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Abstract
Vortex beams carrying orbital angular momentum (OAM) with the doughnut-shaped intensity distribution can be employed in free-space optical (FSO) communication links to circumvent obstructions. However, the size of the receiver aperture is proportional to the size of obstructions, which seriously constrains the application of OAM beams in this scenario. In this paper, we propose a method to generate bottle vortex beams (BVBs) with a parabolic trajectory by manipulating the radial phase distribution of conventional OAM beams. Meanwhile, the trajectory of BVBs generated are highly compatible with the predesigned trajectory by using this method. Moreover, we evaluate the free-space transmission performance of BVBs under atmospheric turbulence and limited receiving aperture. The results show that BVBs have better OAM FSO communication link performance compared with conventional OAM beams and Bessel beams. In addition, the performance of the BVBs circumventing obstructions is further investigated. The simulation results show that when setting the atmospheric turbulence strength D/r0 = 2 and the obstruction size of 40 mm, the average received optical power of the BVBs captured by a limited receiving aperture diameter (d = 40 mm) is improved about 7 dB and 3 dB compared to conventional OAM beams and Bessel beams, respectively.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Free-space optical (FSO) communication links have received a lot attention compared with radio-frequency communication links due to the larger communication capacity, faster transmission rate and higher confidentiality [1–4]. However, with the continuous development of FSO communication technology, the exploitation of resources such as wavelength, frequency, complex amplitude and polarization of optical waves has almost reached its limit and FSO communication is facing the great challenge of capacity crisis [5]. In recent years, the development of spatial dimensional resources of optical waves has provided new ways for sustainable capacity expansion of FSO communication links, such as structured beams with different spatial structure characteristics have been widely used in FSO systems to increase the communication capacity [6,7]. Typical structured beams include vortex beams carrying orbital angular momentum (OAM), Bessel beams, vector beams, Hermite-Gaussian beams, and others. Moreover, the vortex beam carrying OAM is a structured beam with an azimuthal phase term exp(ilφ), where l is the OAM order and φ is the azimuthal angle [8–11]. High capacity and spectrally efficient OAM optical communications can be realized by multiplexing the OAM modes using their own unbounded states and orthogonal properties [12–14]. Additionally, vortex beams carrying OAM have demonstrated great potential for high-dimensional free-space quantum communications [15–18].
Due to the special spiral phase structure of the OAM beam, the divergence speed of the vortex beam in free space is proportional to the order of the angular momentum mode. When the receiver aperture is limited, it is difficult for the receiver to capture the high-order OAM beam, which will cause serious power loss [19–21]. In the past, lenses were usually used in OAM FSO links to suppress beam divergence so that the OAM beam could be captured efficiently at the receiver side [22,23]. However, the parameters of the lens need to vary with the transmission distance, which will cause a lot of inconvenience. Additionally, atmospheric turbulence can also cause the mode expansion of the OAM beam and thus degrade the BER performance of FSO links [24,25]. To improve the anti-turbulence performance of OAM FSO links, various methods such as adaptive optics (AO), multiple-input multiple-output (MIMO) equalization and deep learning-based wavefront correction have been proposed, etc., but this leads to the increase of system complexity and cost [26–30]. Recently, the performance of FSO communication links under limited receiver aperture and atmospheric turbulence can be effectively improved by manipulating the trajectory of OAM beams [31].
OAM beams with the doughnut-shaped intensity distribution can be employed in FSO communication links to circumvent obstructions. However, the size of receiver aperture is proportional to the size of obstructions, which seriously constraints the application of OAM beams in this scenario. Since the analytical solution of the Airy wave packet in the fluctuation equation was obtained, the propagation properties of Airy vortex beams carrying OAM have been widely studied, where an important physical feature of such Airy vortex beams is that their accelerated trajectories are an example of diffraction catastrophe [32–34]. Meanwhile, the accelerated trajectory of the Airy vortex beam is a localized region of concentrated intensity associated with the envelope caustic to a family of the tangential geometric rays and the principle of radial phase design for generating the accelerated beam is given in the literature [35]. Moreover, the radial phase required to generate the accelerated beam with an arbitrary convex trajectory is calculated by the geometric Legendre transformation, which is used to interpret the connection between the desired radial phase and the target trajectory [36]. Therefore, by the geometric Legendre transform, the radial phase can be designed to generate an OAM beam with an arbitrary convex trajectory, which may broaden the application of OAM beams in FSO communication systems [37].
In this paper, we propose a method to generate bottle vortex beams (BVBs) with a parabolic trajectory by modulating only the phase of the incident Gaussian beam. Meanwhile, the trajectory of BVBs generated are highly compatible with the predesigned trajectory by using this method. To illustrate the performance of the proposed BVBs for FSO communication links, we simulate the transmission of BVBs under atmospheric turbulence, limited receiving apertures and obstructions, respectively. Firstly, we simulate the received optical power of conventional OAM beams, Bessel beams and BVBs under weak to strong turbulence (D/r0 = 1, 2, 4). Simulation results show that under atmospheric turbulence strength D/r0 = 2, the average received optical power of BVBs improved about 5 dB and 2 dB compared to conventional OAM beams and Bessel beams, respectively. Secondly, we simulate the received optical power of conventional OAM beams, Bessel beams and BVBs captured by a limited receiving aperture diameter (d = 10, 40, 80 mm) at atmospheric turbulence strength D/r0 = 2. The simulation results show that the average received optical power of the BVBs is improved about 73 dB and 14 dB compared to conventional OAM beam and Bessel beam at a limited receiving aperture d = 10 mm, respectively. Finally, we simulate the transmission characteristics of BVBs circumventing obstructions under atmospheric turbulence and different receiving apertures. The simulation results show that when setting the atmospheric turbulence strength D/r0 = 2 and the obstruction size of 40 mm, the average received optical power of the BVBs captured by a limited receiving aperture diameter (d = 40 mm) is improved about 7 dB and 3 dB compared to conventional OAM beams and Bessel beams, respectively.
2. Concept and principle
The concept and principle of FSO communications of BVBs are shown in Fig. 1. Figure 1(a) illustrates the free-space atmosphere turbulence transmission of a conventional OAM beam which is usually generated by modulating an incident Gaussian beam with a helical phase plate. The performance of OAM FSO communication links is primarily influenced by atmospheric turbulence and self-divergence, particularly undergoing long distances transmission. The OAM beam is difficult to be fully captured by the limited receiver aperture, which can substantially degrade the performance of OAM FSO communication links. OAM beams with the doughnut-shaped intensity distribution can be employed in FSO communication links to circumvent obstructions. However, the size of receiver aperture is proportional to the size of obstructions, which seriously constraints the application of OAM beams in this scenario. To improve the communication performance of OAM FSO communications circumventing obstructions under atmospheric turbulence and limited receiving aperture, this paper designs BVBs propagating along a predefined parabolic trajectory g(z) by associating the trajectory of the OAM beam with the optical caustics, as shown in Fig. 1(b). Comparing with conventional OAM beams, one can design and generate BVBs to circumvent obstructions, which can significantly reduce power loss. In addition, BVBs have a smaller beam diameter at the receiver side, which can further reduce the power loss under limited receiving aperture. Moreover, BVBs have better turbulence resistance performance, which can effectively reduce the OAM mode spreading caused by atmospheric turbulence. It can be seen from Fig. 1(b) that the modulation phase of BVBs is obtained by combining a spiral phase and a radial phase function φ(r) in the z = 0 plane which generates the trajectory of BVBs as a caustic. The caustic is defined as an envelope of a family of tangents such that every point on the caustic can be related to each point r in the radial phase function φ(r) via a tangent of slope θ, where tan θ = ǵ(z) = dg(z)/dz. After further geometric operations, the relationship between r and the predefined trajectory can be obtained as ǵ(z) = [g(z) – r]/z. Since the OAM beam continuously diverges as the transmission distance increases and the higher the mode of the OAM beam, the faster the divergence. Therefore, to enable the trajectory of the BVB highly coincident with the predefined trajectory, the radial phase function φ(r) needs to be designed considering the divergence trajectory of the OAM beam. At this point, the needed radial phase distribution φ(r) can be defined as:
Fig. 1. Concept and principle of FSO communications of BVBs circumventing obstructions under atmospheric turbulence and limited receiving aperture. (a) Conventional OAM beam transmission; (b) BVB transmission.
3. Simulation results
Utilizing Eqs. (1)–(3), the modulation phase for generating the desired BVBs can be designed based on the predefined trajectory. We generated the transmission trajectory of BVBs with different OAM modes (l=±1 and l=±3) by modulating an incident Gaussian beam with a radius of 0.1 m and a wavelength of 1550 nm, respectively, as shown in Fig. 2. The white dashed curve marked in each figure is the predefined trajectory of the BVBs. Figures 2(a) and (b) show the transmission trajectory of the BVBs (l = 1 and l = 3) without considering the inherent divergence h(z) of the OAM beam. The actual transmission trajectory of the BVBs with mode l = 3 deviate from the predefined trajectory, while the BVBs with mode l = 1 appear to have little difference, which may be accounted for the small inherent divergence of the OAM beam with mode l = 1. Due to the inherent divergence of the OAM mode, BVBs with high OAM mode states will deviate more from the preset trajectory. Therefore, the inherent divergence of the OAM mode needs to be taken into account when designing the modulation phase for generating high mode BVBs, as shown in Eq. (3). Under considering the inherent divergence h(z) of the OAM, we simulated the transmission trajectory of the BVB with OAM modes l=±1 and l=±3, as shown in Fig. 2(c)-(f). Simulation results show that the transmission trajectory of the BVBs generated are highly compatible with the predefined trajectory by the method proposed in this paper and the BVBs with opposite OAM modes have the same transmission trajectory.
Fig. 2. (a), (b) Generated BVBs transmission trajectories and corresponding phase pattern without considering the inherent divergence with OAM modes l=+1 and l=+3, respectively. (c)–(f) Generated BVBs transmission trajectories and corresponding phase pattern with considering the inherent divergence with OAM modes l=+1, l=+3, l = −1, and l = −3, respectively.
To illustrate the effectiveness of the proposed method, we generate three BVBs of the OAM mode l = 3, which have the same spot diameter size (d = 0.0072 m) at the receiver for different transmission distances. The transmission distances are designed as 400 m, 500 m and 600 m, respectively, as shown in Fig. 3. The first to third columns of Fig. 3 correspond to the modulation phase patterns, transmission trajectories, and intensity distributions (z = 300 m, 400 m and 500 m) marked by the white dashed line of the three BVBs, respectively. The incident Gaussian beam with a radius of 0.1 m and a wavelength of 1550 nm. The illustrated results demonstrate that the spot diameter of BVBs at different transmission distances can easily be controlled by designing the modulation phase.
Fig. 3. (a)–(c) Simulation results of three BVBs (l=+3) for achieving the same spot diameter at different transmission distances of 300 m, 400 m, and 500 m, respectively.
In order to illustrate the performance of the proposed BVBs for FSO communication links under atmospheric turbulence, we measure the received optical power and the corresponding cumulative probability of conventional OAM beams, Bessel beams and BVBs under weak to strong turbulence (D/r0 = 1, 2, 4), as shown in Fig. 4. The simulation results show that the average received power of all three OAM-carrying beams decreases when the atmospheric turbulence intensity ranges from weak turbulence to strong turbulence, which is because the stronger the turbulence affects the OAM modes more. When the atmospheric turbulence intensity is set to D/r0 = 2, the average received optical power of BVBs improved about 5 dB and 2 dB compared to conventional OAM beams and Bessel beams, respectively. It is obvious that BVBs have better performance of turbulence resistance OAM FSO communication links.
Fig. 4. Received optical power and cumulative probability of three different beams carrying OAM under different atmospheric turbulence (D/r0 = 1, 2, 4). (a) and (d) D/r0 = 1; (b) and (e) D/r0 = 2; (c) and (f) D/r0= 4.
Then, we also measure the received optical power and the corresponding cumulative probability of conventional OAM beams, Bessel beams and BVBs captured by a limited receiving aperture diameter (d = 10, 40, 80 mm) at atmospheric turbulence strength D/r0 = 2, as shown in Fig. 5. It can be seen that the BVBs and Bessel beams have higher received optical power compared to the conventional OAM beams, where the BVBs and Bessel beams have almost the same received optical power when the receiver aperture is larger. When the receiver aperture is reduced to 10 mm, the average received optical power of the BVBs is improved about 73 dB and 14 dB compared to the conventional OAM beam and Bessel beam, respectively. Thus, BVBs have better OAM FSO communication link performance under atmospheric turbulence and limited receiving apertures.
Fig. 5. Received optical power and cumulative probability of three different beams carrying OAM captured by a limited receiving aperture diameter (d = 10, 40, 80 mm) at atmospheric turbulence strength D/r0 = 2. (a) and (d) 10 mm; (b) and (e) 40 mm; (c) and (f) 80 mm.
Additionally, to evaluate the mode multiplexing characteristics of BVBs (l = 1and l = 3), we measure the mode crosstalk and the corresponding cumulative probability of conventional OAM beams, Bessel beams, and BVBs under different atmospheric turbulence (D/r0 = 1, 2, 4), as shown in Fig. 6. It can be seen that the stronger the turbulence, the higher the mode crosstalk of three different beams carrying OAM. Under atmospheric turbulence D/r0 = 2, the average mode crosstalk of BVBs improved about 13 dB and 8 dB compared to conventional OAM beams and Bessel beams, respectively. Therefore, BVBs can effectively improve the performance of OAM multiplexed FSO communication links under atmospheric turbulence.
Fig. 6. Mode crosstalk and cumulative probability of three different beams carrying OAM under different atmospheric turbulence (D/r0 = 1, 2, 4). (a) and (d) D/r0 = 1; (b) and (e) D/r0= 2; (c) and (f) D/r0= 4.
By using the proposed method, we generated two different BVBs with the same OAM mode (l = 3) over a transmission distance of 500 m, as shown in Fig. 7. The incident Gaussian beam with a radius of 0.08 m and the wavelength of 1550 nm. Figure 7(a) shows the simulation results of the generated BVBs with a maximum beam diameter of 0.0552 m as well as a beam diameter of 0.0072 m at the receiver, where the modulation phase patterns are on the left of Fig. 7(a) and the corresponding light intensity distribution at the white dashed lines is shown in the inset of Fig. 7(a). In addition, we can change the maximum beam diameter of BVBs without changing the size of the beam diameter at the receiver, as shown in Fig. 7(b). At this point, the resulting BVBs have a maximum beam diameter of 0.0696 m and a beam diameter of 0.0072 m at the receiver. The results show that BVBs of different bottle sizes with the same beam diameter at the receiver can be easily generated by modulating the trajectory, which will further broaden the application of BVBs in OAM FSO communication systems.
Fig. 7. Simulation results of two BVBs with different bottle sizes and the same beam diameter at the receiver. (a) Maximum beam diameter of 0.0552 m and a beam diameter of 0.0072 m at the receiver; (b) Maximum beam diameter of 0.0696 m and a beam diameter of 0.0072 m at the receiver.
To further illustrate the transmission characteristics of BVBs over obstructions, we simulated the free-space transmission trajectories of the conventional OAM beam, the Bessel beam and BVBs over obstructions size of 40 mm, respectively, as shown in Fig. 8. The incident Gaussian beam with a radius of 0.05 m and the wavelength of 1550 nm. The first to third columns of Fig. 8 correspond to the modulation phase patterns, transmission trajectories and intensity distributions (z = 300 m) of the three different beams carrying OAM, respectively. The simulation results show that the main lobe of both the conventional OAM beams and Bessel beams are moderately blocked compared to BVBs. Therefore, BVBs can effectively improve the performance of circumventing obstructions OAM FSO communication links.
Fig. 8. Free-space transmission trajectories circumventing obstructions of three different beams carrying OAM. (a) Conventional OAM beam; (b) Bessel beam; (c) BVB.
Finally, we measure the transmission performances of BVBs circumventing obstructions compare with conventional OAM beams and Bessel beams under atmospheric turbulence and limited receiving apertures. The received optical power and the corresponding cumulative probability of three different beams carrying OAM are shown in Fig. 9, which corresponds to the simulation result of limited receiving apertures d = 10, 25, 40 mm under the atmospheric turbulence strength D/r0 = 2 and the obstruction size of 40 mm, respectively. Simulation results show that when the receiver aperture is set to 40 mm, the average received optical power of the BVB is improved about 7 dB and 3 dB compared to the conventional OAM beam and Bessel beam, respectively. Meanwhile, as the received aperture decreases, BVBs still have the lowest received optical power. As a result, BVBs have better performance of circumventing obstructions under atmospheric turbulence and limited receiver apertures.
Fig. 9. Received optical power and cumulative probability of three different beams carrying OAM captured by a limited receiving aperture diameter (d = 10, 25, 40 mm) under the atmospheric turbulence strength D/r0 = 2 and the obstruction size of 40 mm. (a) and (d) 10 mm; (b) and (e) 25 mm; (c) and (f) 40 mm.
4. Conclusion
In brief, we propose a method to generate BVBs with a parabolic trajectory by modulating only the phase of the incident Gaussian beam. Meanwhile, the trajectory of BVBs generated are highly compatible with the predesigned trajectory by using this method. Moreover, we evaluate the free-space transmission performance of BVBs under atmospheric turbulence and limited receiving aperture. The results show that BVBs have better OAM FSO communication link performance compared with conventional OAM beams and Bessel beams. In addition, the performance of the BVBs circumventing obstructions is further investigated. The simulation results show that when setting the atmospheric turbulence strength D/r0 = 2 and the obstruction size of 40 mm, the average received optical power of the BVBs captured by a limited receiving aperture diameter (d = 40 mm) is improved about 7 dB and 3 dB compared to conventional OAM beams and Bessel beams, respectively. Therefore, BVBs can effectively improve the performance of OAM FSO communication links in different scenarios. Additionally, BVBs can be employed in OAM mode multiplexed FSO communication links to circumvent obstructions, which will be explored in the future.
Funding
National Natural Science Foundation of China (12104078); China Postdoctoral Science Foundation (2021M700561); Natural Science Foundation of Chongqing (cstc2021jcyj-bshX0223); Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202000622).
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.
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