## 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 [16–18]. Toselli *et al*. presented a non-Kolmogorov power spectrum using generalized exponent and amplitude factor [19, 20]. When the exponent value $\alpha $ 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 ${l}_{0}$, refractive-index structure parameter of turbulence ${C}_{n}^{2}$, wavelength of beam $\lambda $ and propagation distance *z* on OAM modes of the HyGG beams are discussed in detail. In the case of the hollowness parameter $p=-\left|{l}_{0}\right|$, we find that the received power ${p}_{l}{}_{{}_{0}}$ drops with the decrease of *p*. On the other hand, there are little effects on the received power ${p}_{l}{}_{{}_{0}}$ when *p* is positive. The smaller OAM quantum number ${l}_{0}$ and longer wavelength $\lambda $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]

*p*is the hollowness parameter of the HyGG beam [8]; ${l}_{0}$ corresponds to the number of OAM. $r=\left|r\right|,r=(x,y)$ is the two-dimensional position vector in the source plane. $\phi $ denotes the azimuthally angle and ${\omega}_{0}$ is the beam waist. The normalized constant ${C}_{pl0}$ is given by ${C}_{p{l}_{0}}=\sqrt{{2}^{p+\left|{l}_{0}\right|+1}/\left[\pi \Gamma (p+\left|{l}_{0}\right|+1)\right]}\Gamma (p/2+\left|{l}_{0}\right|+1)/\Gamma (\left|{l}_{0}\right|+1)$, which ensures that the HyGG laser beam carries a finite power as long as $p\ge -\left|{l}_{0}\right|$; $\Gamma (x)$ 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

*z*is the propagation distance of the launch beam. ${\psi}_{1}(r,\phi ,z)$ is a complex phase perturbation of the wave propagating through turbulence. ${E}_{p,{l}_{0}}\left(r,\phi ,z\right)$ denotes the HyGG laser mode at the plane in free space. In the paraxial case, ${E}_{p,{l}_{0}}\left(r,\phi ,z\right)$ takes the form [10]

*k*is the wave number of light related to the wavelength $\lambda $ by $k=2\pi /\lambda $. ${F}_{1}(a,b;x)$ 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 $\mathrm{exp}(il\phi )$ [23]

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

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

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

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

Based on the integral expression [24]

*n*order. We have the mode probability density of finding one photon in the signal mode $|l\u3009$

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

The received power ${p}_{{l}_{0}}$ is defined as relative power of the OAM mode ${l}_{0}$ in the receiver plane (i.e. $l=l{}_{0}$) and the crosstalk power ${p}_{\Delta l}$ is the relative power in the receiver plane that is found to be in OAM mode $l=l{}_{0}+\Delta l$ [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 $\alpha =11/3$, $z=5{z}_{R}$, $\lambda =1550$ nm, ${\omega}_{0}=0.03$ m, ${C}_{n}^{2}={10}^{-16}$ ${m}^{3-\alpha}$, ${l}_{0}=1$ and *p* = 3, respectively.

The characteristics of the HyGG beams brought by two parameters *p* and ${l}_{0}$ as well as the corresponding effects on ${p}_{{l}_{0}}$ and ${p}_{\Delta l}$ are discussed in detail and shown in Figs. 1-3. The variations of ${p}_{{l}_{0}}$ and ${p}_{\Delta l}$, 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 $z=5{z}_{R}$. $p\ge -\left|l{}_{0}\right|$ 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, ${l}_{0}=0$ is meaningless. The sign of ${l}_{0}$ 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 ${p}_{{l}_{0}}$ is symmetric with ${l}_{0}=0$. The received power ${p}_{{l}_{0}}$ decreases along the positive ${l}_{0}$ axis. As *p* increases, the received power ${p}_{{l}_{0}}$ increases monotonically with fixed ${l}_{0}$ except for ${l}_{0}=0$. In the case of ${l}_{0}=0$, ${p}_{{l}_{0}}$ 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 ${l}_{0}=0$ [8, 10]. Figure 1(b) shows that the crosstalk power (the noise of receive signals in the receiver plane) ${p}_{\Delta l}$ is symmetric with $l={l}_{0}$. The larger value of *p* begins to offer smaller ${p}_{\Delta l}$ for $\Delta l>0$. As expected, ${p}_{\Delta l}$ decreases rapidly and becomes zero quickly with the increase of $\Delta l$ ($\Delta l>0$) 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 ${p}_{{l}_{0}}$ and the crosstalk power ${p}_{\Delta l}$ 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 ${p}_{{l}_{0}}$ 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 ${p}_{{l}_{0}}$ reaches about 10% as $l={l}_{0}=3$. As shown in Fig. 2(b), the crosstalk power ${p}_{\Delta l}$ occurs mainly between two adjacent OAM states, that is, $\Delta l=1$. Thus, for $\Delta l=2,3,4\cdot \cdot \cdot $, the influence of crosstalk in the receiver plane is negligible under the weak turbulence.

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

The wavelength selection of source light in free space communication is an important issue for the improvement of ${p}_{{l}_{0}}$. Thus, Fig. 4 illustrates the effects of wavelength $\lambda $ on ${p}_{{l}_{0}}$ and ${p}_{\Delta l}$ for the HyGG beams. Because the Rayleigh range is a function of wavelength $\lambda $, 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 ${p}_{{l}_{0}}$ of the HyGG beams with $\lambda $ *=* 1550 nm is 1.23 times higher than that in the case of $\lambda =532$ 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 ${p}_{\Delta l}$ with respect to *z* is flatter in the longer $\lambda $ cases. This conclusion provides a significant guidance for choosing the light source.

Figures 5 and 6 present how do the various parameters of atmospheric turbulence affect the received power ${p}_{{l}_{0}}$ and the crosstalk power ${p}_{\Delta l}$. Figure 5(a) is a plot of the received power ${p}_{{l}_{0}}$ versus the propagation distance *z* for the different non-Kolmogorov turbulence parameters $\alpha $. The curve associated with $\alpha =3.67$ corresponds to Kolmogorov turbulence. As$\alpha $ approaches 3, the ${p}_{{l}_{0}}$keeps a constant with the increasing *z*, but the received power ${p}_{{l}_{0}}$has a clear reduction if $\alpha $tends toward 4. Figure 5(a) is also consistent with [15, 28] that the smaller $\alpha $ comes with the larger ${p}_{{l}_{0}}$ in the receiver plane for any given propagation distance. The reason for this is that the spatial coherence radius ${\rho}_{0}$ approaches infinity when $\alpha $ gets close to 3. It implies that the interruption of the turbulence on the OAM modes is very weak, and a high ${p}_{{l}_{0}}$ is acquired. However, when $\alpha $ 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 ${p}_{{l}_{0}}$ decreases with $\alpha $ approaching to 4. The relationship between the crosstalk power ${p}_{\Delta l}$ and the index $\alpha $ of non-Kolmogorov turbulence is revealed by Fig. 5(b). For any given *z*, the crosstalk power ${p}_{\Delta l}$ of the HyGG beams increases rapidly when $\alpha $ changes from 3.07 to 3.97. As $\alpha $ increases, the crosstalk power increases at the given propagation distance *z* for the same reason that the received power decreases.

Physically, ${C}_{n}^{2}$ is a measurement of the strength of the fluctuations in the refractive index. It typically ranges from ${10}^{-17}$${m}^{3-\alpha}$ to ${10}^{-14}$${m}^{3-\alpha}$representing the conditions of “weak turbulence” to “strong turbulence” [22]. The variations of ${p}_{{l}_{0}}$ and ${p}_{\Delta l}$ of the HyGG beams versus the propagation distance *z* for different ${C}_{n}^{2}$ are plotted in Fig. 6. As shown in Fig. 6(a), ${p}_{{l}_{0}}$ almost remains unchanged in the case of ${C}_{n}^{2}={10}^{-17}$ ${m}^{3-\alpha}$(weak turbulence) and descends gradually for ${C}_{n}^{2}={10}^{-16}$and ${10}^{-15}$${m}^{3-\alpha}$(intermediate turbulence). At ${C}_{n}^{2}={10}^{-14}$${m}^{3-\alpha}$(strong turbulence), ${p}_{{l}_{0}}$ drops quickly with the increasing propagation distance *z*. For $z>1.2{z}_{R}$, the downtrend of ${p}_{{l}_{0}}$ develops slowly. In Fig. 6(b), the crosstalk power ${p}_{\Delta l}$ is almost zero when the HyGG beams travel through the weak turbulence region (${C}_{n}^{2}={10}^{-17}$${m}^{3-\alpha}$). In the cases of ${C}_{n}^{2}={10}^{-16}$ and ${10}^{-15}$${m}^{3-\alpha}$, ${p}_{\Delta l}$ is enhanced by the increase of *z*. However, there is a distinct feature in the case of ${C}_{n}^{2}={10}^{-14}$${m}^{3-\alpha}$. ${p}_{\Delta l}$ 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 $\Delta l=2,3,4\cdot \cdot \cdot $, the conspicuous increase of ${p}_{\Delta l}$, which makes the decline in ${p}_{\Delta l}$ at $\Delta l=1$, occurs in atmospheric link. ${p}_{{l}_{0}}$ and ${p}_{\Delta l}$ 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 ${p}_{{l}_{0}}$~z curve at $p=-\left|{l}_{0}\right|$. In the case of *p*>0, ${p}_{{l}_{0}}$remains almost unchanged with the increase of *p*. The received power ${p}_{{l}_{0}}$ increases with the decrease of ${l}_{0}$ at the same propagation distance. The increases of ${C}_{n}^{2}$ and $\alpha $ result in the degradation of the received power ${p}_{{l}_{0}}$. In the strong turbulent and long distance region, the decline tendency of ${p}_{{l}_{0}}$ curve is weakened. The longer wavelength of the HyGG beam contributes more to the received power ${p}_{{l}_{0}}$ 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]