into Eq. (7), and making use of the following integral formula and the property of Hermite function [71]

exp[(xy)2]Hn(ax)dx=π(1a2)n2Hn(ay(1a2)1/2),
xnexp[(xβ)2]dx=(2i)nπHn(iβ),
Hn(x+y)=12n/2k=0n(kn)Hk(2x)Hnk(2y),
Hn(x)=m=0[n/2](1)mn!m!(n2m)!(2x)n2m,
the tedious but straightforward integral calculations lead to analytical expressions for the cross-spectral density function of circular edge dislocations beams propagating through atmospheric turbulence, which is given by
W(ρ1,ρ2,z)=(k2πz)2AxAyexp[(ρ1ρ2)2ρ02]exp[ik2z(ρ12ρ22)]×124n(n!)2t1=0nt2=0n(t1n)(t2n)BC,
where
Ax=exp[14D(ρ1xρ2xρ02ikρ2xz)2]exp(Fx24G),
B=c1=0t1d1=02t2e1=0[d12](d12t2)(1)c1+e1(2i)(2t22c1+d12e1)(2t1)!d1!c1!(2t12c1)!e1!(d12e1)!(22w0)2t12c1×πD(12w02D)t22t2[4ρ02w02D22D]d12e1(1G)2t12c1+d12e1+1×H2t2d1[(ρ1xρ2x)zikρ2xρ02ρ02zw02D22D]H2t12c1+d12e1(iFx2G),
C=c2=0nt1d2=02n2t2e2=0[d22](d22n2t2)(1)c2+e2(2i)(2n2t12c2+d22e2)(2n2t1)!d2!c2!(2n2t12c2)!e2!(d22e2)!×πD(12w02D)nt12(nt1)[4ρ02w02D22D]d22e2(1G)2n2t12c2+d22e2+1×(22w0)2n2t12c2H2n2t2d2[(ρ1yρ2y)zikρ2yρ02ρ02zw02D22D]H2n2t12c2+d22e2(iFy2G),
D=1w02ik2z+1ρ02,
Fx=ikρ1xzρ1xρ2xρ02+1Dρ02(ρ1xρ2xρ02ikρ2xz),
G=1w02+ik2z+1ρ021Dρ04.
According to the symmetry, Ay and Fy can be obtained by replacement of ρ1x and ρ2x in Ax and Fx with ρ1y and ρ2y. For Cn2 = 0, Eq. (13) degenerates to the analytical expressions for the cross-spectral density function of circular edge dislocations beams in free space.

The spectral degree of coherence is defined as [72]

μ(ρ1,ρ2,z)=W(ρ1,ρ2,z)[I(ρ1,z)I(ρ2,z)]1/2,
where I(ρi, z) = W(ρi, ρi, z) (i = 1, 2) stands for the spectral intensity of the point (ρi, z). The position of phase singularities is given by [73]
Re[μ(ρ1,ρ2,z)]=0,
Im[μ(ρ1,ρ2,z)]=0.
where Re and Im denote the real and imaginary parts of μ(ρ1,ρ2,z). The topological charge and its sign of coherent vortices are determined by the vorticity of phase contours around singularities [74].

3. Dynamic evolution of circular edge dislocations in free space

Figure 2 gives curves of Reμ = 0 (solid curves) and Imμ = 0 (dashed curves) and contour lines of phase of circular edge dislocations beams at the source plane z = 0 and at the propagation distance z = 2km and 5km though free space. The calculation parameters are λ = 632.8nm, w0 = 3cm, n = 1, ρ1 = (8cm, 8cm). Figure 2(a) indicates that only there exist the curves of Reμ = 0 (solid curves) of circular edge dislocations beams, whereas no curves of Imμ = 0 (dashed curves) of circular edge dislocations beams. From Eq. (6) we know that the cross-spectral density function of circular edge dislocations beams haven’t image at the source plane. As suggested from Fig. 2(a) to Fig. 2(f), the phase distribution of circular edge dislocations beams changes with increasing propagation distance z. There exists one circular edge dislocation (marked as A) both at the source plane and on propagation, and the center of circular edge dislocations located at the point (0, 0). It also can be seen that the longer the propagation distance z is, the larger the radius of circular edge dislocation RA is, e.g. for z = 0, 2 and 5km, RA = 2.15, 2.35 and 3.23cm, respectively.

 figure: Fig. 2

Fig. 2 (a)-(c) Curves of Reμ = 0 (red solid curves) and Imμ = 0 (blue dashed curves) and (d)-(f) contour lines of phase of circular edge dislocations beams at different propagation distance (a) and (d) z = 0, (b) and (e) z = 2km, (c) and (f) z = 5km. The abscissa represents ρ2x direction, ordinate represents ρ2y direction, and their units are cm.

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The curves of Reμ = 0 (solid curves) and Imμ = 0 (dashed curves) of circular edge dislocations beams in free space are plotted in Fig. 3 for different values of 3(a)-3(c) n = 2 and 3(d)-3(f) n = 3. The other calculation parameters are the same as in Fig. 2. From Figs. 3(a)-3(c) it is seen that there exist two circular edge dislocations (marked as B and C) both at the source plane and on propagation, and the radii of circular edge dislocations RB and RC increase with increasing z. Figures 3 (d)-3(f) appear three circular edge dislocations (marked as D, E and F), the dynamic evolution of D, E and F are similar to B and C in Figs. 3 (a)-3(c). From Fig. 2 and Fig. 3 we see that the number of circular edge dislocations on propagation equals that at the source plane propagating in free space, the center of circular edge dislocations located at the point (0, 0), and their radii increase with increasing propagation distance z.

 figure: Fig. 3

Fig. 3 Curves of Reμ = 0 (red solid curves) and Imμ = 0 (blue dashed curves) of circular edge dislocations beams at different propagation distance (a) and (d) z = 0, (b) and (e) z = 2km, (c) and (f) z = 5km for different values of (a)-(c) n = 2, (d)-(f) n = 3.

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4. Dynamic evolution of circular edge dislocations in atmospheric turbulence

Figure 4 represents curves of Reμ = 0 (solid curves) and Imμ = 0 (dashed curves) and contour lines of phase of circular edge dislocations beams propagating through atmospheric turbulence for different propagation distance. Cn2 = 10−14m-2/3, the other calculation parameters are the same as in Fig. 2, where the black and white dots denotes the topological charge + 1 and −1 of the optical vortices, respectively. From Figs. 4(a)-4(c) and 4(e)-4(g) we see that the circular edge dislocation A (in Figs. 2(a) and 2(d)) vanishes and transforms to a pair of optical vortices with the opposite topological charge (marked as A- and A+) when circular edge dislocations beams propagate through atmospheric turbulence. The position of the pair of optical vortices varies with increasing propagation distance z. For z = 0.2km, 1km and 4km, the position of optical vortices A- and A+ located at (−1.18cm,1.77cm) and (1.77cm,-1.18cm), (−0.82cm,5.53cm) and (5.53cm,-0.82cm), (5.40cm,8.03cm) and (8.03cm,5.40cm), respectively, from which it is seen that the position of optical vortices A- and A+ are symmetric about the y = x axis. Figures 4(d) and 4(h) infer that the pair of optical vortices A- and A+ annihilate as soon as the propagation distance becomes large enough.

 figure: Fig. 4

Fig. 4 Curves of Reμ = 0 (red solid curves) and Imμ = 0 (blue dashed curves) and (e)-(h) contour lines of phase of circular edge dislocations beams propagating through atmospheric turbulence for different propagation distance (a) and (e) z = 0.2km, (b) and (f) z = 1km, (c) and (g) z = 4km, (d) and (h) z = 8km. Their units are cm, “” topological charge is −1, “” topological charge is + 1.

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The curves of Reμ = 0 (solid curves) and Imμ = 0 (dashed curves) of circular edge dislocations beams propagating though atmospheric turbulence are depicted in Fig. 5 at the different propagation distance z and for different values of 5(a)-5(d) n = 2, 5(e)-5(h) n = 3. Cn2 = 10−14m-2/3, the other calculation parameters are the same as in Fig. 2. From Figs. 5(a)-5(c), we find that circular edge dislocations B, C (in Fig. 3(a)) vanish and transform to two pairs of optical vortices (marked as B- and B+, C- and C+) when circular edge dislocations beams with n = 2 propagate through atmospheric turbulence, and the position of optical vortices B- and B+, C- and C+ are symmetric about the y = x axis. In the longer propagation distance, two pairs of optical vortices B- and B+, C- and C+ will annihilate, as shown in Fig. 5(d). Figures 5 (e)-5(h) appear three pairs of optical vortices D- and D+, E- and E+, F- and F+, and their dynamic evolution are similar to B- and B+, C- and C+ in Figs. 5 (a)-5(d). From Fig. 4 and Fig. 5 we see that the n-circular edge dislocation vanish and transform to n pairs of optical vortices with the opposite topological charge when circular edge dislocations beams propagate through atmospheric turbulence. The position of the n pairs of optical vortices will move with increasing propagation distance z, and they are symmetric about the y = x axis. As soon as the propagation distance becomes large enough, all optical vortices will annihilate. From [52, 54, 56, 75, 76] we can know that the position of optical vortices will move and the wavefront will change with increasing propagation distance, which is consistent with the obtained results in the present paper. Additionally, the result that the pairs of optical vortices are symmetric about the y = x axis is a new phenomena. It is attributed to that each pair of optical vortices is transformed from a same circular edge dislocation, respectively. Both of them connect with each other, namely, create, move in the same pattern and annihilate synchronously.

 figure: Fig. 5

Fig. 5 Curves of Reμ = 0 (red solid curves) and Imμ = 0 (blue dashed curves) of circular edge dislocations beams propagating through atmospheric turbulence at different propagation distance (a) and (e) z = 0.2km, (b) and (f) z = 1km, (c) and (g) z = 4km, (d) and (h) z = 8km for different values of (a)-(d) n = 2, (e)-(h) n = 3.

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Figure 6 gives the 3D trajectory of the pairs optical vortices in atmospheric turbulence versus the propagation distance z for the number of circular edge dislocations with 6(a) n = 1, 6(b) n = 2, 6(c) n = 3. The other calculation parameters are the same as in Fig. 4. Figure 6(a) implies that with increasing propagation distance z the distance between the optical vortices A- and A+ increases firstly and decreases sequentially, when the propagation distance z increases to 6.1km, the pairs of optical vortices A- and A+ with opposite topological charge −1 and + 1 will annihilate. From Fig. 6(b) we can see that the distance between the optical vortices B- and B+, C- and C+ also increases firstly and decreases sequentially with the increment of propagation distance z, respectively. The pairs of optical vortices B- and B+ annihilate when z = 5.55km, and the pairs of optical vortices C- and C+ annihilate when z = 6.16km. From the above it is seen that the pairs of optical vortices B- and B+ annihilate earlier than C- and C+. Similarly, the pairs of optical vortices D- and D+, E- and E+, F- and F+ annihilate at z = 5.13, 5.66 and 6.13km, respectively, from which it can be seen that the annihilation distance is different for the different pairs of optical vortices, and the smaller radius of circular edge dislocation corresponds to the sooner annihilation of optical vortices.

 figure: Fig. 6

Fig. 6 3D trajectory of the pairs optical vortices in atmospheric turbulence versus the propagation distance z for different values of (a) n = 1, (b) n = 2, (c) n = 3, “” topological charge is −1, “” topological charge is + 1.

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The 3D trajectory of the pairs optical vortices in atmospheric turbulence versus the propagation distance z are plotted in Fig. 7 for the structure constant Cn2, 7(a) Cn2 = 5 × 10−15 m-2/3, 7(b) Cn2 = 10−15 m-2/3, n = 2, the other calculation parameters are the same as in Fig. 4. As can be seen, the pairs of optical vortices B- and B+, C- and C+ annihilate at z = 6.89km and 7.76km for Cn2 = 5 × 10−15 m-2/3, respectively. While the pairs of optical vortices B- and B+, C- and C+ annihilate at z = 10.29km and 11.52km for Cn2 = 10−15 m-2/3, respectively. A comparison among Fig. 6(b) and Figs. 7(a)-7(b) shows that the structure constant Cn2 affects the annihilation distance of the pairs of optical vortices, the bigger the Cn2 is, the stronger the atmospheric turbulence is, the smaller the annihilation distance of the pairs of optical vortices is. Therefore, with the decrement of the structure constant Cn2, the annihilation distance of the pairs of optical vortices will increase.

 figure: Fig. 7

Fig. 7 3D trajectory of the pairs optical vortices in atmospheric turbulence versus the propagation distance z for different structure constant (a) Cn2 = 5 × 10−15 m-2/3, (b) Cn2 = 10−15 m2/3.

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5. Conclusion

In this paper, by using the extended Huygens-Fresnel principle, the analytical expressions for the cross-spectral density function of circular edge dislocations beams propagating through free space and atmospheric turbulence have been derived, and used to study the dynamic evolution of circular edge dislocations. It is shown that the number of circular edge dislocations on propagation in free space equals that at the source plane, the center of circular edge dislocations located at the point (0, 0). The radius of circular edge dislocation increases with increasing propagation distance z. n-circular edge dislocation vanish and transform to n pairs of optical vortices with the opposite topological charge when propagating through atmospheric turbulence, and the position of each pair of optical vortices are symmetric about the y = x axis. All the optical vortices will annihilate as soon as the propagation distance becomes large enough. The annihilation distance is different for the different pairs of optical vortices, and the smaller radius of circular edge dislocation corresponding to the sooner annihilation of optical vortices. The structure constant Cn2 affects the annihilation distance of the pairs of optical vortices, and with the decrement of the structure constant Cn2, the annihilation distance of the pairs of optical vortices will increase. The results obtained in this paper are useful for understanding the evolution behavior of circular edge dislocations in atmospheric turbulence and have potential applications in free space optical communications.

Funding

National Natural Science Foundation of China (NSFC) (61405136, 61505075).

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25. L. B. Ju, T. W. Huang, K. D. Xiao, G. Z. Wu, S. L. Yang, R. Li, Y. C. Yang, T. Y. Long, H. Zhang, S. Z. Wu, B. Qiao, S. C. Ruan, and C. T. Zhou, “Controlling multiple filaments by relativistic optical vortex beams in plasmas,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 94(3), 033202 (2016). [CrossRef]   [PubMed]  

26. E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83(1), 123–135 (1991). [CrossRef]  

27. M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular-momentum,” Opt. Commun. 96(1–3), 123–132 (1993). [CrossRef]  

28. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef]   [PubMed]  

29. S. Li and Z. Wang, “Generation of optical vortex based on computer-generated holographic gratings by photolithography,” Appl. Phys. Lett. 103(14), 141110 (2013). [CrossRef]  

30. C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, and S. G. Lipson, “Adjustable spiral phase plate,” Appl. Opt. 43(12), 2397–2399 (2004). [CrossRef]   [PubMed]  

31. Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. 38(24), 5418–5421 (2013). [CrossRef]   [PubMed]  

32. V. V. Kotlyar and A. A. Kovalev, “Fraunhofer diffraction of the plane wave by a multilevel (quantized) spiral phase plate,” Opt. Lett. 33(2), 189–191 (2008). [CrossRef]   [PubMed]  

33. M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464(7289), 737–739 (2010). [CrossRef]   [PubMed]  

34. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006). [CrossRef]   [PubMed]  

35. G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27(21), 1875–1877 (2002). [CrossRef]   [PubMed]  

36. Ya. Izdebskaya, V. Shvedov, and A. Volyar, “Generation of higher-order optical vortices by a dielectric wedge,” Opt. Lett. 30(18), 2472–2474 (2005). [CrossRef]   [PubMed]  

37. X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007). [CrossRef]  

38. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). [CrossRef]   [PubMed]  

39. E. Rueda, D. Muñetón, J. A. Gómez, and A. Lencina, “High-quality optical vortex-beam generation by using a multilevel vortex-producing lens,” Opt. Lett. 38(19), 3941–3944 (2013). [CrossRef]   [PubMed]  

40. S. Mochizuki, X. Gu, K. Tanabe, A. Matsutani, M. Ahmed, A. Bakry, and F. Koyama, “Generation of vortex beam using Bragg reflector waveguide,” Appl. Phys. Express 7(2), 022502 (2014). [CrossRef]  

41. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express 14(7), 3039–3044 (2006). [CrossRef]   [PubMed]  

42. R. K. Tyson, M. Scipioni, and J. Viegas, “Generation of an optical vortex with a segmented deformable mirror,” Appl. Opt. 47(33), 6300–6306 (2008). [CrossRef]   [PubMed]  

43. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46(15), 2893–2898 (2007). [CrossRef]   [PubMed]  

44. S. Vyas and P. Senthilkumaran, “Vortex array generation by interference of spherical waves,” Appl. Opt. 46(32), 7862–7867 (2007). [CrossRef]   [PubMed]  

45. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77(22), 4544–4547 (1996). [CrossRef]   [PubMed]  

46. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78(11), 2108–2111 (1997). [CrossRef]  

47. G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 88(2), 023902 (2001). [CrossRef]   [PubMed]  

48. R. P. Singh and S. R. Chowdhury, “Noncanonical vortex transformation and propagation in a two-dimensional optical system,” J. Opt. Soc. Am. A 20(3), 573–576 (2003). [CrossRef]   [PubMed]  

49. F. S. Roux, “Paraxial modal analysis technique for optical vortex trajectories,” J. Opt. Soc. Am. B 20(7), 1575–1580 (2003). [CrossRef]  

50. F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242(1), 45–55 (2004). [CrossRef]  

51. A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(4), 046609 (2009). [CrossRef]   [PubMed]  

52. J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A 11(4), 045710 (2009). [CrossRef]  

53. J. Li and B. Lü, “Composite coherence vortices in superimposed partially coherent vortex beams and their propagation through atmospheric turbulence,” J. Opt. A 11(7), 075401 (2009). [CrossRef]  

54. J. Li, H. Zhang, and B. Lü, “Partially coherent vortex beams propagating through slant atmospheric turbulence and coherence vortex evolution,” Opt. Laser Technol. 42(2), 428–433 (2010). [CrossRef]  

55. J. Li, H. Zhang, and B. Lü, “Composite coherence vortices in a radial beam array propagating through atmospheric turbulence along a slant path,” J. Opt. 12(6), 065401 (2010). [CrossRef]  

56. J. Li, J. Zeng, and M. Duan, “Classification of coherent vortices creation and distance of topological charge conservation in non-Kolmogorov atmospheric turbulence,” Opt. Express 23(9), 11556–11565 (2015). [CrossRef]   [PubMed]  

57. J. Zeng and J. Li, “Dynamic evolution and classification of coherent vortices in atmospheric turbulence,” Opt. Appl. 45(3), 299–308 (2015).

58. A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54(1), 870–879 (1996). [CrossRef]   [PubMed]  

59. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beams in nonlinear media: Snake instability and creation of optical vortices,” Phys. Rev. Lett. 76(13), 2262–2265 (1996). [CrossRef]   [PubMed]  

60. M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam,” J. Opt. Soc. Am. A 25(6), 1279–1286 (2008). [CrossRef]   [PubMed]  

61. S. G. Reddy, S. Prabhakar, A. Aadhi, J. Banerji, and R. P. Singh, “Propagation of an arbitrary vortex pair through an astigmatic optical system and determination of its topological charge,” J. Opt. Soc. Am. A 31(6), 1295–1302 (2014). [CrossRef]   [PubMed]  

62. M. Vasnetsov, V. Gorshkov, I. Marienko, and M. Soskin, “Wavefront motion in the vicinity of a phase dislocation:“optical vortex,” Opt. Spectrosc. 88(2), 260–265 (2000). [CrossRef]  

63. C. Ding, L. Pan, and B. Lü, “Phase singularities and spectral changes of spectrally partially coherent higher-order Bessel-Gauss pulsed beams,” J. Opt. Soc. Am. A 26(12), 2654–2661 (2009). [CrossRef]   [PubMed]  

64. H. Yan and B. Lü, “Dynamic evolution of an edge dislocation through aligned and misaligned paraxial optical ABCD systems,” J. Opt. Soc. Am. A 26(4), 985–992 (2009). [CrossRef]   [PubMed]  

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

66. E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986). [CrossRef]  

67. J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010). [CrossRef]  

68. I. Kimel and L. R. Elias, “Relations between hermite and laguerre gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]  

69. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

70. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979). [CrossRef]  

71. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

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

73. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003). [CrossRef]  

74. I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994). [CrossRef]   [PubMed]  

75. M. Chen and F. S. Roux, “Influence of the least-squares phase on optical vortices in strongly scintillated beams,” Phys. Rev. A 80(1), 013824 (2009). [CrossRef]  

76. M. Chen, C. Dainty, and F. S. Roux, “Speckle evolution with multiple steps of least-squares phase removal,” Phys. Rev. A 84(2), 023846 (2011). [CrossRef]  

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  53. J. Li and B. Lü, “Composite coherence vortices in superimposed partially coherent vortex beams and their propagation through atmospheric turbulence,” J. Opt. A 11(7), 075401 (2009).
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  54. J. Li, H. Zhang, and B. Lü, “Partially coherent vortex beams propagating through slant atmospheric turbulence and coherence vortex evolution,” Opt. Laser Technol. 42(2), 428–433 (2010).
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  55. J. Li, H. Zhang, and B. Lü, “Composite coherence vortices in a radial beam array propagating through atmospheric turbulence along a slant path,” J. Opt. 12(6), 065401 (2010).
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  56. J. Li, J. Zeng, and M. Duan, “Classification of coherent vortices creation and distance of topological charge conservation in non-Kolmogorov atmospheric turbulence,” Opt. Express 23(9), 11556–11565 (2015).
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  57. J. Zeng and J. Li, “Dynamic evolution and classification of coherent vortices in atmospheric turbulence,” Opt. Appl. 45(3), 299–308 (2015).
  58. A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54(1), 870–879 (1996).
    [Crossref] [PubMed]
  59. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beams in nonlinear media: Snake instability and creation of optical vortices,” Phys. Rev. Lett. 76(13), 2262–2265 (1996).
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  60. M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam,” J. Opt. Soc. Am. A 25(6), 1279–1286 (2008).
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  61. S. G. Reddy, S. Prabhakar, A. Aadhi, J. Banerji, and R. P. Singh, “Propagation of an arbitrary vortex pair through an astigmatic optical system and determination of its topological charge,” J. Opt. Soc. Am. A 31(6), 1295–1302 (2014).
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  62. M. Vasnetsov, V. Gorshkov, I. Marienko, and M. Soskin, “Wavefront motion in the vicinity of a phase dislocation:“optical vortex,” Opt. Spectrosc. 88(2), 260–265 (2000).
    [Crossref]
  63. C. Ding, L. Pan, and B. Lü, “Phase singularities and spectral changes of spectrally partially coherent higher-order Bessel-Gauss pulsed beams,” J. Opt. Soc. Am. A 26(12), 2654–2661 (2009).
    [Crossref] [PubMed]
  64. H. Yan and B. Lü, “Dynamic evolution of an edge dislocation through aligned and misaligned paraxial optical ABCD systems,” J. Opt. Soc. Am. A 26(4), 985–992 (2009).
    [Crossref] [PubMed]
  65. J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
    [Crossref]
  66. E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986).
    [Crossref]
  67. J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
    [Crossref]
  68. I. Kimel and L. R. Elias, “Relations between hermite and laguerre gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
    [Crossref]
  69. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
  70. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979).
    [Crossref]
  71. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).
  72. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  73. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003).
    [Crossref]
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2016 (5)

J. Li, W. Wang, M. Duan, and J. Wei, “Influence of non-Kolmogorov atmospheric turbulence on the beam quality of vortex beams,” Opt. Express 24(18), 20413–20423 (2016).
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J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4(5), B14–B28 (2016).
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R. Paez-Lopez, U. Ruiz, V. Arrizon, and R. Ramos-Garcia, “Optical manipulation using optimal annular vortices,” Opt. Lett. 41(17), 4138–4141 (2016).
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A. Abulikemu, T. Yusufu, R. Mamuti, S. Araki, K. Miyamoto, and T. Omatsu, “Octave-band tunable optical vortex parametric oscillator,” Opt. Express 24(14), 15204–15211 (2016).
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L. B. Ju, T. W. Huang, K. D. Xiao, G. Z. Wu, S. L. Yang, R. Li, Y. C. Yang, T. Y. Long, H. Zhang, S. Z. Wu, B. Qiao, S. C. Ruan, and C. T. Zhou, “Controlling multiple filaments by relativistic optical vortex beams in plasmas,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 94(3), 033202 (2016).
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2015 (5)

A. Abulikemu, T. Yusufu, R. Mamuti, K. Miyamoto, and T. Omatsu, “Widely-tunable vortex output from a singly resonant optical parametric oscillator,” Opt. Express 23(14), 18338–18344 (2015).
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M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22(1), 162–165 (2015).
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J. Li, J. Zeng, and M. Duan, “Classification of coherent vortices creation and distance of topological charge conservation in non-Kolmogorov atmospheric turbulence,” Opt. Express 23(9), 11556–11565 (2015).
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J. Zeng and J. Li, “Dynamic evolution and classification of coherent vortices in atmospheric turbulence,” Opt. Appl. 45(3), 299–308 (2015).

2014 (4)

S. G. Reddy, S. Prabhakar, A. Aadhi, J. Banerji, and R. P. Singh, “Propagation of an arbitrary vortex pair through an astigmatic optical system and determination of its topological charge,” J. Opt. Soc. Am. A 31(6), 1295–1302 (2014).
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S. Mochizuki, X. Gu, K. Tanabe, A. Matsutani, M. Ahmed, A. Bakry, and F. Koyama, “Generation of vortex beam using Bragg reflector waveguide,” Appl. Phys. Express 7(2), 022502 (2014).
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B. Tang, Y. Luo, Y. Zhang, S. Zheng, and Z. Gao, “Analytical vectorial structure of Gaussian beams carrying mixed screw–edge dislocations in the far field,” Opt. Commun. 324(1), 182–187 (2014).
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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).
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2013 (4)

2011 (4)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
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J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
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B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
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M. Chen, C. Dainty, and F. S. Roux, “Speckle evolution with multiple steps of least-squares phase removal,” Phys. Rev. A 84(2), 023846 (2011).
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2010 (5)

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010).
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M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464(7289), 737–739 (2010).
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J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
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J. Li, H. Zhang, and B. Lü, “Partially coherent vortex beams propagating through slant atmospheric turbulence and coherence vortex evolution,” Opt. Laser Technol. 42(2), 428–433 (2010).
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J. Li, H. Zhang, and B. Lü, “Composite coherence vortices in a radial beam array propagating through atmospheric turbulence along a slant path,” J. Opt. 12(6), 065401 (2010).
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2009 (7)

C. Ding, L. Pan, and B. Lü, “Phase singularities and spectral changes of spectrally partially coherent higher-order Bessel-Gauss pulsed beams,” J. Opt. Soc. Am. A 26(12), 2654–2661 (2009).
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J. Li and B. Lü, “Composite coherence vortices in superimposed partially coherent vortex beams and their propagation through atmospheric turbulence,” J. Opt. A 11(7), 075401 (2009).
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G. A. Swartzlander., “The optical vortex coronagraph,” J. Opt. A 11(9), 094022 (2009).
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2008 (3)

2007 (4)

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46(15), 2893–2898 (2007).
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S. Vyas and P. Senthilkumaran, “Vortex array generation by interference of spherical waves,” Appl. Opt. 46(32), 7862–7867 (2007).
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2006 (3)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
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K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express 14(7), 3039–3044 (2006).
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2005 (3)

2004 (3)

2003 (4)

2002 (1)

2001 (4)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
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2000 (1)

M. Vasnetsov, V. Gorshkov, I. Marienko, and M. Soskin, “Wavefront motion in the vicinity of a phase dislocation:“optical vortex,” Opt. Spectrosc. 88(2), 260–265 (2000).
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1997 (1)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78(11), 2108–2111 (1997).
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1996 (3)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. 77(22), 4544–4547 (1996).
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A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54(1), 870–879 (1996).
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1994 (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
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1993 (3)

I. Kimel and L. R. Elias, “Relations between hermite and laguerre gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
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1992 (1)

1991 (1)

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83(1), 123–135 (1991).
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1986 (1)

1979 (1)

1974 (1)

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E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83(1), 123–135 (1991).
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Agrawal, A.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
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Ahluwalia, B.

X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
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Ahmed, M.

S. Mochizuki, X. Gu, K. Tanabe, A. Matsutani, M. Ahmed, A. Bakry, and F. Koyama, “Generation of vortex beam using Bragg reflector waveguide,” Appl. Phys. Express 7(2), 022502 (2014).
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Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
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Allen, L.

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular-momentum,” Opt. Commun. 96(1–3), 123–132 (1993).
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Anderson, D. Z.

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54(1), 870–879 (1996).
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Anderson, I. M.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
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Araki, S.

Arita, Y.

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22(1), 162–165 (2015).
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M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38(22), 4919–4922 (2013).
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Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
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Arrizon, V.

Bakry, A.

S. Mochizuki, X. Gu, K. Tanabe, A. Matsutani, M. Ahmed, A. Bakry, and F. Koyama, “Generation of vortex beam using Bragg reflector waveguide,” Appl. Phys. Express 7(2), 022502 (2014).
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Barnett, S.

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular-momentum,” Opt. Commun. 96(1–3), 123–132 (1993).
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Berry, M.

J. Nye and M. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(2), 165–190 (1974).
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L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
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X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
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Buchleitner, A.

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91(1), 012345 (2015).
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X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
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Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. 38(24), 5418–5421 (2013).
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J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
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N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
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Chen, H.

X. Yuan, B. Ahluwalia, H. Chen, J. Bu, J. Lin, R. Burge, X. Peng, and H. Niu, “Generation of high-quality optical vortex beams in free-space propagation by microfabricated wedge with spatial filtering technique,” Appl. Phys. Lett. 91(5), 051103 (2007).
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Chen, M.

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22(1), 162–165 (2015).
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M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38(22), 4919–4922 (2013).
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M. Chen, C. Dainty, and F. S. Roux, “Speckle evolution with multiple steps of least-squares phase removal,” Phys. Rev. A 84(2), 023846 (2011).
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M. Chen and F. S. Roux, “Influence of the least-squares phase on optical vortices in strongly scintillated beams,” Phys. Rev. A 80(1), 013824 (2009).
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M. Chen and F. S. Roux, “Accelerating the annihilation of an optical vortex dipole in a Gaussian beam,” J. Opt. Soc. Am. A 25(6), 1279–1286 (2008).
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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).
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Chowdhury, S. R.

Courtial, J.

Cui, Z.

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
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M. Chen, C. Dainty, and F. S. Roux, “Speckle evolution with multiple steps of least-squares phase removal,” Phys. Rev. A 84(2), 023846 (2011).
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Dholakia, K.

M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Creating and probing of a perfect vortex in situ with an optically trapped particle,” Opt. Rev. 22(1), 162–165 (2015).
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M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38(22), 4919–4922 (2013).
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Ding, C.

Dipankar, A.

A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(4), 046609 (2009).
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Dong, Y.

Duan, M.

Elias, L. R.

I. Kimel and L. R. Elias, “Relations between hermite and laguerre gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
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J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
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N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
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Gao, Z.

B. Tang, Y. Luo, Y. Zhang, S. Zheng, and Z. Gao, “Analytical vectorial structure of Gaussian beams carrying mixed screw–edge dislocations in the far field,” Opt. Commun. 324(1), 182–187 (2014).
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N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
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Gómez, J. A.

Gorshkov, V.

M. Vasnetsov, V. Gorshkov, I. Marienko, and M. Soskin, “Wavefront motion in the vicinity of a phase dislocation:“optical vortex,” Opt. Spectrosc. 88(2), 260–265 (2000).
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S. Mochizuki, X. Gu, K. Tanabe, A. Matsutani, M. Ahmed, A. Bakry, and F. Koyama, “Generation of vortex beam using Bragg reflector waveguide,” Appl. Phys. Express 7(2), 022502 (2014).
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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).
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L. B. Ju, T. W. Huang, K. D. Xiao, G. Z. Wu, S. L. Yang, R. Li, Y. C. Yang, T. Y. Long, H. Zhang, S. Z. Wu, B. Qiao, S. C. Ruan, and C. T. Zhou, “Controlling multiple filaments by relativistic optical vortex beams in plasmas,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 94(3), 033202 (2016).
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N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
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S. Mochizuki, X. Gu, K. Tanabe, A. Matsutani, M. Ahmed, A. Bakry, and F. Koyama, “Generation of vortex beam using Bragg reflector waveguide,” Appl. Phys. Express 7(2), 022502 (2014).
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J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97(5), 053901 (2006).
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Figures (7)

Fig. 1
Fig. 1 (a, b) The intensity and (c, d) phase distribution of (a, c) circular edge dislocation beam and (b, d) vortex beam at the source plane.
Fig. 2
Fig. 2 (a)-(c) Curves of Reμ = 0 (red solid curves) and Imμ = 0 (blue dashed curves) and (d)-(f) contour lines of phase of circular edge dislocations beams at different propagation distance (a) and (d) z = 0, (b) and (e) z = 2km, (c) and (f) z = 5km. The abscissa represents ρ2x direction, ordinate represents ρ2y direction, and their units are cm.
Fig. 3
Fig. 3 Curves of Reμ = 0 (red solid curves) and Imμ = 0 (blue dashed curves) of circular edge dislocations beams at different propagation distance (a) and (d) z = 0, (b) and (e) z = 2km, (c) and (f) z = 5km for different values of (a)-(c) n = 2, (d)-(f) n = 3.
Fig. 4
Fig. 4 Curves of Reμ = 0 (red solid curves) and Imμ = 0 (blue dashed curves) and (e)-(h) contour lines of phase of circular edge dislocations beams propagating through atmospheric turbulence for different propagation distance (a) and (e) z = 0.2km, (b) and (f) z = 1km, (c) and (g) z = 4km, (d) and (h) z = 8km. Their units are cm, “” topological charge is −1, “” topological charge is + 1.
Fig. 5
Fig. 5 Curves of Reμ = 0 (red solid curves) and Imμ = 0 (blue dashed curves) of circular edge dislocations beams propagating through atmospheric turbulence at different propagation distance (a) and (e) z = 0.2km, (b) and (f) z = 1km, (c) and (g) z = 4km, (d) and (h) z = 8km for different values of (a)-(d) n = 2, (e)-(h) n = 3.
Fig. 6
Fig. 6 3D trajectory of the pairs optical vortices in atmospheric turbulence versus the propagation distance z for different values of (a) n = 1, (b) n = 2, (c) n = 3, “” topological charge is −1, “” topological charge is + 1.
Fig. 7
Fig. 7 3D trajectory of the pairs optical vortices in atmospheric turbulence versus the propagation distance z for different structure constant (a) C n 2 = 5 × 10−15 m-2/3, (b) C n 2 = 10−15 m2/3.

Equations (22)

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E(s,θ,0)= ( 2 s w 0 ) m L n m ( 2 s 2 w 0 2 )exp( s 2 w 0 2 )exp(imθ),
exp(imθ) s 2 L n m ( s 2 )= (1) n 2 2n+m n! t=0 n r=0 m i r ( t n ) ( r m ) H 2t+mr ( s x ) H 2n2t+r ( s y ),
E(s,0)= (1) n 2 2n+m n! t=0 n r=0 m i r ( t n ) ( r m ) H 2t+mr ( 2 s x w 0 ) H 2n2t+r ( 2 s y w 0 )exp( s 2 w 0 2 ).
E(s,0)= 1 2 m r=0 m i r ( r m ) H mr ( 2 s x w 0 ) H r ( 2 s y w 0 )exp( s 2 w 0 2 ),
E(s,0)= (1) n 2 2n n! t=0 n ( t n ) H 2t ( 2 s x w 0 ) H 2n2t ( 2 s y w 0 )exp( s 2 w 0 2 ).
W 0 ( s 1 , s 2 ,0)= 1 2 4n (n!) 2 t 1 =0 n t 2 =0 n ( t 1 n ) ( t 2 n ) H 2 t 1 ( 2 s 1x w 0 ) H 2 t 2 ( 2 s 2x w 0 ) × H 2n2 t 1 ( 2 s 1y w 0 ) H 2n2 t 2 ( 2 s 2y w 0 )exp( s 1 2 + s 2 2 w 0 2 ).
W( ρ 1 , ρ 2 ,z)= ( k 2πz ) 2 W 0 ( s 1 , s 2 ,0)exp{ ik 2z [ ( s 1 ρ 1 ) 2 ( s 2 ρ 2 ) 2 ]} ×exp[ψ( s 1 , ρ 1 )+ ψ * ( s 2 , ρ 2 )]d s 1 d s 2 ,
exp[ψ( s 1 , ρ 1 )+ ψ * ( s 2 , ρ 2 )]=exp[ ( s 1 s 2 ) 2 + ( ρ 1 ρ 2 ) 2 +( s 1 s 2 )( ρ 1 ρ 2 ) ρ 0 2 ],
exp[ (xy) 2 ] H n (ax)dx= π (1 a 2 ) n 2 H n ( ay (1 a 2 ) 1/2 ),
x n exp[ (xβ) 2 ]dx= (2i) n π H n (iβ),
H n (x+y)= 1 2 n/2 k=0 n ( k n ) H k ( 2 x) H nk ( 2 y),
H n (x)= m=0 [n/2] (1) m n! m!(n2m)! (2x) n2m ,
W( ρ 1 , ρ 2 ,z)= ( k 2πz ) 2 A x A y exp[ ( ρ 1 ρ 2 ) 2 ρ 0 2 ]exp[ ik 2z ( ρ 1 2 ρ 2 2 )] × 1 2 4n (n!) 2 t 1 =0 n t 2 =0 n ( t 1 n ) ( t 2 n )BC ,
A x =exp[ 1 4D ( ρ 1x ρ 2x ρ 0 2 ik ρ 2x z ) 2 ]exp( F x 2 4G ),
B= c 1 =0 t 1 d 1 =0 2 t 2 e 1 =0 [ d 1 2 ] ( d 1 2 t 2 ) (1) c 1 + e 1 (2i) (2 t 2 2 c 1 + d 1 2 e 1 ) (2 t 1 )! d 1 ! c 1 !(2 t 1 2 c 1 )! e 1 !( d 1 2 e 1 )! ( 2 2 w 0 ) 2 t 1 2 c 1 × π D (1 2 w 0 2 D ) t 2 2 t 2 [ 4 ρ 0 2 w 0 2 D 2 2D ] d 1 2 e 1 ( 1 G ) 2 t 1 2 c 1 + d 1 2 e 1 +1 × H 2 t 2 d 1 [ ( ρ 1x ρ 2x )zik ρ 2x ρ 0 2 ρ 0 2 z w 0 2 D 2 2D ] H 2 t 1 2 c 1 + d 1 2 e 1 (i F x 2 G ) ,
C= c 2 =0 n t 1 d 2 =0 2n2 t 2 e 2 =0 [ d 2 2 ] ( d 2 2n2 t 2 ) (1) c 2 + e 2 (2i) (2n2 t 1 2 c 2 + d 2 2 e 2 ) (2n2 t 1 )! d 2 ! c 2 !(2n2 t 1 2 c 2 )! e 2 !( d 2 2 e 2 )! × π D (1 2 w 0 2 D ) n t 1 2 (n t 1 ) [ 4 ρ 0 2 w 0 2 D 2 2D ] d 2 2 e 2 ( 1 G ) 2n2 t 1 2 c 2 + d 2 2 e 2 +1 × ( 2 2 w 0 ) 2n2 t 1 2 c 2 H 2n2 t 2 d 2 [ ( ρ 1y ρ 2y )zik ρ 2y ρ 0 2 ρ 0 2 z w 0 2 D 2 2D ] H 2n2 t 1 2 c 2 + d 2 2 e 2 (i F y 2 G ),
D= 1 w 0 2 ik 2z + 1 ρ 0 2 ,
F x = ik ρ 1x z ρ 1x ρ 2x ρ 0 2 + 1 D ρ 0 2 ( ρ 1x ρ 2x ρ 0 2 ik ρ 2x z ),
G= 1 w 0 2 + ik 2z + 1 ρ 0 2 1 D ρ 0 4 .
μ( ρ 1 , ρ 2 ,z)= W( ρ 1 , ρ 2 ,z) [ I( ρ 1 ,z)I( ρ 2 ,z) ] 1/2 ,
Re[ μ( ρ 1 , ρ 2 ,z) ]=0,
Im[ μ( ρ 1 , ρ 2 ,z) ]=0.

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