The probability densities of orbital angular momentum (OAM) modes of the autofocusing Airy beam (AAB) carrying power-exponent-phase vortex (PEPV) after passing through the weak anisotropic non-Kolmogorov turbulent atmosphere are theoretically formulated. It is found that the AAB carrying PEPV is the result of the weighted superposition of multiple OAM modes at differing positions within the beam cross-section, and the mutual crosstalk among different OAM modes will compensate the distortion of each OAM mode and be helpful for boosting the anti-jamming performance of the communication link. Based on numerical calculations, the role of the wavelength, waist width, topological charge and power order of PEPV in the probability density distribution variations of OAM modes of the AAB carrying PEPV is explored. Analysis shows that a relatively small beam waist and longer wavelength are good for separating the detection regions between signal OAM mode and crosstalk OAM modes. The probability density distribution of the signal OAM mode does not change obviously with the topological charge variation; but it will be greatly enhanced with the increase of power order. Furthermore, it is found that the detection region center position of crosstalk OAM mode is an emergent property resulting from power order and topological charge. Therefore, the power order can be introduced as an extra steering parameter to modulate the probability density distributions of OAM modes. These results provide guidelines for the design of an optimal detector, which has potential application in optical vortex communication systems.
© 2017 Optical Society of America
The atmosphere can be divided into the following two parts:1) the free atmosphere (above the atmospheric boundary layer) and 2) the atmospheric boundary layer (up to 1–2 km in altitude) . In the beginning, optical turbulence inside the atmospheric boundary layer is considered isotropic, and the isotropic Kolmogorov spectrum model is generally correct within the inertial subrange. However, a wealth of experimental data and literatures [2–5] have shown that atmospheric turbulence is not flat and horizontal due to the presence of very strong positive temperature gradients in the atmosphere . When a laser beam propagates through the anisotropic atmospheric turbulence, it undergoes extra beam spreading, scintillation, beam wander, and angle-of-arrival fluctuation. These unwanted effects strongly degrade the performance in numerous applications such as remote optical sensing, imaging, laser radar and free-space laser communications [6, 7]. Therefore, the persistent challenge driving most research activities is to mitigate the turbulence-induced degradation.
The key point in investigating these issues is determining how to reduce the diffraction. As of today, various methods have been proposed [8–11]. One method is to use a type of beams with special propagation characteristics (i.e., non-diffracting and self-healing) that leaves their transverse intensity distribution quasi-invariant during propagation despite of the severity of the imposed perturbations. Examples of such non-diffracting and self-healing beams include the Bessel, Laguerre-Gaussian and Airy beams.Among these non-diffracting and self-healing beams , has shown that the Airy beam can be controlled with ease, and the peak beam intensity can be delivered to and repositioned on a given target after propagating through disordered or turbulent media. This result opens the possibilities for controlling beam propagation and navigation in turbulent media .
It is well known that phase plays an important role in the quality of beam propagation. Optical vortex beams described by a phase, which carries an orbital angular momentum (OAM) of per photon, can resist turbulence-induced distortions [13, 14]. The vortex Airy beam, an optical vortex embedded within an Airy beam, has therefore attracted the attention of many scientists. In recent years, considerable work has been done to investigate the properties of the vortex Airy beam. The beam spreading of a ring Airy beam vortex beam traveling through an anisotropic vertical atmospheric turbulence is analytically explored and numerically calculated . The beam wander of an Airy beam with a spiral phase in a turbulent atmosphere is investigated in detail. Furthermore, the expressions for the second-order moment, the second central moments are derived . The performance of an vortex Airy beam propagating through moderate-to-strong maritime turbulence is simulated and the spatial coherence radius is derived based on the modified Rytov approximation . The propagation properties of a sharply autofocused ring Airy Gaussian vortex beams is numerically investigated by appropriately selecting the distribution factor . The properties of partially coherent Airy beam propagating in turbulent media is studied . The imposed vortices are able to take advantage of the energy of an Airy beam’s intensity peak . In the papers mentioned above, the optical vortices imposed on an Airy beam are always the canonical vortices in which the phase gradient is constant. Since the propagation dynamics of an optical vortex are influenced by the phase function itself, type of noncanonical vortex that carries a spiral phase that varies nonuniformly with the azimuthal angle is investigated to explore the properties and applications of OAM [21, 22]. More interesting than canonical vortex, the anisotropy of the phase profile can be used as a steering parameter of optical vortex .
The power-exponent-phase vortex (PEPV) beam is a kind of noncanonical vortex that is characterized by a power-exponent-phase. PEPV was proposed by Li , and the autofocusing Airy beam (AAB) carrying PEPV have been experimentally demonstrated. More recently, Lao imposed a PEPV into a Gaussian beam and investigated the properties of the intensity of such a beam on propagation in free space . However, to the best of our knowledge, the AAB carrying PEPV has not yet been used in anisotropic atmospheric turbulence.
In this paper, the Rytov theory is employed to derive an analytical formula for the radial probability density of the OAM mode of the AAB carrying PEPV propagating in the weak anisotropic non-Kolmogorov turbulent atmosphere. Considering the needs for precise tracking and signal reception in certain atmospheric experiments , the radial probability density distribution of signal OAM mode and crosstalk OAM modes of the AAB carrying PEPVs are investigated in detail. We also explore the influence of the power order n on the changes in the mode probability density of the OAM modes. We expect that the work will be particularly meaningful in practical applications for free-space optics communication. In Section 2, the model of the probability density for the AAB carrying PEPV in weak non-Kolmogorov turbulence is established. In Section 3, numerical simulations and graphical outputs related to the effects of topological charges, power order, waist width, and wavelength on the probability densities of signal and crosstalk OAM modes of the AAB carrying PEPV along the radial direction are investigated. Conclusions are presented in Section 4.
2. Theoretical model of the AAB carrying PEPV and its propagation in the turbulent atmosphere
Optical vortex beam for communication is to use OAM modes to encode information for communication . However recent works have also shown that [27, 28], the transmitted signal carrying the particular OAM mode launched by the source may actually be susceptible to spatial aberrations after propagation through atmospheric turbulence. The reason is the turbulence induces OAM-mode scattering and leaks the power contained in a given OAM mode from other neighboring OAM modes. So, the received signal actually includes multiple OAM modes, as indicated in Fig. 1. Therefore, to known the precise probability density distributions of OAM modes at the receiver plane is of great significance. In this section, the probability densities of the OAM modes of the AAB carrying PEPV propagating in the weak anisotropic non-Kolmogorov turbulent atmosphere are formulated under a given generalized anisotropic von Karman spectrum based on the Rytov theory.
The electric fields of the AABs carrying PEPV at the source plane can be expressed as :Fig. 2. Figure 2(d) is the same as that given in , which verifies the results validity.
Within the validity of the paraxial approximation, the electric field distribution of the beam in the receiver plane can be determined, based on the Rytov theory , from the knowledge of the initial electric field distribution:30].
On substituting from Eq. (1) into Eq. (2), we found the diffraction integral in Eq. (2) cannot be exactly evaluated. P. Zhang  develops an accurate description of this integral by introduce a delta ring of amplitude , e.g., . In this situation, at, the amplitude can be expressed as :
By the recall of the formulas of Eqs. (1)-(4), and taking the integrations, we finally obtain the AAB carrying PEPV, at the receiver plane in turbulence-free channel and in the paraxial channel, has the analytical expression:24]
Note that in Eq. (5), the AAB carrying PEPV with a given topological charge v is, in fact, the result of the weighted superposition of multiple propagating OAM beams with phase, where can take a series of integer values, at differing positions within the beam cross-section. As beam wave propagates through the turbulence, the effect of the refractive index fluctuations perturbs the phase of the wave so that it cannot be studied deterministically. Only statistical averages can be considered. In the present cases, we are keen on the average probability density distribution at the receiver plane, which will be discussed below.
The function can be written as a superposition of plane waves with phase as follows :
By combining Eqs. (5)-(8), the mode probability density of the AAB carrying PEPV in paraxial channel is given by:Eq. (9) is given by the expression [17, 34, 35]:30]. In this paper, we use the generalized anisotropic von Karman spectrum to describe the anisotropic non-Kolmogorov turbulence, and takes the form :34].
Equation (13) provides a quantitative description to the anisotropic turbulence induced changes in the compositions of OAM modes of the received AAB carrying PEPV.
In our study, v represents the signal OAM mode. For the designated OAM mode m = v, is defined as the detection probability density of the signal OAM mode at the receiver. A larger value of means more power remains in the signal OAM mode with index m . For , is defined as the crosstalk probability density, which represents the probability that the transmitted OAM state changed from the transmitted channel to the other channels. (This disagrees with , which argues that denotes the OAM state change from the signal channel to the other channels. It is necessary to be clear that , here, has the same meaning as shown in . As described in the literature , OAM modes with different l are mutually orthogonal. As is shown in Eq. (5), the AAB carrying PEPV is the result of the weighted superposition of multiple propagating OAM beams at differing positions within the beam cross-section. On this basis, the propagation behavior of the AAB carrying PEPV in turbulence can be modeled by the superposition of the components with appropriate weights. For an isolated signal OAM mode of the basis set, as illustrated in Fig. 1, the signal purity will be reduced by intermodal crosstalk among data channels induced by turbulence. However, for all components, the mutual crosstalk among different channels will compensate for the distortion of each signal OAM mode and will be helpful in allowing the distributive characteristics of these components to get close to the initial transmitted state. The schematic diagram of the propagation behavior of the AAB carrying PEPV is shown in Fig. 3. We just mention a few components of the basis as an examples (l = 1, 2, 3, 11 and 20) and other values can also be adopted. The component weight mappings are colored differently in this figure to indicate the different component channels.
3. Numerical experiment and discussion
In this section, we will analyze the propagation characteristics of the AAB carrying PEPV in anisotropic atmospheric turbulence using the aforementioned formulae. There is a universal phenomenon that the longer propagating distance and larger structure parameter lead to lower signal OAM mode power and higher crosstalk OAM modes power received. Therefore, we do not discuss the influences of the propagation distance and structure parameter. Instead we mainly devote ourselves to studying the radial probability density distributions of the signal OAM mode and the crosstalk OAM modes of the AAB carrying PEPV under variations in wavelength, waist width, topological charge and power order.
For description convenience in the following, be used to represents the signal OAM mode probability density, and take as the crosstalk OAM modes probability densities, where can take a series positive integer values, (i.e., denotes the probability densities of the crosstalk OAM modes adjacent to the signal OAM mode). In addition, The following parameters of numerical calculations are assumed unless otherwise specified:, , , , and ;turbulence outer scale ; inner scale ; propagation altitude ; wind speed; and propagation distance . The parameters adopted here set as an illustration for theoretical analyses and other values can also be adopted.
The probability density curves in Fig. 4 are plotted as functions of the beam radius. Figure 4(a) shows the probability density curves of the signal OAM modes with n = 1 and 3 and fixed v = 3. There are multiple-peaks in the curves, but most of the energy is restricted to a small region between = 0 and = 0.05 m. We call it effective detection region of the signal OAM mode, and the effective detection region radius corresponds to the radius of the main lobe of the AAB. Under the condition n = 1, the result reduces to that of the canonical Airy vortex beam. Figure 4(b) shows the curves with = 1, 2, 3, 4 and 5, for fixed n = 3 and v = 3. The curves have strong oscillations and the fluctuations fade as increases, which is the same as fractional Bessel Gauss vortex beams given in ,we attribute this result to the nature of Airy beam as a Bessel beam of fractional order 1/3 . Due to the strong oscillations, in the following section, we focus on the propagation characteristics of the envelope of the curve for a convenient description. Furthermore, the magnitude and position of the envelope peak denote the power and the detected region center of the crosstalk OAM modes. In Fig. 4(b), the peaks of the envelope curves move outward and downward clearly as becomes larger. Therefore, the noise OAM modes that matter the most to the signal OAM mode are the immediately adjacent OAM modes ( = 1). Similar to [34, 40], the other cross talk ( = 2, 3, 4 and 5) effects on the detection of the signal OAM mode are ignored. In addition, the peak position of the envelope curve is located at = 0.159 m with = 1, which means there is an observable distance between the effective detection region center of and the detection region center of . This observable distance is helpful for improving the signal detection accuracy at the receiver plane by choosing a proper position.
The changes of the normalized probability density curves and are plotted in Figs. 5(a) and 5(b) for different beam waists = 0.03, 0.05, 0.08 and 0.10 m. It is shown that the normalized and values all increase as the transverse scale increases, which is similar to the performance of the Airy Schell beam given in the fractional Bessel Gauss vortex beams given in . It is worth noting that as larger increases, the slower the curves decays; meanwhile the effective detection region of the signal OAM mode enlarged. In Fig. 5(b), the peak heights of the envelope curves move upward clearly as gets larger, but the peak positions outward slightly. When larger than a certain value (i.e. >0.05), the detection region of signal OAM mode overlaps with that of crosstalk OAM modes and further influences the detection accuracy of the signal OAM mode.
The variation of the normalized and curves against for prescribed = 1550, 980, 632.8 and 532 nm are shown in Fig. 6. Figure 6(a) shows that as the increases, the decay of the curves increase, the peak heights of the curves decrease, the peak positions of the curves drift toward the outward side of , and the widths of the first rings of the curves broaden. As shown in Fig. 6(b), the variation tendency of the curves is similar to that of , but the difference is that it is not just the first ring that is broader but the entire curve that is broader. Therefore, a relatively small and longer are good for separate the detection regions between signal OAM mode and crosstalk OAM mode. On the whole, the curves’ variation tendency in Fig. 6 is consistent with that of the Airy Schell beam given in .
The influence of power order n on the and curves of the receiver plane is examined. In this context, for fixed v = 3, it is clear from Fig. 7(a) that as the power order n is increased, also increases. Figure 7(b) demonstrates that rises in n values will weaken the originally appearing oscillations of the curve tails. Furthermore, these rises in n will initially reduce the magnitudes of the envelope curve peaks, while at n values exceeding 3.5, the magnitudes of the peaks will begin to rise again. We attribute this effect to the balance between the two following aspects.1) As the parameter n increases, the light spot size converges , which can compensate, to a certain extent, for the beam spreading effect  and the smaller probability densities of crosstalk OAM modes are achieved. 2) However when n is increases further (i.e. ), the light concentrates faster , which makes the beam speckle effect induced by atmospheric turbulence increases in strength. Therefore, the effects of turbulence on the probability densities of the OAM modes are worsen. Consequently, taking both of these effects into account, the decreases to a minimum value and then increases with the increase of n. It is worth noting that as the power order n increases, the envelope curve peaks of the curves drift toward the inward side of .
The changes in the curves profile against topological charges are investigated in Fig. 8, by fixing n to 3. As shown in Fig. 8, increases lightly with rising values of topological charges, and these rising values of topological charges will cause the envelope peak values of curves to be reduced obviously; at the same time, the peak positions will drift toward the inward side of .
Based on Figs. 7(b) and 8(b), we find that the envelop curve peak position of curve is an emergent property that depends on the power order and topological charge. We will now examine the influences of the different v and n values on the curves of the AAB carrying PEPV. Figure 9 shows that, for a fixed value of topological charge, the envelop curve peak position of curve moves to the inward side of with increasing power order until it approaches a nearly stable value at approximately = 0.126 m. And the moving rate of the envelope peak position will become slower as the topological charges increases.
In this paper, the Rytov theory is employed to derive an analytic formula for the radial probability densities of the OAM modes of the AAB carrying PEPV propagating in the anisotropic non-Kolmogorov turbulent atmosphere. Numerical calculations have shown that there are multiple-peaks in the signal OAM mode and crosstalk OAM mode probability density curves, but most of the energy is restricted in a really small region between = 0 and = 0.05 m, which corresponds to the radius of the main lobe of the AAB. There is an observable distance between the detection region centers of the signal OAM mode and crosstalk OAM modes. This distance is helpful for improving the signal detection accuracy at the receiver by choosing a proper position. But the detection region of signal OAM mode broadens clearly as the gets larger and becomes shorter, which would lead to the detection regain of signal OAM mode overlaps with that of the crosstalk OAM modes and adversely affect the detection accuracy for the signal OAM mode. Therefore, a relatively small and longer should be used in real applications. The radial probability density distribution of the signal OAM mode does not change obviously with the topological charge variation; but it will be greatly enhanced with the increase of power order. And rises in the power order values will weaken the originally appearing oscillations of the curve tails. Furthermore, it is found that, for a fixed value of topological charge, the detection region center of crosstalk OAM mode with = 1 moves to the inward side of with increasing values of power order until it approaches a nearly stable value at approximately = 0.126 m, and the moving rate will become slower as the topological charge increases. It has been revealed that power order can be used as a steering parameter to modulate propagation characteristics of the AAB carrying PEPV.
National Natural Science Foundation of China (NSFC) (61431010, 61621005).
The authors would like to thank Xiuxiang Chu in School of Science in Zhejiang Agriculture and Forestry University, Peng Li in School of Science in Northwestern Polytechnical University, and Xinzhong Li in School of Physics and Engineering in Henan University of Science and Technology for the helpful discussions.
References and links
2. G. Grechko, A. Gurvich, V. Kan, S. Kireev, and S. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992). [CrossRef]
3. C. Robert, J.-M. Conan, V. Michau, J.-B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisotropic refractive index fluctuations spectrum in the stratosphere from balloon-borne observations of stellar scintillation,” J. Opt. Soc. Am. A 25(2), 379–393 (2008). [CrossRef] [PubMed]
6. M. A. Cox, C. Rosales-Guzmán, M. P. Lavery, D. J. Versfeld, and A. Forbes, “On the resilience of scalar and vector vortex modes in turbulence,” Opt. Express 24(16), 18105–18113 (2016). [CrossRef] [PubMed]
7. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015). [CrossRef]
8. J. Wang, S. Zhu, H. Wang, Y. Cai, and Z. Li, “Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence,” Opt. Express 24(11), 11626–11639 (2016). [CrossRef] [PubMed]
9. Y. Ren, Z. Wang, G. Xie, L. Li, Y. Cao, C. Liu, P. Liao, Y. Yan, N. Ahmed, Z. Zhao, A. Willner, N. Ashrafi, S. Ashrafi, R. D. Linquist, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, “Free-space optical communications using orbital-angular-momentum multiplexing combined with MIMO-based spatial multiplexing,” Opt. Lett. 40(18), 4210–4213 (2015). [CrossRef] [PubMed]
10. H. Jian, D. Ke, L. Chao, Z. Peng, J. Dagang, and Y. Zhoushi, “Effectiveness of adaptive optics system in satellite-to-ground coherent optical communication,” Opt. Express 22(13), 16000–16007 (2014). [CrossRef] [PubMed]
11. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53(7), 537–578 (2003). [CrossRef]
14. T. Wang, J. Pu, and Z. Chen, “Beam-spreading and topological charge of vortex beams propagating in a turbulent atmosphere,” Opt. Commun. 282(7), 1255–1259 (2009). [CrossRef]
15. D. Zhi, R. Tao, P. Zhou, Y. Ma, W. Wu, X. Wang, and L. Si, “Propagation of ring Airy Gaussian beams with optical vortices through anisotropic non-Kolmogorov turbulence,” Opt. Commun. 387, 157–165 (2017). [CrossRef]
19. H. T. Eyyuboğlu and E. Sermutlu, “Partially coherent Airy beam and its propagation in turbulent media,” Appl. Phys. B 110(4), 451–457 (2013). [CrossRef]
20. Y. Zhang, Q. Zhang, X. Ma, Z. Jiang, T. Xu, S. Wu, and X. Wu, “Measurement of Airy-vortex beam topological charges based on a pixelated micropolarizer array,” Appl. Opt. 55(32), 9299–9304 (2016). [CrossRef] [PubMed]
21. G.-H. Kim, H. J. Lee, J.-U. Kim, and H. Suk, “Propagation dynamics of optical vortices with anisotropic phase profiles,” J. Opt. Soc. Am. B 20(2), 351–359 (2003). [CrossRef]
25. Y. Zhu, L. Zhang, and Y. Zhang, “Spiral spectrum of Airy–Schell beams through non-Kolmogorov turbulence,” Chin. Opt. Lett. 14(4), 042101 (2016). [CrossRef]
26. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]
27. M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20(12), 13195–13200 (2012). [CrossRef] [PubMed]
28. Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38(20), 4062–4065 (2013). [CrossRef] [PubMed]
29. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005)
30. L. G. Wang, W. W. Zheng, and L. Q. Wang, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A, Pure Appl. Opt. 11(6), 065703 (2009). [CrossRef]
31. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011). [CrossRef] [PubMed]
32. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Courier Corporation, 1964)
33. 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]
34. M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016). [CrossRef] [PubMed]
35. Y. Zhu, L. Zhang, Z. Hu, and Y. Zhang, “Effects of non-Kolmogorov turbulence on the spiral spectrum of Hypergeometric-Gaussian laser beams,” Opt. Express 23(7), 9137–9146 (2015). [CrossRef] [PubMed]
36. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic, 2015).
37. C. Chen, H. Yang, S. Tong, and Y. Lou, “Changes in orbital-angular-momentum modes of a propagated vortex Gaussian beam through weak-to-strong atmospheric turbulence,” Opt. Express 24(7), 6959–6975 (2016). [CrossRef] [PubMed]
38. J. Gao, Y. Zhang, W. Dan, and Z. Hu, “Turbulent effects of strong irradiance fluctuations on the orbital angular momentum mode of fractional Bessel Gauss beams,” Opt. Express 23(13), 17024–17034 (2015). [CrossRef] [PubMed]
39. J. Rogel-Salazar, H. A. Jiménez-Romero, and S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A 89(2), 204–208 (2014). [CrossRef]
40. J. Gao, Y. Zhu, D. Wang, Y. Zhang, Z. Hu, and M. Cheng, “Bessel–Gauss photon beams with fractional order vortex propagation in weak non-Kolmogorov turbulence,” Photonics Res. 4(2), 30 (2016). [CrossRef]
41. Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014). [CrossRef] [PubMed]