## Abstract

Orbital angular momentum (OAM) mode crosstalk induced by atmospheric turbulence is a challenging phenomenon commonly occurring in OAM-based free-space optical (FSO) communication. Recent advances have facilitated new practicable methods using abruptly autofocusing light beams for weakening the turbulence effect on the FSO link. In this work, we show that a circular phase-locked Airy vortex beam array (AVBA) with sufficient elements has the inherent ability to form an abruptly autofocusing light beam carrying OAM, and its focusing properties can be controlled on demand by adjusting the topological charge values and locations of these vortices embedded in the array elements. The performance of a tailored Airy vortex beam array (TAVBA) through atmospheric turbulence is numerically studied. In a comparison with the ring Airy vortex beam (RAVB), the results indicate that TAVBA can be a superior light source for effectively reducing the intermodal crosstalk and vortex splitting, thus leading to improvement in the FSO system performance.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Light, carrying an orbital angular momentum (OAM) of $l\hslash $ per photon, has a helical phase pattern with an azimuthal phase that ranges from 0 to 2π*l*, where *l* denotes the topological charge with any integer value. This infinite set contains mutually orthogonal OAM states corresponding to different values of *l*. Such an OAM based on multiplexing can potentially increase the system capacity [1–3].

However, refractive index fluctuations in the atmospheric channel associated with turbulence give rise primarily to irradiance fluctuations, beam wander, beam spreading and loss of spatial coherence of the optical wave [4–6], and the energy received from the desired OAM state will be coupled to other neighboring OAM states. Moreover, a pioneering study indicates that the vortex splitting induced by turbulence results in an offset in the average OAM measured after propagation over the FSO link [7], which will affect the precision of signal reception. Therefore, one of the persistent challenges in improving the OAM-based FSO link performance is to mitigate the crosstalk and vortex splitting induced by atmospheric turbulence. Recently, an effective scheme based on a spherical concave mirror (SCM) to mitigate the turbulence effects has been introduced [8]. In this experiment, the SCM at the receiver side of the FSO link plays a role in collecting the Gaussian beams refracted by the turbulence and focusing them onto the photodetector. Conceiving, the possibility of a beam is able to focus its power right before a target without the presence of an external focusing element, the FSO communication system can potentially be simplified. Fortunately, such beam known as abruptly autofocusing beams have been demonstrated [9–12]. To generate an abruptly autofocusing beam, a programmable spatial light modulator (SLM) is required to modulate the phase distribution of the input laser beam with a Gaussian spatial distribution at the transmitter. Therefore, for a given link span, the SCM-based and autofocusing beam-based systems are no less complex than each other. However, with the variation in link length, in the SCM scheme, the SCM radius should be accordingly changed. However, the auto-focusing scheme is easy to adopt by redesigning the computer-generated holograms, and moving or replacing any system components is not required. In addition, a previous work [13] has also shown that the focusing property of the light wave in the turbulence channel helps to weaken turbulence-induced spreading of the beams. It is therefore reasonable to infer that the abruptly autofocusing beams can be used to mitigate the turbulence effects in an FSO communication system with relatively low complexity.

A ring Airy vortex beam (RAVB), as a typical representative of the abruptly autofocusing beam family, has been systematically investigated in the area of atmospheric propagation [14, 15]. It has been shown that the abruptly autofocusing property, which weakens the beam spreading effect induced by atmospheric turbulence, is what increases the strength of the beam wander effect which aggravates the OAM channel crosstalk [15]. Earlier studies have also shown that the beam wander ratio of the focused beam is larger than that of the collimated beam in the case of a finite outer scale presence [16]. Therefore, for an abruptly autofocusing light source, therefore, balancing the beam wander and beam spreading effects is of great significance in mitigating crosstalk in FSO communication.

In this work, a new strategy is proposed to resolve the problem by controlling the focusing properties of the autofocusing light. On the one hand, the beam’s effective detectable region is enlarged to mitigate the beam wander effect, and on the other hand, the central local focusing improvement is achieved to weaken the beam spreading effect to a certain extent. In what follows, by using the self-accelerating property of the Airy vortex beam [17, 18], an abruptly autofocusing beam carrying orbital angular momentum produced by a circular phased-locked coherent Airy vortex beam array (AVBA) with sufficient beamlets is proposed. There are two types of optical vortices in the proposed light beam. The first vortex, as the information carrier, is constructed by combining array elements with different initial phases in a well-ordered distribution; the second type of vortex is that embedded in the beamlet to tailor the cross-section power distribution of the resultant light beam without energy loss. When the tailored Airy vortex beam array (TAVBA) propagates, under the combined effects of self-heal and self-focusing, a concentric multi-ring structure is generated in the focal plane, where different rings represent different degrees of local focusing. By employing the multiple-phase-screen method, we demonstrate that the transmission channel crosstalk and vortex splitting of TAVBA is much lower than that of RAVB, creating a superior information carrier.

## 2. Theoretical background

Analytic solutions are rare in many practical problems, particularly when optical waves propagate through randomly fluctuating media [19]. Numerical experiments thus have to play the role of an efficient and important tool to quantitatively investigate the propagation effects [20]. Modelling optical beam propagation in atmosphere turbulence using phase screen method is an effective numerical approach, on the condition of ignoring the amplitude fluctuations [21], which is obedient to the statistics postulated by Kolmogorov turbulence theory.

In this section, we will model the propagation characteristics of the RAVB and AVBA in atmospheric turbulence using the Fourier multiple-phase-screen method which provides the best match to experimental data [22, 23].

#### 2.1 Paraxial propagation of the optical vortex beams

- (1) Airy vortex beam array
The optical field of the

*j*-th AVBA beamlet, represented as a scalar wave field at the source plane $z=0$, takes the form as [24–26]$${E}_{j}(x,y,z=0)={\left[(x-{x}_{c})+isign(l)(y-{y}_{c})\right]}^{\left|l\right|}Ai(\frac{X}{{x}_{0}})Ai(\frac{Y}{{y}_{0}})\mathrm{exp}\left[a\cdot (\frac{X}{{x}_{0}}+\frac{Y}{{y}_{0}})\right],$$where

$$\left[\begin{array}{c}X\\ Y\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}(\frac{j-1}{n}\cdot 2\pi +\pi )& \mathrm{sin}(\frac{j-1}{n}\cdot 2\pi +\pi )\\ -\mathrm{sin}\frac{j-1}{n}\cdot 2\pi +\pi )& \mathrm{cos}(\frac{j-1}{n}\cdot 2\pi +\pi )\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}d\\ d\end{array}\right],$$_{$(x,y,z)$}denote the Cartesian coordinates, ${x}_{c}$ and ${y}_{c}$ are the center of the optical vortex from the origin along the*x*and*y*axes, respectively; $sign(\cdot )$ is the sign function;*l*is the topological charge of the optical vortex and its sign determines whether the vorticity is clockwise or counter clockwise [24];$i=\sqrt{-1}$and*j*= 1,2,3,…,*n*,*n*is beamlet number; $Ai(\cdot )$ represents the Airy function; $X$ and $Y$ are dimensionless transverse coordinates, with ${x}_{0}$ and ${y}_{0}$ being the transverse scales; $0\le a\le 1$ is the exponential truncation factor;*d*is the transverse displacement parameter.Using Eq. (1), the optical field of the circular phased-locked coherent AVBA consisting of

*n*beamlets in the source plane is found to be:$$E(x,y,z=0)={\displaystyle \sum _{j=1}^{n}{\left[(x-{x}_{c})+isign(l)(y-{y}_{c})\right]}^{\left|l\right|}Ai(\frac{X}{{x}_{0}})Ai(\frac{Y}{{y}_{0}})\mathrm{exp}\left[a\cdot (\frac{X}{{x}_{0}}+\frac{Y}{{y}_{0}})\right]}\mathrm{exp}(i{\varphi}_{j}{}^{\prime}),$$where ${\varphi}_{j}{}^{\prime}=m\cdot 2j\pi /n$, and

*m*is the topological charge of the resultant beam. - (2) Ring Airy vortex beam
For RAVB, the optical field carrying vortex at the source plane can be written as [9,27]

$$E(x,y,z=0)=Ai(\frac{{r}_{0}-\sqrt{{x}^{2}+{y}^{2}}}{\omega})\mathrm{exp}\left[a\cdot \frac{{r}_{0}-\sqrt{{x}^{2}+{y}^{2}}}{\omega}\right]\mathrm{exp}\left[imarc\mathrm{tan}\left(\frac{y}{x}\right)\right],$$where, $arctan\left(\frac{y}{x}\right)$ ranges from 0 to 2π; ${r}_{0}$ and $\omega $ represent, respectively, the radius and width of the primary ring; $x$, $y$, $Ai(\cdot )$, $a$, $i$ and $m$ are the same as those given in Eqs. (1) and (2).

- (3) Split–step beam propagation through free space
Operator notation is useful for describing the Fresnel diffraction integral equation simply and compactly. Here, four operators will be used, $\Re \{\cdot \}$, $FFT\{\cdot \}$, $IFFT\{\cdot \}$, and ${\Re}_{2}\{\cdot \}$, and they are defined by [19, 28]:

$$\Re \left\{b;x,y\right\}E(x,y)=\mathrm{exp}\left[i\frac{k}{2}b\left({x}^{2}+{y}^{2}\right)\right]E(x,y),$$$$FFT\left\{x,y;{f}_{x},{f}_{y}\right\}E(x,y)={\displaystyle {\int}_{-\infty}^{\infty}{\displaystyle {\int}_{-\infty}^{\infty}E(x,y)\mathrm{exp}\left[-i2\pi \left({f}_{x}x+{f}_{y}y\right)\right]dxdy}},$$$$IFFT\left\{{f}_{x},{f}_{y};x,y\right\}E(x,y)={\displaystyle {\int}_{-\infty}^{\infty}{\displaystyle {\int}_{-\infty}^{\infty}E(x,y)\mathrm{exp}\left[i2\pi \left({f}_{x}x+{f}_{y}y\right)\right]dxdy}},$$The operators’ parameters, $b$, ${f}_{x}$, and ${f}_{y}$, are given in curly braces. $k=2\pi /\lambda $ is the wavenumber, and $\lambda $ is the wavelength of the optical beam in vacuum.

The non–adaptive coordinate transformation is used to calculate the focused optical field [29]. Using multiple partial propagation with the help of the well-known angular-spectrum theory, the free space propagation formula from [19] can be rewritten as:

$$\begin{array}{c}E(x,y,z)=B\mathrm{exp}\left[iC\left(x,y,z\right)\right]\Re \left[\frac{{\tilde{m}}_{n-1}-1}{{\tilde{m}}_{n-1}{z}_{n-1}};{x}_{n},{y}_{n}\right]\times {\displaystyle \prod _{j=1}^{{N}_{l}-1}\{S\left({x}_{j+1},{y}_{j+1}\right)IFFT\left[{f}_{{x}_{j}},{f}_{{y}_{j}};\frac{{x}_{j+1}}{{\tilde{m}}_{j}},\frac{{y}_{j+1}}{{\tilde{m}}_{j}}\right]}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\times {\Re}_{2}\left[-\frac{\Delta {z}_{j}}{{\tilde{m}}_{j}};{f}_{{x}_{j}},{f}_{{y}_{j}}\right]FFT\left[{x}_{j+1},{y}_{j+1};{f}_{{x}_{j}},{f}_{{y}_{j}}\right]\frac{1}{{\tilde{m}}_{j}}\}\times \left\{\Re \left[\frac{1-{\tilde{m}}_{1}}{{z}_{1}};{x}_{1},{y}_{1}\right]E({x}_{1},{y}_{1})\right\},\end{array}$$where [29]

_{$B={D}_{F}/{D}_{s}$}, ${D}_{s}$ and ${D}_{F}$ are the characteristic scales of the optical field in the source plane and focal plane respectively; $E({x}_{1},{y}_{1})$ is the input optical field; ${z}_{j}$ is the distance of the*j*-th layer; ${N}_{l}$ is the number of layers; ${\tilde{m}}_{j}=\left[\left(1-\frac{{z}_{j+1}}{z}\right){\delta}_{1}+\left(\frac{{z}_{j+1}}{z}\right){\delta}_{n}\right]/\left[\left(1-\frac{{z}_{j}}{z}\right){\delta}_{1}+\left(\frac{{z}_{j}}{z}\right){\delta}_{n}\right]$ is the scaling factor from plane*j*to plane*j*+ 1, ${\delta}_{1}$ and ${\delta}_{n}$ are grid spacing in the source plane and observation plane respectively; $S\left({x}_{j},{y}_{j}\right)=\mathrm{exp}\left[-{\left(\sqrt{{x}_{j}{}^{2}+{y}_{j}{}^{2}}/{\sigma}_{j}\right)}^{16}\right]$ is super-Gaussian function (SGF) [19, 30], and ${\sigma}_{j}=\left[\left(1-\frac{{z}_{j}}{z}\right){\delta}_{1}+\left(\frac{{z}_{j}}{z}\right){\delta}_{n}\right]$ is considered as the “width” of SGF, which make the optical field not attenuate in the center of the*j*-th plane but close to zero at the edge of the*j*-th plane.

#### 2.2 Multiple random phase screen model of atmospheric turbulence

In the Fourier multiple-phase-screen method, the free space propagation and the phase modulation induced by phase screen are regarded as two processes that are independent and simultaneously complete. Thus, the turbulent atmosphere can be represented as a series of thin random phase screens which are equally spaced between source and receiver. As illustrated in Fig. 1, the optical field after a $\Delta z$ free space transmission is phase-modulated by a phase screen, and the resulting optical field is diffracted another distance of $\Delta z$ before being phase-modulated again and then diffracted, and so on until the final output field is obtained at the receiver. Based on the above arguments, the change of field induced by multiple-phase-screen is described as a modification to the free space propagation equation:

_{${\phi}_{j}\left(x,y\right)$}represents the

*j*-th random complex phase. Considerable effort has been put into generating a random phase screen on a finite grid, by implementing the sub–harmonic compensate [31], and the random phase screen can be written as a Fourier series

*j*-th layer Fourier-series coefficients given by

*j*-th layer phase screen, given by [19], where $k=2\pi /\lambda $ is the wave vector and $\lambda $ is the light wavelength; ${C}_{n}^{2}\left({z}_{j}\right)$ is the structure parameter of the refractive index, and a constant value for ${C}_{n}^{2}$ along the path is assumed; ${f}_{0}=1/{L}_{0}$ and ${f}_{m}=5.92/\left(2\pi {l}_{0}\right)$ with ${l}_{0}$ and ${L}_{0}$ being the inner-scale and outer-scale sizes of turbulence; ${f}_{{x}_{{n}^{\prime}}}={n}^{\prime}/{L}_{x}$ and ${f}_{{y}_{{m}^{\prime}}}={m}^{\prime}/{L}_{y}$ are the spatial frequencies; ${{f}^{\prime}}_{{x}_{{n}^{\u2033}}}={3}^{-p}{n}^{\u2033}/{L}_{x}$ and ${{f}^{\prime}}_{{y}_{{m}^{\u2033}}}={3}^{-p}{m}^{\u2033}/{L}_{y}$ are the frequency grids, where

*p*is the sub-harmonic level (

*p*= 1,2,…${N}_{p}$) with ${N}_{p}=3$ according to [31].

## 3. Numerical experiment and discussion

The effects of a turbulent atmosphere on the propagation of different types of beams, including RAVB, AVBA and TAVBA, are numerically quantified by using the multiple-phase-screen method described in Sect. 2. Moreover, TAVBA will be proven as a superior information carrier for the FSO communication system.

#### 3.1 Modeling verification

To verify the above split-step propagation model, we compare our simulation results with the previous analytical and simulation results [32, 33] under the same parameters. Our simulation results [Fig. 2] are in good agreement with previous results [32, 33], indicating that our multiple-phase-screen method can accurately represent the phase accumulated along the turbulent atmosphere propagation path. The key simulation parameters are given in Table 1.

#### 3.2 Vortex splitting and crosstalk mitigation in turbulent atmosphere

By using Eq. (1), the effects of the topological charge values ($l$) and positions (${x}_{c},{y}_{c}$) of the vortices on the intensity distributions of the array beamlets are numerically calculated; the results are shown in Fig. 3. Figure 3(a) shows the intensity distribution of an Airy beam, which is a special case of the Airy vortex beam with *l* = 0. Figure 3(b) shows a vortex with *l* = 2 centered at the *x*-axis (${x}_{c}=0.1\text{m},{y}_{c}=0$), and the diagonal (45°axis) symmetry is broken with more energy accumulation along the *y*–axis. Similarly, for the vortex placed at *y*-axis (${x}_{c}=0,{y}_{c}=0.1\text{m}$), the energy accumulates in the area around the *x*-axis [Fig. 3(c)]. When the vortex centers at the 45°axis [Figs. 3(d) and 3(e)]; the intensity distribution still maintains a diagonal symmetry. Moreover, as shown in Fig. 3(f), the greater value of *l*, the smaller are the maximum intensities of the subsequent oscillations [Fig. 3(f)]. Equivalently, more energy converges on the main lobe.

Implementing Eqs. (1)–(3) with *m* = 3 and 5, we find that, the formation of the given OAM modes strongly depends on the ratio of the number of array elements ($n$) to the value of topological charge ($m$). As illustrated in Fig. 4, in the case of $n<2m$ (the first column), there are no distinct rules in the phase distributions; in the case of $n=2m$ (the second column), phase topological structures are evident, which correspond to the engineered *m*. The insets give the magnified details of the single $2\pi $ phase shifts corresponding to the regions marked by the dashed squares. In comparison with the phase variation of the canonical vortex, we notice the main difference being that the values of the phase change discontinuously from 0 to $\pi $, then to $2\pi $. Fortunately, the unique identification that the number of $2\pi $ phase shifts is equal to *m* in one cycle around the singularity is preserved. As $n$ increases (the third and fourth columns), a continuous phase variation from 0 to $2\pi $ is achieved and the intensity distribution present better cylinder symmetry. Therefore, we infer that the number of beamlets should be at least twice the value of the OAM topological charge, or greater, to transmit the signal precisely.

Implementing Eqs. (1)–(10) with *m* = 3, the normalized intensity distributions and phase patterns at the input and focal planes for RAVB, AVBA, and TAVBA are illustrated in Fig. 5. AVBA and TAVBA comprise the beamlets shown in Figs. 3(a) and 3(d), respectively. To realize the same radius for the main rings of the three vortex beams, the following parameters are selected for numerical calculations: $d=0.066\text{\hspace{0.17em}}\text{m}$, ${r}_{0}=\sqrt{2}\cdot d$, and $\omega =0.012\text{m}$ to enable the three vortex beams have the same radius of their main rings. It is clear that the AVBA has a ring Airy-like vortex pattern with plenty of beamlets (*n* = 150). Moreover, the maximum intensities of its subsequent oscillations are smaller than that of the RAVB [Fig. 5(a1) versus Fig. 5 (b1)], which is consistent with that given in [9]. When the RAVB and AVBA propagate to the focal plane, their energy focuses mostly on their central lobes so that their focusability can be characterized by the full-width at half maximum (FWHM) of their central lobes, and they have the same FWHM = 28mm.

Figure 5(c1) shows that, under the action of vortices embedded in the beamlets, the original multiple rings profile is changed and most of the energy flows to the central ring forming the TAVBA. From a comparison of Figs. 5(a2), 5(b2), and 5(c2), we find that the three vortex beams have a major difference lying in the rotation angle of the spiral phase wavefront, although they have a similar structure. When the TAVBA propagates to the focal plane, the energy mainly resides not in a single ring at the center, but in multiple rings, and central local-focusing improvement with FWHM = 17mm is achieved [Fig. 5(c3)]. In Fig. 5(c4), even though the phase distribution is mixed with the cusp of the moon, the topological charge information is still clear. The intensity images of the TAVBA at various distances are plotted to explicitly illustrate the formation of the focal fields. As shown in Fig. 5(d), the beam recovers its multi-ring profile to a great extent during propagation due to the self-healing effect. In this process, the energy flows outward spirally, which perturbs the phase distribution and carves out the cusp of the moon in the phase pattern, but the main topological feature is preserved. Then, under the action of the auto-focusing effect, the resulting optical field forms the concentric multi-ring pattern in the focal plane.

There are two types of optical vortices in TAVBA. *m*, as the information carrier, denotes the topological charge of the first vortex; *l* is the topological charge of the second type of vortex embedded in the array element. As described above, fixed *m* = 3, the optical focal field of TAVBA can be controlled by adjusting $l$, ${x}_{c}$, and ${y}_{c}$. As shown in Fig. 6, their control behavior is studied in two design strategies: TAVBAs with a constant ${x}_{c}$(${y}_{c}$) and variable *l* [Figs. 6(a)–6(c)], and TAVBAs with a constant *l* and variable ${x}_{c}$(${y}_{c}$) [Figs. 6(d)–6(f)]. Note that ${x}_{c}={y}_{c}$ is required to keep the symmetry. The normalized intensity cross sections of TAVBAs corresponding to Figs. 6(a)–6(c) and Figs. 6(d)–6(f) are shown in Figs. 6(g) and 6(h), respectively. It can be noticed that the subsequent oscillations of TAVBAs decrease and the energy relatively converges as *l* increases. On the other hand, the intensities of the subsequent oscillations of TAVBAs increase as ${x}_{c}$ (${y}_{c}$) increases. Moreover, when the value of ${x}_{c}$ (${y}_{c}$) becomes greater than 0.11 m, the intensities of subsequent oscillations will be stronger than that of the central ring. In addition, *d* being the transverse displacement parameter, for different transmission distances of TAVBA beams in free–space optical communications, ${x}_{c}$, ${y}_{c}$, $l$and $d$should be harmonically chosen.

To illustrate the anti-jamming performance of the three vortex beams, the far-field spiral phase distributions with *m* = 3 and *m* = 5 under different turbulence conditions at a distance of 3.3 km are plotted in Fig. 7. Here, the outer scales are chosen to 10m. In the case of ${C}_{n}^{2}=5\times {10}^{-16}{\text{m}}^{-2/3}$ (corresponding to weak turbulence condition), over 1000 samples being averaged, there is a distinct character that the topological charge of these vortex beams with $m>1$ breaks up to give *m* individual charge-one vortices indicated by the white dotted circles, which shows an agreement with that of previous works [34–37]. The vortex splitting ratio can be written as $V=\frac{\Delta {V}_{\u3008r\u3009}}{{\omega}_{0}}$, where $\Delta {V}_{\u3008r\u3009}$ is the average radial distance from the beam origin for the individual vortices [7]. The study [7] also indicates that a greater $V$ is undesired for the free-space transfer of high-fidelity OAM modes. Fortunately, as shown in Fig. 7(a) distinctly, for a given transmitted OAM mode, the vortex splitting ratio of the TAVBA (*V* = 0.058, *l* = 3; *V* = 0.094, *l* = 5) is smaller than that of the RAVB (*V* = 0.086, *l* = 3; *V* = 0.159, *l* = 5) and AVBA (*V* = 0.107, *l* = 3; *V* = 0.188, *l* = 5) due to the stronger central local focusability. It also indicates that OAM modes with larger quantum numbers are more sensitive to the effects of weak atmospheric turbulence than those with smaller quantum numbers. The reason for this phenomenon is that when vortex splitting occurs, these individual vortices wander the transverse plane quasi-independently [37], but they are confined into the central ring. However, as the quantum number of OAM modes increases, the FWHM of the central ring increases, and accordingly the individual vortices achieve an even larger wandering area. The second and fourth columns in Fig. 7(b) show the aberrant phase maps with ${C}_{n}^{2}=5\times {10}^{-15}{\text{m}}^{-2/3}$ (corresponding to moderate turbulence condition). Comparing the totally distorted phase distribution of RAVB and AVBA with that of TAVBA, one can be seen that the TAVBA still preserves some of its spiral phase structures, marked as colored lines [the bottom in Fig. 7(b)].

For further quantitative comparison of the crosstalk performance among the RAVB, AVBA, and TAVBA, the received power proportions over different OAM channels are depicted in Fig. 8. For the input OAM channel with *m* = 3 and 5 under the turbulence ${C}_{n}^{2}=5\times {10}^{-15}{\text{m}}^{-2/3}$, the power proportions are calculated across 10 channel regions corresponding to the ranges $m\in \left[-1,0,1,2,3,4,5,6,7,8\right]$ and $\left[1,2,3,4,5,6,7,8,9,10\right]$, respectively. The power proportion of each OAM channel can be acquired, referring to [15, 38]. It can be seen that the crosstalk of RAVB and AVBA is more serious than that of TAVBA. We attribute this result to the fact that both the beam wandering and vortex splitting will result in crosstalk, compared with RAVB and AVBA, TAVBA has not only a larger light spot to mitigate the beam wandering, but also an improved central local focusing ability to weaken vortex splitting.

## 4. Conclusions

In conclusion, an abruptly autofocusing light beam carrying OAM based on the superposition of multiple phased-locked two-dimensional Airy vortex beams arranged in a ring configuration is presented. The formation of a given OAM mode strongly depends on the ratio of the number of array elements ($n$) to the value of topological charge ($m$), and this ratio ($n/m$) is found to be 2, or greater. The optical focal field distribution of such a beam can be effectively controlled by appropriately adjusting the parameters ${x}_{c}$,${y}_{c}$, and $l$. The effects of atmospheric turbulence on the vortex splitting and crosstalk for different beams, including RAVB, AVBA, and TAVBA, have been investigated. It is found that OAM modes with larger quantum numbers are more vulnerable to splitting induced by atmospheric turbulence than those with smaller quantum number. Furthermore, with our controlling method, the vortex splitting ratio of TAVBA is smaller than that of the RAVB and AVBA, and TAVBA is a profoundly superior in reducing the inter-mode crosstalk and vortex splitting. When implementing this scheme in practice, similar to the prior demonstration [27], we can generate TAVBAs by using an expanded Gaussian beam to illuminate the SLM. The specific phase hologram [e.g., Fig. 5(c2)] is displayed on the surface of the SLM. Although further research will be needed for applications in strong turbulent media, the present study provides an effective scheme to optimize the light source. Finally, TAVBA, as a superior light source, can be used together with the innovative methods, including adaptive optics [39], multi-input multi-output [40] and channel coding [6], for a more robust OAM-based FSO communication system.

## Funding

Key Industrial Innovation Chain Project in Industrial Domain (2017ZDCXL–GY–06–02); Foundation for Innovative Research Groups of the National Natural Science Foundation of China (61621005); Open Research Fund of State Key Laboratory of Pulsed Power Laser Technology (SKL2016KF05).

## Acknowledgments

We thank Prof. J. J. Da, from Shaanxi Normal University, for her great contribution on English–language polishing. We would like to thank the three anonymous reviewers for their detailed and valuable comments.

## References and links

**1. **Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. **5**, 4876 (2014). [CrossRef] [PubMed]

**2. **S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. **2**(4), 299–313 (2008). [CrossRef]

**3. **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]

**4. **H. Kaushal and G. Kaddoum, “Optical Communication in Space: Challenges and Mitigation Techniques,” IEEE Trans. Vehicular Technol. **19**(1), 57–96 (2017).

**5. **D. Vasylyev, A. A. Semenov, and W. Vogel, “Atmospheric Quantum Channels with Weak and Strong Turbulence,” Phys. Rev. Lett. **117**(9), 090501 (2016). [CrossRef] [PubMed]

**6. **Y. Zhang, P. Wang, L. Guo, W. Wang, and H. Tian, “Performance analysis of an OAM multiplexing-based MIMO FSO system over atmospheric turbulence using space-time coding with channel estimation,” Opt. Express **25**(17), 19995–20011 (2017). [CrossRef] [PubMed]

**7. **M. P. J. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. **3**(10), e1700552 (2017). [CrossRef] [PubMed]

**8. **M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in FSO communications,” J. Lightwave Technol. **32**(7), 1323–1328 (2014). [CrossRef]

**9. **N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. **35**(23), 4045–4047 (2010). [CrossRef] [PubMed]

**10. **J. A. Davis, D. M. Cottrell, and D. Sand, “Abruptly autofocusing vortex beams,” Opt. Express **20**(12), 13302–13310 (2012). [CrossRef] [PubMed]

**11. **Y. Jiang, Z. Cao, H. Shao, W. Zheng, B. Zeng, and X. Lu, “Trapping two types of particles by modified circular Airy beams,” Opt. Express **24**(16), 18072–18081 (2016). [CrossRef] [PubMed]

**12. **Y. Jiang, X. Zhu, W. Yu, H. Shao, W. Zheng, and X. Lu, “Propagation characteristics of the modified circular Airy beam,” Opt. Express **23**(23), 29834–29841 (2015). [CrossRef] [PubMed]

**13. **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]

**14. **Y. Zhu, Y. Zhang, and Z. Hu, “Spiral spectrum of Airy beams propagation through moderate-to-strong turbulence of maritime atmosphere,” Opt. Express **24**(10), 10847–10857 (2016). [CrossRef] [PubMed]

**15. **X. Yan, L. Guo, M. Cheng, J. Li, Q. Huang, and R. Sun, “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] [PubMed]

**16. **J. Recolons, L. C. Andrews, and R. L. Phillips, “Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case,” Opt. Eng. **46**(8), 086002 (2007). [CrossRef]

**17. **H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation properties of an optical vortex carried by an Airy beam: experimental implementation,” Opt. Lett. **36**(9), 1617–1619 (2011). [CrossRef] [PubMed]

**18. **Y. Hu, Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating Airy Beams: Generation, control, and applications,” in *Nonlinear Photonics and Novel Optical Phenomena*, Z. Chen and R. Morandotti, eds, (Springer, 2012).

**19. **J. D. Schmidt, *Numerical Simulation of Optical Wave Propagation With Examples in MATLAB* (SPIE, 2010).

**20. **R. Rao, “Statistics of the fractal structure and phase singularity of a plane light wave propagation in atmospheric turbulence,” Appl. Opt. **47**(2), 269–276 (2008). [CrossRef] [PubMed]

**21. **L. C. Andrews and R. L. Phillips, *Laser beam propagation through random media* (SPIE, 2005).

**22. **S. M. Flatté and J. S. Gerber, “Irradiance-variance behavior by numerical simulation for plane-wave and spherical-wave optical propagation through strong turbulence,” J. Opt. Soc. Am. A **17**(6), 1092–1097 (2000). [CrossRef] [PubMed]

**23. **J. P. Bos, M. C. Roggemann, and V. S. Rao Gudimetla, “Anisotropic non-Kolmogorov turbulence phase screens with variable orientation,” Appl. Opt. **54**(8), 2039–2045 (2015). [CrossRef] [PubMed]

**24. **M. Mansuripur, *Classical optics and its applications* (Cambridge University, 2002).

**25. **H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett. **35**(23), 4075–4077 (2010). [CrossRef] [PubMed]

**26. **R. P. Chen, K. H. Chew, and S. He, “Dynamic control of collapse in a vortex Airy beam,” Sci. Rep. **3**(1), 1406 (2013). [CrossRef] [PubMed]

**27. **P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express **22**(7), 7598–7606 (2014). [CrossRef] [PubMed]

**28. **M. Nazarathy and J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. A **70**(2), 150–159 (1980). [CrossRef]

**29. **F. Zhang, “Non-adaptive transformation to calculate the propagation of the focused laser beams,” Chin. J. Quantum Elect. **20**(6), 656–660 (2003).

**30. **S. M. Flatté, J. Martin, and G.-Y. Wang, “Irradiance variance of optical waves through atmospheric turbulence by numerical simulation and comparison with experiment,” J. Opt. Soc. Am. A **10**(11), 2363–2370 (1993). [CrossRef]

**31. **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]

**32. **Y. Gu and G. Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett. **35**(20), 3456–3458 (2010). [CrossRef] [PubMed]

**33. **C. Chen, H. Yang, M. Kavehrad, and Z. Zhou, “Propagation of radial Airy array beams through atmospheric turbulence,” Opt. Lasers Eng. **52**, 106–114 (2014). [CrossRef]

**34. **A. Mamaev, M. Saffman, and A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. **78**(11), 2108–2111 (1997). [CrossRef]

**35. **A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. **47**, 291–391 (2005). [CrossRef]

**36. **X. L. Ge, B. Y. Wang, and C. S. Guo, “Evolution of phase singularities of vortex beams propagating in atmospheric turbulence,” J. Opt. Soc. Am. A **32**(5), 837–842 (2015). [CrossRef] [PubMed]

**37. **G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A **25**(1), 225–230 (2008). [CrossRef] [PubMed]

**38. **S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express **20**(1), 452–461 (2012). [CrossRef] [PubMed]

**39. **Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre-and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica **1**(6), 376–382 (2014). [CrossRef]

**40. **Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. **41**(11), 2406–2409 (2016). [CrossRef] [PubMed]