Vortex beams such as Laguerre–Gaussian (LG) beams and Bessel beams are intensively investigated for their unique performance. It was shown by Allen et al. that the complex amplitude of a vortex beam comprises an azimuthal phase term exp(ilφ), where φ is the azimuthal angle, and l is the topological charge ^[1]. And each photon in vortex beams carries the orbital angular momentum (OAM) of lℏ (ℏ is Planck’s constant divided by 2π).

There have been recent growing interest in applying vortex beams in free-space optical communications ^[2,3]. The topological charge l of a vortex beam, in principle, is an infinite integer value. In addition, the OAM of an optical beam can carry an unlimited number of bits. Therefore, vortex beams can tremendously increase the capacity of optical communication systems through mode multiplexing. Thus terabit free-space data transmission has been realized ^[2]. In general, LG beams are used as the vortex beams in free-space optical communication systems. However, another kind of vortex beam, the Bessel beam, also can be employed in the free-space multicasting ^[4]. Compared with LG beams, Bessel beams have the characteristics of nondiffraction ^[5], which means they are capable of recovering by themselves in the face of obstruction.

In a free-space optical communication system based on vortex beams, the effects caused by atmospheric turbulence should not be ignored. It is well known that the temperature and pressure’s inhomogeneities of the atmosphere will contribute to the variations of the refractive index along the transmission path, which may distort the helical wavefront and increase the bit error rate (BER) of data transmission in optical communication systems ^[6]. The aberrant wavefront distorted by turbulence will also result in the OAM spectrum’s dispersion ^[6,7]. The stronger the turbulence is, the more dispersive the spectrum will be. Hence, it is significant to evaluate the atmospheric turbulence effect on vortex beams by studying the variation of their OAM spectra.

Researchers have done a lot in assessing the atmospheric turbulence influence on vortex beams ^[6–10]. LG beams and Bessel beams can be utilized in high-capacity free-space data transmission through mode-division multiplexing. When they propagate through turbulent atmosphere simultaneously under the same condition (beam size, topological charge, turbulence strength, and so on), the influences of turbulent atmosphere on the OAM spectra of LG beams and Bessel beams have not been discussed simultaneously.

In this paper, we numerically study the variation of OAM spectrum when LG beams and Bessel beams propagate through atmospheric turbulence under the same condition in different cases. In addition, the case of transmission through the same turbulence strength at different propagation distances and the variation of the desired channel proportion of the two kinds of vortex beams are also investigated.

To study the influence on vortex beams caused by turbulent atmosphere, the first thing to do is generate phase gratings that can simulate the turbulent atmosphere. The turbulence can be modeled by an Nth order matrix of random complex numbers with statistics that match Kolmogorov turbulence theory ^[11]. In our simulation, the turbulence model developed by von Karman is used ^[12,13]. It can be shown from modified von Karman spectrum, which includes the inner scale introduced by Tatarskii, that the spectrum of fluctuations in refractive index φn(κ) can be written as where k0=2π/L0 and km=5.92/l0. Parameters L0 and l0 are the outer and inner scale of turbulence, respectively. In Eq. (1), κ denotes the spatial wavenumber, and Cn2 is the structure constant of the refractive index, which represents the turbulence strength. Under the Markov approximation, the phase spectrum can be expressed as

In Eq. (2), k denotes the wavenumber of the incident beam, and z can be regarded as the propagation distance in the turbulence. By this time, the phase distribution of turbulent atmosphere can be obtained according to Eq. (2) and ^[14] as where C is an N×N array of complex random numbers with zero mean and variance 1. Δx is the grid spacing and F denotes the Fourier transformation.

However, the approach shown above does not produce strict phase gratings because we cannot sample the spatial-frequency grid low enough to accurately represent low-order modes such as tilt. It was reported by Lane et al. that, by implementing the subharmonic method, can we acquire the accurate turbulent simulations ^[15]. The phase distribution of low spatial frequency by this method is a sum of M different screens, which is written as where each value of the index P denotes a different grid, and the sums over n and m are over discrete frequencies. cn,m,P is the random draws of Fourier coefficients, and its derivation refers to ^[15]. fxn and fym are the frequency grid. In our simulation, a 3×3 grid of frequencies is used for each value of P, and M=3 different grids are used. Hence, the frequency grid spacing for each P is 1/(3PD), where D is the screen’s size. By this method, the original sample grid of spatial frequency is replaced by 32P small samples.

We do the simulation under three different turbulence strengths, the structure constant of the refractive index Cn2 of which is 1×10−13, 1×10−14, and 1×10−15m−2/3 separately. All of the basic parameters of these three phase screens are identical except the structure constant. In the process of generating the phase screen, the outer scale of the turbulence is 100 m, and the inner scale is 0.01 m. The wavelength of the incident beams is chosen as 1550 nm. The size of the screen is set as 0.44 m, and N is 1000. Notably, each propagation distance z is 100 m, and the simulations are repeated 10 times in order to compute the optical fields after propagating a distance of 1 km in turbulence.

The single-mode or multiplexed LG beams and Bessel beams with the same topological charge are generated in our simulation. To compare the turbulence effect on these two kinds of vortex beams, the waist size of the inner ring of the Bessel beam is set the same as the LG beam.

We first investigate the difference of the turbulence effect on single-mode LG beams and Bessel beams under the same condition. In calculations, both of the two optical vortices carry the OAM of 3ℏ, and the waist size is 0.06 m (for LG beams, the fundamental mode waist size ω0=0.03 m; for Bessel beams, the radial wave vector kr=91.01m−1). Figure 1 illustrates the intensity and the phase distribution before and after propagating through different atmospheric turbulence at the distance of 1 km. The optical fields after the turbulence are calculated through the scalar diffraction theory in our simulation. It is easy to find from Fig. 1 that, under the same condition, the stronger the turbulence is, the more distorted the beams will be. When Cn2=1×10−13m−2/3, the phases of the two vortex beams are totally chaotic.

In order to quantitatively study the effect of atmospheric turbulence on different kinds of optical vortices, we analyze the OAM spectrum before and after propagating through the atmospheric turbulence. The weight of each OAM channel can be obtained, if we project the coherent beam to a series of spiral harmonics in the form of exp(ilφ) and gain the relative power corresponding to all these harmonics ^[16]. Then the OAM spectrum that consists of the weight of power is acquired. The OAM spectra of single-mode LG beams and Bessel beams before and after turbulence are calculated and displayed in Fig. 2.

Under the strong turbulence (Cn2=1×10−13m−2/3), it can be seen from Fig. 2(b) that a large cross-talk arises for the LG beams and the Bessel beams. We even cannot find how much OAM the incident beams carry. The OAM spectra shown in Figs. 2(c) and 2(d) illustrate that, in medium (Cn2=1×10−14m−2/3) or weak (Cn2=1×10−15m−2/3) turbulence, the spectrum of a single-mode Bessel beam is more dispersive than the LG beam, when they passed through the atmospheric turbulence with all the factors such as the turbulence strength, the beam size and so on are identical. It also means that the wavefront of the Bessel beam will suffer more than the LG beam in a turbulent atmosphere, resulting in the cross talk and bigger BER of a data transmission, although Bessel beams are free of obstructions.

Moreover, we study the difference of the turbulence effect on multiplexed vortex beams. Both of the multiplexed LG beams and multiplexed Bessel beams consist of four channels. The topological charges of the four channels are −6, −2, +3, and +7, respectively. And the intensity proportion of the four channels is identical. In addition, the fundamental waist size of each LG beam is 0.03 m, and the radial wave vector of the four Bessel beams are 112.54, 92.63, 91.01, and 118.94m−1, respectively, which makes the inner ring waist size of each channel of Bessel beams identical to the corresponding channel of LG beams, to create the same collating condition. The intensity and the phase profile of the two multiplexed vortex beams before and after propagating a distance of 1 km in different atmospheric turbulence are displayed in Fig. 3. The OAM spectra are shown in Fig. 4. The simulation results of multiplexed vortex beams are in agreement with the cases of single mode. It can be found from the OAM spectrum in Fig. 4 that the turbulence effect on the LG beams and Bessel beams is in relation to the turbulence strength. The OAM spectrum of Bessel beams is more dispersive, and the cross talk is more serious than LG beams. The desired channel proportion of Bessel beams is lower than LG beams even in weak turbulence (Cn2=1×10−15m−2/3).

Under the same turbulence strength, the OAM spectra of beams that pass through the turbulence with different propagation distance are further investigated. We choose single-mode LG beams and Bessel beams with topological charge of +3 as the incident rays. The waist radii of the LG beams and the inner ring of Bessel beams are identical, the parameters of which are the same as the previous simulation of single mode. Ten phase screens of turbulence, whose Cn2 is 1×10−14m−2/3 and propagation distance is 100 m, are used in the calculation to cover the varying propagation distance. The intensity proportion of the desired OAM channel (l=+3) is computed through analyzing the OAM spectrum. We figure out a group of proportions per 100 m and measure for 1 km. Then a curve to describe the relationship between the desired channel proportion of the two vortex beams and the propagating distance is obtained, which is depicted at Fig. 5(a).

It can be found from Fig. 5(a) that the curve of Bessel beams undulates stronger than that of LG beams. We compute the average and the variance of the two groups of channel proportions. As for LG beams, the average x¯LG=0.8653, the variance σLG2=0.0186. And for Bessel beams, the average x¯BS=0.6049, the variance σBS2=0.0686. The larger variance and the lower average of Bessel beams mean their wavefront distortion is more serious than LG beams when they pass through turbulent atmosphere with all the factors being identical.

Based on the OAM spectra discussed above, we can conclude that Bessel beams will suffer more than LG beams in passing through atmospheric turbulence under the same condition. It also can be understood with the help of Fried’s parameter ^[17], which can be calculated from Cn2 by the formula:

The turbulence strength is imposed on the beams, which can be characterized by d/r0, with d being the beam diameter at the turbulence ^[6]. In each of our calculations, r0 is a constant according to Eq. (5). Although the waist radii of the inner ring of Bessel beams we set is identical with the waist of LG beams, the side rings of Bessel beams enlarge the beam diameter indeed. Therefore, compared with LG beams, Bessel beams will suffer more from turbulent atmosphere.

We have analyzed the OAM spectra difference of LG beams and Bessel beams. While all of the above are the cases of single-ring LG beams (the radial index of LG beams p=0). When p≠0, multiring LG beams will appear. Under the same topological charge and fundamental mode waist, the waist size of multiring LG beams can be expressed as (2p+|l|+1)/(|l|+1)·ωl, where ωl are the waist size of single-ring LG beams. With the increasing of p, the optical fields will be more similar with those of Bessel beams. We also make a simulation to show the relationship between the desired channel proportion and the radial index of LG beams. The simulation is based on the work shown in Figs. 1 and 2. We do the simulation under weak turbulence (Cn2=1×10−15m−2/3), and all the factors are the same as those in the previous single-mode simulation. The desired channel proportions of 11 different p are figured out, as shown in Fig. 5(b). In Fig. 5(b), the blue dashed line corresponds to the channel proportion of Bessel beams. And the red dots denote the measured proportion of LG beams with different radial index p. It is easy to find that the proportions are monotone decreasing as the increasing of p, and they will be very close to the dashed line.

The scatter diagram shown in Fig. 5(b) also can be understood by Eq. (5). We know the increasing of p will contribute to a larger beam diameter d. According to Eq. (4), larger d means larger influence. Hence the desired mode proportion will decrease.

One thing should be noticed is that our study is based on the assumption that the diameters of the Bessel beam’s inner ring and LG beams are identical, resulting in the disagreement with some other work such as in ^[18]. In addition, a Bessel beam has infinitely extending sidelobes, which means it is not realistic. Hence, in the simulation, we clip the plane by a square for the initial plane of Bessel beams, leading to results that may depend on the size of the square considered to calculate. While in practical applications, the center part of Bessel beams is merely utilized. Therefore, our work is significant.

In brief, we have numerically demonstrated the atmospheric turbulence effects on different kinds of vortex beams, including LG beams and Bessel beams through analyzing their OAM spectra. The computed OAM spectra summarize that the cross talk of Bessel beams is larger, and the desired channel proportion is lower than LG beams. The simulation results mean Bessel beams suffer more from turbulence than LG beams under the same condition. We also explain the reason of the spectrum difference between Bessel beams and LG beams, which can be understood, as the side rings of Bessel beams enlarge the beam diameter.

Our work can be a reference for building a free-space optical data transmission link based on OAM. As for a short transmission distance, such as the system reported in ^[4], Bessel beams will be more dominant than LG beams because of their characteristic of being free from obstructions. But, for a long transmission distance without an adaptive optical system, the use of single-ring LG beams will be better.