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Abstract
Vortex beam carrying angular momentum (OAM) will be disturbed by the random fluctuation of the refraction index of turbulent atmosphere, resulting in intermodal crosstalk among the different OAM modes. Recent advances have demonstrated that the employment of the abruptly autofocusing vortex beams can potentially mitigate the crosstalk effect. In this paper, a new type of abruptly autofocusing vortex beams, called Airy Gaussian vortex beam array (AGVBA) is proposed. By means of multi-plane wave optics simulation, the degradation of signal mode for AGVBA propagating through isotropic atmospheric turbulence is studied. In a comparison with the conventional abruptly autofocusing vortex beams, such as the ring Airy vortex beam (RAVB) and the Airy vortex beam array (AVBA), it is shown that AGVBA achieves more centralized intensity as well as a larger spot at the focal plane, thus can effectively balance the beam spreading and beam wander effect, resulting in mitigation of intermodal crosstalk.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Orbital angular momentum (OAM) is a property of light associated with the helicity of a photon’s wavefront. Unlike spin angular momentum, for which only two states are possible, the OAM states of a photon are infinite [1]. The vortex beams carrying OAM keep a subject of intense research because they have important applications in optical manipulation and trapping [2], imaging [3], quantum entanglement [4], free-space optical (FSO) communication [5] and so on. In particular, the property of OAM allow simultaneous transmission of information from different users, which can remarkably enhance the channel information capacity of FSO communication without a corresponding increase in spectral bandwidth [6–9]. Despite the promising aspects, a fundamental concern is that the light beam will be disturbed by the random fluctuation of the refraction index of turbulent atmosphere, resulting in intermodal crosstalk among the different OAM modes. The crosstalk mostly reduces the link performance, therefore, one of the persistent challenges is to lessen the intermodal crosstalk.
Various techniques have been employed to mitigate the crosstalk, including adaptive optics (AO) [10–13], multiple-input multiple-output (MIMO) equalization [14]. AO has been widely used to correct the phase distortions in previous work. In [10], M. Li et al. show that the crosstalk of OAM modes induced by turbulence can be significantly compensated by the real-time correction of the properly designed AO. In [11], S. Li et al. demonstrate that Shack-Hartmann methods can efficiently compensate for 1 km moderate atmospheric turbulence of OAM multicasting links. In [12], G. Xie et al. show SPGD algorithm can correct multiple OAM beams propagating through the same turbulence, and the crosstalk among these modes is reduced by more than 5 dB. In [13], S. Fu et al. indicate that the Gerchberg-Saxton algorithm shows favorable compensation performance for distorted optical vortex beams carrying orbital angular momentum. A $4 \times 4$ MIMO equalizer with four-channel OAM multiplexed was demonstrated in [14]. Results indicate that MIMO equalization could be helpful to mitigate the crosstalk caused by turbulence. Recently, a simple and effective scheme based on a spherical concave mirror (SCM) to focus the light onto a small area photodetector has been introduced [15]. The results testify that SCM can effectively compensate for the optical spot scattering and wandering effect arising from atmospheric turbulences, thus leading to the improved performance of the system. In [16], a scheme based on a focusing mirror for the reduction of crsootalk in a free-space optical communication link was also introduced. However, when the length of the optical link changes, the mirror parameters and experimental apparatus should be reset accordingly. To further simplify such systems, the abruptly autofocusing light beams have been investigated widely due to that the computer-generated holograms modifications are only required in case of variable link length [17,18].
Abruptly autofocusing light beam is one in which energy suddenly focuses while maintaining a low-intensity profile before the focal point. In [19], the expression of the OAM spiral spectrum of autofocusing Hypergeometric-Gaussian (HyGG) beams propagating through moderate-to-strong anisotropic non-Kolmogorov turbulence is derived. Results show that the autofocusing property of HyGG beams has a significant impact on the propagation of the OAM mode probability density. As a typical representative of the abruptly autofocusing beam family, ring Airy vortex beam (RAVB) carrying power-exponent-phase vortex (PEPV) has been shown the ability to weakens the beam wander effect induced by atmospheric turbulence, and thereby reducing the OAM channel crosstalk [20]. However, all systems that utilize abruptly autofocusing beams must take into account the beam wander effect arising from focusing property. Recently, X. Yan et al. facilitated new feasible methods to balance the beam spreading and beam wander effect on the OAM in atmospheric turbulence by using a tailored Airy vortex beam array (TAVBA) [21]. In the study, 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. It is, therefore, reasonable to infer that the beam with centralized intensity as well as a larger spot at the focal plane can potentially mitigate the crosstalk effects.
In this work, a new strategy is proposed by superposing Gaussian component over circular phase-locked Airy vortex beam array (AVBA), known as Airy Gaussian vortex beam array (AGVBA). Due to the Gaussian component, the side lobe of the beam is significantly suppressed, meanwhile, the centralized intensity is achieved because of the self-focusing properties of the Airy beam. Therefore, it is rational to infer that the AGVBA can be engaged to overcome the crosstalk effects of OAM. To thoroughly investigate the anti-crosstalk performance of RAVB, AVBA and AGVBA, the intensity and phase distribution, beam spreading and OAM spiral spectrum are compared for various turbulence strengths and systems configurations.
The remainder of this paper is organized as follows. In Section II, we introduce three vortex beams, including RAVB, AVBA, and AGVBA. We also review the wave optics simulation (WOS) theory and the probability density of signal OAM mode theory. Section VI presents the numerical results of three vortex beams, and Section VII concludes the paper.
2. Concept description
2.1 Optical field of vortex beams
2.1.1 Airy vortex beam array (AVBA)
As shown in Fig. 1, the optical field of the jth Airy vortex beam array (AVBA) beamlet at the source plane can be expressed as [21]:
2.1.2 Airy Gaussian vortex beam array (AGVBA)
For generating the AGVBA, we superimpose each beamlet in AVBA by a Gaussian component in cylindrical coordinate, the (1) can be rewritten as:
2.1.3 Ring Airy vortex beam (RAVB)
The optical field of the RAVB at the source plane can be expressed as [22]:
2.2 Multistep wave optics simulation for angular-spectrum propagation
The wave optics simulation uses the split-step approach where the continuous turbulent path is subdivided into segments and a phase screen is placed at the center of each segment to account for the effect of turbulence within the segment volume. To simulate the propagation of an optical beam through oceanic turbulence we resort to a multistep form of the Fresnel diffraction integral. For this purpose, the propagation path is divided into $n$ equal intervals by ${n-1}$ transmittance planes, so the optical field in the $nth$ plane can be written as [23] :
2.3 Distribution function of the random phase screen
To obtain ${h\left ( {{\kappa _x},{\kappa _y}} \right )}$, we firstly generate complex Gaussian random matrix, then it is filtered by the power spectrum function of oceanic turbulence, finally, it is obtained by inverse Fourier transform, the discrete formula can be expressed as [23]:
2.4 Detection probability density of signal OAM mode
The optical field of beams propagating through atmospheric turbulence can be written as a superposition of plane waves with phase ${\exp (im\theta )}$ [26] :
where ${{\beta _m}(\rho ,z)}$ is given by the integral:3. Numerical results and discussion
In this section, we presented the numerical results for the effects of atmospheric turbulence on the propagation of different types of vortex beams. Additionally, the corresponding crosstalk versus the different beams parameters and atmospheric turbulence strength was addressed. Unless otherwise stated, all the necessary links and turbulence parameters of interest are shown in Table 1. To realize the same radius for the main rings of the three vortex beams, the transverse displacement parameter satisfied the criteria ${r_0} = \sqrt 2 d$ [21]. For computing each result data point, 1000 realizations were used. The grid size for the source and observation planes was ${N_G} \times {N_G} = 1024 \times 1024$ points and 10 turbulence phase screens were used for simulation.
Table 1. Simulation Parameters
By using Eq. (4), the effects of the distribution factor $b$ on the intensity distributions of the single Airy beam of the array are presented in Fig. 2. It is observed that the proper distribution factor $b$ can make the initial beams tend to an Airy vortex beam with a smaller value, such as ${b=0.1}$ in Fig. 2(a), or a Gaussian vortex beam with the larger one, such as ${b=0.5}$ in Fig. 2(c). In other words, the greater the value of $b$, the smaller is the maximum intensities of the subsequent oscillations. Equivalently, more energy converges on the main lobe.
Next, we implement Eq. (3) with the number of beamlets ${n=150}$, the normalized intensity profiles and spiral phase images of the AGVBA are illustrated in Fig. 3. As expected, the AGVBA beams tend to be a ring Airy vortex beam with ${b=0.1}$ and a hollow Gaussian vortex beam with the larger $b$. For example, when ${b=0.5}$, the beams become a hollow Gaussian vortex beam, more or less. It can be noticed that the subsequent oscillations of AGVBAs decrease as $b$ increases. Moreover, the phase topological structures of three AGVBA are evident, which correspond to the topological $m$.
Fig. 3. Normalized intensity profiles (a) and spiral phase images (b) of the AGVBA with different distribution factor $b$.
Figure 4 show the normalized intensity distributions and phase patterns at the input planes for RAVB, AVBA, and AGVBA. It is clear that the AVBA has a ring Airy-like vortex pattern with plenty of beamlets (${n = 150}$). Moreover, the intensities of its subsequent oscillations are smaller than that of the RAVB. The third column of Fig. 4 show under the action of the Gaussian component superimposed in the beamlets, the side lobe of AGVBA is smaller than that of RAVB. Although the phase topological structures of three vortex beams are noticeable, the rotation angle of the spiral phase wavefront for RAVB is different from AVBA and AGVBA.
Fig. 4. Normalized intensity distributions and phase patterns at the input planes for RAVB, AVBA, and AGVBA.
By implementing Eq. (6), the normalized intensity distributions at different locations along the propagation in free-space are illustrated in Fig. 5. It is quite clear that the beams sharply focus after a certain distance of propagation. Due to the existence of distribution factor $b$, the focal distance of AGVBA (10.5km) is shortest as compared to RAVB (13.5km) and AVBA (12km). In addition, AGVBA provides a considerable increase in focal beam radius, which can potentially alleviate the effect of beam wander.
Fig. 5. Normalized intensity images at different locations along the propagation direction in free-space: (a) RAVB; (b) AVBA; (c) AGVBA.
Figure 6 demonstrate the phase distributions at different locations along the propagation in free-space. We can find that the center of RAVB and AVBA present some "grid" at the beginning, then the phase distribution is mixed with the "cusp of the moon" for a longer propagation distance. On the contrary, AGVBA still presents remarkably clear topological charge information. Thus it is certain to say that the phase wavefront of AGVBA keep the best spiral structure in free-space.
Fig. 6. Phase images at different locations along the propagation direction in free-space: (a) RAVB; (b) AVBA; (c) AGVBA.
Figure 7 depict the intensity distributions and spiral phase of three vortex beams propagating through turbulence with Figs. 7(a)-(b) ${C_n^2 = 1 \times {10^{ - 15}}{m^{ - 2/3}}}$ and Figs. 7(c)-(d) ${C_n^2 = 5 \times {10^{ - 15}}{m^{ - 2/3}}}$. Here we set path distance $z=1000m$, all the cases are weak fluctuation conditions. For ${C_n^2 = 1 \times {10^{ - 15}}{m^{ - 2/3}}}$ (Figs. 7(a)-(b)), three vortex beams preserve some of their spiral phase structures. It indicates that AGVBA is less sensitive to the effects of atmospheric turbulence than RAVB and AVBA. When atmospheric turbulence gets stronger (Figs. 7(c)-(d)), we note that the spiral phase wavefront of AGVBA can be explicitly recognized as compared to RAVB and AVBA with badly distorted wavefront. This is due to the fact that the effects of turbulence on the beam quality is determined by $\textrm {d}/{r_\textrm {c}}$ [27], with $d$ being the beam diameter at the turbulence and the $r_c$ the Fried parameter with ${{r_\textrm {c}} = {\left ( {0.423{k^2}C_n^2z} \right )^{ - 3/5}}}$. The beams with larger $d$ will be more easily affected by turbulence. Although the radius of the main ring of three vortex beams is identical, the side rings of RAVB and AVBA are bigger than that of AGVBA (Fig. 4(a)). The radius of the side ring enlarges the beam diameter $d$, which virtually aggravates the effect of turbulence. Generally, the beam diameter parameter d indicates an area containing 86.5$\%$ of the power of the optical field. The $\textrm {d}/{r_\textrm {c}}$ of three vortex beams have been marked in Fig. 7, the $\textrm {d}/{r_\textrm {c}}$ of AGVBA is the smallest as expected.
Fig. 7. Intensity distributions and spiral phase map after propagation through atmospheric turbulence with (a)-(b) ${C_n^2 = 1 \times {10^{ - 15}}{m^{ - 2/3}}}$ and (c)-(d) ${C_n^2 = 5 \times {10^{ - 15}}{m^{ - 2/3}}}$.
To show the advantage of AGVAB in reducing the effect of beam spreading, the variation of the beam spreading in the condition of atmospheric turbulence is shown in Fig. 8. The beam spreading is defined as [28]
Fig. 8. Beam spreading for the RAVB, AVBA and AGVBA with different $b$ versus distance on ${C_n^2 = 1 \times {10^{ - 15}}{m^{ - 2/3}}}$.
For the quantitative comparison of the intermodal crosstalk among the RAVB, AVBA, and AGVBA in weak atmospheric turbulence, the power proportions over different OAM modes are depicted in Fig. 9. For ${C_n^2 = 1 \times {10^{ - 15}}{m^{ - 2/3}}}$ in Fig. 9(a), the spectrum of RAVB and AVBA is more dispersive than the AGVBA, which means that the wavefront of the AGVBA suffers less aberration in turbulent atmosphere. The results of Fig. 9(b) imply that when atmospheric turbulence gets stronger, the crosstalk increases significantly. It is seen that AGVBA still suffer the smallest crosstalk among three vortex beams.
Fig. 9. Power proportion for signal OAM modes =3 of the RAVB, AVBA, AGVBA: (a) ${C_n^2 = 1 \times {10^{ - 15}}{m^{ - 2/3}}}$ and (b) ${C_n^2 = 5 \times {10^{ - 15}}{m^{ - 2/3}}}$.
The power proportion results of OAM modes are given in Fig. 10. The first thing to notice is that as distance increases, the crosstalk becomes larger, resulting in the power proportion drops sharply. This is easy to interpret that the wavefront suffers more distortion as distance increases. The other behavior is the power proportion of signal mode increases steadily as $b$ increases. This is because the AGVBAs with greater $b$ alleviate the effect of both beam wander and beam spreading, resulting in mitigation of intermode crosstalk.
Fig. 10. The power proportion of the signal OAM modes plotted against distance and distribution factor $b$ with ${C_n^2 = 5 \times {10^{ - 15}}{m^{ - 2/3}}}$.
4. Conclusion
In this paper, we have proposed a new type of abruptly autofocusing vortex beams, called Airy Gaussian vortex beam array (AGVBA). The optical field distribution of beams can be effectively controlled by appropriately adjusting the distribution factor $b$ from a ring Airy vortex beam to a hollow Gaussian vortex beam. By a suitable choice of distribution factor $b$, the focal distance and intensity of the AGVBA beams can be also easily modulated. The effects of atmospheric turbulence for AGVBA is compared to that of RAVB and AVBA in details. It is found that AGVBA with bigger $b$ achieves more centralized intensity and a larger spot at the focal plane, resulting in less effect of beam spreading and beam wander. Furthermore, the phase wavefront aberration of AGVBA is not significant both in free-space and in atmospheric turbulence. Numerical results reveal that AGVBA achieves a narrower spiral spectrum, and thus greatly reducing intermode crosstalk induced by atmospheric turbulence. In addition, for a fixed path distance, the anti-crosstalk properties can be improved as $b$ increases. Our results in this paper will prove to be helpful in the design and performance improvement optical communication systems.
Funding
National Natural Science Foundation of China (61505155, 61571367); Higher Education Discipline Innovation Project (B08038); Fundamental Research Funds for the Central Universities (JB160110, XJS16051).
Disclosures
The authors declare no conflicts of interest.
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