## Abstract

In free-space optical (FSO) communications, the orbital angular momentum (OAM) multiplexing/demultiplexing of Bessel beams perturbed by atmospheric turbulence is of great significance. We used the Gerchberg-Saxton algorithm without a beacon beam to compensate for the aberrant helical phase of the Bessel beam distorted by the turbulent atmosphere. The optical vortex Dammann axicon grating was applied for the simultaneous measurement of the intensities of the demodulated spectra of the OAM modes of the Bessel beams disturbed by atmospheric turbulence. The experimental results demonstrate that the distorted phase of the Bessel beam can be compensated and the mode purity of the target OAM mode is enhanced from 0.85 to 0.92 in case of weak turbulence. Our results will improve the quality of the OAM modes of Bessel beam (de)multiplexing in FSO communication systems.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Vortex beams have helical wavefronts with phase exp(*ilφ*), where *φ* denotes the azimuthal angle, *l* an integer denoting the topological charge, and a photon carries an amount of *lℏ* of orbital angular momentum (OAM) with ℏ being the Planck constant [1–3]. Because the OAM modes carried by the vortex beams are mutually orthogonal, these OAM modes become the new degrees of freedom of the beam, which can be used by the mode-division multiplexing technique in free-space or underwater optical (FSO) communications to increase the capacity of the information transport [4–7]. In FSO communications, the main challenge is atmospheric turbulence [8]. The helical wavefronts of the vortex beams propagating in free space are distorted by the turbulent atmosphere. Some of the power of the incident OAM state of the vortex beam leakage to the neighboring states generate the spectra of the OAM states at the receiver plane. Thereafter, the purity of the OAM modes decreases, and the quality of the optical communications is significantly influenced by the generated spectra of the OAM modes [9]. The effect of random aberrations in atmospheric turbulence on the OAM state of a single photon in FSO communications has been investigated theoretically and experimentally, and the probabilities of the signal OAM states and the spectra of the OAM states distorted by atmospheric turbulence are both quantified and measured [9,10]. The OAM modes of the 11-dimensional space perturbed by Kolmogorov turbulence are studied experimentally in FSO communication systems. It is shown that the degradation of the channel capacity induced by turbulence can be mitigated by increasing the spacing between the signal OAM modes [11]. In kilometer-length FSO communications, the power remains in the signal OAM states. The power spreading to the neighboring states increases as the OAM states induced by the turbulence become moderately strong [12].

Recently, an adaptive compensation system was applied in FSO communications to reduce the disturbance caused by the turbulent atmosphere [13–17]. The OAM mode was applied in both channels in the unidirectional and bidirectional cases in the free-space quantum communication link, the entanglement of twisted photons was protected, and the quantum-symbol-error-rate was increased by the adaptive optics system [13,14]. In addition, the methods of post-correction, pre-correction and phase conjugation using adaptive optics were applied to mitigate the distortion of the OAM states by atmospheric turbulence [15]. The wavefront of the data-carrying OAM beam distorted by atmospheric turbulence was corrected by the hybrid input–output algorithm of the adaptive optics system, assisted by a probe beam of the vortex beam. The simulation results showed that the mode purity was improved, and that the crosstalk of the adjacent OAM states were decreased [16]. Using another method, the distorted phases of a collection of collinear multiple OAM beams, propagating through atmospheric turbulence, were corrected by an adaptive optics system. The experimental results demonstrated that the crosstalk induced by the turbulence effects on the neighboring OAM states was significantly reduced [17].

Bessel beams carrying OAM states have two unique properties: self-healing and diffraction-free [18,19]. Therefore, high-order Bessel beams can be applied in FSO communication systems to increase their capacity. Bessel beams carrying different OAM states were realized in FSO and millimeter-wave communication links to overcome the obstruction at the transmission link [20–24]. The spectra of the OAM states were generated when the single-mode Bessel beam carrying the OAM state propagated through the turbulent atmosphere or oceanic turbulence [25–28]. The OAM modes can be discriminated using different optical elements when the collinear high-order Bessel beams are used for (de)multiplexing in FSO communications [20,22,29]. The qualities of the signals of high-order Bessel beams, carrying a high-speed signal and propagating through the turbulent atmosphere and/or obstruction, are improved by closed-loop adaptive optics [30,31]. The helical wavefront of the Bessel beam, distorted by the turbulent atmosphere, was corrected by the pre-correction procedure. Moreover, assisted by the probe Gaussian beams, the mode purity and crosstalk of the inter-channels are improved [32]. In this letter, we use the method of adaptive correction, without the use of a probe beam, to reconstruct the vortex phase of the Bessel beam disturbed by the turbulent atmosphere. The optical vortex Dammann axicon grating was applied to simultaneously measure the spectra of the OAM modes of the Bessel beams in a turbulent atmosphere.

## 2. Principle

#### 2.1 Bessel beam

The high-order Bessel beam can be approximately generated using the holograph method by multiplying the phase of an incoming Gaussian beam by an axicon helical phase [20,33]. Using cylindrical coordinates, the phase function of the Bessel beam can be expressed as $f(r,\varphi ) = \exp ( - 2\pi ir/{r_0} + il\varphi )$. When a Gaussian beam with a wavelength of 1550nm is scattered by a holograph loaded at the spatial light modulator (SLM) [33,34], a Bessel beam is generated. The electric field function of the Bessel beam can be expressed as follows:

where ($\rho ,\varphi$) denotes the radial coordinate at the received plane,*J*(·) the

_{l}*l*th-order Bessel function, and the radius of the Bessel beam is determined by the parameter

*r*

_{0}with

*w*=0.766

_{l}*lr*

_{0}. When the high-order Bessel beam propagates through the turbulent atmosphere, the helical vortex phase evolves from a single OAM state to the spectra of the OAM states (see Fig. 1). The quality of the (de)multiplexing of the collinear OAM states carrying the signal is significantly influenced by this evolution. An adaptive compensation system was used to mitigate the disturbance caused by the turbulent atmosphere. Adaptive compensation system is a system that use machine and algorithm to correct for random media induced the wavefront aberration in real time [8].

Figure 1 schematically shows simultaneous measurement of the mode purity of the spectra of the OAM states of the Bessel beam in a turbulent atmosphere. A single Bessel beam carrying the OAM state propagated through atmospheric turbulence. The mode purity decreased as the power of the original OAM state spread to the neighboring OAM states. An adaptive compensation system is used to increase the mode purity of the input Bessel beam. As shown in Fig. 1(a), the shape of the intensity distribution is non-Gaussian when the Bessel beam propagates through a turbulent atmosphere. The demodulated beam shape of the Bessel beam is distorted and the position of the central spot is shifted, if the beam, without adaptive compensation, is demodulated by the phase conjugate holograph of the input Bessel beam. This phenomenon implies that the mode purity of the OAM state, induced by the turbulent atmosphere, decreases. As shown in Fig. 1(b), the intensity distribution of the demodulated beam at the receiver plane has a Gaussian beam shape when adaptive phase compensation with the Gerchberg-Saxton (GS) algorithm is used. Clearly, the mode purity of the OAM state, disturbed by the atmospheric turbulence, can be improved by adaptive compensation system.

#### 2.2 Phase screen

We used the phase-screen method to simulate the Bessel beam, propagating through a turbulent atmosphere [35]. The evolution characteristics of the phase and intensity of the beam can be obtained using this method. The power spectral density function for the von Karman turbulence spectrum can be expressed as follows [8]:

where $h({\kappa _x},{\kappa _y})$ denotes a mean value of 0 and variance of 1. The phase screen was divided into grids with ${N_x} \times {N_y}$. The values of the discretized parameters in Eq. (3) are ${x_i} = {n_x}\Delta x$, ${y_i} = {n_y}\Delta y$, ${\kappa _x} = {m_x}\Delta {\kappa _x}$, ${\kappa _y} = {m_y}\Delta {\kappa _y}$, $\Delta {\kappa _x} = 2\pi /({N_x}\Delta x)$, and $\Delta {\kappa _y} = 2\pi /({N_y}\Delta y)$, where ${n_x}$, ${n_y}$, ${m_x}$ and ${m_y}$ are integers, $\Delta x$ and $\Delta y$ denote the sampling intervals in the *x* and *y* directions in the spatial domain, $\Delta {\kappa _x}$ and $\Delta {\kappa _y}$ denote the sampling intervals in the *x* and $y$ directions in the wavenumber domain, respectively, and $\Delta x = X/{N_x}$ $\Delta y = Y/{N_y}$ with *X* and *Y* for the size of the phase screen

#### 2.3 Adaptive optics

The helical phase is distorted when the Bessel beam propagates under atmospheric turbulence. The mode purity of the OAM states of the Bessel beam cannot remain the same as it has been distorted when the additional phase is coupled to the original phase [9], and adaptive technology can be used to reduce the phase disturbance by the turbulent atmosphere. Wavefront sensors is/isn’t used in an adaptive correction technology system. We adopted the GS phase recovery algorithm without a wavefront sensor to deal with the distortion of the phase [36]. Moreover, the GS algorithm is not complicated and is based on an iterative scheme, and a beacon light is not necessary [37].

Figure 2 shows the schematic diagram of the GS algorithm. The disturbed phase is corrected by reversing the phase information from the intensity distribution at the input and output planes. First, the amplitude distribution carrying the spiral phase of the target light field is selected, thereafter, the Fourier transform of the target light is performed. Second, in the frequency domain, the target amplitude spectrum $U({\kappa _x},{\kappa _y})$ is replaced by a distorted light field, and the phase spectrum $\psi ({\kappa _x},{\kappa _y})$ is unchanged. Third, the amplitude spectrum ${U^{\prime}}(x,y)$ and phase spectrum $\varphi (x,y)$ are obtained by IFFT from the results of the second step. Finally, in the spatial domain, the generated amplitude spectrum ${U^{\prime}}(x,y)$ is replaced by the target amplitude. The light field, disturbed by atmospheric turbulence, was then obtained from the IFFT of the last step. Thereafter, the disturbed field starts in the next cycle. The reconstructed phase $\varphi ({x,y} )$ with the compensation phase of atmospheric turbulence is then obtained through multiple iterations.

## 3. Experimental results

Bessel beams with OAM states can be used for (de)multiplexing in FSO communications to increase spectral efficiency and capacity. The OAM state is disturbed by atmospheric turbulence when the Bessel beam propagates; thus, the quality of the communication system decreases. However, the OAM state distortion of the Bessel beam can be compensated by an adaptive compensation system, and the corrected Bessel beam is then detected at the receiver plane. The mode purity of the OAM state carried by the Bessel beam was enhanced.

Figure 3 shows the experimental setup for the correction of the distorted wavefront, and the measurements of the intensities of the spectra of the OAM modes of the Bessel beam. A laser at a wavelength of 1550 nm was sent to a high-power erbium-doped fiber amplifier (EDFA) to amplify the beam. The amplified beam propagates through the polarization controller (PC) and is fed into the collimator (Col.) to produce a collimated Gaussian beam. A high-order Bessel beam was generated when the collimated Gaussian beam propagating through the SLM1 loaded the spiral phase multiplication axicon phases. The distortion OAM state was detected when the Bessel beam was sent to SLM2 and the various phase screens of the atmospheric turbulence were loaded with different atmospheric structure constants at the propagation distance z=1000m. The distorted phase of the Bessel beam was then corrected by SLM3, which enables the Bessel beam to load the compensation phase of the distorted wavefront phase calculated by the GS algorithm. Thereafter, the improved Bessel beam was demodulated to the Gaussian beam by the conjugate spiral phase loaded on SLM4. The mode purity and crosstalk from the neighboring OAM states can be analyzed by the intensity of the demodulated Gaussian beam recorded by the CCD.

Figure 4 shows the intensity of the spectra of the OAM states with and without compensation demodulated by Bessel beams with various topological charges. The spectra of the OAM states are generated when the signal Bessel beam with a topological charge of 2 propagates in the turbulent atmosphere. Subsequently the quality of the FSO communication system is influenced by the crosstalk that comes from the spectra of the OAM states. We use the method of phase compensation for the reduction of crosstalk using an adaptive compensation system. The mode purity of the spectra of the OAM states was measured using Bessel beams with topological charges of 2, 1, 0, -1, -2, -3, -4, -5, and -6, respectively.

From Fig. 4, we observe that the Bessel beam, carrying the topological charge of 2 and propagating in the turbulent atmosphere, has the main power at the receiver plane. The shape of the intensity of the demodulated beam, according to the conjugate phase method, is not a standard Gaussian beam without further phase compensation (see the first line of a1, b1 and c1 in Fig. 4 at the opposite spiral phase for topological charge 2). The coupling efficiency of the single-mode fiber decreases when the shape of the Gaussian beam is irregular. The quality of the demodulated Gaussian beam can be improved by phase compensation (see the second line of a1, b1 and c1 in Fig. 4). The power of the signal Bessel beam with topological charge 2 in a turbulent atmosphere leak to the adjacent OAM states (see the charts of a2, b2 and c2 in the right column in Fig. 4). Thereafter, the single OAM state of the Bessel beam generates the spectra of the OAM states of the Bessel beams. The quality of optical communications is significantly influenced by this phenomenon. Fortunately, the mode purity can be enhanced by phase compensation, which enables a decrease in the power of the adjacent OAM states of the Bessel beams. After the correction by the adaptive system, the applied phase compensation leads to the following calculated values for the mode purity of the target OAM: 92%, 92% and 90%, respectively. These values correspond to the following structural constants of the turbulent atmosphere: $C_n^2 = 1 \times {10^{ - 16}}{\textrm{m}^{ - 2/3}}$, $C_n^2 = 1 \times {10^{ - 15}}{\textrm{m}^{ - 2/3}}$ and $C_n^2 = 1 \times {10^{ - 14}}{\textrm{m}^{ - 2/3}}$, respectively. The corrected mode purities were comparable with those without phase compensation at 87%, 85% and 83%, respectively.

Figure 5 shows the simultaneous measurement of the intensity of the spectra of the OAM states with and without phase compensation demodulated by the optical vortex Dammann axicon grating. All the diffraction spots of the Dammann grating carry equal power. Because of this property, the redesigned Dammann grating can be used to (de)multiplex the collinear OAM states in FSO communications [12,38,39]. The optical vortex Dammann axicon grating, resulting from the redesigned grating is the phase of the Dammann grating multiplied by the vortex and axicon phase. The resulting phase function can be expressed as $g(r,\varphi ) = \sum\nolimits_{ - m}^{ + m} {{C_m}[im(2\pi x/d) + l\varphi ]\exp ( - i2\pi r/{r_0})}$, where *m* denotes the grating diffraction order, *d* the period of the blazed grating, and $|{{C_m}} |$ the weight coefficient with equal values. From Fig. 5, the Bessel beam carrying topological charges of -3, -2, -1, 0, 1, 2, and 3 can be simultaneously demodulated by this redesigned grating. Therefore, this system can be used for the simultaneous measurement of the spectra of the OAM of the Bessel beams in a turbulent atmosphere with and without phase compensation. Figure 5 shows the intensity spectra of the OAM states carried by the signal Bessel beam, and the crosstalk Bessel beams are simultaneously detected when the Bessel beam with topological charge 1 propagates through the turbulent atmosphere. The spot of the demodulated signal Bessel beam with phase compensation becomes brighter than the spot without phase compensation. The mode purities of the signal OAM with phase compensation were 92%, 92% and 90%, for values $C_n^2\textrm{ = }1 \times {10^{ - 16}}{\textrm{m}^{ - 2/3}}$, $C_n^2\textrm{ = }1 \times {10^{ - 15}}{\textrm{m}^{ - 2/3}}$ and $C_n^2\textrm{ = }1 \times {10^{ - 14}}{\textrm{m}^{ - 2/3}}$ of the strength of the turbulent atmosphere, respectively. These values are comparable to those without phase compensation, that is, 90%, 88% and 88%, respectively. The bar chart shows that the mode purity of the signal Bessel beam is enhanced when an adaptive compensation system is used. From Fig. 4 and 5, we observe that the optical vortex Dammann axicon grating can be used for the simultaneous measurement of the spectra of the OAM states carried by the Bessel beams, if the adaptive compensation system is used to mitigate the wavefront distortion of the Bessel beam induced by the turbulent atmosphere.

## 4. Conclusion

In conclusion, we used the vortex Dammann axicon grating and an adaptive compensation system for the simultaneous measurement of the intensity distributions of the spectra of the OAM carried by the Bessel beams propagating in a turbulent atmosphere. The adaptive technology uses a GS algorithm to mitigate or compensate for the wavefront aberration induced by atmospheric turbulence. This adaptive system, without beacon light, is a simple platform that is easy to implement. The experimental results demonstrate that the power of the signal Bessel beam can be increased by an adaptive compensation system. Moreover, the mode purity of the signal Bessel beam can be enhanced at the receiver plane, and the crosstalk coming from the neighboring OAM states of the Bessel beams can be decreased. These results may be very useful for FSO communications.

## Funding

National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (NSFC) 91750201, 11974219, 11974218, 12174227,12074005); Innovation Group of Jinan (2018GXRC010); Shandong Provincial Natural Science Foundation of China (ZR2019MA028); Local Science and Technology Development Project of the Central Government (YDZX20203700001766); Shandong Provincial Key Laboratory of Optics and Photonic Devices (K202010).

## 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, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef]

**2. **I. Gianani, A. Suprano, T. Giordani, N. Spagnolo, F. Sciarrino, D. Gorpas, V. Ntziachristos, K. Pinker, N. Biton, and J. Kupferman, “Transmission of vector vortex beams in dispersive media,” Adv. Photonics **2**(3), 1–8 (2020). [CrossRef]

**3. **S. Fu, Y. Zhai, J. Zhang, X. Liu, R. Song, H. Zhou, and C. Gao, “Universal orbital angular momentum spectrum analyzer for beams,” PhotoniX **1**(1), 19 (2020). [CrossRef]

**4. **J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics **6**(7), 488–496 (2012). [CrossRef]

**5. **A. Forbes, M. de Oliveira, and M. R. Dennis, “Structured light,” Nat. Photonics **15**(4), 253–262 (2021). [CrossRef]

**6. **Y. Chen, W. Shen, Z. Li, C. Hu, Z. Yan, Z. Jiao, J. Gao, M. Cao, K. Sun, and X. Jin, “Underwater transmission of high-dimensional twisted photons over 55 meters,” PhotoniX **1**(1), 5–11 (2020). [CrossRef]

**7. **Z. Qiao, Z. Wan, G. Xie, J. Wang, L. Qian, and D. Fan, “Multi-vortex laser enabling spatial and temporal encoding,” PhotoniX **1**(1), 13 (2020). [CrossRef]

**8. **L. C. Andrews and R. L. Philips, *Laser Beam Propagation through Random Media* (SPIE, 2005).

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

**10. **K. Wang, M. Zhang, J. Tang, L. Wang, L. Hu, X. Wu, W. Li, J. Di, G. Liu, and J. Zhao, “Deep learning wavefront sensing and aberration correction in atmospheric turbulence,” PhotoniX **2**(1), 8 (2021). [CrossRef]

**11. **M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. 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]

**12. **T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. **4**(3), e257 (2015). [CrossRef]

**13. **C. Liu, K. Pang, Z. Zhao, P. Liao, R. Zhang, H. Song, Y. Cao, J. Du, L. L. Li, H. Song, Y. Ren, G. Xie, Y.-F. Zhao, J. Zhao, S. M. H. Rafsanjani, A. N. Willner, J. H. Shapiro, R. W. Boyd, M. Tur, and A. E. Willner, “Single-end adaptive optics compensation for emulated turbulence in a bi-directional 10-mbit/s per channel free-space quantum communication link using orbital-angular-momentum encoding,” Research **2019**, 1–10 (2019). [CrossRef]

**14. **N. Leonhard, G. Sorelli, V. N. Shatokhin, C. Reinlein, and A. Buchleitner, “Protecting the entanglement of twisted photons by adaptive optics,” Phys. Rev. A **97**(1), 012321 (2018). [CrossRef]

**15. **M. P. J. Lavery, “Vortex instability in turbulent free-space propagation,” New J. Phys. **20**(4), 043023 (2018). [CrossRef]

**16. **X. Yin, H. Chang, X. Cui, J. Ma, Y. Wang, G. Wu, L. Zhang, and X. Xin, “Adaptive turbulence compensation with a hybrid input–output algorithm in orbital angular momentum-based free-space optical communication,” Appl. Opt. **57**(26), 7644–7650 (2018). [CrossRef]

**17. **Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. **39**(10), 2845–2848 (2014). [CrossRef]

**18. **J. Durnin, J. J. Miceli, and J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**(15), 1499–1501 (1987). [CrossRef]

**19. **M. Mclaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. **5**(1), 3248 (2014). [CrossRef]

**20. **Y. Yuan, T. Lei, S. Gao, and X. Yuan, “Parallel detection of OAM states carried by coaxial Bessel beams,” IEEE Photonics Technol. Lett. **28**(3), 315–318 (2016). [CrossRef]

**21. **N. Mphuthi, L. Gailele, I. Litvin, A. Dudley, Roelf Botha, and A. Forbes, “Free-space optical communication link with shape-invariant orbital angular momentum Bessel beams,” Appl. Opt. **58**(16), 4258–4264 (2019). [CrossRef]

**22. **S. Li, S. Chen, C. Gao, A. E. Willner, and J. Wang, “Atmospheric turbulence compensation in orbital angular momentum communications: Advances and perspectives,” Opt. Commun. **408**, 68–81 (2018). [CrossRef]

**23. **N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. **6**(1), 22082 (2016). [CrossRef]

**24. **J. Du, S. Li, Y. Zhao, Z. Xu, L. Zhu, P. Zhou, J. Liu, and J. Wang, “Demonstration of M-ary encoding/decoding using visible-light Bessel beams carrying orbital angular momentum (OAM) for free-space obstruction-free optical communications,” in Optical Fiber Communication Conference & Exhibition (Optical Society of America, 2015), paper M2F.4.

**25. **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]

**26. **M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photon. J. **8**(1), 7901411 (2016). [CrossRef]

**27. **H. Zhang, W. Ding, P. Fu, X. Liu, Y. Gao, Y. Gao, Y. Cai, and Y. Yuan, “Reducing orbital angular momentum crosstalk of the Bessel gaussian beam for underwater optical communications,” J. Opt. **22**(6), 065702 (2020). [CrossRef]

**28. **S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photon. Res. **4**(5), B1–B4 (2016). [CrossRef]

**29. **A. Gatto, M. Tacca, P. Martelli, P. Boffi, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. **13**(6), 064018 (2011). [CrossRef]

**30. **S. Chen, S. Li, Y. Zhao, J. Liu, L. Zhu, A. Wang, J. Du, L. Shen, and J. Wang, “Demonstration of 20-Gbit/s high-speed Bessel beam encoding/decoding link with adaptive turbulence compensation,” Opt. Lett. **41**(20), 4680–4683 (2016). [CrossRef]

**31. **S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. **7**(1), 43233 (2017). [CrossRef]

**32. **S. Fu, T. Wang, Z. Zhang, Y. Zhai, and C. Gao, “Pre-correction of distorted Bessel–Gauss beams without wavefront detection,” Appl. Phys. B **123**(12), 275 (2017). [CrossRef]

**33. **J. A. Davis, J. Guertin, and D. M. Cottrell, “Diffraction-free beams generated with programmable spatial light modulators,” Appl. Opt. **32**(31), 6368–6370 (1993). [CrossRef]

**34. **D. Peng, Z. Huang, Y. Liu, Y. Chen, F. Wang, S. A. Ponomarenko, and Y. Cai, “Optical coherence encryption with structured random light,” PhotoniX **2**(1), 6 (2021). [CrossRef]

**35. **J. Recolons and F. Dios, “Accurate calculation of phase screens for the modeling of laser beam propagation through atmospheric turbulence,” Proc. SPIE **5891**, 51–62 (2005). [CrossRef]

**36. **S. Fu, S. Zhang, T. Wang, and C. Gao, “Pre-turbulence compensation of orbital angular momentum beams based on a probe and the Gerchberg-Saxton algorithm,” Opt. Lett. **41**(14), 3185–3188 (2016). [CrossRef]

**37. **J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature **491**(7423), 232–234 (2012). [CrossRef]

**38. **N. Zhang, X. C. Yuan, and R. E. Burge, “Extending the detection range of optical vortices by Dammann vortex gratings,” Opt. Lett. **35**(20), 3495–3497 (2010). [CrossRef]

**39. **Y. Yuan, T. Lei, Z. Li, Y. Li, S. Gao, Z. Xie, and X. Yuan, “Beam wander relieved orbital angular momentum communication in turbulent atmosphere using Bessel beams,” Sci. Rep. **7**(1), 42276 (2017). [CrossRef]