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Abstract
The obstacle in a channel is a typical scenario for free-space optical (FSO) communications, however, it will destroy the information in channels, especially for the orbital angular momentum (OAM) multiplexing systems and cause performance degradation. Motivated by the feature of predefining intensity profile, here we propose to use frozen wave (FW) carrying OAM for the FSO communications to mitigate the influence of obstacles on the beam propagation. The key idea is to design the longitudinal intensity profile of FW to distribute the beam energy of the location where the obstacle exists over a large region and focus again on the central region after the obstacle for propagation. By analyzing the cases under different sizes, positions, and shapes of the obstacles with on-axis and off-axis scenarios, it has been demonstrated that the detection probability of OAM mode carried by FW can be improved by 0.35 and 0.15 in short-distance and long-distance transmission scenarios, respectively, when compared to that carried by Bessel-Gaussian beam. It demonstrates the FWs have great potential in the OAM-based FSO communications, especially for the obstacle channels.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Bessel beams are the most popular nondiffracting beams carrying orbital angular momentum (OAM) with helical phase factor exp(ilφ), where φ is the azimuthal angle and l denotes the OAM state of the topological charge [1,2]. Bessel beams with different OAM states are theoretically orthogonal to each other, and thus have been widely studied in free-space optical (FSO) communications to increase channel capacity via multiplexing technology [3,4]. Another feature of Bessel beams is the self-healing property that allows the beams to reconstruct the optical field disturbed by obstacles along the propagation axis. To date, much work has been done on the applications, such as OAM multiplexing technology [5,6], and the propagation property [7,8] of individual Bessel beams.
Recently, frozen wave (FW), which is a superposition of multiple copropagating Bessel beams, has received lots of attention [9,10]. The superposed Bessel beams have the same frequency but different longitudinal wave numbers, which enables FW to predesign the transverse and longitudinal intensity profile along the beam axis. In addition, FW also possesses the inherent non-diffracting and self-healing properties of Bessel beams.
With the advantage of predefining intensity profile, FW with an increasing intensity profile along the beam axis is proposed to resist propagation loss in the absorbing fluid [11]. In addition, several efforts have been paid to the superposition of multiple FWs with different OAM topological charges, intensity distributions, longitudinal wave numbers and polarizations to control the rotation, radius and polarization of the beam as well as the OAM mode along the beam axis on demands [12–15]. To overcome the crosstalk induced by turbulence for FSO communications, a longitudinal OAM multiplexing (LOAMM) system is also proposed [16], which controls different OAM modes to appear in different space intervals along the beam axis as carriers. Compared to the traditional OAM multiplexing system, the physical-layer security of the LOAMM system is much higher, as all the transmitted channels cannot be wiretapped simultaneously [17]. Furthermore, considering both turbulence and absorption effects of seawater, the transmission characteristics of FWs in ocean turbulence are investigated as well [18,19]. Other potential applications of FWs can also be found in the fields of optical trapping and micromanipulation [20,21].
In this paper, FW is proposed to mitigate the effect of obstacles on the beam propagation by carefully designing the longitudinal intensity profile, according to the position and size of the obstacle. The maximum transmission distance of FW is analyzed first. Next, two examples of FWs with different longitudinal intensity profiles have been designed for short-distance transmission to analyze the detection probability of OAM mode in the presence of different sizes’ square and circle obstacles lying in different positions and with on-axis and off-axis scenarios. Finally, the characteristics of FW for long-distance transmission with atmospheric turbulence (AT) are studied. The results demonstrate the superiority of using FW for the channel in the presence of obstacles by comparing with the traditional case with a single Bessel-Gaussian (BG) beam (an approximate Bessel beam).
2. Principle of frozen wave
Bessel beams are the most common non-diffracting beams. In cylindrical coordinate (ρ, φ, z), their intensity distribution can be expressed as [1]
FW is a superposition of 2N + 1 Bessel beams, which can be written as [9]
For the transverse spot size of FW, it depends on the value of Q [10]. For FW with l = 0, the radius of the beam is approximately given by ${\rho _0} \approx {{\textrm{2}\textrm{.4}} / {\sqrt {{k^2} - {Q^2}} }}$. For FW with l ≠ 0, the radius of the beam is ${\rho _l} \approx {{{u_l}} / {\sqrt {{k^2} - {Q^2}} }}$, where ulcorresponds to the value of u when Jl(u) approaches its maximum.
In this paper, we have considered nonzero-order (l ≠ 0) FW that carries the desired longitudinal intensity profile. The longitudinal intensity profile F(z) is carefully designed under the assumption that the position and size of the obstacles are known.
3. Frozen wave channel with obstacles
It is well known that the existence of obstacles will destroy the beam propagation and degrade the performance of FSO communication systems. With the ability of predefining longitudinal intensity profile along the beam axis, FW makes it possible to design such a beam to mitigate the effect of obstacles on the beam propagation. Figure 1 shows the schematic diagram of the transmission system in the presence of an obstacle.
Fig. 1. Schematic diagram of the transmission system. (a) The transverse intensity and phase distributions of the generated FW; Examples for the longitudinal intensity distribution of FW (b) without and (c) with the obstacle.
At the transmitter, a Gaussian beam with wavelength λ is sent onto a spatial light modulator (SLM1) encoded with a complex transmission function. Phase masks are loaded on the SLM1 to modulate the Gaussian beam into the FW carrying the OAM. The transverse intensity and phase distributions of the generated FW are shown in Fig. 1(a). Besides, a finite Gaussian aperture with waist radius ω0 is superimposed on SLM1 to generate the desired waveform over a certain distance [12].
During the free-space transmission of the beam, there is an obstacle located at a propagation distance z. To avoid disturbing from the obstacle, we predefine the intensity of FW in the region where the obstacle exists to be zero (with the knowledge of the position and size of the obstacle in advance). That is, the energy of FW around the obstacle is distributed over a larger space and will be restored back to the beam center for propagation after passing the obstacle. Figures 1(b) and 1(c) give an example for the comparison of the longitudinal intensity of FW in the absence and presence of the obstacle, respectively.
The received light field U (x, y, z + Δz) after the obstacle can be written as [22]
where O(x, y) is the complex amplitude transmittance function of an obstacle, which is a binary moment array only consisting of zeros and ones. H(f) is the transfer function propagated by Fresnel diffraction, given by where fx and fy represent the frequencies in the x and y directions, respectively.At the receiver, a conjugate spiral phase hologram is used on SLM2 (exp(–ilφ)) to obtain the energy El of the target OAM mode. The processed FW is then passed through a Fourier lens with focal length f. A pinhole plate with the radius of 3 pixels is placed at the focal plane (the Fourier plane) to filter out the diffraction pattern except for the central intensity. Finally, the received light intensity is recorded by a CCD camera, and the computer is used to calculate the energy El of the target OAM mode, which can be expressed as:
where p(x, y) represents the pinhole with the radius of r0, given bySince the spiral wavefront of the vortex beam is damaged because of obstacles, partial power of the target OAM mode will be unavoidably coupled to other OAM modes. Therefore, other holograms (exp(–inφ), n ≠ l) will be used on SLM2 to detect the energy En (n ≠ l) of other OAM modes. The detection probability pl denotes the ratio of the energy of the target OAM mode over all possible received OAM modes, which can be written as [23]
4. Simulation results and discussions
In this section, we first analyze the FW’s maximum transmission distance and then investigate the characteristics of short-distance and long-distance transmission of FW in different obstacle channels. The wavelength of the beam we used is λ = 532 nm with ω = 3.54 × 1015 Hz and the topological charge in Eq. (2) is l = 5.
4.1 Maximum transmission distance of FW
In principle, the theoretical model of FW yields the periodic waveform with infinite power. However, this is not realistic. In practice, Bessel beams are often truncated by a finite Gaussian aperture with waist radius ω0, which can only maintain the nondiffracting property within a certain distance (field depth), as given by [9]
As a result, the maximum transmission distance Zmax of FW is limited by the smallest longitudinal wave number of the superposed Bessel beams, i.e., the field depth zmax of the Bessel beam with kz = Q – 2πN/L. Therefore, in addition to waist radius ω0, the parameters of Q, N and L all will affect the maximum transmission distance of FW. Due to L is in fact limited by Zmax (L ≤ Zmax), we will set L = 100 cm for the FWs in the following contents and discuss the influence of parameters ω0, Q and N on the maximum transmission distance of FW.
Figure 2 shows the transverse and longitudinal intensity profiles of FWs with N = 7 for Q = 0.99997 k and Q = 0.9999958 k. The waist radius is ω0 = 9 mm. The longitudinal intensity profile is given by
Fig. 2. (a) Transverse and (b) longitudinal intensity profiles of FW with N = 7, ω0 = 9 mm and Q = 0.99997 k; (c) Transverse and (d) longitudinal intensity profiles of FW with N = 7, ω0 = 9 mm and Q = 0.9999958 k.
The resulted maximum transmission distances of FWs for Q = 0.99997 k and Q = 0.9999958 k are 1.0959 m and 2.2608 m, respectively. Figure 2 shows that when the transmission distance z exceeds Zmax, the intensity of FW will decay rapidly. On the other hand, we can see that larger Q will result in larger transverse spot size of FW.
Figure 3 shows the FW’s maximum transmission distance against the waist radius under different values of Q and N. It can be seen from Fig. 3(a) (with N = 7) that the maximum transmission distance increases with the waist radius ω0 and the central longitudinal wave number Q. When Q is determined (see Fig. 3(b) with Q = 0.9999958 k), the smaller value of N results in a larger maximum transmission distance. However, as the smaller N will make the generated FW distortion, N is usually set to Nmax. Therefore, flexibly adjusting the values of parameters ω0 and Q is usually used to meet the need of different transmission distances.
Fig. 3. The FW’s maximum transmission distance Zmax against waist radius ω0 with different values of (a) Q and (b) N.
4.2 FW for short-distance transmission
To achieve high-speed and low-cost indoor wireless communication, free space optics is a promising medium in which the transmission distance is typically within the range of several meters. Considering the indoor FSO communication scenario, the transmission characteristics of vortex FWs for short-distance communication are thus analyzed. In this scenario, two FWs with longitudinal intensity profiles F2(z) and F3(z) are designed for 1 m transmission with Q = 0.9999958 k and ω0 = 4 mm. The yielded maximum value of N is Nmax = 7 that results in 2Nmax + 1 = 15 Bessel beams’ superpositions in Eq. (2). The square and circle obstacles with diameters of d = 1.2 mm and d = 1.6 mm lie at positions of z = 0.2 m and z = 0.5 m in the FSO channels. It should be noted that the value of d corresponds to the diameter of the circle obstacle, which is slightly larger than the side length of the square obstacle to keep the same area.
For comparison, the transmitting performance of the BG beam is also explored and shown in the following figures. For the sake of fairness, the longitudinal wave number of the BG beam is set as the central longitudinal wave number of FW, i.e., kz = Q, to make the FW and BG beams have a similar radius of the inner ring.
4.2.1 FW with F2(z) for obstacles with d = 1.2 mm at z = 0.2 m
In this case, we assume that the obstacle is located at z = 0.2 m with d = 1.2 mm. Accordingly, the F2(z) can be defined as
The corresponding two-dimensional (2D) and three-dimensional (3D) longitudinal intensity profiles as well as the distribution of the weight factor |Am| for each Bessel order of the FW are shown in Fig. 4. As we can see, the intensity for F2(z) is 0 in the range 0 ≤ z ≤ 0.4 m along the propagation axis, while the transverse intensity at z = 0.2 m in the range –0.6 mm ≤ ρ ≤ 0.6 mm is 0 as well, which demonstrate higher fidelity to our presets.
Fig. 4. The (a) 2D and (b) 3D longitudinal intensity profiles with (c) the corresponding distribution of the weight factor for each Bessel beam of FW with F2(z).
Figure 5(a) shows the detection probability against the propagation distance Δz (after obstacle) for the FW with F2(z) (solid lines) and BG beam (dashed lines), when the obstacle is on-axis. The lines with square and circle markers are the results for square and circle obstacles, respectively. On the one hand, it can be seen that benefiting from the self-healing property of BG beams, both FW and BG beam will reconstruct the optical field upon propagation. This results in a rapid increase in terms of detection probability of the target OAM mode when Δz exceeds the minimum self-recovery distance, i.e., 0.207 m. For the positions beyond the minimum self-recovery distance, the detection probability of OAM mode carried by FW will be slightly higher than that carried by the BG beam, due to the less blocked energy.
Fig. 5. Detection probability comparison of FW with F2(z) and BG beam when passing (a) on-axis and (b) off-axis square or circle obstacle; The transverse intensity profiles with the corresponding detection probabilities at Δz = 0 m and 0.76 m for (c) circle and (d) square obstacles.
When the obstacle is off-axis, the FW’s advantage becomes more significant, as shown in Fig. 5(b). In this case, the off-axis distance of the obstacle is 0.1 mm. As the symmetry of the spiral phase of the OAM mode is destroyed, the off-axis obstacle hinders the self-healing ability of BG beams [24]. Therefore, the minimum reconstruction distance for the off-axis case becomes slightly larger than that for the on-axis case, and the detection probability of the OAM mode for the off-axis case decreases as well. However, it can be found that the reduction of the detection probability of the OAM mode carried by the FW is much smaller than that carried by the BG beam.
On the other hand, the results in Fig. 5(a) show that for the on-axis case, the square obstacle has more serious influence than the circle one on the propagation of either FW or BG beam. This is because the energy of the OAM beam mainly concentrates as a donut shape in transverse, square obstacle blocks more side lobes than circle one. However, this is not the case for off-axis obstacles shown in Fig. 5(b). In the off-axis case, the symmetry of the spiral phase of the OAM mode is destroyed more seriously by the circle obstacle, due to the slightly larger size. This leads to a lower detection probability for off-axis circle obstacle than that for off-axis square obstacle.
Figures 5(c) and 5(d) are the transverse intensity profiles and detection probabilities for the on-axis circle and square obstacles at Δz = 0 m and Δz = 0.76 m, respectively. At the position where the obstacle exists (Δz = 0), the detection probabilities are all quite low due to the obstacle. However, when the beam is reconstructed, the detection probability will increase from 0.84 to 0.95 (with 0.11 improvement) for the circle obstacle by using FW to replace the BG beam to carry OAM mode. For the square obstacle, the improvement of the detection probability brought by FW is up to 0.24.
4.2.2 FW with F3(z) for obstacles with d = 1.6 mm at z = 0.5 m
In this case, we assume that the obstacle is located at z = 0.5 m with d = 1.6 mm. Therefore, we generate an FW with a desired longitudinal intensity profile F3(z), given by
The longitudinal intensity profiles and the distribution of the weight factor |Am| for each Bessel beam of the FW with F3 (z) are shown in Fig. 6. The intensity for F3(z) is 0 in the range 0.2 m ≤ z ≤ 0.8 m along the propagation axis, while the transverse intensity at z = 0.5 m in the range –0.8 mm ≤ ρ ≤ 0.8 mm is 0 as well, which agree well with our presets.
Fig. 6. The (a) 2D and (b) 3D longitudinal intensity profiles with (c) the corresponding distribution of the weight factor for each Bessel beam of FW with F3(z).
Figures 7(a) and 7(b) show the comparison of detection probability for the OAM mode carried by the FW with F3(z) and the BG beam in the circumstances of on-axis and off-axis obstacles, respectively. Due to the increase of obstacle size, the minimum self-recovery distance in this case increases to 0.276 m. Therefore, we can see that the detection probability rises rapidly when the propagation distance Δz exceeds around 0.276 m. In particular, for the positions beyond the minimum self-healing distance, the improvements of detection probability can be up to 0.13 and 0.35 for circle and square obstacles at Δz = 0.48 m, respectively, as shown in Figs. 7(c) and 7(d).
Fig. 7. Detection probability comparison of FW with F3(z) and BG beam when passing (a) on-axis and (b) off-axis square or circular obstacle; The transverse intensity profiles with the corresponding detection probabilities at Δz = 0 m and 0.46 m for (c) circle and (d) square obstacles.
4.3 FW for long-distance transmission
In the long-distance case, FW with Q = 0.9999958 k and ω0 = 4 m is considered to achieve 1000 m transmission. The distribution of the longitudinal intensity is shown in Fig. 8, which is a periodic waveform with the longitudinal intensity profile of F2(z). The square and circle obstacles with d = 1.2 mm diameter are placed at positions of z = 100.2 m.
Fig. 8. The distribution of the longitudinal intensity of FW with F2(z), Q = 0.9999958 k, N = 7 and ω0 = 4 m.
Figures 9(a) and 9(b) are the comparisons of detection probabilities of FW and BG beam when passing on-axis and off-axis obstacles, respectively. We can see that the detection probability of the target OAM mode carried by the BG beam and FW remains stable after long-distance transmission. And the detection probability of FW is 0.10 and 0.15 larger than that of the BG beam for circle and square obstacles, respectively, at Δz = 900.6 m in the case of on-axis obstacles. When the obstacles are off-axis, 0.14 and 0.13 improvements of detection probability are realized by using FW.
Fig. 9. Detection probability comparison of FW with F2(z) and ω0 = 4 m and BG beam for long-distance transmission when passing (a) on-axis square or circle obstacles without AT, (b) off-axis square or circle obstacles without AT and (c) circle obstacle with AT.
Since vortex beams are susceptible to atmospheric turbulence during long-distance transmission in free space, the transmission performance of FW in the obstacle channels with weak, medium and strong turbulences is studied as well. The results are shown in Fig. 9(c). The AT is simulated using the Hill-Andrews model [25,26], in which the inner and outer AT scales are set as 1e-4 m and 100 m, respectively, and the phase screens are with M = 1000 size and 100 m interval. The obstacle is circular with the off-axis distance of 1 mm. It can be seen that with the increase of turbulence intensity, the detection probabilities of FW and BG beam both decrease rapidly, due to the serious phase distortion. However, the performance of FW always outperforms the BG beam. These results prove the advantage of FW for long-distance transmission in different-obstacle channels.
But one thing needs to be noted that when FW is used for the OAM-based FSO communications, there is a limitation on the location of the receiver. That is, only when the receiver is located at the stable intensity distribution area, i.e., the region with F(z) ≠ 0, the energy of the OAM mode can be detected. Otherwise, the receiver gets nothing. However, on the other hand, this property of FW will bring inherent security for the FSO communication, as the eavesdropper has no knowledge on the longitudinal intensity profile of FW and thus is hard to eavesdrop any information. Therefore, flexibly designing longitudinal intensity profile of FW to achieve secure FSO communications would be an interesting topic for future works.
5. Conclusions
In this paper, we have demonstrated the superiority of using frozen waves (instead of Bessel-Gaussian beams) in FSO communications with the presence of obstacles in the transmission channels. This benefit comes from the flexible control of the longitudinal intensity profiles of FWs to reduce the blocked energy, according to the position and size of the obstacles. For short-distance and long-distance transmission, up to 0.35 and 0.15 performance improvements have been obtained, respectively, in terms of the detection probability of the OAM mode. The results well indicate that FW provides a new alternative for OAM-based FSO communications to effectively address the performance degradation caused by obstacles.
Funding
National Natural Science Foundation of China (62001151, 61775050).
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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