Abstract

We investigate the atmospheric turbulence effects on the propagation of vortex modes carried by Lommel beam. The analytic expression of the received signal and crosstalk mode intensity is derived based on the weak-to-strong non-Kolmogorov turbulence theory. The simulation results show that turbulence with small non-Kolmogorov spectrum parameter, small inner-scale factor and large outer-scale factor is more likely to induce modal crosstalk. With the increment of turbulence strength, the crosstalk spreads from adjacent modes to peripheral modes. The received signal intensity can be improved by use of Lommel beam with small asymmetry parameter, low orbital angular momentum quantum number and long wavelength. The results are helpful to the design of orbital angular momentum based free-space optical communication link.

© 2017 Optical Society of America

1. Introduction

Free-space optical communications based on various vortex beams have attracted a lot of attention in recent years because of their high channel capacity [1,2]. In addition to amplitude, phase, frequency and polarization, the mode of vortex beam carries orbital angular momentum (OAM), which provides a new degree of freedom for information encoding [3]. Since the vortex modes with different OAM quantum numbers are orthogonal, vortex beams are widely used in mode division multiplexing. However, the beam’s wavefront distortions caused by atmospheric turbulence always induce the modal crosstalk and increase the bit error rate [4]. The influence of turbulence effects on the propagation of vortex mode has been analyzed in some pervious reports, mainly focused on the classical Laguerre-Gaussian (LG) beam [5,6]. Further studies reveal that some nondiffracting vortex beams, such as Bessel-Gaussian beam [7], Hankel-Bessel beam [8], Airy beam [9], may be good alternatives to LG beam. The transverse intensity profiles of nondiffracting beams are structurally preserved upon propagation and able to reconstruct after encountering obstacles, which help to mitigate the adverse effects of turbulence [10].

As a new type of nondiffracting beam, Lommel beam is essentially an infinite linear superposition of Bessel modes whose wave vectors have identical axial projections [11]. In contrast with the radial symmetry of Bessel modes, the intensity profile of Lommel beam is symmetric about the Cartesian coordinate axes. The symmetry can be adjusted conveniently by only one parameter. Since proposed by Kovalev and Kotlyar, Lommel beam attracted the academic concern rapidly. Belafhal et al. analyzed its scattering properties by a rigid and isolated sphere for applications such as optical trapping [12]. Ez-zariy et al. simulated the axial intensity of a Lommel modulated Gaussian beam propagating in turbulence [13]. Zhao et al. made the first experimental realization of Lommel beam by use of binary amplitude masks [14]. However, the OAM propagation properties of Lommel beam in turbulence haven’t been investigated yet. Considering its specialty of continuous changeable OAM [11], Lommel beam is more likely to realize the essential mode division multiplexing, and is worthy of research in further details.

In this paper, we discuss the turbulence effects on the propagation of vortex modes for Lommel beam. The effective power spectrum model of non-Kolmogorov atmospheric turbulence in weak-to-strong region is established by use of the spatial filter. The analytic expressions are derived to analyze the received intensity of vortex modes including signal and crosstalk. Finally, numerical simulations are used to demonstrate the relationships between the intensity of vortex modes and the parameters of beam or turbulence, so as to obtain the optimal parameters of free-space OAM communication using Lommel beam.

2. Theoretical model

In the cylindrical coordinate system, the electric field of Lommel beam is expressed as [11]

E0(r,φ,z)=cn0exp(izk2kρ2)un0[ckρrexp(iφ),kρr],
wherezis the propagation distance, r and φare the radial and angular coordinate respectively in the z plane, kρ is the transverse component of the beam’s wavenumber k (k=2π/λ), the variables c, λ and n0 are the beam’s asymmetry parameter, wavelength and OAM quantum number respectively. To ensure the convergence of Lommel function un0[ckρrexp(iφ),kρr], the modulus of c should be less than unity. If expanding un0in the paraxial case (kρk), Eq. (1) can be rewritten as

E0(r,φ,z)=exp(ikz)p=0(c2)pexp[i(n0+2p)φ]Jn0+2p(kρr).

Figure 1 gives the transverse intensity distribution of Lommel beam at the source plane (z=0). The beam parameters are set as n0=1, kρ=0.001k, λ=1550nmand the coordinates are limited in the range 0.01mx,y0.01m. Obviously, the asymmetry parameter c determines the shape of intensity profile in Fig. 1, and can be used for beam control. With the increment of the modulus of c, the circular symmetry gradually degenerates into two crescents with axial symmetry. The direction of symmetric axis is decided by the argument of c. In particular, as for pure real and imaginary number of c, the intensity profile of Lommel beam is symmetric about x axis and y axis respectively.

 

Fig. 1 The transverse intensity pattern of Lommel beam with different asymmetry parameters at the source plane.

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When Lommel beam propagates in atmospheric turbulence, the refractive index fluctuation of atmosphere contributes an additional complex phase perturbation factor ψ(r,φ,z) to E0, which results in the deviation of vortex modes from the original OAM eigenstates. In order to obtain the weight of new vortex mode component, the disturbed electric field E is decomposed into a series of spiral harmonics exp(inφ) with corresponding coefficient βn as follows [15]:

E(r,φ,z)=E0(r,φ,z)exp[ψ(r,φ,z)]=12πn=βnexp(inφ),
where βnis given by the integral
βn=12π02πE(r,φ,z)exp(inφ)dφ.
The ensemble average of |βn|2 represents the probability density of vortex mode in atmospheric turbulence, which has the form of
|βn|2=12π02π02πE0(r,φ,z)E0*(r,φ,z)exp[ψ(r,φ,z)+ψ*(r,φ,z)]exp[in(φφ)]dφdφ.
According to the definition of complex phase structure function [16], we can get
exp[ψ(r,φ,z)+ψ*(r,φ',z)]=exp[13π2k2z[2r22r2cos(φφ')]0κ3Φneff(κ)dκ],=exp[2r22r2cos(φφ)ρ02]
where ρ0 is the spatial coherence radius, κand Φneffare the spatial frequency and the effective power spectrum of refractive index fluctuations respectively. Substituting Eq. (2) and Eq. (6) into Eq. (5), and simplifying the expression by some integral calculations, the final result of mode probability density can be written as
|βn|2=12πp=0p=0(1)p+p02π02πc2p+2pexp[i2(p-p)φ]Jn0+2p(kρr)Jn0+2p(kρr)×exp[2r2ρ02]exp[2r2cos(φφ)ρ02i(nn02p)(φφ)]dφdφ,=2πp=0c4p[Jn0+2p(kρr)]2exp[2r2ρ02]Inn0-2p(2r2ρ02)
where Jn0+2p and Inn02p denote Bessel function of the first kind and the modified Bessel function of the first kind respectively. The difference between n and n0 determines whether the received vortex mode is a signal mode or a crosstalk mode. If we define Δn=nn0, Eq. (7) is transformed into
|βΔn|2=2πp=0c4p[Jn0+2p(kρr)]2exp[2r2ρ02]IΔn2p(2r2ρ02).
The probability density of signal mode (Δn=0) and crosstalk mode (Δn0) can then be calculated using Eq. (8). As for a finite-aperture receiver with diameter D, the relative received intensity PΔn of vortex mode is then given by
PΔn=0D/2|βΔn|2rdrΔn=0D/2|βΔn|2rdr.
According to Eq. (8) and Eq. (9), it is noticeable that ρ0 is a major factor determining the propagation characteristics of vortex mode in atmospheric turbulence. To obtain the value ofρ0, the key problem lies in constructing the turbulence model to describe Φneff in Eq. (6).

Here the extended Rytov approximation is used to construct the non-Kolmogorov turbulence model in weak-to-strong turbulence region, with consideration of both its inner-scale and outer-scale effects. By introducing the spatial filter functions, Φneff can be written as

Φneff(κ)=A(α)Cn2κα[f(κ,α,l0)g(κ,L0)GX(κ,α)+GY(κ,α)]A(α)=Γ(α1)4π2cos(π2α),
where α is the non-Kolmogorov spectrum parameter with value in the region from 3 to 4, Cn2 is the refractive-index structure constant with unit m3α, Γ(α1) denotes the Gamma function. The termsf(κ,α,l0) and g(κ,L0) describe the inner-scale and outer-scale effect of turbulence respectively with corresponding scale factor l0 and L0, while GX(κ,α) and GY(κ,α) represent the large-scale and small-scale spatial filter function respectively. Their expressions are given by [17]
f(κ,α,l0)=exp(κ2κl2),g(κ,L0)=1exp(κ2κL2).Gx(κ,α)=exp[κ2κx2],Gy(κ,α)=κα[κ2+κy2]α/2
In Eq. (11), κl=c(α)/l0 and κL=8π/L0 are the spatial frequency corresponding to the inner-scale and outer-scale of turbulence eddies respectively, κx=kηx/z and κy=kηy/z are the spatial cutoff frequency for large-scale and small-scale filter function respectively. The parameters ηx, ηy and c(α) can be further expressed as [18]
c(α)=[2π3A(α)Γ(5α2)]1/(α5),β(α)=4Γ(1α2)sinπα4Γ2(α/2)Γ(α),ηx=11+fx(α)σR4/(α2)[7.35β(α)Γ(3α/2)]2/(6α),fx(α)=[1.02r(α)I(α)]2/(α6),ηy=[0.06375(α2)β(α)]2/(2α)[1+fy(α)σR4/(α2)],fy(α)=(ln20.51)2/(2α),σR2=β(α)A(α)Cn2π2k3α/2zα/2,I(α)=(α1)6αα2Γ2(α3)Γ(2α6),r(α)=1α22(3α)(α10)α2[Γ(1α/2)Γ(α/2)]α6α2Γ(6αα2)[β(α)]82αα2.
Substituting Eq. (11) and Eq. (12) into Eq. (10), the final expression of Φneff is written as
Φneff(κ)=A(α)Cn2κα{exp(κ2κlx2)exp(κ2κLlx2)+κα[κ2+κy2]α/2},1κlx2=1κl2+1κx2,1κLlx2=1κL2+1κl2+1κx2.
Thus according to Eq. (6) and Eq. (13), ρ0 has the form of
ρ0=[13π2k2z0κ3Φneff(κ)dκ]12={Γ(α1)Cn2k2z48cos(πα2)[2Γ(2α2)(κlx4ακLlx4α)+κl4κyαF21(α2,2;3,κl2κy2)]}12.
where F21 denotes the confluent hypergeometric function. Then the received vortex mode’s intensity PΔn of Lommel beam can be calculated by combining Eq. (8), Eq. (9) and Eq. (14), so as to analyze the turbulence effects on the propagation of vortex mode quantitatively.

3. Simulation and analysis

In this section, numerical simulations are used to demonstrate the vortex mode intensity of signal and crosstalk under different beam parameters and turbulence conditions. Unless otherwise mentioned, the parameters used in simulations are set as n0=1, c=0.1, α=3.67, λ=1550nm, l0=1mm, L0=1m, D=0.05mand z=1km.The intensity distributions of the received vortex modes carried by Lommel beam in weak-to-strong turbulence are shown in Fig. 2(a) to Fig. 2(c). Obviously the turbulence causes the energy migration from the initial OAM eigenstate to others. When the turbulence is weak, the crosstalk occurs mainly between adjacent modes (i.e. the difference of OAM quantum number Δn=±1) with negligible intensity. While in strong turbulence, the crosstalk intensity increases greatly and spreads to more peripheral modes. As a comparison, the received mode intensities of Laguerre-Gaussian (LG) beam under the same turbulence conditions are shown in Fig. 2(d) to Fig. 2(f). The waist radius, radial index and azimuthal index (i.e. OAM quantum number) of LG beam are set as 0.01 m, 0 and 1 respectively. No matter in weak or strong turbulence, the received signal mode intensity of LG beam is always lower than that of Lommel beam. Moreover, the total energy of LG beam tends to distribute equally among the signal mode and each crosstalk mode in strong turbulence, which is more likely to cause the severe performance degeneration or even failure of OAM multiplexing communication. So from the view of anti-turbulence interference, Lommel beam is superior to the traditional LG beam.

 

Fig. 2 The received mode intensity distribution of Lommel beam and Laguerre-Gaussian beam in (a, d) weak, (b, e) moderate and (c, f) strong turbulence. (a)-(c): Lommel beam; (d)-(f): Laguerre-Gaussian beam.

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Figure 3 displays the received vortex mode intensity of Lommel beam under different propagation distances and turbulent strengths. With the increment of propagation distance z, the signal mode (Δn=0) intensity Ps keeps decreasing due to the accumulation of turbulence effect. Since the variation trend of the total crosstalk intensity is just contrary to Ps without the need for further study, we discuss the chief crosstalk mode (Δn=1) intensity Pc instead in the following text. In weak turbulence (Cn2=1015m3α) and moderate turbulence (Cn2=1014m3α),Pc is monotone increasing. As for strong turbulence (Cn2=1013m3α), Pc gradually decreases after reaching its maximum, because the enhancement of high-order crosstalk (|Δn|>1) reduces the weight of Pc. Since nondiffracting property is the key to mitigating the turbulence effects and maintaining high signal intensity, a long nondiffracting region is important for long distance communication. Aruga et al. used a telescope to form a distorted concave spherical wave front, and thus obtained a nondiffracting region up to a few kilometers for Bessel beam [19]. Birch et al. proposed immersing axicon in an index-matching material to reduce the radial wave vector of Bessel beam to low values, so the nondiffracting region could reach to tens of kilometers [20]. Given that Lommel beam is essentially a linear superposition of Bessel modes and their structural similarity, using the above methods to generate long-range nondiffracting Lommel beam can also be expected.

 

Fig. 3 The received (a) signal intensity Ps and (b) crosstalk intensity Pc under different propagation distances and turbulence strengths.

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To optimize the parameters of Lommel beam propagating in atmospheric turbulence, the relationships between the received mode intensity and the modulus of asymmetry parameter c under different OAM quantum numbers n0 are shown in Fig. 4. No matter in weak or strong turbulence, Ps keeps decreasing with the increment of |c|, while Pc first increases slightly and then decreases rapidly. This is because for a large value of |c|, the high-order modal crosstalk (|Δn|>1) becomes obvious which in turn decreases the weight of low-order crosstalk Pc. The variation of n0 has little influence on Ps and Pc in weak turbulence. However, in conditions of strong turbulence, a small n0 can obtain a slightly higher Ps and lowerPc. So the values of |c| and n0 should be selected small in Lommel beam based communication to mitigate the turbulence induced crosstalk and obtain high signal intensity.

 

Fig. 4 In (a, b) weak and (c, d) strong turbulence, the received signal intensityPsand crosstalk intensityPcversus the OAM quantum number and the modulus of beam asymmetry parameter.

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Figure 5 shows the influence of beam wavelength λ on the signal intensity Ps and crosstalk intensity Pc . Five typical wavelengths are selected and the range of Cn2 covers the weak-to-strong turbulence. Ps decreases with the increment of turbulence strength and the signal mode with a longer wavelength suffers less intensity loss. In conditions of long wavelength, Pc has a contrary variation trend to Ps because the low-order crosstalk (|Δn|=1) dominates the whole crosstalk. However, as for short wavelength, the significant enhancement of other crosstalk modes (|Δn|>1) finally cause the decrease of Pc in strong turbulence. So the long-wavelength Lommel beam is more suitable for OAM communication if considering its less susceptibility to turbulence.

 

Fig. 5 The received (a) signal intensity Ps and (b) crosstalk intensity Pc under different beam wavelengths and turbulence strengths.

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To obtain the propagation performance of Lommel beam under different turbulent conditions, the impact of non-Kolmogorov spectrum parameter α on Ps and Pc are investigated. Figure 6 shows that a larger α corresponds to a higher Ps, which can be attributed to the beam scintillation effect. In general, the turbulence eddies with large wavenumber induce relatively strong scintillation. Since those eddies are fewer in turbulence when α moves towards 4, less scintillation and higher Ps are achieved [21]. With the increment of Cn2, the curve’s inflection point of Pc occurs earlier in the turbulence with small α. This is unfavorable to channel multiplexing, meaning that the crosstalk is easier to spread from adjacent modes to other peripheral modes. Given that α is relevant to the atmospheric layer altitude [22], an appropriate altitude selected for communication may help to reduce the crosstalk.

 

Fig. 6 The received (a) signal intensityPsand (b) crosstalk intensityPc under different values of non-Kolmogorov spectrum parameter and turbulence strengths.

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Fig. 7 shows the relationships between Ps,Pc and the turbulence scale l0, L0. No matter in weak (Cn2=1015m3α) or strong (Cn2=1013m3α) turbulence, Ps decreases if the inner-scale factor l0 decreases or the outer-scale factor L0 increases, partly owing to the increment of the effective turbulent eddies’ number within the scale range [l0, L0]. On the other hand, the relatively large beam wander caused by a large L0 induces the beam pointing error, which further decreasesPs [23]. The variation trends of Pc with l0 and L0 are contrary to those of Ps in weak turbulence. However, this rule is not satisfied in strong turbulence with small l0 or large L0. The whole crosstalk energy, originally concentrated on OAM modes with |Δn|=1, starts improving injection into other modes (|Δn|>1). So Pc begins to decrease when l0 is smaller or L0 is larger than some certain values. Overall, compared with L0, the effects of l0 on Ps and Pc are more obvious.

 

Fig. 7 In (a, b) weak and (c, d) strong turbulence, the received signal intensityPsand crosstalk intensityPcversus the turbulence’s inner-scale factor and outer-scale factor.

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4. Conclusion

We develop a theoretical model to investigate the vortex mode intensity of Lommel beam propagating in atmospheric turbulence. Based on the extended Rytov approximation and spatial filter functions, the model is effective for non-Kolmogorov turbulence in weak-to-strong region. With the increment of propagation distance and turbulence strength, the received signal mode intensity Ps decreases and the OAM crosstalk spreads from adjacent modes to peripheral modes. A smaller beam asymmetry parameter, a lower OAM quantum number and a longer wavelength are helpful to improve Ps. The influences of turbulence parameters on Ps mainly include two aspects. The decrease of non-Kolmogorov spectrum parameter and inner-scale factor, or the increment of outer-scale factor, can cause the reduction of Ps. The effect of inner-scale is more obvious than outer-scale. The results provide useful reference for the optimal design of OAM communication in both weak and strong atmospheric turbulence.

Funding

The Fundamental Research Funds for the Central Universities of China (JUSRP11721, JUSRP51721B, JUSRP51716A).

References and links

1. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015). [CrossRef]  

2. S. C. Mi, T. J. Wang, G. S. Jin, and C. Wang, “High-capacity quantum secure direct communication with orbital angular momentum of photons,” IEEE Photonics J. 7(5), 7600108 (2015). [CrossRef]  

3. A. Trichili, A. B. Salem, A. Dudley, M. Zghal, and A. Forbes, “Encoding information using Laguerre Gaussian modes over free space turbulence media,” Opt. Lett. 41(13), 3086–3089 (2016). [CrossRef]   [PubMed]  

4. 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]  

5. V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre-Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15(4), 044007 (2013). [CrossRef]  

6. S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express 20(1), 452–461 (2012). [CrossRef]   [PubMed]  

7. J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel-Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014). [CrossRef]  

8. Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014). [CrossRef]   [PubMed]  

9. P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22(7), 7598–7606 (2014). [CrossRef]   [PubMed]  

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

11. A. A. Kovalev and V. V. Kotlyar, “Lommel modes,” Opt. Commun. 338, 117–122 (2015). [CrossRef]  

12. A. Belafhal, L. Ez-zariy, and Z. Hricha, “A study of nondiffracting Lommel beams propagating in a medium containing spherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 184, 1–7 (2016). [CrossRef]  

13. L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016). [CrossRef]  

14. Q. Zhao, L. Gong, and Y. M. Li, “Shaping diffraction-free Lommel beams with digital binary amplitude masks,” Appl. Opt. 54(25), 7553–7558 (2015). [CrossRef]   [PubMed]  

15. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef]   [PubMed]  

16. 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]  

17. L. C. Andrews, R. L. Philips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).

18. L. Cui, B. Xue, S. Zheng, W. Xue, X. Bai, X. Cao, and F. Zhou, “Atmospheric spectral model and theoretical expressions of irradiance scintillation index for optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 29(6), 1091–1098 (2012). [CrossRef]   [PubMed]  

19. T. Aruga, S. W. Li, S. Yoshikado, M. Takabe, and R. Li, “Nondiffracting narrow light beam with small atmospheric turbulence-influenced propagation,” Appl. Opt. 38(15), 3152–3156 (1999). [CrossRef]   [PubMed]  

20. P. Birch, I. Ituen, R. Young, and C. Chatwin, “Long-distance Bessel beam propagation through Kolmogorov turbulence,” J. Opt. Soc. Am. A 32(11), 2066–2073 (2015). [CrossRef]   [PubMed]  

21. P. Deng, X. H. Yuan, and D. X. Huang, “Scintillation of a laser beam propagation through non- Kolmogorov strong turbulence,” Opt. Commun. 285(6), 880–887 (2012). [CrossRef]  

22. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008). [CrossRef]   [PubMed]  

23. Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam wander of partially coherent array beams through non-Kolmogorov turbulence,” Opt. Lett. 40(8), 1619–1622 (2015). [CrossRef]   [PubMed]  

References

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  1. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
    [Crossref]
  2. S. C. Mi, T. J. Wang, G. S. Jin, and C. Wang, “High-capacity quantum secure direct communication with orbital angular momentum of photons,” IEEE Photonics J. 7(5), 7600108 (2015).
    [Crossref]
  3. A. Trichili, A. B. Salem, A. Dudley, M. Zghal, and A. Forbes, “Encoding information using Laguerre Gaussian modes over free space turbulence media,” Opt. Lett. 41(13), 3086–3089 (2016).
    [Crossref] [PubMed]
  4. 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]
  5. V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre-Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15(4), 044007 (2013).
    [Crossref]
  6. S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express 20(1), 452–461 (2012).
    [Crossref] [PubMed]
  7. J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel-Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
    [Crossref]
  8. Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014).
    [Crossref] [PubMed]
  9. P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22(7), 7598–7606 (2014).
    [Crossref] [PubMed]
  10. Z. Y. Qin, R. M. Tao, P. Zhou, X. J. Xu, and Z. J. Liu, “Propagation of partially coherent Bessel-Gaussian beams carrying optical vortices in non-Kolmogorov turbulence,” Opt. Laser Technol. 56, 182–188 (2014).
    [Crossref]
  11. A. A. Kovalev and V. V. Kotlyar, “Lommel modes,” Opt. Commun. 338, 117–122 (2015).
    [Crossref]
  12. A. Belafhal, L. Ez-zariy, and Z. Hricha, “A study of nondiffracting Lommel beams propagating in a medium containing spherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 184, 1–7 (2016).
    [Crossref]
  13. L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016).
    [Crossref]
  14. Q. Zhao, L. Gong, and Y. M. Li, “Shaping diffraction-free Lommel beams with digital binary amplitude masks,” Appl. Opt. 54(25), 7553–7558 (2015).
    [Crossref] [PubMed]
  15. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008).
    [Crossref] [PubMed]
  16. 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]
  17. L. C. Andrews, R. L. Philips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
  18. L. Cui, B. Xue, S. Zheng, W. Xue, X. Bai, X. Cao, and F. Zhou, “Atmospheric spectral model and theoretical expressions of irradiance scintillation index for optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 29(6), 1091–1098 (2012).
    [Crossref] [PubMed]
  19. T. Aruga, S. W. Li, S. Yoshikado, M. Takabe, and R. Li, “Nondiffracting narrow light beam with small atmospheric turbulence-influenced propagation,” Appl. Opt. 38(15), 3152–3156 (1999).
    [Crossref] [PubMed]
  20. P. Birch, I. Ituen, R. Young, and C. Chatwin, “Long-distance Bessel beam propagation through Kolmogorov turbulence,” J. Opt. Soc. Am. A 32(11), 2066–2073 (2015).
    [Crossref] [PubMed]
  21. P. Deng, X. H. Yuan, and D. X. Huang, “Scintillation of a laser beam propagation through non- Kolmogorov strong turbulence,” Opt. Commun. 285(6), 880–887 (2012).
    [Crossref]
  22. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
    [Crossref] [PubMed]
  23. Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam wander of partially coherent array beams through non-Kolmogorov turbulence,” Opt. Lett. 40(8), 1619–1622 (2015).
    [Crossref] [PubMed]

2016 (3)

A. Trichili, A. B. Salem, A. Dudley, M. Zghal, and A. Forbes, “Encoding information using Laguerre Gaussian modes over free space turbulence media,” Opt. Lett. 41(13), 3086–3089 (2016).
[Crossref] [PubMed]

A. Belafhal, L. Ez-zariy, and Z. Hricha, “A study of nondiffracting Lommel beams propagating in a medium containing spherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 184, 1–7 (2016).
[Crossref]

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016).
[Crossref]

2015 (6)

Q. Zhao, L. Gong, and Y. M. Li, “Shaping diffraction-free Lommel beams with digital binary amplitude masks,” Appl. Opt. 54(25), 7553–7558 (2015).
[Crossref] [PubMed]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

S. C. Mi, T. J. Wang, G. S. Jin, and C. Wang, “High-capacity quantum secure direct communication with orbital angular momentum of photons,” IEEE Photonics J. 7(5), 7600108 (2015).
[Crossref]

P. Birch, I. Ituen, R. Young, and C. Chatwin, “Long-distance Bessel beam propagation through Kolmogorov turbulence,” J. Opt. Soc. Am. A 32(11), 2066–2073 (2015).
[Crossref] [PubMed]

A. A. Kovalev and V. V. Kotlyar, “Lommel modes,” Opt. Commun. 338, 117–122 (2015).
[Crossref]

Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam wander of partially coherent array beams through non-Kolmogorov turbulence,” Opt. Lett. 40(8), 1619–1622 (2015).
[Crossref] [PubMed]

2014 (4)

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel-Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014).
[Crossref] [PubMed]

P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22(7), 7598–7606 (2014).
[Crossref] [PubMed]

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

2013 (1)

V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre-Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15(4), 044007 (2013).
[Crossref]

2012 (3)

2008 (3)

2004 (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]

1999 (1)

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Aksenov, V. P.

V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre-Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15(4), 044007 (2013).
[Crossref]

Alfano, R. R.

Anguita, J. A.

Aruga, T.

Ashrafi, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ashrafi, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Bai, X.

Bao, C.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Belafhal, A.

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016).
[Crossref]

A. Belafhal, L. Ez-zariy, and Z. Hricha, “A study of nondiffracting Lommel beams propagating in a medium containing spherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 184, 1–7 (2016).
[Crossref]

Birch, P.

Boufalah, F.

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016).
[Crossref]

Cao, X.

Cao, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Chatwin, C.

Cui, L.

Dalil-Essakali, L.

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016).
[Crossref]

Deng, P.

P. Deng, X. H. Yuan, and D. X. Huang, “Scintillation of a laser beam propagation through non- Kolmogorov strong turbulence,” Opt. Commun. 285(6), 880–887 (2012).
[Crossref]

Ding, J.

Dudley, A.

Ez-zariy, L.

A. Belafhal, L. Ez-zariy, and Z. Hricha, “A study of nondiffracting Lommel beams propagating in a medium containing spherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 184, 1–7 (2016).
[Crossref]

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016).
[Crossref]

Forbes, A.

Gan, X.

Gao, J.

Gao, Z.

Golbraikh, E.

Gong, L.

Gong, L. Y.

He, Y. T.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel-Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Hricha, Z.

A. Belafhal, L. Ez-zariy, and Z. Hricha, “A study of nondiffracting Lommel beams propagating in a medium containing spherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 184, 1–7 (2016).
[Crossref]

Huang, D. X.

P. Deng, X. H. Yuan, and D. X. Huang, “Scintillation of a laser beam propagation through non- Kolmogorov strong turbulence,” Opt. Commun. 285(6), 880–887 (2012).
[Crossref]

Huang, H.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Huang, Y.

Ituen, I.

Jiang, Y. S.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel-Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Jin, G. S.

S. C. Mi, T. J. Wang, G. S. Jin, and C. Wang, “High-capacity quantum secure direct communication with orbital angular momentum of photons,” IEEE Photonics J. 7(5), 7600108 (2015).
[Crossref]

Kolosov, V. V.

V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre-Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15(4), 044007 (2013).
[Crossref]

Kopeika, N. S.

Kotlyar, V. V.

A. A. Kovalev and V. V. Kotlyar, “Lommel modes,” Opt. Commun. 338, 117–122 (2015).
[Crossref]

Kovalev, A. A.

A. A. Kovalev and V. V. Kotlyar, “Lommel modes,” Opt. Commun. 338, 117–122 (2015).
[Crossref]

Lavery, M. P. J.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Leach, J.

Li, L.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Li, P.

Li, R.

Li, S. W.

Li, Y. M.

Liao, J. L.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel-Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Liu, S.

Liu, X.

Liu, Z. J.

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

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]

Mi, S. C.

S. C. Mi, T. J. Wang, G. S. Jin, and C. Wang, “High-capacity quantum secure direct communication with orbital angular momentum of photons,” IEEE Photonics J. 7(5), 7600108 (2015).
[Crossref]

Molisch, A. F.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Neifeld, M. A.

Ou, J.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel-Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Peng, T.

Pogutsa, C. E.

V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre-Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15(4), 044007 (2013).
[Crossref]

Qin, Z. Y.

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

Ramachandran, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ren, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Salem, A. B.

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]

Sztul, H. I.

Takabe, M.

Tang, H.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel-Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Tao, R. M.

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

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]

Trichili, A.

Tur, M.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Vasic, B. V.

Wang, C.

S. C. Mi, T. J. Wang, G. S. Jin, and C. Wang, “High-capacity quantum secure direct communication with orbital angular momentum of photons,” IEEE Photonics J. 7(5), 7600108 (2015).
[Crossref]

Wang, J.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Wang, S. H.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel-Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Wang, T. J.

S. C. Mi, T. J. Wang, G. S. Jin, and C. Wang, “High-capacity quantum secure direct communication with orbital angular momentum of photons,” IEEE Photonics J. 7(5), 7600108 (2015).
[Crossref]

Willner, A. E.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Xie, G.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22(7), 7598–7606 (2014).
[Crossref] [PubMed]

Xu, X. J.

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

Xue, B.

Xue, W.

Yan, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[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]

Yoshikado, S.

Young, R.

Yuan, X. H.

P. Deng, X. H. Yuan, and D. X. Huang, “Scintillation of a laser beam propagation through non- Kolmogorov strong turbulence,” Opt. Commun. 285(6), 880–887 (2012).
[Crossref]

Zeng, A.

Zghal, M.

Zhang, B.

Zhang, J. H.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel-Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Zhang, Y.

Zhao, F.

Zhao, J.

Zhao, Q.

Zhao, S. M.

Zhao, Z.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Zheng, B. Y.

Zheng, S.

Zhou, F.

Zhou, P.

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

Zhu, Y.

Zilberman, A.

Adv. Opt. Photonics (1)

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Appl. Opt. (4)

IEEE Photonics J. (1)

S. C. Mi, T. J. Wang, G. S. Jin, and C. Wang, “High-capacity quantum secure direct communication with orbital angular momentum of photons,” IEEE Photonics J. 7(5), 7600108 (2015).
[Crossref]

J. Opt. (1)

V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre-Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15(4), 044007 (2013).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transf. (1)

A. Belafhal, L. Ez-zariy, and Z. Hricha, “A study of nondiffracting Lommel beams propagating in a medium containing spherical scatterers,” J. Quant. Spectrosc. Radiat. Transf. 184, 1–7 (2016).
[Crossref]

Opt. Commun. (3)

A. A. Kovalev and V. V. Kotlyar, “Lommel modes,” Opt. Commun. 338, 117–122 (2015).
[Crossref]

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel-Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

P. Deng, X. H. Yuan, and D. X. Huang, “Scintillation of a laser beam propagation through non- Kolmogorov strong turbulence,” Opt. Commun. 285(6), 880–887 (2012).
[Crossref]

Opt. Express (4)

Opt. Laser Technol. (1)

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

Opt. Lett. (2)

Optik (Stuttg.) (1)

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016).
[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 (1)

L. C. Andrews, R. L. Philips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).

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Figures (7)

Fig. 1
Fig. 1 The transverse intensity pattern of Lommel beam with different asymmetry parameters at the source plane.
Fig. 2
Fig. 2 The received mode intensity distribution of Lommel beam and Laguerre-Gaussian beam in (a, d) weak, (b, e) moderate and (c, f) strong turbulence. (a)-(c): Lommel beam; (d)-(f): Laguerre-Gaussian beam.
Fig. 3
Fig. 3 The received (a) signal intensity P s and (b) crosstalk intensity P c under different propagation distances and turbulence strengths.
Fig. 4
Fig. 4 In (a, b) weak and (c, d) strong turbulence, the received signal intensity P s and crosstalk intensity P c versus the OAM quantum number and the modulus of beam asymmetry parameter.
Fig. 5
Fig. 5 The received (a) signal intensity P s and (b) crosstalk intensity P c under different beam wavelengths and turbulence strengths.
Fig. 6
Fig. 6 The received (a) signal intensity P s and (b) crosstalk intensity P c under different values of non-Kolmogorov spectrum parameter and turbulence strengths.
Fig. 7
Fig. 7 In (a, b) weak and (c, d) strong turbulence, the received signal intensity P s and crosstalk intensity P c versus the turbulence’s inner-scale factor and outer-scale factor.

Equations (14)

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E 0 ( r , φ , z ) = c n 0 exp ( i z k 2 k ρ 2 ) u n 0 [ c k ρ r exp ( i φ ) , k ρ r ] ,
E 0 ( r , φ , z ) = exp ( i k z ) p = 0 ( c 2 ) p exp [ i ( n 0 + 2 p ) φ ] J n 0 + 2 p ( k ρ r ) .
E ( r , φ , z ) = E 0 ( r , φ , z ) exp [ ψ ( r , φ , z ) ] = 1 2 π n = β n exp ( i n φ ) ,
β n = 1 2 π 0 2 π E ( r , φ , z ) exp ( i n φ ) d φ .
| β n | 2 = 1 2 π 0 2 π 0 2 π E 0 ( r , φ , z ) E 0 * ( r , φ , z ) exp [ ψ ( r , φ , z ) + ψ * ( r , φ , z ) ] exp [ i n ( φ φ ) ] d φ d φ .
exp [ ψ ( r , φ , z ) + ψ * ( r , φ ' , z ) ] = exp [ 1 3 π 2 k 2 z [ 2 r 2 2 r 2 cos ( φ φ ' ) ] 0 κ 3 Φ n e f f ( κ ) d κ ] , = exp [ 2 r 2 2 r 2 cos ( φ φ ) ρ 0 2 ]
| β n | 2 = 1 2 π p = 0 p = 0 ( 1 ) p + p 0 2 π 0 2 π c 2 p + 2 p exp [ i 2 ( p - p ) φ ] J n 0 + 2 p ( k ρ r ) J n 0 + 2 p ( k ρ r ) × exp [ 2 r 2 ρ 0 2 ] exp [ 2 r 2 cos ( φ φ ) ρ 0 2 i ( n n 0 2 p ) ( φ φ ) ] d φ d φ , = 2 π p = 0 c 4 p [ J n 0 + 2 p ( k ρ r ) ] 2 exp [ 2 r 2 ρ 0 2 ] I n n 0 - 2 p ( 2 r 2 ρ 0 2 )
| β Δ n | 2 = 2 π p = 0 c 4 p [ J n 0 + 2 p ( k ρ r ) ] 2 exp [ 2 r 2 ρ 0 2 ] I Δ n 2 p ( 2 r 2 ρ 0 2 ) .
P Δ n = 0 D / 2 | β Δ n | 2 r d r Δ n = 0 D / 2 | β Δ n | 2 r d r .
Φ n e f f ( κ ) = A ( α ) C n 2 κ α [ f ( κ , α , l 0 ) g ( κ , L 0 ) G X ( κ , α ) + G Y ( κ , α ) ] A ( α ) = Γ ( α 1 ) 4 π 2 cos ( π 2 α ) ,
f ( κ , α , l 0 ) = exp ( κ 2 κ l 2 ) , g ( κ , L 0 ) = 1 exp ( κ 2 κ L 2 ) . G x ( κ , α ) = e x p [ κ 2 κ x 2 ] , G y ( κ , α ) = κ α [ κ 2 + κ y 2 ] α / 2
c ( α ) = [ 2 π 3 A ( α ) Γ ( 5 α 2 ) ] 1 / ( α 5 ) , β ( α ) = 4 Γ ( 1 α 2 ) sin π α 4 Γ 2 ( α / 2 ) Γ ( α ) , η x = 1 1 + f x ( α ) σ R 4 / ( α 2 ) [ 7.35 β ( α ) Γ ( 3 α / 2 ) ] 2 / ( 6 α ) , f x ( α ) = [ 1.02 r ( α ) I ( α ) ] 2 / ( α 6 ) , η y = [ 0.06375 ( α 2 ) β ( α ) ] 2 / ( 2 α ) [ 1 + f y ( α ) σ R 4 / ( α 2 ) ] , f y ( α ) = ( ln 2 0.51 ) 2 / ( 2 α ) , σ R 2 = β ( α ) A ( α ) C n 2 π 2 k 3 α / 2 z α / 2 , I ( α ) = ( α 1 ) 6 α α 2 Γ 2 ( α 3 ) Γ ( 2 α 6 ) , r ( α ) = 1 α 2 2 ( 3 α ) ( α 10 ) α 2 [ Γ ( 1 α / 2 ) Γ ( α / 2 ) ] α 6 α 2 Γ ( 6 α α 2 ) [ β ( α ) ] 8 2 α α 2 .
Φ n e f f ( κ ) = A ( α ) C n 2 κ α { exp ( κ 2 κ l x 2 ) exp ( κ 2 κ L l x 2 ) + κ α [ κ 2 + κ y 2 ] α / 2 } , 1 κ l x 2 = 1 κ l 2 + 1 κ x 2 , 1 κ L l x 2 = 1 κ L 2 + 1 κ l 2 + 1 κ x 2 .
ρ 0 = [ 1 3 π 2 k 2 z 0 κ 3 Φ n e f f ( κ ) d κ ] 1 2 = { Γ ( α 1 ) C n 2 k 2 z 48 cos ( π α 2 ) [ 2 Γ ( 2 α 2 ) ( κ l x 4 α κ L l x 4 α ) + κ l 4 κ y α F 2 1 ( α 2 , 2 ; 3 , κ l 2 κ y 2 ) ] } 1 2 .

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