Effects of non-Kolmogorov turbulence on the spiral spectrum of Hypergeometric-Gaussian laser beams Yun Zhu,1,2 Licheng Zhang,1,2 Zhengda Hu2, and Yixin Zhang2,* 1
School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China 2 School of Science, Jiangnan University, Wuxi 214122, China *[email protected].
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 OCIS codes: (010.1330) Atmospheric turbulence; (010.3310) Laser beam transmission; (060.2605) Free-space optical communication.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
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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). 18. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a nonKolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). 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). 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). 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). 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). 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). 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). 28. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
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 [16–18]. 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
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Received 26 Nov 2014; revised 17 Feb 2015; accepted 23 Feb 2015; published 2 Apr 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.009137 | OPTICS EXPRESS 9138
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 nonKolmogorov 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 pl 0 drops with the decrease of p. On the other hand, there are little effects on the received power pl 0 when p is positive. The smaller OAM quantum number l0 and longer wavelength λ of 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] r E p ,l0 ( r , ϕ , 0 ) = C pl0 ω0
p + l0
r2 exp − 2 + il0ϕ , ω0
(1)
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 C pl 0 is p + 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.
given by C pl0 = 2
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 E p ( r , ϕ , z ) = E p ,l0 ( r , ϕ , z ) exp ψ 1 ( r , ϕ , z ) ,
(2)
where z is the propagation distance of the launch beam. ψ 1 (r , ϕ , z ) is a complex phase perturbation of the wave propagating through turbulence. E p ,l0 ( r , ϕ , z ) denotes the HyGG
laser mode at the z plane in free space. In the paraxial case, E p ,l0 ( r , ϕ , z ) takes the form [10] E p ,l0 ( r , ϕ , z ) = C pl0 p
z +i zR
p − 1+ l0 + 2
l0
i
l0 +1
r exp ( il0ϕ ) ω0
p z 2R r 2 z 2 iz r 2 , × exp − 2 R − + F l , 1; 1 1 0 2 2 2 + z izz ω ω0 ( z + iz R ) ( ) zR 0 R
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(3)
Received 26 Nov 2014; revised 17 Feb 2015; accepted 23 Feb 2015; published 2 Apr 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.009137 | OPTICS EXPRESS 9139
where the Rayleigh range z R is given by z R = 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] 1
E p ( r,ϕ , z ) =
2π
β ( r , z ) exp ( −ilϕ ), l
(4)
l
where coefficient β1 (r , z ) is given by the integral 1
βl ( r , z ) =
2π
2π
0
E p ( r , ϕ , z ) exp ( ilϕ ) dϕ .
(5)
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
1 2π 1 = 2π
=
2π
2π
0
0
2π
2π
0
0
E p ( r , ϕ , z ) E *p ( r , ϕ ′, z ) exp −il (ϕ − ϕ ′ ) dϕ ′dϕ E p ,l0 ( r , ϕ , z ) E *p ,l0 ( r , ϕ ′, z )
(6)
× exp ψ 1 ( r , ϕ , z ) + ψ 1* ( r , ϕ ′, z ) exp −il (ϕ − ϕ ′ ) dϕ ′dϕ ,
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 2r 2 − 2r 2 cos (ϕ − ϕ ′ ) exp ψ 1 ( r , ϕ , z ) + ψ 1* ( r , ϕ ′, z ) = exp − , ρ02
(7)
where ρ 0 is the spatial coherence radius of a spherical wave propagating in the nonKolmogorov turbulence [15]. It is given by 1
(α − 2 ) (α − 2 ) 2 3 α 8 2 − 2 (α − 1) Γ Γ 2 α − 2 α − 2 ρ0 = 3 < α < 4, 1 2 −α 2 2 2 π Γ k Cn z 2
(8)
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
=
1 2π
2π
2π
0
0
E p ,l0 ( r , ϕ , z ) E *p ,l0 ( r , ϕ ′, z )
2r 2 − 2r 2 cos (ϕ ′ − ϕ ) × exp −il (ϕ − ϕ ′ ) exp − dϕ ′dϕ . ρ 02
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(9)
Received 26 Nov 2014; revised 17 Feb 2015; accepted 23 Feb 2015; published 2 Apr 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.009137 | OPTICS EXPRESS 9140
Substituting Eq. (3) into Eq. (9), we obtain
βl (r , z ) r × ω0
2
2 l0
2 p + l +1 Γ 2 ( p / 2 + l0 + 1) z 2 0 1 + = 2π 2 Γ( p + l0 + 1) Γ 2 ( l0 + 1) z R
− ( p / 2 + l0 +1)
p p z 2r 2 z R2 z R2 r 2 exp F , l 1; − − + 1 1 0 2 2 2 2 2 2 ω0 ( z + izzR ) ω0 ( z R + z ) zR
2
(10)
2r 2 cos (ϕ ′ − ϕ ) 2 r 2 2π 2π × exp − 2 exp −i ( l − l0 )(ϕ − ϕ ′ ) + dϕ ′dϕ . 0 ρ02 ρ0 0 Based on the integral expression [24] 2π
exp[−inϕ
1
+ η cos(ϕ1 − ϕ2 )]dϕ1 = 2π exp(−inϕ 2 ) I n (η ),
(11)
0
where I n (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 p p + l0 + 1) 2 − 2 + l0 +1 2 l0 p 2 z r z 2 2 1 + βl (r , z ) = Γ( p + l0 + 1) Γ 2 ( l0 + 1) z R ω z 0 R (12) 2 2 2 2 2 2 2 p 2r 2r 2r z z r × exp − 2 2 R 2 1 F1 − , l0 + 1; 2 R exp − 2 I l − l0 2 . ω0 z ( z + iz R ) ω0 ( z R + z ) ρ0 ρ0 2
p + l0 + 2
Γ2 (
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 )
∞
l =−∞ 0
2
rd r
β l (r , z )
2
rdr
.
(13)
The received power pl0 is defined as relative power of the OAM mode l0 in the receiver plane (i.e. l = l 0 ) and the crosstalk power pΔl is the relative power in the receiver plane that is found to be in OAM mode l = l 0 +Δl [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 = 5 z R ,
λ = 1550 nm, ω0 = 0.03 m, Cn2 = 10−16 m3−α , l0 = 1 and p = 3, respectively.
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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 .
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 = 5 z R . p ≥ − l 0 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. 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 5 z R , 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.
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Received 26 Nov 2014; revised 17 Feb 2015; accepted 23 Feb 2015; published 2 Apr 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.009137 | OPTICS EXPRESS 9142
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.
For p ≥ − l0 , the variations of pl0 and pΔl with 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.
Fig. 3. 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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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.
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 z R 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.
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 pl . 0
Figures 5-6 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
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parameters α . The curve associated with α = 3.67 corresponds to Kolmogorov turbulence. As α approaches 3, the pl0 keeps a constant with the increasing z, but the received power pl0 has a clear reduction if α tends 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 ρ 0 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 nonKolmogorov 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.
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 = 10−17 ,10−16 ,10−15 and 10 −14 m3−α ) with α = 11 / 3 . The pl0 almost remains unchanged in the weak turbulence ( Cn2 = 10−17 m3−α ), descends gradually in the intermediate turbulence ( Cn2 = 10−16 and 10−15 m3−α ) and in the strong turbulence ( Cn2 = 10−14 m3−α ) drops quickly with the increasing propagation distance z..
Physically, Cn2 is a measurement of the strength of the fluctuations in the refractive index. It typically ranges from 10−17 m3−α to 10−14 m3−α representing 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 = 10−17 m3−α (weak turbulence) and
descends C = 10 2 n
−14
gradually m
3 −α
for
Cn2 = 10−16 and
10−15 m3−α (intermediate
turbulence).
At
(strong turbulence), pl0 drops quickly with the increasing propagation
distance z. For z > 1.2 z R , 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 = 10−17 m3−α ). In the cases of Cn2 = 10−16 and 10−15 m3−α , pΔl is enhanced by the
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increase of z. However, there is a distinct feature in the case of Cn2 = 10−14 m3−α . pΔl increases first and reaches the maximum value at z = 1.2 z R . 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, pl0 remains 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).
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Received 26 Nov 2014; revised 17 Feb 2015; accepted 23 Feb 2015; published 2 Apr 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.009137 | OPTICS EXPRESS 9146
School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China 2 School of Science, Jiangnan University, Wuxi 214122, China *[email protected].
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 OCIS codes: (010.1330) Atmospheric turbulence; (010.3310) Laser beam transmission; (060.2605) Free-space optical communication.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
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Received 26 Nov 2014; revised 17 Feb 2015; accepted 23 Feb 2015; published 2 Apr 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.009137 | OPTICS EXPRESS 9137
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). 18. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a nonKolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). 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). 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). 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). 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). 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). 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). 28. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
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 [16–18]. 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
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Received 26 Nov 2014; revised 17 Feb 2015; accepted 23 Feb 2015; published 2 Apr 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.009137 | OPTICS EXPRESS 9138
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 nonKolmogorov 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 pl 0 drops with the decrease of p. On the other hand, there are little effects on the received power pl 0 when p is positive. The smaller OAM quantum number l0 and longer wavelength λ of 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] r E p ,l0 ( r , ϕ , 0 ) = C pl0 ω0
p + l0
r2 exp − 2 + il0ϕ , ω0
(1)
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 C pl 0 is p + 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.
given by C pl0 = 2
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 E p ( r , ϕ , z ) = E p ,l0 ( r , ϕ , z ) exp ψ 1 ( r , ϕ , z ) ,
(2)
where z is the propagation distance of the launch beam. ψ 1 (r , ϕ , z ) is a complex phase perturbation of the wave propagating through turbulence. E p ,l0 ( r , ϕ , z ) denotes the HyGG
laser mode at the z plane in free space. In the paraxial case, E p ,l0 ( r , ϕ , z ) takes the form [10] E p ,l0 ( r , ϕ , z ) = C pl0 p
z +i zR
p − 1+ l0 + 2
l0
i
l0 +1
r exp ( il0ϕ ) ω0
p z 2R r 2 z 2 iz r 2 , × exp − 2 R − + F l , 1; 1 1 0 2 2 2 + z izz ω ω0 ( z + iz R ) ( ) zR 0 R
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(3)
Received 26 Nov 2014; revised 17 Feb 2015; accepted 23 Feb 2015; published 2 Apr 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.009137 | OPTICS EXPRESS 9139
where the Rayleigh range z R is given by z R = 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] 1
E p ( r,ϕ , z ) =
2π
β ( r , z ) exp ( −ilϕ ), l
(4)
l
where coefficient β1 (r , z ) is given by the integral 1
βl ( r , z ) =
2π
2π
0
E p ( r , ϕ , z ) exp ( ilϕ ) dϕ .
(5)
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
1 2π 1 = 2π
=
2π
2π
0
0
2π
2π
0
0
E p ( r , ϕ , z ) E *p ( r , ϕ ′, z ) exp −il (ϕ − ϕ ′ ) dϕ ′dϕ E p ,l0 ( r , ϕ , z ) E *p ,l0 ( r , ϕ ′, z )
(6)
× exp ψ 1 ( r , ϕ , z ) + ψ 1* ( r , ϕ ′, z ) exp −il (ϕ − ϕ ′ ) dϕ ′dϕ ,
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 2r 2 − 2r 2 cos (ϕ − ϕ ′ ) exp ψ 1 ( r , ϕ , z ) + ψ 1* ( r , ϕ ′, z ) = exp − , ρ02
(7)
where ρ 0 is the spatial coherence radius of a spherical wave propagating in the nonKolmogorov turbulence [15]. It is given by 1
(α − 2 ) (α − 2 ) 2 3 α 8 2 − 2 (α − 1) Γ Γ 2 α − 2 α − 2 ρ0 = 3 < α < 4, 1 2 −α 2 2 2 π Γ k Cn z 2
(8)
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
=
1 2π
2π
2π
0
0
E p ,l0 ( r , ϕ , z ) E *p ,l0 ( r , ϕ ′, z )
2r 2 − 2r 2 cos (ϕ ′ − ϕ ) × exp −il (ϕ − ϕ ′ ) exp − dϕ ′dϕ . ρ 02
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(9)
Received 26 Nov 2014; revised 17 Feb 2015; accepted 23 Feb 2015; published 2 Apr 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.009137 | OPTICS EXPRESS 9140
Substituting Eq. (3) into Eq. (9), we obtain
βl (r , z ) r × ω0
2
2 l0
2 p + l +1 Γ 2 ( p / 2 + l0 + 1) z 2 0 1 + = 2π 2 Γ( p + l0 + 1) Γ 2 ( l0 + 1) z R
− ( p / 2 + l0 +1)
p p z 2r 2 z R2 z R2 r 2 exp F , l 1; − − + 1 1 0 2 2 2 2 2 2 ω0 ( z + izzR ) ω0 ( z R + z ) zR
2
(10)
2r 2 cos (ϕ ′ − ϕ ) 2 r 2 2π 2π × exp − 2 exp −i ( l − l0 )(ϕ − ϕ ′ ) + dϕ ′dϕ . 0 ρ02 ρ0 0 Based on the integral expression [24] 2π
exp[−inϕ
1
+ η cos(ϕ1 − ϕ2 )]dϕ1 = 2π exp(−inϕ 2 ) I n (η ),
(11)
0
where I n (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 p p + l0 + 1) 2 − 2 + l0 +1 2 l0 p 2 z r z 2 2 1 + βl (r , z ) = Γ( p + l0 + 1) Γ 2 ( l0 + 1) z R ω z 0 R (12) 2 2 2 2 2 2 2 p 2r 2r 2r z z r × exp − 2 2 R 2 1 F1 − , l0 + 1; 2 R exp − 2 I l − l0 2 . ω0 z ( z + iz R ) ω0 ( z R + z ) ρ0 ρ0 2
p + l0 + 2
Γ2 (
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 )
∞
l =−∞ 0
2
rd r
β l (r , z )
2
rdr
.
(13)
The received power pl0 is defined as relative power of the OAM mode l0 in the receiver plane (i.e. l = l 0 ) and the crosstalk power pΔl is the relative power in the receiver plane that is found to be in OAM mode l = l 0 +Δl [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 = 5 z R ,
λ = 1550 nm, ω0 = 0.03 m, Cn2 = 10−16 m3−α , l0 = 1 and p = 3, respectively.
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Received 26 Nov 2014; revised 17 Feb 2015; accepted 23 Feb 2015; published 2 Apr 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.009137 | OPTICS EXPRESS 9141
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 .
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 = 5 z R . p ≥ − l 0 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. 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 5 z R , 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.
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Received 26 Nov 2014; revised 17 Feb 2015; accepted 23 Feb 2015; published 2 Apr 2015 6 Apr 2015 | Vol. 23, No. 7 | DOI:10.1364/OE.23.009137 | OPTICS EXPRESS 9142
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.
For p ≥ − l0 , the variations of pl0 and pΔl with 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.
Fig. 3. 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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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.
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 z R 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.
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 pl . 0
Figures 5-6 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
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parameters α . The curve associated with α = 3.67 corresponds to Kolmogorov turbulence. As α approaches 3, the pl0 keeps a constant with the increasing z, but the received power pl0 has a clear reduction if α tends 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 ρ 0 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 nonKolmogorov 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.
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 = 10−17 ,10−16 ,10−15 and 10 −14 m3−α ) with α = 11 / 3 . The pl0 almost remains unchanged in the weak turbulence ( Cn2 = 10−17 m3−α ), descends gradually in the intermediate turbulence ( Cn2 = 10−16 and 10−15 m3−α ) and in the strong turbulence ( Cn2 = 10−14 m3−α ) drops quickly with the increasing propagation distance z..
Physically, Cn2 is a measurement of the strength of the fluctuations in the refractive index. It typically ranges from 10−17 m3−α to 10−14 m3−α representing 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 = 10−17 m3−α (weak turbulence) and
descends C = 10 2 n
−14
gradually m
3 −α
for
Cn2 = 10−16 and
10−15 m3−α (intermediate
turbulence).
At
(strong turbulence), pl0 drops quickly with the increasing propagation
distance z. For z > 1.2 z R , 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 = 10−17 m3−α ). In the cases of Cn2 = 10−16 and 10−15 m3−α , pΔl is enhanced by the
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increase of z. However, there is a distinct feature in the case of Cn2 = 10−14 m3−α . pΔl increases first and reaches the maximum value at z = 1.2 z R . 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, pl0 remains 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).
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