Abstract

We study the effects of non-Kolmogorov turbulence on the orbital angular momentum (OAM) of Hypergeometric-Gaussian (HyGG) beams in a paraxial atmospheric link. The received power and crosstalk power of OAM states of the HyGG beams are established. It is found that the hollowness parameter of the HyGG beams plays an important role in the received power and crosstalk power. The larger values of hollowness parameter give rise to the higher received power and lower crosstalk power. The results also show that the smaller OAM quantum number and longer wavelength of the launch beam may lead to the higher received power and lower crosstalk power.

© 2015 Optical Society of America

1. Introduction

Recently, Interests have been paid to the diffracting-free beams carrying an intrinsic orbital angular moment (OAM) because of their potential applications in optical trapping, image processing and optical communications, etc. As a solution of the paraxial wave equation, a new family laser beam called hypergeometric beam (HyG) was introduced in [1]. The HyG beam carries an OAM like Bessel and Laguerre beam and can be an alternative candidate for communication purposes. Furthermore, the HyG beams possess a number of properties which are similar to Bessel beams [2]. Compared with the Bessel beams, the HyG beams possess larger angular momentum [3]. According to their potential applications, the study of the HyG beams is a newly thriving field. The HyG beams have been generated in experiments by diffractive optical elements [2, 3] or computer-synthesized holograms [4]. The analytical expressions for propagation of the HyG beams in different media, such as a hyperbolic-index medium [5], an uniaxial crystal [6], a parabolic refractive index medium [7], and turbulent atmosphere [8, 9], have been derived. To achieve a practically realizable beam, karimi et al proposed hypergeometric-Gaussian (HyGG) beam and hypergeometric-GaussianII [10, 11]. Modulating the HyG beam with a Gaussian factor is one way of obtaining the HyGG beam to ensure the HyGG beam carrying a finite power. The Gaussian, modified Bessel Gauss, modified Exponential Gauss and modified Laguerre-Gauss beams can be regarded as the special cases of the HyGG beams when the hollowness parameter and the OAM quantum number are changed [10].

The OAM states of light have attracted many attentions because they provide infinite dimensional basis sets of orthonormal states to describe the transverse structure of beams [12] and can be used to quantum communication. For atmospheric optical communication, however, the OAM states may be susceptible to atmospheric turbulence [13]. The crosstalk among the OAM states of single photons arise, and consequently reduce the information capacity of the communication channel in Kolmogorov atmospheric turbulence [13]. In addition, there are some significant deviations of atmospheric turbulence from the Kolmogorov model as revealed by experiments [14, 15]. In the free troposphere and stratosphere, the power spectrum of turbulence exhibits non-Kolmogorov properties and its spectral exponent depends on altitude [1618]. Toselli et al. presented a non-Kolmogorov power spectrum using generalized exponent and amplitude factor [19, 20]. When the exponent value α is equal to 11/3, the non-Kolmogorov spectrum transforms to the conventional Kolmogorov spectrum. Based on this model, the effects of non-Kolmogorov turbulence tilt, coma, astigmatism and defocus aberration on the probability models of the OAM crosstalk for single photons have been analyzed [21]. Furthermore, the non-diffraction characteristics of a beam may reduce the interference of atmospheric turbulence on the orbital angular momentum states. However, to the best of our knowledge, the effects of non-Kolmogorov turbulence on OAM modes of the non-diffracting HyGG beams have not been reported elsewhere.

In this study, we characterize the effects of non-Kolmogorov turbulence on the OAM modes of the HyGG beams. The effects of the hollowness parameter p, OAM quantum number l0, refractive-index structure parameter of turbulence Cn2, wavelength of beam λ and propagation distance z on OAM modes of the HyGG beams are discussed in detail. In the case of the hollowness parameter p=|l0|, we find that the received power pl0 drops with the decrease of p. On the other hand, there are little effects on the received power pl0 when p is positive. The smaller OAM quantum number l0 and longer wavelength λof the launch beam have contributed to the increase of the received power and decrease of the crosstalk power.

2. Relative power

The electric field distribution of the HyGG beam in the source plane (z = 0) can be described, in cylindrical coordinates, as [10]

Ep,l0(r,φ,0)=Cpl0(rω0)p+|l0|exp(r2ω02+il0φ),
where p is the hollowness parameter of the HyGG beam [8]; l0 corresponds to the number of OAM. r=|r|,r=(x,y) is the two-dimensional position vector in the source plane. φ denotes the azimuthally angle and ω0 is the beam waist. The normalized constant Cpl0 is given by Cpl0=2p+|l0|+1/[πΓ(p+|l0|+1)]Γ(p/2+|l0|+1)/Γ(|l0|+1), which ensures that the HyGG laser beam carries a finite power as long as p|l0|; Γ(x) is the Gamma function.

As a beam propagates, a phase aberration caused by atmospheric turbulence disturbs the complex amplitude of the wave. In the weak fluctuation atmosphere [22] and in the half-space z>0, the complex amplitude of the HyGG laser beam can be expressed

Ep(r,φ,z)=Ep,l0(r,φ,z)exp[ψ1(r,φ,z)],
where z is the propagation distance of the launch beam. ψ1(r,φ,z) is a complex phase perturbation of the wave propagating through turbulence. Ep,l0(r,φ,z) denotes the HyGG laser mode at the plane in free space. In the paraxial case, Ep,l0(r,φ,z) takes the form [10]
Ep,l0(r,φ,z)=Cpl0(zzR+i)[1+|l0|+p2]i|l0|+1(rω0)|l0|exp(il0φ)×(zzR)p2exp[izRr2ω02(z+izR)]F11(p2,|l0|+1;zR2r2ω02(z2+izzR)),
where the Rayleigh range is given by zR=kω02; k is the wave number of light related to the wavelength λ by k=2π/λ. F1(a,b;x) is the confluent hypergeometric function.

The refractive index fluctuations disturb the complex amplitude of the propagating wave, which is no longer guaranteed to be in the original eigenstate of OAM. The resulting wave now can be written as a superposition of the plane waves with new OAM modes and phase exp(ilφ) [23]

Ep(r,φ,z)=12πlβl(r,z)exp(ilφ),
where coefficient β1(r,z) is given by the integral

βl(r,z)=12π02πEp(r,φ,z)exp(ilφ)dφ.

Substituting Eq. (2) into Eq. (5) and taking the ensemble average of the turbulence, we have the mode probability density for beams in the paraxial channel

|βl(r,z)|2=12π02π02πEp(r,φ,z)Ep*(r,φ,z)exp[il(φφ)]dφdφ=12π02π02πEp,l0(r,φ,z)Ep,l0*(r,φ,z)×exp[ψ1(r,φ,z)+ψ1*(r,φ,z)]exp[il(φφ)]dφdφ,
where denotes the average over the ensemble of the turbulent atmosphere.

Applying the quadratic approximation of the wave structure function [22], the middle term in the above formula can be given by

exp[ψ1(r,φ,z)+ψ1*(r,φ,z)]=exp[2r22r2cos(φφ)ρ02],
where ρ0 is the spatial coherence radius of a spherical wave propagating in the non-Kolmogorov turbulence [15]. It is given by
ρ0={2(α1)Γ(3α2)[8α2Γ(2α2)](α2)2π12Γ(2α2)k2Cn2z}1(α2)3<α<4,
where α is the non-Kolmogorov turbulence parameter and Cn2 is the refractive-index structure constant over the path for horizontal homogeneous atmospheric propagation.

Substituting Eq. (7) into Eq. (6) we have

|βl(r,z)|2=12π02π02πEp,l0(r,φ,z)Ep,l0*(r,φ,z)×exp[il(φφ)]exp[2r22r2cos(φφ)ρ02]dφdφ.

Substituting Eq. (3) into Eq. (9), we obtain

|βl(r,z)|2=2p+|l0|+12π2Γ(p+|l0|+1)Γ2(p/2+|l0|+1)Γ2(|l0|+1)[1+(zzR)2](p/2+|l0|+1)×(rω0)2|l0|(zzR)pexp[2r2zR2ω02(zR2+z2)]|F11(p2,|l0|+1;zR2r2ω02(z2+izzR))|2×exp(2r2ρ02)02π02πexp[i(ll0)(φφ)+2r2cos(φφ)ρ02]dφdφ.

Based on the integral expression [24]

02πexp[inφ1+ηcos(φ1φ2)]dφ1=2πexp(inφ2)In(η),
where In(η) is the Bessel function of second kind with n order. We have the mode probability density of finding one photon in the signal mode |l

|βl(r,z)|2=2p+|l0|+2Γ(p+|l0|+1)Γ2(p2+|l0|+1)Γ2(|l0|+1)[1+(zzR)2](p2+|l0|+1)(rω0)2|l0|(zzR)p×exp[2r2zR2ω02(zR2+z2)]|F11(p2,|l0|+1;zR2r2ω02z(z+izR))|2exp(2r2ρ02)Ill0(2r2ρ02).

The relative power of the spiral harmonics marked with l in the paraxial regime of light propagation is determined by [25]

pl=0|βl(r,z)|2rdrl=0|βl(r,z)|2rdr.

The received power pl0 is defined as relative power of the OAM mode l0 in the receiver plane (i.e. l=l0) and the crosstalk power pΔl is the relative power in the receiver plane that is found to be in OAM mode l=l+0Δl [13, 26].

3. Numerical results

In this section, by using the analytical formulas derived in the previous section, we investigate the properties of the HyGG beams through atmospheric turbulence. Unless otherwise stated, the main parameters of numerical calculations in this paper are taken as α=11/3, z=5zR, λ=1550 nm, ω0=0.03 m, Cn2=1016 m3α, l0=1 and p = 3, respectively.

The characteristics of the HyGG beams brought by two parameters p and l0 as well as the corresponding effects on pl0 and pΔl are discussed in detail and shown in Figs. 1-3. The variations of pl0 and pΔl, which is due to the change of p, are investigated in Figs. 1(a) and 1(b) respectively, when the distance of propagation is kept at z=5zR. p|l0| must be satisfied in Eq. (1) to ensure the finite power of the HyGG beams. Thus, in Fig. 1(a), the bar of p = −1, l0=0 is meaningless. The sign of l0 just changes the direction of phase distribution and has no influence on the properties of the spiral spectrum of the HyGG beams. Therefore, the value of pl0 is symmetric with l0=0. The received power pl0 decreases along the positive l0 axis. As p increases, the received power pl0 increases monotonically with fixed l0 except for l0=0. In the case of l0=0, pl0 increases first and then decreases with the increase of p, exhibiting a marked peak at p = 2. It is known that the HyGG beams are the pure Gaussian beams for l0=0 [8, 10]. Figure 1(b) shows that the crosstalk power (the noise of receive signals in the receiver plane) pΔl is symmetric with l=l0. The larger value of p begins to offer smaller pΔl for Δl>0. As expected, pΔl decreases rapidly and becomes zero quickly with the increase of Δl (Δl>0) at the same hollowness parameter p.

 figure: Fig. 1

Fig. 1 The spiral spectrum of the HyGG beams propagating in non-Kolmogorov turbulence for different hollowness parameters p. (a) The received power pl0 observed on OAM mode l=l0. (b) The crosstalk power pΔl observed on OAM mode l=l0+Δl with OAM number l0=1. The bars of Δl=0 correspond to the original signals. The crosstalk power is symmetric with the OAM mode l=l0.

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When a beam propagates in turbulent atmosphere, it suffers signal attenuation, noise and wrong code caused by turbulence. Figures 2(a)-2(b) exhibit the received power pl0 and the crosstalk power pΔl against the propagation distance z for the different quantum numbers l of OAM in atmospheric turbulence. For any given propagation distance z, Fig. 2(a) shows that the attenuation of the received power pl0 is increased by the increasing quantum number l of OAM. When the propagation distance of the HyGG beams reaches up to 5zR, the attenuation of pl0 reaches about 10% as l=l0=3. As shown in Fig. 2(b), the crosstalk power pΔl occurs mainly between two adjacent OAM states, that is, Δl=1. Thus, for Δl=2,3,4, the influence of crosstalk in the receiver plane is negligible under the weak turbulence.

 figure: Fig. 2

Fig. 2 (a) The received power pl0 of the HyGG beams in non-Kolmogorov turbulence as a function of the propagation distance z for the OAM numbers l=l0=0,1,2 and 3. The lower OAM number gives the higher received power. (b) The crosstalk power pΔl versus the propagation distance z for Δl=1,2,3, and 4 with l0=1. The crosstalk power occurs mainly between two adjacent OAM states under the weak turbulence.

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For p|l0|, the variations of pl0 and pΔlwith the propagation distance z are explored in Fig. 3. In the case of p=|l0|, Fig. 3(a) shows that no matter p is odd (modified Bessel Gauss modes) or even (modified exponential Gauss modes) [10], pl0 drops rapidly when p decreases. It has almost no effect on pl0 for p>0, which is similar to that in Fig. 1(a). As seen from Fig. 3(b), when z increases, pΔl increases quickly at p = −1. Besides, for p>0, the changes of pΔl are almost negligible with the increase of p at the given propagation z.

 figure: Fig. 3

Fig. 3 (a) The received power pl0 of the HyGG beams in non-Kolmogorov turbulence as a function of the propagation distance z with the different hollowness parameters p. (b) The crosstalk power pΔl calculated for l0=1,Δl=1 versus the propagation distance z when the hollowness parameter p changes from −1 to 3.

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The wavelength selection of source light in free space communication is an important issue for the improvement of pl0. Thus, Fig. 4 illustrates the effects of wavelength λ on pl0 and pΔl for the HyGG beams. Because the Rayleigh range is a function of wavelength λ, in Fig. 4, we choose the propagation distance z, in kilometer, as the abscissa. From Fig. 4(a), we find indications for less attenuation at the wavelength of 1550 nm. The received power pl0 of the HyGG beams with λ = 1550 nm is 1.23 times higher than that in the case of λ=532 nm. The longer wavelength is benefit to the propagation of optical signal which is consistent with [27]. It is shown in Fig. 4(b) that the curve of pΔl with respect to z is flatter in the longer λ cases. This conclusion provides a significant guidance for choosing the light source.

 figure: Fig. 4

Fig. 4 The received power pl0 (a) and the crosstalk power pΔl calculated for l0=1,Δl=1 (b) of the HyGG beams propagating in non-Kolmogorov turbulence versus the propagation distance z for the different wavelengths λ=532,632.8,850 and 1550 nm. The longer wavelength is benefit to the propagation of optical signal with OAM.

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Figures 5 and 6 present how do the various parameters of atmospheric turbulence affect the received power pl0 and the crosstalk power pΔl. Figure 5(a) is a plot of the received power pl0 versus the propagation distance z for the different non-Kolmogorov turbulence parameters α. The curve associated with α=3.67 corresponds to Kolmogorov turbulence. Asα approaches 3, the pl0keeps a constant with the increasing z, but the received power pl0has a clear reduction if αtends toward 4. Figure 5(a) is also consistent with [15, 28] that the smaller α comes with the larger pl0 in the receiver plane for any given propagation distance. The reason for this is that the spatial coherence radius ρ0 approaches infinity when α gets close to 3. It implies that the interruption of the turbulence on the OAM modes is very weak, and a high pl0 is acquired. However, when α is close to 4, the wave aberration caused by turbulence is a pure wavefront tilt, which shifts the beam off axis and results in the beam center away from the receiver plane [15, 28]. Thus, the received power pl0 decreases with α approaching to 4. The relationship between the crosstalk power pΔl and the index α of non-Kolmogorov turbulence is revealed by Fig. 5(b). For any given z, the crosstalk power pΔl of the HyGG beams increases rapidly when α changes from 3.07 to 3.97. As α increases, the crosstalk power increases at the given propagation distance z for the same reason that the received power decreases.

 figure: Fig. 5

Fig. 5 The received power pl0 (a) and the crosstalk power pΔl calculated for l0=1,Δl=1 (b) of the HyGG beams propagating in non-Kolmogorov turbulence versus the propagation distance z for the different non-Kolmogorov turbulence parameters α=3.07, 3.37, 3.67 and 3.97. The curves associated with α=3.67 correspond to Kolmogorov turbulence. The smaller α comes with the larger pl0.

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 figure: Fig. 6

Fig. 6 The received power pl0 (a) and the crosstalk power pΔl calculated for l0=1,Δl=1 (b) of the HyGG beams in non-Kolmogorov turbulence as a function of z under different turbulent conditions (Cn2=1017,1016,1015 and 1014m3α) with α=11/3. The pl0 almost remains unchanged in the weak turbulence (Cn2=1017m3α), descends gradually in the intermediate turbulence (Cn2=1016 and 1015m3α) and in the strong turbulence (Cn2=1014m3α) drops quickly with the increasing propagation distance z.

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Physically, Cn2 is a measurement of the strength of the fluctuations in the refractive index. It typically ranges from 1017m3α to 1014m3αrepresenting the conditions of “weak turbulence” to “strong turbulence” [22]. The variations of pl0 and pΔl of the HyGG beams versus the propagation distance z for different Cn2 are plotted in Fig. 6. As shown in Fig. 6(a), pl0 almost remains unchanged in the case of Cn2=1017 m3α(weak turbulence) and descends gradually for Cn2=1016and 1015m3α(intermediate turbulence). At Cn2=1014m3α(strong turbulence), pl0 drops quickly with the increasing propagation distance z. For z>1.2zR, the downtrend of pl0 develops slowly. In Fig. 6(b), the crosstalk power pΔl is almost zero when the HyGG beams travel through the weak turbulence region (Cn2=1017m3α). In the cases of Cn2=1016 and 1015m3α, pΔl is enhanced by the increase of z. However, there is a distinct feature in the case of Cn2=1014m3α. pΔl increases first and reaches the maximum value at z=1.2zR. Then, it decreases gradually when z further increases. A reasonable cause is that, for strong turbulence, OAM states are further redistributed and crosstalk is in all observed links. For Δl=2,3,4, the conspicuous increase of pΔl, which makes the decline in pΔl at Δl=1, occurs in atmospheric link. pl0 and pΔl in all observed links have a similar intensity distribution for further increasing the propagation distance z.

4. Conclusions

We have evaluated the transmission features of the HyGG beams through non-Kolmogorov turbulence. It is found that the decrease of the hollowness parameter p of the HyGG beams will cause the increase of the absolute value of the slope for pl0~z curve at p=|l0|. In the case of p>0, pl0remains almost unchanged with the increase of p. The received power pl0 increases with the decrease of l0 at the same propagation distance. The increases of Cn2 and α result in the degradation of the received power pl0. In the strong turbulent and long distance region, the decline tendency of pl0 curve is weakened. The longer wavelength of the HyGG beam contributes more to the received power pl0 than that of the shorter one. The results provide a convenient way of controlling the properties of the HyGG beams through atmospheric turbulence by choosing initial parameters properly.

Acknowledgment

This work is supported by the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20140128), the National Natural Science Foundation of Special Theoretical Physics (Grant No. 11447174) and the Fundamental Research Funds for the Central Universities of China (Grant No. 1142050205135370).

References and links

1. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. 32(7), 742–744 (2007). [CrossRef]   [PubMed]  

2. V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, S. N. Khonina, and J. Turunen, “Generating hypergeometric laser beams with a diffractive optical element,” Appl. Opt. 47(32), 6124–6133 (2008). [CrossRef]   [PubMed]  

3. R. V. Skidanov, S. N. Khonina, and A. A. Morozov, “Optical rotation of microparticles in hypergeometric beams formed by diffraction optical elements with multilevel microrelief,” J. Opt. Technol. 80(10), 585–589 (2013). [CrossRef]  

4. J. Chen, “Production of confluent hypergeometric beam by computer-generated hologram,” Opt. Eng. 50(2), 024201 (2011). [CrossRef]  

5. B. de Lima Bernardo and F. Moraes, “Data transmission by hypergeometric modes through a hyperbolic-index medium,” Opt. Express 19(12), 11264–11270 (2011). [CrossRef]   [PubMed]  

6. J. Li and Y. Chen, “Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 44(5), 1603–1610 (2012). [CrossRef]  

7. V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Propagation of hypergeometric laser beams in a medium with a parabolic refractive index,” J. Opt. 15(12), 125706 (2013). [CrossRef]  

8. H. T. Eyyuboğlu and Y. Cai, “Hypergeometric Gaussian beam and its propagation in turbulence,” Opt. Commun. 285(21–22), 4194–4199 (2012). [CrossRef]  

9. H. T. Eyyuboğlu, “Scintillation analysis of hypergeometric Gaussian beam via phase screen method,” Opt. Commun. 309, 103–107 (2013). [CrossRef]  

10. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32(21), 3053–3055 (2007). [CrossRef]   [PubMed]  

11. E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Improved focusing with hypergeometric-gaussian type-II optical modes,” Opt. Express 16(25), 21069–21075 (2008). [CrossRef]   [PubMed]  

12. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001). [CrossRef]   [PubMed]  

13. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005). [CrossRef]   [PubMed]  

14. E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Process. Geophys. 13(3), 297–301 (2006). [CrossRef]  

15. C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000). [CrossRef]  

16. 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]  

17. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008). [CrossRef]  

18. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). [CrossRef]   [PubMed]  

19. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007). [CrossRef]  

20. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026001 (2008).

21. Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011). [CrossRef]  

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

23. L. Torner, J. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005). [CrossRef]   [PubMed]  

24. A. Jeffrey and D. Zwillinger, Table of Integrals, Series, and Products (Academic, 2007).

25. Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008). [CrossRef]  

26. G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34(2), 142–144 (2009). [CrossRef]   [PubMed]  

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

28. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995). [CrossRef]  

References

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  1. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. 32(7), 742–744 (2007).
    [Crossref] [PubMed]
  2. V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, S. N. Khonina, and J. Turunen, “Generating hypergeometric laser beams with a diffractive optical element,” Appl. Opt. 47(32), 6124–6133 (2008).
    [Crossref] [PubMed]
  3. R. V. Skidanov, S. N. Khonina, and A. A. Morozov, “Optical rotation of microparticles in hypergeometric beams formed by diffraction optical elements with multilevel microrelief,” J. Opt. Technol. 80(10), 585–589 (2013).
    [Crossref]
  4. J. Chen, “Production of confluent hypergeometric beam by computer-generated hologram,” Opt. Eng. 50(2), 024201 (2011).
    [Crossref]
  5. B. de Lima Bernardo and F. Moraes, “Data transmission by hypergeometric modes through a hyperbolic-index medium,” Opt. Express 19(12), 11264–11270 (2011).
    [Crossref] [PubMed]
  6. J. Li and Y. Chen, “Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 44(5), 1603–1610 (2012).
    [Crossref]
  7. V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Propagation of hypergeometric laser beams in a medium with a parabolic refractive index,” J. Opt. 15(12), 125706 (2013).
    [Crossref]
  8. H. T. Eyyuboğlu and Y. Cai, “Hypergeometric Gaussian beam and its propagation in turbulence,” Opt. Commun. 285(21–22), 4194–4199 (2012).
    [Crossref]
  9. H. T. Eyyuboğlu, “Scintillation analysis of hypergeometric Gaussian beam via phase screen method,” Opt. Commun. 309, 103–107 (2013).
    [Crossref]
  10. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32(21), 3053–3055 (2007).
    [Crossref] [PubMed]
  11. E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Improved focusing with hypergeometric-gaussian type-II optical modes,” Opt. Express 16(25), 21069–21075 (2008).
    [Crossref] [PubMed]
  12. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
    [Crossref] [PubMed]
  13. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
    [Crossref] [PubMed]
  14. E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Process. Geophys. 13(3), 297–301 (2006).
    [Crossref]
  15. C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
    [Crossref]
  16. 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]
  17. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
    [Crossref]
  18. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010).
    [Crossref] [PubMed]
  19. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
    [Crossref]
  20. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026001 (2008).
  21. Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
    [Crossref]
  22. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
  23. L. Torner, J. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
    [Crossref] [PubMed]
  24. A. Jeffrey and D. Zwillinger, Table of Integrals, Series, and Products (Academic, 2007).
  25. Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
    [Crossref]
  26. G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34(2), 142–144 (2009).
    [Crossref] [PubMed]
  27. J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
    [Crossref]
  28. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
    [Crossref]

2014 (1)

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

2013 (3)

R. V. Skidanov, S. N. Khonina, and A. A. Morozov, “Optical rotation of microparticles in hypergeometric beams formed by diffraction optical elements with multilevel microrelief,” J. Opt. Technol. 80(10), 585–589 (2013).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Propagation of hypergeometric laser beams in a medium with a parabolic refractive index,” J. Opt. 15(12), 125706 (2013).
[Crossref]

H. T. Eyyuboğlu, “Scintillation analysis of hypergeometric Gaussian beam via phase screen method,” Opt. Commun. 309, 103–107 (2013).
[Crossref]

2012 (2)

H. T. Eyyuboğlu and Y. Cai, “Hypergeometric Gaussian beam and its propagation in turbulence,” Opt. Commun. 285(21–22), 4194–4199 (2012).
[Crossref]

J. Li and Y. Chen, “Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 44(5), 1603–1610 (2012).
[Crossref]

2011 (3)

J. Chen, “Production of confluent hypergeometric beam by computer-generated hologram,” Opt. Eng. 50(2), 024201 (2011).
[Crossref]

B. de Lima Bernardo and F. Moraes, “Data transmission by hypergeometric modes through a hyperbolic-index medium,” Opt. Express 19(12), 11264–11270 (2011).
[Crossref] [PubMed]

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

2010 (1)

2009 (1)

2008 (6)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026001 (2008).

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, S. N. Khonina, and J. Turunen, “Generating hypergeometric laser beams with a diffractive optical element,” Appl. Opt. 47(32), 6124–6133 (2008).
[Crossref] [PubMed]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Improved focusing with hypergeometric-gaussian type-II optical modes,” Opt. Express 16(25), 21069–21075 (2008).
[Crossref] [PubMed]

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]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

2007 (3)

2006 (1)

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Process. Geophys. 13(3), 297–301 (2006).
[Crossref]

2005 (2)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

L. Torner, J. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
[Crossref] [PubMed]

2001 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

2000 (1)

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

1995 (1)

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026001 (2008).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Boyd, R. W.

Branover, H.

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Process. Geophys. 13(3), 297–301 (2006).
[Crossref]

Cai, Y.

H. T. Eyyuboğlu and Y. Cai, “Hypergeometric Gaussian beam and its propagation in turbulence,” Opt. Commun. 285(21–22), 4194–4199 (2012).
[Crossref]

Carrasco, S.

Chen, J.

J. Chen, “Production of confluent hypergeometric beam by computer-generated hologram,” Opt. Eng. 50(2), 024201 (2011).
[Crossref]

Chen, Y.

J. Li and Y. Chen, “Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 44(5), 1603–1610 (2012).
[Crossref]

de Lima Bernardo, B.

Eyyuboglu, H. T.

H. T. Eyyuboğlu, “Scintillation analysis of hypergeometric Gaussian beam via phase screen method,” Opt. Commun. 309, 103–107 (2013).
[Crossref]

H. T. Eyyuboğlu and Y. Cai, “Hypergeometric Gaussian beam and its propagation in turbulence,” Opt. Commun. 285(21–22), 4194–4199 (2012).
[Crossref]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026001 (2008).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Gao, C.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Gao, M.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Golbraikh, E.

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]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Process. Geophys. 13(3), 297–301 (2006).
[Crossref]

Guo, H.

He, Y.

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

Jia, J.

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

Jiang, W.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

Jiang, Y.

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

Karimi, E.

Khonina, S. N.

Kopeika, N. S.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

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]

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Process. Geophys. 13(3), 297–301 (2006).
[Crossref]

Kotlyar, V. V.

Kovalev, A. A.

V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Propagation of hypergeometric laser beams in a medium with a parabolic refractive index,” J. Opt. 15(12), 125706 (2013).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, S. N. Khonina, and J. Turunen, “Generating hypergeometric laser beams with a diffractive optical element,” Appl. Opt. 47(32), 6124–6133 (2008).
[Crossref] [PubMed]

Kupershmidt, I.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Li, F.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Li, J.

J. Li and Y. Chen, “Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 44(5), 1603–1610 (2012).
[Crossref]

Liao, J.

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

Ling, N.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

Liu, Y.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Luo, B.

Marrucci, L.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Moraes, F.

Morozov, A. A.

Nalimov, A. G.

V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Propagation of hypergeometric laser beams in a medium with a parabolic refractive index,” J. Opt. 15(12), 125706 (2013).
[Crossref]

Ou, J.

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

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026001 (2008).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Piccirillo, B.

Rao, C.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Santamato, E.

Shtemler, Y.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Skidanov, R. V.

Soifer, V. A.

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Tang, H.

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

Torner, L.

L. Torner, J. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
[Crossref] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Torres, J.

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026001 (2008).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Turunen, J.

Tyler, G. A.

Virtser, A.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Wang, J.

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

Wang, S.

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

Wang, Y.

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Wu, G.

Xu, J.

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

Yu, S.

Zhang, J.

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

Zhang, Y.

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

Zilberman, A.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

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]

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Process. Geophys. 13(3), 297–301 (2006).
[Crossref]

Zito, G.

Appl. Opt. (2)

Atmos. Res. (1)

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

J. Mod. Opt. (1)

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

J. Opt. (1)

V. V. Kotlyar, A. A. Kovalev, and A. G. Nalimov, “Propagation of hypergeometric laser beams in a medium with a parabolic refractive index,” J. Opt. 15(12), 125706 (2013).
[Crossref]

J. Opt. Technol. (1)

Nonlinear Process. Geophys. (1)

E. Golbraikh, H. Branover, N. S. Kopeika, and A. Zilberman, “Non-Kolmogorov atmospheric turbulence and optical signal propagation,” Nonlinear Process. Geophys. 13(3), 297–301 (2006).
[Crossref]

Opt. Commun. (5)

H. T. Eyyuboğlu and Y. Cai, “Hypergeometric Gaussian beam and its propagation in turbulence,” Opt. Commun. 285(21–22), 4194–4199 (2012).
[Crossref]

H. T. Eyyuboğlu, “Scintillation analysis of hypergeometric Gaussian beam via phase screen method,” Opt. Commun. 309, 103–107 (2013).
[Crossref]

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

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

Opt. Eng. (2)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026001 (2008).

J. Chen, “Production of confluent hypergeometric beam by computer-generated hologram,” Opt. Eng. 50(2), 024201 (2011).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

J. Li and Y. Chen, “Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 44(5), 1603–1610 (2012).
[Crossref]

Opt. Lett. (4)

Phys. Rev. Lett. (2)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

Proc. SPIE (2)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Other (2)

A. Jeffrey and D. Zwillinger, Table of Integrals, Series, and Products (Academic, 2007).

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

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Figures (6)

Fig. 1
Fig. 1 The spiral spectrum of the HyGG beams propagating in non-Kolmogorov turbulence for different hollowness parameters p. (a) The received power p l 0 observed on OAM mode l= l 0 . (b) The crosstalk power p Δl observed on OAM mode l= l 0 +Δl with OAM number l 0 =1 . The bars of Δl=0 correspond to the original signals. The crosstalk power is symmetric with the OAM mode l= l 0 .
Fig. 2
Fig. 2 (a) The received power p l 0 of the HyGG beams in non-Kolmogorov turbulence as a function of the propagation distance z for the OAM numbers l= l 0 =0,1,2 and 3. The lower OAM number gives the higher received power. (b) The crosstalk power p Δl versus the propagation distance z for Δl=1,2,3, and 4 with l 0 =1 . The crosstalk power occurs mainly between two adjacent OAM states under the weak turbulence.
Fig. 3
Fig. 3 (a) The received power p l 0 of the HyGG beams in non-Kolmogorov turbulence as a function of the propagation distance z with the different hollowness parameters p. (b) The crosstalk power p Δl calculated for l 0 =1,Δl=1 versus the propagation distance z when the hollowness parameter p changes from −1 to 3.
Fig. 4
Fig. 4 The received power p l 0 (a) and the crosstalk power p Δl calculated for l 0 =1,Δl=1 (b) of the HyGG beams propagating in non-Kolmogorov turbulence versus the propagation distance z for the different wavelengths λ=532,632.8,850 and 1550 nm. The longer wavelength is benefit to the propagation of optical signal with OAM.
Fig. 5
Fig. 5 The received power p l 0 (a) and the crosstalk power p Δl calculated for l 0 =1,Δl=1 (b) of the HyGG beams propagating in non-Kolmogorov turbulence versus the propagation distance z for the different non-Kolmogorov turbulence parameters α=3.07 , 3.37, 3.67 and 3.97. The curves associated with α=3.67 correspond to Kolmogorov turbulence. The smaller α comes with the larger p l 0 .
Fig. 6
Fig. 6 The received power p l 0 (a) and the crosstalk power p Δl calculated for l 0 =1,Δl=1 (b) of the HyGG beams in non-Kolmogorov turbulence as a function of z under different turbulent conditions ( C n 2 = 10 17 , 10 16 , 10 15 and 10 14 m 3α ) with α=11/3 . The p l 0 almost remains unchanged in the weak turbulence ( C n 2 = 10 17 m 3α ), descends gradually in the intermediate turbulence ( C n 2 = 10 16 and 10 15 m 3α ) and in the strong turbulence ( C n 2 = 10 14 m 3α ) drops quickly with the increasing propagation distance z.

Equations (13)

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E p, l 0 ( r,φ,0 )= C p l 0 ( r ω 0 ) p+| l 0 | exp( r 2 ω 0 2 +i l 0 φ ),
E p ( r,φ,z )= E p, l 0 ( r,φ,z )exp[ ψ 1 ( r,φ,z ) ],
E p, l 0 ( r,φ,z )= C p l 0 ( z z R +i ) [ 1+| l 0 |+ p 2 ] i | l 0 |+1 ( r ω 0 ) | l 0 | exp( i l 0 φ ) × ( z z R ) p 2 exp[ i z R r 2 ω 0 2 ( z+i z R ) ] F 1 1 ( p 2 ,| l 0 |+1; z R 2 r 2 ω 0 2 ( z 2 +iz z R ) ),
E p ( r,φ,z )= 1 2π l β l ( r,z )exp( ilφ ) ,
β l ( r,z )= 1 2π 0 2π E p ( r,φ,z ) exp( ilφ )dφ.
| β l (r,z) | 2 = 1 2π 0 2π 0 2π E p ( r,φ,z ) E p * ( r, φ ,z ) exp[ il( φ φ ) ]d φ dφ = 1 2π 0 2π 0 2π E p, l 0 ( r,φ,z ) E p, l 0 * ( r, φ ,z ) × exp[ ψ 1 ( r,φ,z )+ ψ 1 * ( r, φ ,z ) ] exp[ il( φ φ ) ]d φ dφ,
exp[ ψ 1 ( r,φ,z )+ ψ 1 * ( r, φ ,z ) ] =exp[ 2 r 2 2 r 2 cos( φ φ ) ρ 0 2 ],
ρ 0 = { 2( α1 )Γ( 3α 2 ) [ 8 α2 Γ( 2 α2 ) ] ( α2 ) 2 π 1 2 Γ( 2α 2 ) k 2 C n 2 z } 1 ( α2 ) 3<α<4,
| β l (r,z) | 2 = 1 2π 0 2π 0 2π E p, l 0 ( r,φ,z ) E p, l 0 * ( r, φ ,z ) ×exp[ il( φ φ ) ]exp[ 2 r 2 2 r 2 cos( φ φ ) ρ 0 2 ]d φ dφ.
| β l (r,z) | 2 = 2 p+| l 0 |+1 2 π 2 Γ(p+| l 0 |+1) Γ 2 (p/2+| l 0 |+1) Γ 2 (| l 0 |+1) [ 1+ ( z z R ) 2 ] ( p/2+| l 0 |+1 ) × ( r ω 0 ) 2| l 0 | ( z z R ) p exp[ 2 r 2 z R 2 ω 0 2 ( z R 2 + z 2 ) ] | F 1 1 ( p 2 ,| l 0 |+1; z R 2 r 2 ω 0 2 ( z 2 +iz z R ) ) | 2 ×exp( 2 r 2 ρ 0 2 ) 0 2π 0 2π exp[ i( l l 0 )( φ φ )+ 2 r 2 cos( φ φ ) ρ 0 2 ]d φ dφ .
0 2π exp[in φ 1 +ηcos( φ 1 φ 2 )] d φ 1 =2πexp(in φ 2 ) I n (η),
| β l (r,z) | 2 = 2 p+| l 0 |+2 Γ(p+| l 0 |+1) Γ 2 ( p 2 +| l 0 |+1) Γ 2 (| l 0 |+1) [ 1+ ( z z R ) 2 ] ( p 2 +| l 0 |+1 ) ( r ω 0 ) 2| l 0 | ( z z R ) p ×exp[ 2 r 2 z R 2 ω 0 2 ( z R 2 + z 2 ) ] | F 1 1 ( p 2 ,| l 0 |+1; z R 2 r 2 ω 0 2 z( z+i z R ) ) | 2 exp( 2 r 2 ρ 0 2 ) I l l 0 ( 2 r 2 ρ 0 2 ).
p l = 0 | β l (r,z) | 2 rdr l= 0 | β l (r,z) | 2 rdr .

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