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Abstract
We demonstrate that the spiral spectrum (also known as orbital angular momentum spectrum) of a Laguerre-Gaussian (LG) beam with topological charge (TC) l is asymmetrically broadened propagating through moderate-to-strong atmospheric turbulence, even the statistics of turbulence is isotropic. This phenomenon is quite different from that predicted in weak turbulence where the spiral spectrum of a disturbed LG beam is symmetric with respect to its TC number l. An explicit analytical expression of the spiral spectrum of the LG beam with l = 1 is derived based on the extend Huygens-Fresnel integral and quadratic approximation, which is used to illustrate the transition scenarios of the spiral spectrum from symmetry to asymmetry in weak-to-strong turbulence. The physical mechanism for the asymmetric spiral spectrum in moderate-to-strong turbulence is thoroughly discussed. Our results are confirmed by the multi-phase screen numerical simulations and are consistent with the experimental results reported in Phys. Rev. A 105, 053513 (2022) [CrossRef] and Opt. Lett. 38, 4062 (2013) [CrossRef] .
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Vortex beams refer to light beams carrying a helical phase front exp(${\rm i}l\varphi$), where $l$ is the topological charge and $\varphi$ is the azimuthal angle. It was demonstrated by Allen $et$ $al$. in 1992 that Laguerre-Gaussian (LG) beams with the topological charge $l$, which is the most known class of the vortex beams, possess a well-defined orbital angular momentum (OAM) $l\hbar$ per photon, where $\hbar$ is the reduced Plank constant [1]. Therefore, the vortex beams are commonly referred as OAM beams. Owing to the unique phase structure, the vortex beams display doughnut intensity profile with zero on-axis intensity. Driven by their distinctive characteristics, they have made substantial developments in optical imaging, optical trapping, optical sensing and quantum science [2–10]. The vortex beams of different OAM modes are inherently orthogonal, providing a new degree of freedom for information multiplexing. Therefore, different data streams can be loaded into corresponding OAM modes with the same wavelength, dramatically increasing the information transmission capacity in optical communications [11–16].
However, when the OAM beams are applied to free-space optical communications, the presence of atmospheric turbulence will take random perturbations on both the amplitude and phase of the OAM beams, which leads to the modal crosstalk between neighboring OAM modes [17–21]. Hence, the knowledge of the interaction of turbulence and OAM modes is crucial. Paterson studied the diffusion behavior of OAM modes in isotropic Kolmogorov turbulence based on a purely phase perturbation model [17], which treats the effect of turbulence as a random phase screen at the receiver, hereafter, referred to as the single phase screen(SPS) method. Since then, a great deal of work have been devoted to investigating the OAM mode diffusion with different types of vortex beams in isotropic or anisotropic turbulence [22–28]. Recently, a new perspective on the interaction of OAM modes and turbulence by calculating the OAM of atmospheric turbulence itself was studied in [29]. It was found that a pure OAM mode diffuses uniformly to both sides with its eigenmode, which means that the degenerate spiral spectrum of the OAM mode is symmetric with respect to its eigenmode. However, these studies are all carried out under the assumption of the SPS method, which strictly speaking is only valid in the region of weak fluctuations. In practice, the turbulence is gradually accumulated and becomes stronger if the propagation distance is long or if the refractive index of atmosphere fluctuates very violent. A question naturally arises: when turbulence is strong, does it have an extraordinary effect on OAM mode degeneration?
In this paper, we revisit the spiral spectrum of LG beams in isotropic Kolmogorov turbulence based on the extended Huygens-Fresnel (eHF) principle that is applicable to both the weak and strong turbulence cases [30–32]. Closed-form expressions for the mutual coherence function and spiral spectrum of the LG beam with radial index 0 and azimuthal index $l$ = 1 are derived in the receiver after propagating through atmospheric turbulence. Our results show that in moderate-to-strong turbulence, the spiral spectrum turns into asymmetry due to the additional effects caused by the moderate or strong turbulence. The diffused OAM modes on the near-zero side are weighted more than those on the opposite side. The physical mechanism leading to asymmetric spiral spectrum is discussed in depth, and the results are confirmed by the multi-phase screen (MPS) simulation method.
2. Mutual coherence function of a Laguerre-Gaussian beam propagating in weak-to-strong atmospheric turbulence
The electric field of the LG beam with radial index 0 and azimuthal index $l$ in the source plane is expressed as
When the LG beam propagates in atmosphere, the turbulence-induced phase distortion deteriorates the purity of the LG mode. Energy is thus transferred to its adjacent OAM modes, resulting in an expansion of the spiral spectrum. To evaluate the spiral spectrum of a laser beam propagation through the atmospheric turbulence, two propagation modes that are the SPS method and eHF method are commonly used. The former [17,21–27] is assumed that the transmission channel from the transmitter to the receiver is in vacuum without turbulence, and the cumulative effect of the turbulence on the channel is equivalent to random phase perturbations on the beam at the receiver, and the diagram is shown in Fig. 1(a). Note that this model is restricted to weak turbulence regime. The latter [33,34] considers the interactions of the beam and turbulence during propagation, as shown in Fig. 1(b). This model is known to be valid from weak to strong turbulence.
According to the eHF principle, the mutual coherence function (MCF) of a LG beam in the output plane after propagating in atmospheric turbulence is given by [33,34]
To evaluate Eq. (2), we assume that the turbulence is isotropic and obeys the Kolmogorov spectrum, i.e., $\Phi \left (\kappa \right )=0.033C_{n}^{2}\kappa ^{-11/3}$ , where $C_{n}^{2}$ is the structure constant describing the strength of the turbulence, and $\kappa$ is the spatial frequency. By applying the quadratic approximation within the observation distance being far greater than the outer scale, the last exponential term in Eq. (2) can be expressed as [34]
It is hard to find the explicit analytical expression for high-order LG beams ($|l|>1$) from the integral formula of Eq. (2), while for $l$ = 1 we obtain an explicit expression for the MCF in the receiving plane, after integrating over ${\bf r}_{1}$ and ${\bf r}_{2}$,
In Eq. (4), we divide the MCF into two terms. The first term $\Gamma _{\rm LG}$ can be regarded as the MCF of the LG beam with $l$ = 1 propagating in relatively weak turbulence. In fact, if we neglect the turbulence-induced beam spreading in $\Gamma _{\rm LG}$, i.e., removing the third term of Eq. (7) in $w(z)$ and $R(z)$, and further assuming $\delta (z)\approx \rho _{0}$, $\Gamma _{\rm LG}$ reduces to the form that obtained from the SPS model [35]. The second term $\Gamma _{\rm t}$ is an additional effect with appreciation only in moderate-to-strong turbulence. This is because the coherence length $\rho _{0}$ in the denominator of Eq. (6) is relatively large in weak turbulence, resulting in a small effect of $\Gamma _{\rm t}$ on the MCF. Nevertheless, as the turbulence strength increases, $\Gamma _{\rm t}$ gradually becomes an important role in the MCF.
The spiral spectrum refers to the energy distribuition (or the detection probability) of OAM modes contained in a light field. The parameter $a_m$ that is proportional to the energy of a specified OAM mode $m$ can be calculated by the following integral [17]
Substituting Eqs. (5) and (6) into Eq. (11) and setting $\rho _{1}=\rho _{2}=\rho$, and integrating over $\rho$, $\theta _{1}$ and $\theta _{2}$, $a_m$ has found the analytical form (see the appendix A)
with3. Numerical results
In this section, we numerically investigate the propagation behavior of the spiral spectrum of the LG beam with $l$ = 1 in weak-to-strong turbulence based on the formula derived in Section 2, and reveal the physical mechanism of the asymmetric spectrum expansion when the turbulence is relatively strong. Then, the results obtained in Eqs. (12)–(15) are compared by the MPS simulation method. Finally, our results extend to other high-order LG beams, demonstrating that the turbulence-induced the asymmetric spiral spectrum is universal.
When using the Kolmogorov spectrum to study a laser beam propagating on a path of length $z$, it is customary to distinguish the strength of the turbulence by values of the $Rytov$ $variance$ $\sigma _{\rm R}^{2}=1.23C_{n}^{2}k^{7/6}z^{11/6}$, which is the intensity scintillation index of a plane wave. Weak fluctuations fall in the region $\sigma _{\rm R}^{2}<1$. Moderate fluctuations are characterized by $\sigma _{\rm R}^{2}\sim 1$, and strong fluctuations are associated with $\sigma _{\rm R}^{2}>1$.
Figures 2(a1)–(a4) illustrate the spiral spectra of the LG beam with $l$ = 1 at the propagation distance $z$ = 1 km for different strengths of turbulence, calculated from Eq. (12). For comparison, the corresponding results calculated by Eq. (13) (similar with the results obtained from the SPS method) and the additional contribution calculated by Eq. (14) are shown in Figs. 2(b1)-(b4) and Figs. 2(c1)-(c4), respectively. The spectra are normalized by the total energy $\pi \omega _{0}^{4}/4$ carried by the beam. Other parameters used in the calculations are $\omega _{0}$ = 0.02 m, $\lambda$ = 1550 nm. When the turbulence is weak, i.e., $\sigma _{\rm R}^{2}=0.1$, the spiral spectra obtained from the eHF method and SPS method are almost identical, symmetric about $l$ = 1. This is because the additional effect caused by the strong turbulence can be ignored. As shown in Fig. 2(c1), the value of the major mode weight $m$ = 0 is about 0.002. The effect on the spiral spectrum is less than 2%. The presence of the negative mode weight in $c_{m}$ implies that the second term in Eq. (4) does not satisfy the nonnegative condition. Actually, the two terms are separated in the MCF is only for convenience of analysis.
Fig. 2. (a1)-(a4) Sprial spectra of the LG beam with $l$ = 1 at the receiver ($z$ = 1 km) after propagating in isotropic atmospheric turbulence from weak to strong fluctuations using the eHF method. (b1)-(b4) Corresponding spiral spectra calculated from the SPS model, but the turbulence-induced beam spreading is considered. (c1)-(c4) Additional effects of the spiral spectra induced due to strong turbulence.
As the turbulence strength increases, the spiral spectra are more dispersive [see Fig. 2(a2)-(a4)]. Nevertheless, the spectra predicted by the eHF method become asymmetric. The energy spread on both sides with respect to $l$ = 1 is non-uniform, resulting in the mode weight $m$ = 0 greater than $m$ = 2. This phenomenon is quite different from that predicated by the SPS method, where the spiral spectra are always symmetric about $l$ = 1, no matter how strong the turbulence. The physical mechanism leading to the asymmetric distribution is shown in Figs. 2(c2)-(c4). The additional effect caused by the strong turbulence is symmetric about $m$ = 0, not $m$ = 1. Furthermore, this effect becomes more pronounced with increasing strength of turbulence. As a result, the spiral spectra becomes asymmetric in moderate or strong turbulence. For an instance, in the case of $\sigma _{\rm R}^{2}=1.99$, the additional values of mode weights $m$ = 0, 1 and 2 are about 0.04, 0.015, 0.002, respectively, resulting in the maximum mode weight at $m$ = 0.
To assess the asymmetry of the spiral spectrum, a deviation parameter $D_n$ defined as $D_n=(a_{l-n}-a_{l+n})/a_{l+n}$ is introduced. The zero value of $D_n$ means that the two pair components $l-n$ and $l+n$ are symmetric. Figure 3 shows the variation of $D_n$ with the Rytov variance. The other parameters are the same as in Fig. 2(a4). One can see that the value of $D_n$ for the two nearest pairs $n=1$ or 2 first increases with the increase of $\sigma _{\rm R}^2$, reaches a maximum value, then decreases as $\sigma _{\rm R}^2$ further increases. For the other three pairs $n = 3$, 4 or 5, the value of $D_n$ first decreases from a value of zero, reaches a minimum value. After that, the behavior of $D_n$ is similar to that of $n = 1$ or 2 for the further increase of $\sigma _{\rm R}^2$. It implies that in weak or relative strong turbulence regime, the average energy of the OAM component $m = -2$, at the left side of $l = 1$, may be smaller than that of $m = 4$ at the right side of $l$, and so does the pairs (-3, 5) and (-4, 6). When the turbulence is very strong, i.e., $\sigma _{\rm R}^2>20$, It can be seen from Fig. 3 that the values of $D_n$, $n = 1, 2 \ldots 5$, are almost the same, and they all decrease and approach zero as the turbulence further increases. This implies that the average energy of the OAM components between two sides of the eigenmode $l$ = 1 will balance in extremely strong turbulence. Under this circumstance, the spiral spectrum becomes uniform distribution. For instance, the theoretical calculation values $a_m$ from $m$ = -5 to 5 for $\sigma _{\rm R}^2=100$ are all about 0.0029.
Fig. 3. Dependence of the deviation parameter $D_{n}$ on the Rytov variance for the nearest five pairs of OAM modes with respect to $l$ = . The varying range of $\sigma _{\rm R}^2$ for (a) [0, 35] and for (b) [35, 135].
To verify the validity of our results, the numerical simulations of the LG beams propagating in atmospheric turbulence are performed using the MPS method to compare the results obtained from the eHF method. In the MPS method [36,37], the turbulence is modeled as a collection of random phase screens that follow the desired turbulent statistics. These phase screens are placed along the transmission channel at equal intervals $\Delta z = z / N_{t}$, where $\Delta z$ is the distance between two adjacent screens and $N_{t}$ is the number of the screens. There is no turbulence in the space between any two adjacent screens. With the help of split-step Fourier algorithm, we can calculate a series of electric field realizations for the LG beams passing through the turbulence. For each realization, the OAM modes are analyzed and the spiral spectrum is obtained by averaging over 1000 realizations. The numerical grid comprises 512 $\times$ 512 elements covering a 40 cm $\times$ 40 cm area transverse to propagation axis. 10 phase screens are inserted in the $z$ = 1 km channel and the Kolmogorov spectrum $\Phi (\kappa )=0.033C_{n}^{2}\kappa ^{-11/3}$ is adopted as the power spectrum of refractive index fluctuations.
In Fig. 4, the spiral spectra of the LG beam with $l$ = 1 propagating in different turbulence strengths are presented using the MPS method (pink histograms). For comparison, the corresponding results calculated from the eHF method are also plotted (green histograms). The spectra in moderate-to-strong turbulence obtained by the two methods look somewhat different. This may be due to the different propagation scenarios between the two methods. In the eHF method, the field at the receiver is considered as a coherent superposition of spherical waves emitted at the transmitter, so the coherence length $\rho _{0}=(0.545C_{n}^{2}k^{2}z)^{-3/5}$ of a spherical wave in turbulence is applied. In addition, a quadratic approximation is applied in the calculation. In the MPS method, the field is decomposed into a superposition of plane waves (angular spectrum) propagating between two phase screens. The coherence length of a plane wave in turbulence is $\rho _{\rm pl}=(0.423C_{n}^{2}k^{2}z)^{-3/5}$, which is different from that of a spherical wave [34]. Despite the differences, both methods demonstrate that the spectra turn into asymmetrical distribution in moderate-to-strong turbulence. In the case $\sigma _{\rm R}^2=1.99$, the maximum value on the OAM spectrum is located at the OAM component $m$ = 0, not the eigenmode $l$ = 1 [see Fig. 4(d)].
Fig. 4. Comparison of the spiral spectra of the LG beam with $l$ = 1 at different turbulence strengths between the eHF method and MPS method. The parameters used in the simulation are $\omega _{0}$ = 0.02 m, $\lambda$ = 1550 nm, $z$ = 1 km.
Although we only acquire the explicit expressions for the MCF and spiral spectrum of the LG beam with $l$ = 1 in weak-to-strong turbulence, for high-order LG beams, it is also possible to numerically calculate their spiral spectra by dividing the MCF into two parts as shown in Eq. (4), where the first term is the contribution from weak turbulence and the second one is the additional effect caused by the moderate-to-strong turbulence. Figure 5 presents the numerical results of the spiral spectra of the LG beams with $l$ = 3 and $l$ = 5 in moderate turbulence using the MPS method (pink histograms) and eHF method (green histograms). The results of the SPS method and the additional effect are also illustrated. One finds that the spiral spectra of the high-order LG beams calculated from the SPS method are always symmetric about $l$. Nevertheless, the spiral spectra introduced by the additional effect are lack of symmetry for $l=3$ and 5. It shows in Fig. 5(a) and 5(d) that the value of am for $m = l$ – 1 or $l$ – 2 are larger than that for $m = l$ + 1 or $l$ + 2, resulting in the asymmetric distribution of the spiral spectra for the high-order LG beams.
Fig. 5. (a), (d) Numerical and simulation results of the spiral spectra of the LG beams with $l$ = 3 and $l$ = 5 in moderate turbulence. (b), (e) The spiral spectra calculated from the SPS model. (c), (f) The additional effects caused by the strong turbulence. The turbulence parameter and the beam parameters used in the calculation are $\omega _{0}$ = 0.02 m, $\lambda$ = 1550 nm, $z$ = 1 km, $C_{n}^{2}$ = 5$\times 10^{-13}$ ${\rm m}^{-2/3}$ and $\sigma _{\rm R}^{2}$ = 1.
In the case of LG beams with negative topological charge, the spiral spectra can be obtained by the flipping the spiral spectra of the corresponding LG beams with positive charge with respect to $m$ = 0. Therefore, the spiral spectra will also become asymmetric in moderate or strong turbulence. An exception is the case $l$ = 0, which means that the light beam reduces to a Gaussian beam. Using Eqs. (2) and (3), the MCF of the Gaussian beam propagating through the atmospheric turbulence has found the analytical form
Hence, using Eq. (11), we obtain the expression for the spiral spectrum
Evidently, it follows from Eq. (17) the spiral spectrum is always symmetric about $l$ =0, no matter how strong the turbulence is.
Based on the analysis stated above, the transition of the spiral spectrum of a LG beam (except for $l = 0$) from weak to strong turbulence is that the spectrum first undergoes symmetric broadening with respect to its eigenmode $l$ in weak turbulence regime. As the turbulence strength increases, the spectrum gradually becomes asymmetric, and the mode energy diffused to the side of the OAM component $m = 0$ is larger than that diffused to the opposite side. However, in the extreme case, i.e., the turbulence is strong enough, the deviation parameter $D_n$ with different $n$ all approach zero, as shown in Fig. 3(b). Therefore, we speculate that the two sides of the spiral spectrum with respect to $l$ will be balanced again and the spectrum becomes uniform distribution, which means that the average energy of different OAM components in the beam is equal to each other, as predicted in [17,18,21]. This is because in extremely strong turbulence, the light beam in the receiver plane is almost spatially incoherent, resulting in the uniform distribution of the spiral spectrum.
4. Discussion and conclusion
We note that some previous experimental reports have shown that the spiral spectra of vortex beams after propagating in relatively strong turbulence are asymmetric [20,21], but the authors have not clarify the physical mechanism and we believe that the results were not caused by the experimental measurement errors. As shown in Fig. 5 in [20], the experimental results demonstrated that in medium-strong turbulence, the spiral spectra of the LG beams with $l=0$ and 3 are biased towards near the zero OAM mode side, whereas in weak turbulence the spectra are almost symmetric with respect to their topological charges $l$. It is emphasized here that although a phase screen was used in Ref. [20] to simulate the atmospheric turbulence, the beams passed through the phase screen twice. Our numerical results (not shown here) demonstrate that the spiral spectrum of a LG beam will still become asymmetric after passing through at least two phase screens. The results in Fig. 4(b) in Ref. [21] also showed that the mode weight $m$ =$-2$ is larger than $m$ =$-4$ after the vortex beam with $l$ =$-3$ propagates in oceanic turbulence.
Why are there additional effects leading to the asymmetric distribution of the spiral spectrum of a LG beam in moderate-strong turbulence? We note that in the SPS method the on-axis zero intensity of the LG beam is undisturbed and the average intensity maintains a doughnut shape regardless of how strong the structure constant $C_{n}^{2}$ is. However, in practical cases, the on-axis zero intensity (phase singularity) disappears as the turbulence is strong. The beam profile gradually transforms from a doughnut shape to a Gaussian profile when the turbulence is strong enough, as shown in Fig. 6. Therefore, in this propagation process, we can consider the light field of a LG beam to be composed of two parts. One is the undisturbed LG field and another is field noise induced by the atmospheric turbulence. The field noise is random fluctuated as the fluctuations of the turbulence, however, the average intensity profile of the field noise is nearly Gaussian shape and this part becomes dominant role in strong turbulence. This is the reason why the beam profile of the LG beam turns into Gaussian shape if the turbulence is strong enough. More importantly, the spiral spectrum of the average field noise is not symmetric with respect to the eigenmode of the LG beam, resulting in the asymmetry of the spiral spectrum.
Fig. 6. Density plots of the average intensity of the LG beam with $l$ = 1 for different turbulence strengths at propagation distance $z$ = 1 km. The beam parameters are $\omega _{0}$ = 0.02 m, $\lambda$ = 1550 nm.
As a summary, we have investigated the OAM mode diffusion of LG beams propagating in weak-to-strong isotropic Kolmogorov turbulence based on the eHF method and quadratic approximation. An analytical expression for the spiral spectrum of the LG beam with $l=1$ propagating through weak-to-strong atmospheric turbulence has been derived. The expression divides the effects of turbulence on the spiral spectrum into two parts: one is from weak turbulence and another is from strong turbulence. The results reveal that the symmetry of the spiral spectrum is broken owing to the additional effect induced by the strong turbulence, which is quite different from that predicted the SPS method that is only effective in weak turbulence. For high-order LG beams that analytical expressions for the spiral spectra are hard to found, we have studied the behavior of the LG beams with $l=3$ and $l=5$ using the MPS method and numerical integrating for eHF principle. The results obtained from two methods all show that the symmetries of the spiral spectra are also broken in moderate-to-strong turbulence. Our findings provide a new perspective for understanding the interaction of OAM with random media and will be useful in OAM-based free-space optical communications.
Appendix A: Derivation of the spiral spectrum from Eq. (11)
Let us first pay attention to the integrand of the first term $\Gamma _{\rm LG}\left (\rho,\theta _{1},\rho,\theta _{2},z\right )$ in Eq. (11), which is
It is shown in Eq. (18) that the integrand is only as a function of $\theta _{2}-\theta _{1}$ with respect to two azimuthal angles. For simplicity, the sum and difference angles: $\theta _{\rm d}=\theta _{2}-\theta _{1}$ and $\theta _{\rm s}=\left (\theta _{2}+\theta _{1}\right )/2$ are introduced. Then the Eq. (18) becomes
In order to evaluate Eq. (19), the following expansion formula is applied
By applying the integral formula [38, p.707]
Following the same procedure, the integration of the second term $\Gamma _{\rm t}\left (\rho,\theta _{1},\rho,\theta _{2},z\right )$ can be expressed by making variable changing $\theta _{\rm d}=\theta _{2}-\theta _{1}$ and $\theta _{\rm s}=\left (\theta _{2}+\theta _{1}\right )/2$
With the help of Eq. (20), the result becomes after integrating over $\theta _{\rm s}$ and $\theta _{\rm d}$,
Equation (25) has found the analytical expression as follows
In the derivation of Eq. (26), the following additional integral formula is involved [38, p.703]
Funding
National Key Research and Development Program of China (2019YFA0705000, 2022YFA1404800); National Natural Science Foundation of China (11974218,, 12074310, 12192254, 12304326, 92250304); Local Science and Technology Development Project of the Central Government (YDZX20203700001766); China Postdoctoral Science Foundation (2022M721992); Natural Science Foundation of Shandong Province (ZR202211180168); Open Fund of the Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications (2020GDSGXCG08).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, et al., “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]
2. L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005). [CrossRef]
3. A. Jesacher, S. Furhapter, S. Bernet, et al., “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94(23), 233902 (2005). [CrossRef]
4. L. Paterson, M. P. MacDonald, J. Arlf, et al., “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001). [CrossRef]
5. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef]
6. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, et al., “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013). [CrossRef]
7. W. Zhang, J. Gao, D. Zhang, et al., “Free-space remote sensing of rotation at the photon-counting level,” Phys. Rev. Appl. 10(4), 044014 (2018). [CrossRef]
8. Y. Zhang, M. Agnew, T. Roger, et al., “Simultaneous entanglement swapping of multiple orbital angular momentum states of light,” Nat. Commun. 8(1), 632 (2017). [CrossRef]
9. A. H. Ibrahim, F. S. Roux, M. McLaren, et al., “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88(1), 012312 (2013). [CrossRef]
10. N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic orbital angular momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91(1), 012345 (2015). [CrossRef]
11. J. Wang, J. Y. Yang, I. M. Fazal, et al., “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]
12. Y. Yan, G. Xie, M. P. J. Lavery, et al., “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014). [CrossRef]
13. Y. Li, K. Morgan, W. Li, et al., “Multi-dimensional QAM equivalent constellation using coherently coupled orbital angular momentum (OAM) modes in optical communication,” Opt. Express 26(23), 30969–30977 (2018). [CrossRef]
14. T. Lei, M. Zhang, Y. Li, et al., “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015). [CrossRef]
15. A. E. Willner, H. Huang, Y. Yan, et al., “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015). [CrossRef]
16. J. Wang, J. Liu, S. Li, et al., “Orbital angular momentum and beyond in free-space optical communications,” Nanophotonics 11(4), 645–680 (2022). [CrossRef]
17. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005). [CrossRef]
18. 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]
19. B. Rodenburg, M. P. J. Lavery, M. Malik, et al., “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012). [CrossRef]
20. Y. Ren, H. Huang, G. Xie, et al., “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]
21. D. Yang, Z. D. Hu, S. Wang, et al., “Influence of random media on orbital angular momentum quantum states of optical vortex beams,” Phys. Rev. A 105(5), 053513 (2022). [CrossRef]
22. Y. Wang, L. Bai, J. Xie, et al., “Spiral spectrum of high-order elliptic Gaussian vortex beams in a non-Kolmogorov turbulent atmosphere,” Opt. Express 29(11), 16056–16072 (2021). [CrossRef]
23. J. Zeng, X. Liu, C. Zhao, et al., “Spiral spectrum of a Laguerre-Gaussian beam propagating in anisotropic non-Kolmogorov turbulent atmosphere along horizontal path,” Opt. Express 27(18), 25342–25356 (2019). [CrossRef]
24. X. Yan, L. Guo, M. Cheng, et al., “Probability density of orbital angular momentum mode of autofocusing Airy beam carrying power-exponent-phase vortex through weak anisotropic atmosphere turbulence,” Opt. Express 25(13), 15286–15298 (2017). [CrossRef]
25. Y. Li, Y. Xie, and B. Li, “Probability of orbital angular momentum for square Hermite–Gaussian vortex pulsed beam in oceanic turbulence channel,” Results Phys. 28, 104590 (2021). [CrossRef]
26. Y. Zhu, X. Liu, J. Gao, et al., “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014). [CrossRef]
27. L. Yu, B. Hu, and Y. Zhang, “Intensity of vortex modes carried by Lommel beam in weak-to-strong non-Kolmogorov turbulence,” Opt. Express 25(16), 19538–19547 (2017). [CrossRef]
28. C. H. Acevedo, M. Eshaghi, and A. Dogariu, “Propagation of asymmetric optical vortex beams through turbulence and evolution of their OAM spectra,” J. Opt. Soc. Am. A 40(12), 2135–2145 (2023). [CrossRef]
29. A. Klug, I. Nape, and A. Forbes, “The orbital angular momentum of a turbulent atmosphere and its impact on propagating structured light fields,” New J. Phys. 23(9), 093012 (2021). [CrossRef]
30. F. E. S. Vetelino and R. J. Morgan, “Model validation of turbulence effects on orbital angular momentum of single photons for optical communication,” Proc. SPIE 7685, 76850R (2010). [CrossRef]
31. C. Chen, H. Yang, S. Tong, et al., “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]
32. H. Lv, C. Ren, and X. Liu, “Orbital angular momentum spectrum of partially coherent vortex beams in slant atmospheric turbulence,” Infrared Phys. Technol. 105, 103181 (2020). [CrossRef]
33. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11(6), 1399–1406 (1972). [CrossRef]
34. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, (SPIE, Bellingham, Washington, 1998).
35. M. Cheng, L. Guo, J. Li, et al., “Propagation of an optical vortex carried by a partially coherent Laguerre–Gaussian beam in turbulent ocean,” Appl. Opt. 55(17), 4642–4648 (2016). [CrossRef]
36. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB, (SPIE, Bellingham, Washington, 2010).
37. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008). [CrossRef]
38. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, (Academic Press, New York, 2014).
