Abstract

The turbulent effects of strong irradiance fluctuations on the probability densities and the normalized powers of the orbital angular momentum (OAM) modes are modeled for fractional Bessel Gauss beams in paraxial turbulence channel. We find that the probability density of signal OAM modes is a function of position deviation from the beam center, and the farther away from the beam center the detection position is, the smaller the probability density is. For fractional OAM quantum numbers, the average probability densities of signal/crosstalk modes oscillate along the beam radius except the half-integer. When the beam waist of source decreases or the irradiance fluctuation increases, the average probability density of the signal OAM mode drops. The peak of the average probability density of crosstalk modes shifts to outward of the beam center as beam waist gets larger. In the nearby region of beam center, the larger the quantum number deviation of OAM, the smaller the beam waist and the turbulence fluctuations are, the lower average probability densities of crosstalk OAM modes are. Especially, the increase of turbulence fluctuations can make the crosstalk stronger and more concentrated. Lower irradiance fluctuation can give rise to higher the normalized powers of the signal OAM modes, which is opposite to the crosstalk normalized powers.

© 2015 Optical Society of America

1. Introduction

The progress in optical quantum communication over the last few years have enabled the development of the orbital angular momentum (OAM) of stable vortex beams whose OAM per photon can take an arbitrary value within a continuous range, either integer or non-integer in units of [1–7]. It was demonstrated theoretically that the fractional topological charges associated with center vortices and instantons are necessarily grouped into integral charges on a global scale, but are uncorrelated with each other on any shorter scale [7]. Several experiments have verified that light beam may carry fractional OAM when the beam propagates in free-space [8–13]. Such as, S. H. Tao et al. emphasize that a higher-order fractional Bessel beam will be ideal for investigating the behavior of orbital angular momentum at the opening of the fractional Bessel beam [11]. W. M. Lee et al. examine the evolution of optical beams with a fractional phase step hosted within a Gaussian beam by experimental analysis of both the phase and intensity distribution [12]. S. Vyas et al. experimentally demonstrate the generation of a fractional topological charge beam on the basics of vortex lens [13]. This type of light beam has an additional line singularity from beam center to its outward, where the intensity also vanishes.

The fractional Bessel beams (nondiffracting vortex beams) are of special interest due to their properties of divergence-less propagation and self-repair, and have generated widespread interest in the last decade for optical communication [14–17]. J. B. GÖTTE proposed a quantum mechanical description of the beams with fractional topological charges [14] and a high-order Bessel non-vortex beam of fractional type α (HOBNVBs-Fα), was introduced by F. G. Mitri [16,17]. It was found the light modes with fractional orbital angular momentum can be applied to the field of where range from quantum entanglement [14] to optical manipulation [17]. However few researchers have addressed the turbulence effects of strong irradiance fluctuations on OAM modes of fractional Bessel Gauss (FBG) beams.

To understand the turbulence effects of strong irradiance fluctuations on the receiving signals at each point in the receiving plane and the receiving value of acceptor, for FBG beams with orbital angular momentum l0. In this paper, we outline the influences of atmospheric turbulence on the orbital angular momentum mode probability densities and the normalized powers for FBG beams along the direction of the beam radius. In particular, we draw attention to the models of mode probability density for signal or the crosstalk OAM modes and the normalized power of OAM per photon per unit length of a transverse slice of the beam of a FBG beams in strong turbulence are established. The effects of the fractional order of OAM modes, turbulence strength (in strong fluctuation region), propagation distance and wavelength on the mode probability densities of the signal and crosstalk OAM modes of FBG beams along the direction of r in receiving plane are researched in section 3. Conclusions are presented in Section 4. Our results provide us with new insight into the behavior of the OAM of fractional Bessel Gauss beams in the context of fractional beams and elucidate the connection between traditional optical communication and quantum theory to represent beams with fractional OAM.

2. Ensemble averaging mode probability density

In the strong irradiance fluctuation region [18] and in the half-spacez>0, the normalized complex amplitude of FBG beams in a cylindrical coordinate system (r,φ,z) can be expressed as

E(r,φ,z)=Efree(r,φ,z)exp[ψx(r,φ,z)+ψy(r,φ,z)],
where r=|r|, r=(x,y) is the two-dimensional position vector in the source plane;z is propagation distance; ψx(r,φ,z) and ψy(r,φ,z) are the complex phase perturbations due to large-scale and small-scale turbulence eddies, respectively; Efree(r,φ,z) is the normalized FBG beam at the z plane in turbulence-free channel and, in the paraxial channel, Efree(r,φ,z) has the form [9,19]
Efree(r,φ,z)=1μ(z)exp[ikr2z2kμ(z)r2μ(z)w02]l0=CγJ|l0|[krrμ(z)]exp(il0φ),
where μ(z)=1+iz/zR, zR=kw02/2 is the Rayleigh range, k=2π/λ is the wavenumber, λ is the wavelength, and w0 is the beam waist; J|l0|is the Bessel function of integer orders, l0 describes the helical structure of the wave front around a wave front singularity; kr=ξk is the transverse wavenumber; Cγ=(±i)γsin(±πγ)πi|l0|±γl0, γ is any real (integer or fractional) number [9,19]. The diffractive spread of beams propagation in turbulence-free channel from z = 0 to 1000m is given by Fig. 1.

 figure: Fig. 1

Fig. 1 The figure shows beam intensity simulations for fractional FBG beam along the direction z to describe how the beam spreads when the FBG beam transmits in Free Space Optics (FSO). For short distance transmission (z<200m), the beam is approximate to non-diffracting; for long distance transmission (z>200m), the diffractive spread of beams become obvious (for example, as γ = 5.1, when z increases from 300m to 1000m, the first ring radius of the fractional FBG beam spreads from 0.13m to 0.48m).

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In a quantum description beams with fractional OAM are superpositions of states of different OAM [10], the FBG is expanded as an summation with basis exp(ilφ) which carries OAM of l, then the complex amplitude E(r,φ,z) for FBG can be written as [6,20]

E(r,φ,z)=l=βl(r,z)exp(ilφ),
where βl(r,z) is the mode amplitude of the mode l at the position (r,z), and is given by

βl(r,z)=12π02πE(r,φ,z)exp(ilφ)dφ.

In quantum description, |βl(r,z)|2 is the mode probability density of the mode l at the position (r,z). In turbulent media, the mode probability density is associated with the ensemble average over turbulence medium, i.e.|βl(r,z)|2at, where at represents the ensemble average of the atmospheric turbulence. By Eqs. (1) and (4) and proceeding the ensemble average of the atmospheric turbulence for βl(r,z)βl(r,z), and the denotes complex conjugate. In paraxial channel(kr0), the ensemble averaging mode probability density |βl(r,z)|2at of OAM models l of FBG beam, which is marked withPl(r,z), is given by

Pl(r,z)=|βl(r,z)|2at=(12π)202π02πEfree(r,φ,z)Efree(r,φ,z)exp[il(φφ)]×exp[ψx(r,φ,z)+ψy(r,φ,z)+ψx(r,φ,z)+ψy(r,φ,z)]atdφdφ,
where exp[ψx(r,φ,z)+ψy(r,φ,z)+ψx(r,φ,z)+ψy(r,φ,z)]at is given by
exp[ψx(r,φ,z)+ψy(r,φ,z)+ψx(r,φ,z)+ψy(r,φ,z)]at=exp[12D(r,φ,r,φ;z)],
and D(r,φ,r,φ;z)=Dx(r,φ,r,φ;z)+Dy(r,φ,r,φ;z). The Dx(r,φ,r,φ;z) and Dy(r,φ,r,φ;z) are wave structure functions caused by large-scale and small-scale turbulence eddies, respectively [21].

Based on the modified Rytov method [18,21], the effective atmospheric spectral model is given by

Φn=0.033Cn2κ11/3[f(κli)g(κLo)Gx+Gy],
where κ is the spatial wave number of refractive index fluctuations, Cn2 is the generalized refractive-index structure parameter with units m-2/3; f(κli)=exp(κ2/κm2),g(κLo)=(κ2+κo2)11/6, κm=5.92/li, li denotes the inner scalar of turbulence and κ0=2π/Lo, Lo denotes the outer scaler of turbulence; Gx(κ) and Gy(κ) represent the large and small scale filter functions, respectively. In far field region and strong irradiance fluctuation region [21], the filter function are given by
Gx(κ)=exp[κ2/κx2],Gy(κ)=κ11/3(κ2+κx2)11/6,
where κx2=kηx/zis the large scale frequency cut-off and κy2=kηy/zis the small scale frequency cut-off, ηx=8.561+0.08σR2(10.89z/kli2)1/6, σR2 is the Rytov variance which indicates the strength of strong irradiance fluctuations, and given byσR2=1.23Cn2k7/6z11/6, ηy=9[1+0.69(0.4σR2)6/5] [22].

On spherical wave approximation and by the approximation of generalized hypergeometric functions in [21], we have the large and small scale portions of the wave structure function for a spherical wave

Dx(r,r,φ,φ;z)=0.604Cn2k2zκxm1/3{[r2+r22rrcos(φφ)]×[1+0.033κxm2[r2+r22rrcos(φφ)]]1/6},
Dy(r,r,φ,φ;z)=0.604Cn2k2zκm1/3{[r2+r22rrcos(φφ)]×[1+0.033κm2[r2+r22rrcos(φφ)]]1/6}0.604Cn2k2zκym1/3{[r2+r22rrcos(φφ)]×[1+0.033κym2[r2+r22rrcos(φφ)]]1/6},
where κxm2=35.046kηxkηxli2+35.046z and κym2=35.046kηykηyli2+35.046z.

For strong irradiance fluctuations (σR21) and by approximation (kz[r2+r22rrcos(φφ)])5/6 (kz[r2+r22rrcos(φφ)]) [12], we have

Dx(r,r,φ,φ;z)0.604Cn2k2zκxm1/3[r2+r22rrcos(φφ)],
Dy(r,r,φ,φ;z)0.867σR2kz[r2+r22rrcos(φφ)]0.604Cn2k2zκym1/3[r2+r22rrcos(φφ)].

On substituting the Eqs. (11) and (12) into Eq. (6), we obtain

exp[12D(r,φ,r,φ;z)]=exp{[0.604Cn2k2zκxm1/3+(0.867σR2kz0.604Cn2k2zκym1/3)]×[r2+r22rrcos(φφ)]}=exp[r2+r22rrcos(φφ)ρ02],
where is the new spatial coherence radius of a spherical wave propagating in the strong irradiance fluctuation channel and is given by

ρ0={0.491σR2kz{(35.046ηxzkηxli2+35.046z)1/6+[1.766(35.046ηyzkηyli2+35.046z)1/6]}}1/2.

Based on the integral expression [23]

02πexp[inφ1+ηcos(φ1φ2)]dφ1=2πexp(inφ2)In(η),
Where w(z)=w01+z2/zR2 is the Gaussian spot size; In(η) is the Bessel function of second kind with n order, with the help of Eqs. (2), (5) and (6), we have the mode probability density of the OAM for FBG beams in paraxial turbulence channel
Pl(r,z)=w02w2(z)l0=Cγ2|J|l0|(krrμ(z))|2exp[(1w2(z)+1ρ02)2r2]Ill0(2r2/ρ02),
where Δl=ll0 denotes the value of crosstalk; for Δl=0, Pl0(r,z) is the mode probability density of the signal OAM mode l0 for FBG beams, and for Δl0, Pl(r,z) is the mode probability density of the crosstalk OAM mode, which represents the probability density of the part of the energy launched into the signal OAM mode redistributed into other OAM modes by atmospheric turbulence [24].

The normalized power of OAM per photon per unit length of a transverse slice of the beam is calculated with the following expression [9]

Lz(l)=l=lBPll=BPl.
where l=BPl is the beam power with
BPl=w02w2(z)l0=Cγ20|J|l0|(krrμ(z))|2exp[(1w2(z)1ρ02)2r2]Ill0(2r2/ρ02)rdr,
and BPl is the power of each constituent angular harmonics beam, which is given by [9]. Similarly, for l=l0, Lz(l0) expresses the normalized power of the signal OAM mode for FBG beams, and for ll0, Lz(l) is the normalized powers of the crosstalk OAM modes.

3. Numerical discussion

To investigate the influences of the strong irradiance fluctuations fractional order of OAM modes γ, the deviation of quantum numberΔl, the Rytov variance σR2 and beam waists w0 on the average probability density of signal OAM modes and signal normalized powers, in this section, we employ the method of numerical calculation to analyze the Eqs. (16) and (17). Obviously, the average probability density of signal OAM modes and signal normalized powers decrease, and the average crosstalk probability of OAM modes and crosstalk normalized powers increase when the propagation distance z increases. Then we do not discuss the influences owed to the propagation distance z. To simplify, in figures, we write ensemble average probability density (Pl0(r,z)) of the signal OAM modes to mode probability, and write ensemble average probability density (Pll0(r,z)) of the crosstalk OAM modes to crosstalk probability.

In this section, we first study the effects of γ, Δl, σR2 and w0 on the mode probability densities of the signal (Pl0(r,z)) and crosstalk (Pll0(r,z)) OAM modes for FBG beams along the direction of beam radius r in receiving plane. The simulation results are shown in Figs. 2-5. The parameters are taken as λ = 1550nm, z = 1km, w0 = 0.1m, γ = 5.5, ξ = 0.001, α = 11/3, li = 0.001m, σR2 = 10, Δl = 0, 1, 2 and 3, unless otherwise indicated. Finally, we evaluate the effects of the irradiance fluctuations on the normalized powers Lz(l) in parameter Δl=0 or Δl=1. The simulation results are shown in the final figure.

Figure 2 plots the performance of Pl(r,z) of FBG beam versus r from 0 to 0.2m for the different Δl. Figure 2 demonstrates that the Pl0(r,z) decays rapidly and tends to a stable value when r increases. As can be seen from Fig. 2, the peak of Pll0(r,z) moves outward and downward clearly as Δl gets larger. When r is large enough (in Fig. 2, r>0.05m), Pl0(r,z) and Pll0(r,z) tend to be same. The results show that we can obtain the higher signal-noise ratio (SNR) only in the circle region of radius r<0.05m, and the center of the circle region is beam center. So in the next context, we adopt the circle region of radius r<0.05m.

 figure: Fig. 2

Fig. 2 The average mode probability Pl(r,z) of FBG beam along the direction of the beam radius rfor different values of the quantum number deviation Δl.

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In Fig. 3, Pl(r,z) is plotted as a function of r for mode crosstalk Δl = 0 and 1. Figures 3 (a) and 3(b) is for γ = 5.1, 5.3, 5.5, 5.7 and 5.9, respectively. As is seen from Fig. 3, Pl(r,z)are the vibration curves for γ = 5.1, 5.3, 5.7 and 5.9, and these vibration curves are around the curve of γ = 5.5. Figure 3(a) indicates that Pl0(r,z) decrease with the increasing of r, while Pll0(r,z) decay after the increase in the region close to circle area to circle area in Fig. 3(b). It is interesting to note that when the value of γ is half integer, its curve is without vibration in Figs. 3(c) and 3(d). Obviously, the closer the value of γ to half integer, the closer its curve tends to the image of γ = 5.5. In order to facilitate the beam properties, we set γ = 5.5.

 figure: Fig. 3

Fig. 3 The average probability densities Pl(r,z) of OAM modes for fractional FBG beam along the direction of the beam radius r with different values of γ when (a) Δl=0, signal probability densities; (b) Δl=1, crosstalk probability densities; (c) γ is half-integer, and Δl=0 signal probability densities; (d) γ is half-integer, and Δl=1 crosstalk probability densities.

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Figure 4 indicates the variation of Pl(r,z) against r for selected beam waist w0 = 0.02m, 0.05m and 0.10m. Figure 4 shows that increasing w0 can rise Pl(r,z). From Fig. 4(a), it can be seen that the lower w0 is, the faster Pl0(r,z) decays. And the signal difference of adjacent channel increases as the detection position away from the beam center. As is shown by Fig. 4(b), all the curves pick up and then fall, and the value of Pll0(r,z) keeps growing as w0 gets larger.

 figure: Fig. 4

Fig. 4 The average probability densities Pl(r,z) of OAM modes for fractional FBG beam along the direction of the beam radius r with different values of beam waist w0 when (a) Δl=0, signal probability densities; (b) Δl=1, crosstalk probability densities.

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Figure 5 illustrates that the variation of Pl(r,z) along the radial direction r from 0 to 0.05m. For this example we have chosen σR2 = 5, 10 and 30. According to Fig. 5(a), as σR2 increases, Pl0(r,z) decreases slightly but not obvious. It’s clear that the influence of turbulence on Pl0(r,z) is more apparent as r increases in the strong fluctuation region. Near the central area of the beam, Pll0(r,z) increases with the increasing of σR2 and the main energy still remains at the transmit signal channel. However, as the detection position away from the beam center, the energy of crosstalk channel increases immediately and is almost as strong as the energy that remains in the signal channel. When detection position is far away from the beam center in receiving plane, Pl0(r,z) and Pll0(r,z) tend to be same. For the case of σR2 = 5, the energy of crosstalk channel at r = 0.05m is almost the same as the energy of signal channel. The peak of the Pll0(r,z) shifts to the beam center, and half width of the peak narrows down as σR2 increases, i.e. the increase of σR2 can make the crosstalk noise stronger and more concentrated.

 figure: Fig. 5

Fig. 5 The average probability densities Pl(r,z) of OAM modes for fractional FBG beam along the direction of the beam radius r with different values of Rytov variance σR2 when (a) Δl=0, signal probability densities; (b) Δl=1, crosstalk probability densities.

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Finally, in Fig. 6, we describe the effects of Rytov variance σR2 on the normalized powersLz(l) in the case that the inner scales li tend to 0. Comparing the curves in Fig. 6, as σR2 increases, the normalized power of the signal OAM mode reduces more and the normalized powers of the crosstalk OAM modes increase more. As is seen from this figure, the normalized powers of the signal or crosstalk OAM modes are the complicated functions of beam waist.

 figure: Fig. 6

Fig. 6 The normalized powers Lz(l) of fractional OAM modes for fractional FBG beam along beam waist w0 with different values of turbulence fluctuation σR2 when (a) Δl=0, signal normalized powers Lz(l0); (b) Δl=1, crosstalk normalized powers Lz(ll0).

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4. Conclusions

In conclusion, the model of average probability densities and the normalized powers of the signal/crosstalk OAM modes for FBG beams in the turbulent atmosphere of strong irradiance fluctuations have been developed. This study has revealed that average mode probability Pl0(r,z) and average crosstalk probability Pll0(r,z) are the same when beam radius r is large enough. For FBG beams, when r varies, Pl0(r,z) and Pll0(r,z) oscillate except half-integer, and the larger the parameter γ is, the greater the amplitude of oscillation curves will be. For beams of γ = 5.5, the average mode probability Pl0(r,z) decays with the decreasing of the beam waist w0 or the increasing of irradiance fluctuations σR2 along beam radius, and average crosstalk probability Pll0(r,z) drops rapidly after the slow growth as r increases, namely average crosstalk probability curves have peaks. Near the central region of beam, enlarging Δl or minifying σR2 and w0 can make Pll0(r,z) drop down. The normalized powers of signal or crosstalk OAM modes are the complicated functions of beam waist. Furthermore, lower σR2 can more efficient reduce the turbulence effect on the normalized powerLz(l0) of signal OAM modes, which is opposite to crosstalk normalized powers Lz(ll0).

Acknowledgments

This work is supported by the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20140128) and the National Natural Science Foundation of Special Theoretical Physics (Grant No. 11447174).

References and links

1. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004). [CrossRef]  

2. W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013). [CrossRef]  

3. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004). [CrossRef]  

4. J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Opt. Express 15(10), 6300–6313 (2007). [CrossRef]   [PubMed]  

5. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008). [CrossRef]   [PubMed]  

6. F. G. Mitri, “Vector wave analysis of an electromagnetic high-order Bessel vortex beam of fractional type α,” Opt. Lett. 36(5), 606–608 (2011). [CrossRef]   [PubMed]  

7. J. M. Cornwall, “Center vortices, nexuses, and fractional topological charge,” Phys. Rev. D Part. Fields 61(8), 085012 (2000). [CrossRef]  

8. S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A 21(7), 1192–1197 (2004). [CrossRef]   [PubMed]  

9. J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A: Pure Appl. Opt. 10(1), 015009 (2008). [CrossRef]  

10. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction,” Opt. Express 18(16), 16480–16485 (2010). [CrossRef]   [PubMed]  

11. S. H. Tao, W. M. Lee, and X. C. Yuan, “Dynamic optical manipulation with a higher-order fractional bessel beam generated from a spatial light modulator,” Opt. Lett. 28(20), 1867–1869 (2003). [CrossRef]   [PubMed]  

12. W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004). [CrossRef]  

13. S. Vyas, R. K. Singh, and P. Senthilkumaran, “Fractional vortex lens,” Opt. Laser Technol. 42(6), 878–882 (2010). [CrossRef]  

14. J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum for mulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007). [CrossRef]  

15. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005). [CrossRef]  

16. F. G. Mitri, “High-order Bessel nonvortex beam of fractional type α,” Phys. Rev. A 85(2), 025801 (2012). [CrossRef]  

17. F. G. Mitri, “High-order Bessel non-vortex beam of fractional type α: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67(7), 135 (2013). [CrossRef]  

18. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2th ed. (SPIE, 2005).

19. C. López-Mariscal, D. Burnham, D. Rudd, D. McGloin, and J. C. Gutiérrez-Vega, “Phase dynamics of continuous topological upconversion in vortex beams,” Opt. Express 16(15), 11411–11422 (2008). [CrossRef]   [PubMed]  

20. Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013). [CrossRef]  

21. C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004). [CrossRef]  

22. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2007).

23. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

24. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008). [CrossRef]   [PubMed]  

References

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  1. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004).
    [Crossref]
  2. W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
    [Crossref]
  3. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
    [Crossref]
  4. J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Opt. Express 15(10), 6300–6313 (2007).
    [Crossref] [PubMed]
  5. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008).
    [Crossref] [PubMed]
  6. F. G. Mitri, “Vector wave analysis of an electromagnetic high-order Bessel vortex beam of fractional type α,” Opt. Lett. 36(5), 606–608 (2011).
    [Crossref] [PubMed]
  7. J. M. Cornwall, “Center vortices, nexuses, and fractional topological charge,” Phys. Rev. D Part. Fields 61(8), 085012 (2000).
    [Crossref]
  8. S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A 21(7), 1192–1197 (2004).
    [Crossref] [PubMed]
  9. J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A: Pure Appl. Opt. 10(1), 015009 (2008).
    [Crossref]
  10. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction,” Opt. Express 18(16), 16480–16485 (2010).
    [Crossref] [PubMed]
  11. S. H. Tao, W. M. Lee, and X. C. Yuan, “Dynamic optical manipulation with a higher-order fractional bessel beam generated from a spatial light modulator,” Opt. Lett. 28(20), 1867–1869 (2003).
    [Crossref] [PubMed]
  12. W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
    [Crossref]
  13. S. Vyas, R. K. Singh, and P. Senthilkumaran, “Fractional vortex lens,” Opt. Laser Technol. 42(6), 878–882 (2010).
    [Crossref]
  14. J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum for mulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
    [Crossref]
  15. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
    [Crossref]
  16. F. G. Mitri, “High-order Bessel nonvortex beam of fractional type α,” Phys. Rev. A 85(2), 025801 (2012).
    [Crossref]
  17. F. G. Mitri, “High-order Bessel non-vortex beam of fractional type α: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67(7), 135 (2013).
    [Crossref]
  18. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2th ed. (SPIE, 2005).
  19. C. López-Mariscal, D. Burnham, D. Rudd, D. McGloin, and J. C. Gutiérrez-Vega, “Phase dynamics of continuous topological upconversion in vortex beams,” Opt. Express 16(15), 11411–11422 (2008).
    [Crossref] [PubMed]
  20. Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
    [Crossref]
  21. C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
    [Crossref]
  22. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2007).
  23. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).
  24. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008).
    [Crossref] [PubMed]

2013 (3)

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

F. G. Mitri, “High-order Bessel non-vortex beam of fractional type α: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67(7), 135 (2013).
[Crossref]

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

2012 (1)

F. G. Mitri, “High-order Bessel nonvortex beam of fractional type α,” Phys. Rev. A 85(2), 025801 (2012).
[Crossref]

2011 (1)

2010 (2)

2008 (4)

2007 (2)

J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Opt. Express 15(10), 6300–6313 (2007).
[Crossref] [PubMed]

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum for mulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

2005 (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

2004 (5)

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
[Crossref]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004).
[Crossref]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A 21(7), 1192–1197 (2004).
[Crossref] [PubMed]

C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
[Crossref]

2003 (1)

2000 (1)

J. M. Cornwall, “Center vortices, nexuses, and fractional topological charge,” Phys. Rev. D Part. Fields 61(8), 085012 (2000).
[Crossref]

Anguita, J. A.

Barnett, S. M.

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

Burnham, D.

Cornwall, J. M.

J. M. Cornwall, “Center vortices, nexuses, and fractional topological charge,” Phys. Rev. D Part. Fields 61(8), 085012 (2000).
[Crossref]

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
[Crossref]

Donegan, J. F.

Eastham, P. R.

Fickler, R.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Flossmann, F.

Franke-Arnold, S.

Götte, J. B.

Gutiérrez-Vega, J. C.

Jiang, Y.

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Krenn, M.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004).
[Crossref]

Lee, W. M.

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
[Crossref]

S. H. Tao, W. M. Lee, and X. C. Yuan, “Dynamic optical manipulation with a higher-order fractional bessel beam generated from a spatial light modulator,” Opt. Lett. 28(20), 1867–1869 (2003).
[Crossref] [PubMed]

López-Mariscal, C.

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A: Pure Appl. Opt. 10(1), 015009 (2008).
[Crossref]

C. López-Mariscal, D. Burnham, D. Rudd, D. McGloin, and J. C. Gutiérrez-Vega, “Phase dynamics of continuous topological upconversion in vortex beams,” Opt. Express 16(15), 11411–11422 (2008).
[Crossref] [PubMed]

Lunney, J. G.

Masino, A. J.

C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
[Crossref]

McGloin, D.

Mitri, F. G.

F. G. Mitri, “High-order Bessel non-vortex beam of fractional type α: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67(7), 135 (2013).
[Crossref]

F. G. Mitri, “High-order Bessel nonvortex beam of fractional type α,” Phys. Rev. A 85(2), 025801 (2012).
[Crossref]

F. G. Mitri, “Vector wave analysis of an electromagnetic high-order Bessel vortex beam of fractional type α,” Opt. Lett. 36(5), 606–608 (2011).
[Crossref] [PubMed]

Neifeld, M. A.

O’Dwyer, D. P.

O’Holleran, K.

Ou, J.

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Padgett, M. J.

Phelan, C. F.

Plick, W. N.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Preece, D.

Rakovich, Y. P.

Ramelow, S.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Rudd, D.

Senthilkumaran, P.

S. Vyas, R. K. Singh, and P. Senthilkumaran, “Fractional vortex lens,” Opt. Laser Technol. 42(6), 878–882 (2010).
[Crossref]

Singh, R. K.

S. Vyas, R. K. Singh, and P. Senthilkumaran, “Fractional vortex lens,” Opt. Laser Technol. 42(6), 878–882 (2010).
[Crossref]

Subich, C. J.

C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
[Crossref]

Tang, H.

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Tao, S. H.

Thomas, F. E.

C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
[Crossref]

Vasic, B. V.

Vyas, S.

S. Vyas, R. K. Singh, and P. Senthilkumaran, “Fractional vortex lens,” Opt. Laser Technol. 42(6), 878–882 (2010).
[Crossref]

Wang, S.

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004).
[Crossref]

Yong, C. Y.

C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
[Crossref]

Yuan, X.

Yuan, X. C.

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
[Crossref]

S. H. Tao, W. M. Lee, and X. C. Yuan, “Dynamic optical manipulation with a higher-order fractional bessel beam generated from a spatial light modulator,” Opt. Lett. 28(20), 1867–1869 (2003).
[Crossref] [PubMed]

Zambrini, R.

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum for mulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

Zeilinger, A.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Zhang, J.

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Appl. Opt. (1)

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Eur. Phys. J. D (1)

F. G. Mitri, “High-order Bessel non-vortex beam of fractional type α: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67(7), 135 (2013).
[Crossref]

J. Mod. Opt. (1)

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum for mulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

J. Opt. A: Pure Appl. Opt. (2)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A: Pure Appl. Opt. 10(1), 015009 (2008).
[Crossref]

J. Opt. Soc. Am. A (1)

New J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004).
[Crossref]

Opt. Commun. (2)

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
[Crossref]

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Opt. Express (4)

Opt. Laser Technol. (1)

S. Vyas, R. K. Singh, and P. Senthilkumaran, “Fractional vortex lens,” Opt. Laser Technol. 42(6), 878–882 (2010).
[Crossref]

Opt. Lett. (2)

Phys. Rev. A (2)

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

F. G. Mitri, “High-order Bessel nonvortex beam of fractional type α,” Phys. Rev. A 85(2), 025801 (2012).
[Crossref]

Phys. Rev. D Part. Fields (1)

J. M. Cornwall, “Center vortices, nexuses, and fractional topological charge,” Phys. Rev. D Part. Fields 61(8), 085012 (2000).
[Crossref]

Waves Random Media (1)

C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
[Crossref]

Other (3)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2007).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2th ed. (SPIE, 2005).

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Figures (6)

Fig. 1
Fig. 1 The figure shows beam intensity simulations for fractional FBG beam along the direction z to describe how the beam spreads when the FBG beam transmits in Free Space Optics (FSO). For short distance transmission ( z <200m), the beam is approximate to non-diffracting; for long distance transmission ( z >200m), the diffractive spread of beams become obvious (for example, as γ = 5.1, when z increases from 300m to 1000m, the first ring radius of the fractional FBG beam spreads from 0.13m to 0.48m).
Fig. 2
Fig. 2 The average mode probability P l (r,z) of FBG beam along the direction of the beam radius r for different values of the quantum number deviation Δl .
Fig. 3
Fig. 3 The average probability densities P l (r,z) of OAM modes for fractional FBG beam along the direction of the beam radius r with different values of γ when (a) Δl=0 , signal probability densities; (b) Δl=1 , crosstalk probability densities; (c) γ is half-integer, and Δl=0 signal probability densities; (d) γ is half-integer, and Δl=1 crosstalk probability densities.
Fig. 4
Fig. 4 The average probability densities P l (r,z) of OAM modes for fractional FBG beam along the direction of the beam radius r with different values of beam waist w 0 when (a) Δl=0 , signal probability densities; (b) Δl=1 , crosstalk probability densities.
Fig. 5
Fig. 5 The average probability densities P l (r,z) of OAM modes for fractional FBG beam along the direction of the beam radius r with different values of Rytov variance σ R 2 when (a) Δl=0 , signal probability densities; (b) Δl=1 , crosstalk probability densities.
Fig. 6
Fig. 6 The normalized powers L z (l) of fractional OAM modes for fractional FBG beam along beam waist w 0 with different values of turbulence fluctuation σ R 2 when (a) Δl=0 , signal normalized powers L z ( l 0 ) ; (b) Δl=1 , crosstalk normalized powers L z (l l 0 ) .

Equations (18)

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E( r,φ,z )= E free ( r,φ,z )exp[ ψ x ( r,φ,z )+ ψ y ( r,φ,z ) ],
E free ( r,φ,z )= 1 μ( z ) exp[ i k r 2 z 2kμ( z ) r 2 μ( z ) w 0 2 ] l 0 = C γ J | l 0 | [ k r r μ( z ) ]exp( i l 0 φ ) ,
E( r,φ,z )= l= β l ( r,z ) exp( ilφ ),
β l ( r,z )= 1 2π 0 2π E( r,φ,z ) exp( ilφ )dφ.
P l (r,z)= | β l (r,z) | 2 at = ( 1 2π ) 2 0 2π 0 2π E free ( r,φ,z ) E free ( r , φ ,z ) exp[ il( φ φ ) ] × exp[ ψ x ( r,φ,z )+ ψ y ( r,φ,z )+ ψ x ( r , φ ,z )+ ψ y ( r , φ ,z ) ] at dφd φ ,
exp[ ψ x ( r,φ,z )+ ψ y ( r,φ,z )+ ψ x ( r , φ ,z )+ ψ y ( r , φ ,z ) ] at =exp[ 1 2 D( r,φ, r , φ ;z ) ],
Φ n =0.033 C n 2 κ 11/3 [ f( κ l i )g( κ L o ) G x + G y ],
G x ( κ )=exp[ κ 2 / κ x 2 ], G y ( κ )= κ 11/3 ( κ 2 + κ x 2 ) 11/6 ,
D x ( r, r ,φ, φ ;z )=0.604 C n 2 k 2 z κ xm 1/3 { [ r 2 + r 2 2r r cos( φ φ ) ] × [ 1+0.033 κ xm 2 [ r 2 + r 2 2r r cos( φ φ ) ] ] 1/6 },
D y ( r, r ,φ, φ ;z )=0.604 C n 2 k 2 z κ m 1/3 { [ r 2 + r 2 2r r cos( φ φ ) ] × [ 1+0.033 κ m 2 [ r 2 + r 2 2r r cos( φ φ ) ] ] 1/6 } 0.604 C n 2 k 2 z κ ym 1/3 { [ r 2 + r 2 2r r cos( φ φ ) ] × [ 1+0.033 κ ym 2 [ r 2 + r 2 2r r cos( φ φ ) ] ] 1/6 },
D x ( r, r ,φ, φ ;z )0.604 C n 2 k 2 z κ xm 1/3 [ r 2 + r 2 2r r cos( φ φ ) ],
D y ( r, r ,φ, φ ;z )0.867 σ R 2 k z [ r 2 + r 2 2r r cos( φ φ ) ] 0.604 C n 2 k 2 z κ ym 1/3 [ r 2 + r 2 2r r cos( φ φ ) ].
exp[ 1 2 D( r,φ, r , φ ;z ) ]=exp{ [ 0.604 C n 2 k 2 z κ xm 1/3 +( 0.867 σ R 2 k z 0.604 C n 2 k 2 z κ ym 1/3 ) ] ×[ r 2 + r 2 2r r cos( φ φ ) ] } =exp[ r 2 + r 2 2r r cos( φ φ ) ρ 0 2 ],
ρ 0 = { 0.491 σ R 2 k z { ( 35.046 η x z k η x l i 2 +35.046z ) 1/6 +[ 1.766 ( 35.046 η y z k η y l i 2 +35.046z ) 1/6 ] } } 1/2 .
0 2π exp[in φ 1 +ηcos( φ 1 φ 2 )] d φ 1 =2πexp(in φ 2 ) I n (η),
P l (r,z)= w 0 2 w 2 ( z ) l 0 = C γ 2 | J | l 0 | ( k r r μ( z ) ) | 2 exp[ ( 1 w 2 ( z ) + 1 ρ 0 2 )2 r 2 ] I l l 0 ( 2 r 2 / ρ 0 2 ),
L z (l)= l= lB P l l= B P l .
B P l = w 0 2 w 2 ( z ) l 0 = C γ 2 0 | J | l 0 | ( k r r μ( z ) ) | 2 exp[ ( 1 w 2 ( z ) 1 ρ 0 2 )2 r 2 ] I l l 0 ( 2 r 2 / ρ 0 2 )rdr ,

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