1. Introduction

Orbital angular momentum (OAM), offers the possibility of high security and a large capacity for line-to-sight optical communication by encoding information on larger Hilbert space [1]. However, OAM states are very fragile to turbulent disorders, such as atmospheric turbulence [1–4] and oceanic turbulence [5–11]. Refractive-index power spectrum serves as a statistical description of the randomness of optical turbulence [12]. Tuneable parameters in the power spectrum largely facilitate the theoretical investigation, especially in simulating oceanic environments [5,13–15]. To better predict real-world oceanic turbulence, spectra of oceanic turbulence are gradual including the influence of stratification [13] and outer-scale [5] of turbulence cells. Li’s spectrum [5] demonstrated better agreement with the experimental data than other spectra [14,16], and has been widely considered in underwater OAM communication systems [7,17–19].

The effective reduction of the turbulence effects is a crucial issue for an OAM communication system. Several compensation techniques have been proposed, such as adaptive optics [20–23], deep learning [24], and iterative approaches [25]. Recently, implementing lights with inward acceleration or auto-focusing has been proposed and investigated to assist OAM communication [26,27]. The auto-focusing effect has been reported to produce self-recovery, which would increase the received power of the signal OAM mode and reduce the crosstalk [28]. Besides, auto-focusing enables the vortex beams to be extremely robust against turbulence [29]. Airy beams, which can possess auto-focusing properties, may find considerable potential applications. The propagation behaviors of Airy beams in turbulence have been conducted in turbulence, including both stable and unstable stratified oceanic turbulence [9,17,30–32]. Yet, most of those results for the Airy beams have neglected the self-recovery effects [28] and auto-focusing effects [9,17] on the OAM detection probabilities. Tracing back to the root, the optical field envelope of the Airy beam is hard to analytically derive [33]. Instead, the far-field approximation of the Airy beam was employed, which, however, fails to accurately describe the propagation in turbulence, especially for the near-field propagation in an underwater link case. Despite its widespread usage due to its simple expressions, this far-field approximate method results in the disappearance of the auto-focusing property [9,17]. Fortunately, by employing the extended Huygens-Fresnel principle, this issue can be resolved and provide accurate insight into the Airy envelope [34–36].

To further enhance the quality of detection OAM, the physical settings at the source plane are essential. Excluding the proper beam parameters, selecting the initial beam pattern is an effective way, such as using array beams [37], vector beams [3], and so on. Considering the superpositions of each OAM mode, quantum entanglement between OAM modes plays a central role in the quantum information community [38]. Using the spontaneous parametric down-conversion, the photon pair can be naturally entangled in terms of OAM [38]. The application of OAM entanglement in optical communication is still challenging since turbulence would deteriorate the photonic phase and destroy the OAM entanglement thereof. To relieve the turbulent distortions, controlling the entanglement during propagation through the physical feature of OAM beams is an effective approach. For instance, the self-healing effect of Bessel-Gaussian beams was shown to recover the entanglement after propagating through an obstruction [39]. With the recent advancements of spatially entangled Airy beams [40], there is an urgent need for research to elucidate the underlying physical mechanism behind the co-effects of OAM entanglement and auto-focusing in resisting turbulence distortions.

In this paper, we investigate the co-effects of auto-focusing and OAM entanglement for mitigating and overcoming turbulent distortions. Sec. 2 structures the detection probability of the entangled bi-photon Airy states. The single-phase approximation [1] is implemented to represent the effects of oceanic turbulence. Sec. 3 describes the radial propagation of Airy beams and the oceanic turbulence. The accurate radial distribution of the Airy beam is obtained by the extended Huygens-Fresnel principle. In Sec. 4, we conduct numerical simulations to discuss the influence of oceanic turbulence and beam parameters on auto-focusing and OAM entanglement. Sec. 5 concludes our paper.

2. Entangled OAM states in turbulence

We start from a general single-photon case where the pure phase perturbation model is used. For classical single-photon OAM communication systems, the probability of measuring OAM state $|m \rangle$ at distance z is given as [1]

To extend to the bi-photon entangled OAM communication system, we first assume the initial state of OAM beams to be an entangled Bell-like state as [41,42]

Given the Markov approximation, we consider the turbulent channels 1 and 2 to be independent [41]. The expression of the joint detection probability ${P_{{\textrm{m}_\textrm{1}}{\textrm{m}_\textrm{2}}}}(z )$ that measures $|{{m_\textrm{1}},{m_\textrm{2}}} \rangle$ state is given by [31,43]

Equation (4) can be degraded to the single-photon case when the probability of launching $|{{m_1}} \rangle$ is 1, in which case the detection probability ${P_1}({m_1},z)$ is equivalent to the relative detection probability. Comparing the relative detection probability of the single-photon [see Eq. (1)], an extra item ${P_1}({m_1^ + ,z} )$ in Eq. (4) describes the inner-transition probability of the bi-photon. Taking the OAM state $|{{m_1}} \rangle$ as an example, one has ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )\ge {P_1}({{m_1},z} )$ due to the increment term ${P_1}({m_1^ + ,z} )$ always being a non-negative real quantity.

Note that, the probability ${P_1}({m_1^ + ,z} )$ in Eq. (4) also equals the crosstalk probability between the OAM states whose OAM gap is $m_1^ +$. Physically, ${P_1}({m_1^ + ,z} )$ is the result of the inner transition of the entangled bi-photon $|{{m_1}} \rangle$ and $|{{m_2}} \rangle$, which finally results in the elevation of the relative probability ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$. It is easy to prove that the maximal probability of ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ is acquired for $\mu = {1 / {\sqrt 2 }}$, which corresponds to the maximal entanglement case.

3. Auto-focusing Airy beam and unstable oceanic turbulence

The radial part ${R^{|{{m_0}} |}}({r,z} )$ of Eq. (1) depicts the spatial structure of the OAM mode. For some particular beam geometries, the wave functions at the source plane and the arbitrary distance z are analytically accessible, such as the most employed Laguerre–Gaussian (LG) beams [38,44]. However, for others, only the wave function at the source plane can be given analytically, such as the Airy beam we discussed here. In cylindrical coordinates, the initial radial Airy profile is given as [26]

Figure 1 illustrates the auto-focusing evolution of the Airy beam in the vacuum. With inward acceleration, the primary and side rings of the Airy beam are tightened during its propagation. Around the focus position, the beam intensity quickly rises to a peak. Then after the auto-focusing point, the intensity distribution of the Airy beam resembles that of the Bessel-like beam, implying that the far-field Airy envelope can be approximated by the Bessel function [33].

 

Fig. 1. (a) Side view of the propagation of the Airy beam. (b-e) Snapshots of the transverse intensity patterns taken at the planes marked by the dashed lines in (a). The basic parameter settings are $w = 0.5\textrm{ mm}$, ${r_0} = 0.01\textrm{m}$, $a = 0.05$, ${m_0} = 1$.

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The radial component ${R^{|{{m_0}} |}}({r,z} )$ given in Eq. (1) is displayed in Fig. 1. The second term $\Theta ({r,m - {m_0},z} )$ in Eq. (1) describes the distortions of turbulence on the photonic wavefronts and is generally defined by [1]

Here, the spatial power spectrum of oceanic turbulence ${\Phi _n}(\kappa )$ is related to the magnitude of the wave number vector $\kappa$. In the real ocean circumstance, the eddy diffusivity ratio of temperature to salinity is not equal to unity. When unstable stratification exists and outer-scale effects cannot be ignored in oceanic turbulence, the spectrum is given by [5]

For a spherical wave propagating through unstably stratified oceanic turbulence, the spatial coherence radius is obtained by substituting the modified spectrum ${\Phi _n}(\kappa )$ into Eq. (7) as

Figure 2 provides the variation of spatial coherence radius ${\rho _{\textrm{oc}}}$ versus $\varpi$ for both stable and unstable stratifications. Physically, a larger spatial coherence radius ${\rho _{\textrm{oc}}}$ means that two optical beams can stay correlated within a longer distance [45]. The value of ${\rho _{\textrm{oc}}}$ is a good reminder of the strength of turbulence. In this case, unstable stratified oceanic turbulence that contributes to a larger ${\rho _{\textrm{oc}}}$ is weaker than stable stratified ones for $\varpi \le - 1$. Recalling the physical meaning of $\varpi$ [5], the turbulence effects in the temperature-dominated region are stronger in stable stratified oceanic turbulence compared with unstable stratified oceanic conditions [46]. However, the strength of unstable stratified oceanic turbulence rises quickly when $\varpi$ is approaching $- 1$. As a result, the strength of these two types of turbulence becomes equal when $\varpi ={-} 1$. Indeed, unstable stratified oceanic turbulence would degenerate into stable ones for $\varpi ={-} 1$. When salinity fluctuations account for the dominant effects ($\varpi \ge - 1$), unstable stratification would induce more distortions.

 

Fig. 2. Variation of the spatial coherence radius ${\rho _{\textrm{oc}}}$ versus $\varpi$ in stable and unstable stratified oceanic turbulence.

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To quantitatively characterize the auto-focusing evolution in turbulence, the second-order momentum of the beam width $W(z )$ is given as [28,29]

 

Fig. 3. Side views of the propagation of the Airy beam for different values of ${\rho _{\textrm{oc}}}$. (a-c) ${\rho _{\textrm{oc}}} = 0.06{z^{ - 1/2}}$, (d-f) ${\rho _{\textrm{oc}}} = 0.04{z^{ - 1/2}}$, and (g-i) ${\rho _{\textrm{oc}}} = 0.02{z^{ - 1/2}}$. The dashed lines plot the $W(z )$. (b), (e), and (h) are the snapshots of the intensity at the focus position, (c), (f), and (i) are the snapshots of the phase at the focus position. The basic parameter settings are $w = 0.5\textrm{ mm}$, ${r_0} = 0.01\textrm{ m}$, $a = 0.05$, ${m_0} = 1$.

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4. Evolution of entangled auto-focusing Airy beams in unstable oceanic turbulence

In this section, from the perspective of OAM detection probability, the effects of unstable oceanic turbulence on the entangled auto-focusing Airy beams are numerically simulated and discussed. The key simulation parameters are listed in Table 1, and the values of the simulation parameters are chosen according to experimental studies. For example, the simulation distance is set as 50 m based on a recent experiment of a 55 m underwater link [49].

Table 1. Parameters values used in simulations

View Table

As mentioned in Sec. 1, several studies [30–32] have used the far-field approximated wavefunction of Airy beams to investigate their behaviors in turbulence. In our work, the accurate Airy evolution is considered according to the extended Huygens-Fresnel principle and angular spectrum theory [12]. Thus, Fig. 4(a) compares our results with two previous studies [9,17]. The beam parameters are set the same in these three simulations. Besides, the power exponential phase vortex discussed in Ref. [9] is reduced into a general phase vortex by setting the power of the spiral phase as 1. In this case, the two previous studies have shown the same results. Our result differs from the previous study in that the self-recovery effect is accurately captured by our model. Indeed, the auto-focusing effect is validated to produce the self-recovery of the probability. Yet, using the approximated Airy profile fails to describe the self-recovery effect. This is one of the key points that we want to address.

 

Fig. 4. Validation of the simulation method. Comparison of (a) different beam models and (b) different calculation methods.

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Using multiple phase screen method to investigate the behavior of Airy beams in turbulence has been well-structured [34]. As we have illustrated the difference of our method in Fig. 4(a), the accuracy of our method is validated in Fig. 4(b). By keeping the same simulation condition with Ref. [34], the corresponding OAM spiral spectrum is derived. Basically, the probabilities of different OAM modes are very close. The fine differences in Fig. 4(b) are due to the fact that the quartic phase approximation (Eq. (6)) on the turbulence phase structure function is used in our model.

Figure 5 shows how the detection probability ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ is affected by different beam parameters, including main ring radius ${r_0}$, transverse scale w, exponential truncation parameter a, and wavelength $\lambda$. According to the simulations in Figs. 2 and 3, an elevation period of ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ is found around the focus position, which is called the self-recovery effect [28]. This result verifies that our considered method is accurate in describing the propagation of Airy beams in optical turbulence. With the same inner acceleration, Airy beams with a larger radius ${r_0}$ require more focus time, which ultimately leads to the elevation period of ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ occurring at a longer distance. Meanwhile, the larger the radius ${r_0}$, the larger the effective beam area of Airy beams. In this case, Airy beams “see” more turbulence, resulting in more distortions to be introduced to the spiral phase. After focusing, however, the typical Airy distribution evolves into a Bessel-like distribution, reversing the decreasing law with ${r_0}$ [32]. Similar reversions of the evolution law are also observed in the discussions about w, a and $\lambda$ as depicted in Figs. 5(b-d).

 

Fig. 5. Detection probability ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ versus propagation distance z for different (a) main ring radius ${r_0}$, (b) transverse scale w, (c) exponential parameter a and (d) wavelength $\lambda$.

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The transverse scale w is related to the intensity of the side lobe of the Airy beam. Airy beams with larger w have a stronger intensity of the side lobe [17,36], which means that the transverse distribution can be maintained over a longer distance, and the focal length is therefore longer. However, OAM states would suffer more distortions for propagating a longer distance in oceanic turbulence. Airy beams with larger w thus have a smaller peak value of ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ as illustrated in Fig. 5(b). For short-distance propagation, an appropriate value of w needs to be chosen according to the practical link.

The propagation of Airy beams with different exponential parameters a is discussed in Fig. 5(c). The superiority of Airy beams with smaller a is outstanding around the focus position. With the growing value of a, the Airy beam evolves more like an off-axis Gaussian beam [50] which is a generally considered diffracting beam. As a result, the evolution of ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ is smoother for a larger a. Finally, in Fig. 5(d), the influence of wavelength is discussed. The wavelength $\lambda$ is confined due to the high absorption of the seawater. Here we select three typical communication wavelengths from the “window”. Generally, $\lambda$ has fewer effects at long-distance compared to the other parameters we discussed above. Airy beam with smaller wavelengths has a longer focal length and better performance after auto-focusing. In all, from our results in Fig. 5, to ensure better performance at the receiver, it is necessary to match the focal length with the practical link distance to achieve a higher signal probability.

Figure 6 discusses the influences of different OAM numbers and aperture diameters on the propagation of bi-photon entangled Airy beams. From the results in Fig. 6(a), entangled Airy beams with smaller ${m_0}$ gain larger ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$. In the classical single-photon level, a low-order vortex beam is usually reported to possess stronger resistance to turbulence [1]. Higher order vortex beam tends to spread wider [2] and the vortex is unstable [51]. When we consider the OAM entanglement, the gap (${m_1} \leftrightarrow {m_2}$) between the initial bi-photon states is a crucial term as described in Eq. (4). For example, the gap of the case ${m_0} = 1$ is only 2, while the gap is 6 for ${m_0} = 3$. The larger the gap, the more challenging it becomes to transfer the energy of OAM. The advantages of low-order Airy beams are further enhanced at the bi-photon entangled level. Therefore, as presented in Table 1, we select ${m_0} = 1$ as the optimized OAM mode. The aperture diameter D is a crucial parameter in optical wireless communication. When the total energy is confined by the aperture, the influence of ${m_0}$ gradually loses the dominant part and ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ tends to stabilize with further increase of ${m_0}$ as demonstrated in Fig. 6(a). Although the increment of D actually elevates the received energy of the launched OAM beams, it elevates the received energy of crosstalks. It is worth noting that under the discussed D conditions, most of the side lobes of the Airy beams have been included, as shown in Figs. 1 and 3. Although the energy of the Airy beam is considered to be infinitely divergent [26], we have not taken into account its truncated part, which the detection probability must be normalized. In this case, increasing the aperture diameter primarily enhances the received proportion of crosstalk modes spurred by turbulence, thus contributing more to the overall detection probability than the increments of the signal mode. The final result is that a larger D leads to a lower ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$.

 

Fig. 6. Detection probability ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ versus propagation distance z for different (a) OAM number ${m_0}$ and (b) aperture diameter D.

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Compared to the results in Figs. 5 and 6, the reversions of the evolution law are not observed in Fig. 6. It is easy to learn that D only changes the aperture of the receiver, thus has no relationship to the auto-focusing and has invariant influence in the whole path. The OAM number ${m_0}$ does relate to the auto-focusing of Airy beams [36], while the influence is quite small compared to factors discussed in Fig. 5. These results give some insights into the analytical solution to the detection probability of Airy beams.

The inner scale of turbulence cells varies inversely with the average rate of energy dissipation, $\eta \sim {\varepsilon ^{ - 1/4}}$ [5], showing that stronger turbulence has smaller inner scales. While a large outer scale can induce big beam wander [48]. Thus, unlike the inner scale, the decrease of the outer scale results in weaker turbulence. Figure 7 illustrates the evolution of probability ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ versus the inner and outer scales. We select two different distances to characterize the position of Airy beams before and after auto-focusing. Basically, the effects of $\eta$ and ${L_0}$ are similar under these two distances. A smaller inner scale leads to worse detection at the receiver, while the variation of the outer scale has little influence on the entangled Airy beams. In terms of turbulence cells [12,48], when the values of $\eta$ and ${L_0}$ are closer, the turbulence distortions are weaker.

 

Fig. 7. Detection probability ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ versus (a) inner scale $\eta$ and (b) outer scale ${L_0}$ for different propagation distances.

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Then we will see how auto-focusing Airy beams evolve in different strengths of oceanic turbulence. According to Fig. 3, the distance interval is zoomed out to address the variation of auto-focusing. Figure 8(a) illustrates the evolution of the detection probability ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ versus ${\rho _{\textrm{oc}}}$ and z. Since smaller ${\rho _{\textrm{oc}}}$ corresponds to a stronger strength of oceanic turbulence, the OAM modes of the auto-focusing Airy beams experience more distortions at the same distance. As a result, the signal OAM probability ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ is decreasing with decreasing ${\rho _{\textrm{oc}}}$ for a fixed distance. For comparison, Fig. 8(b) plots the evolution of ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ with ${\rho _{\textrm{oc}}}$ and z of the most studied LG beam, which is a natural diffraction beam. Generally, with the accumulation of turbulence effects, vortex beams would receive more distortions during the propagation. The OAM states of the LG beam would scatter to many crosstalk modes, leading to the reduction of the signal OAM probability as depicted in Fig. 8(b). The auto-focusing property of the Airy beams can compensate for the decay of OAM probability, especially around the focus position.

 

Fig. 8. Contour plot of the detection probability versus ${\rho _{\textrm{oc}}}$ and propagation distance z, (a) entangled Airy beams and (b) entangled LG beams [44].

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From our results, both auto-focusing and OAM entanglement are demonstrated to enhance the detection probability. Airy beams have a significant self-recovery effect on the signal probability around the focus position due to auto-focusing. The increment brought by OAM entanglement, as described in Eq. (4), is also illustrated analytically. We note that the term ${P_\textrm{1}}({m_1^ + ,z} )$ in Eq. (4) actually denotes the conversion between the OAM modes of the initial entangled bi-photon. It is therefore interesting to see the co-effects of auto-focusing and entanglement on the photonic OAM states under a turbulence environment.

We plot the increment ${P_\textrm{1}}({m_1^ + ,z} )$ versus ${\rho _{\textrm{oc}}}$ and z in Fig. 9 to illustrate the co-effects of auto-focusing and OAM entanglement. For a fixed distance, ${P_\textrm{1}}({m_1^ + ,z} )$ first rises to a peak and then decreases as turbulence strength increases, which is the so-called second-order energy transition [52]. When comparing the behaviors of ${P_\textrm{1}}({m_1^ + ,z} )$ at different distances, we note that ${P_\textrm{1}}({m_1^ + ,z} )$ becomes less influenced by the variation of ${\rho _{\textrm{oc}}}$ after auto-focusing. Since the value of ${\rho _{\textrm{oc}}}$ is related to the strength of turbulence, OAM states of the entangled Airy beams may be more stable to the random variation of turbulence after auto-focusing. Then for a fixed value of ${\rho _{\textrm{oc}}}$, OAM entanglement finds the minimal of ${P_\textrm{1}}({m_1^ + ,z} )$ around the focus position. The distance of the minimal of ${P_\textrm{1}}({m_1^ + ,z} )$ is decreased with the increment of turbulence strength, which is a reasonable result as predicted in Fig. 3.

 

Fig. 9. Increment brought by OAM entanglement ${P_\textrm{1}}({m_1^ + ,z} )$ versus z and ${\rho _{\textrm{oc}}}$.

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In Fig. 10, we provide a detailed analysis of the physical mechanism underlying auto-focusing and OAM entanglement. In the common case depicted in Fig. 10(a), after turbulent propagation, part of the energy carried by the signal mode $|{m_0^{}} \rangle$ will be redistributed into the other OAM states (crosstalk modes). To focus on the difference between auto-focusing and entanglement, the redistributions of different crosstalk modes are represented by the same colour-scale. When only OAM entanglement is considered, as explained by Eq. (4), the elevation term ${P_\textrm{1}}({m_1^ + ,z} )$ resulting from the OAM conversion of the initial bi-photon mode $|{ - {m_0}} \rangle$ increases the energy of the signal mode, as shown in Fig. 10(b). However, when only auto-focusing is considered, as depicted in Fig. 10(c), the energy of the signal mode is increased, and the energy into crosstalk modes is reduced due to OAM conversation. In this paper, we investigate the combined effects of auto-focusing and OAM entanglement, as shown in Fig. 10(d). Auto-focusing reduces the energy of crosstalk modes, that contain the initial bi-photon mode $|{ - {m_0}} \rangle$, thereby reducing the increment term ${P_\textrm{1}}({m_1^ + ,z} )$ brought by OAM entanglement. Note that, the elevation of the signal mode arises from the reduction of all crosstalk modes ($\cdots |{ - {m_0} - 1} \rangle ,|{ - {m_0}} \rangle ,|{ - {m_0} + 1} \rangle \cdots$), while the increment brought by entanglement arises only from one crosstalk mode $|{ - {m_0}} \rangle$. Besides, energy transition mainly occurs between the signal mode and the adjacent crosstalk modes $({m_1^ -{\pm} 1} )$ [52], while the minimum of $m_1^ +$ is 2, indicating that the attenuation effect of entanglement on the signal OAM probability is not dominant compared with the elevation effect of auto-focusing. Therefore, the co-effects of auto-focusing and OAM entanglement will reinforce the robustness of OAM beams to turbulence.

 

Fig. 10. OAM inner transition on different scenarios. (a) single-photon diffractive beam; (b) entangled diffractive beams; (c) single-photon auto-focusing beam; (d) entangled auto-focusing beams.

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Finally, Fig. 11 gives a quantitative description of the co-effects of auto-focusing and OAM entanglement. We compare the evolution of the OAM probability of Airy beams with that of a general diffractive vortex beam, the LG beam, in both single-photon and entangled bi-photon cases. The initial beam waist of the LG beam is set to ${w_0} = 14\textrm{ mm}$ in order to have a similar probability evolution with that of the Airy beam before auto-focusing. Compared with the evolution of “LG_si” and “Airy_si” at the focus position, the auto-focusing produces the self-recovery and elevates the signal probability from 0.22 to 0.41 with an $86\%$ increment, where the increment is given by ${{({0.41 - 0.22} )} / {0.22}}$. Then, when only OAM entanglement exists (compared to the evolution of “LG_si” and “LG_bi”), the conversion of the entangled bi-photon raises the signal probability from 0.22 to 0.32 in an increment of $45\%$. Generally, one would expect that the increment term under the co-effects of auto-focusing and OAM entanglement is the summation as $86\%+ 45\%= 131\%$. However, compared to the evolution of “LG_si” and “Airy_bi”, the signal probability increases from 0.22 to 0.48 with only a $118\%$ increment. The reduction from $131\%$ to $118\%$ is exactly a numerical result of the physical mechanism revealed in Fig. 10.

 

Fig. 11. Comparison of the detection probability of LG and Airy beams in different scenarios.

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5. Conclusion

In this study, we have focused on investigating the propagation of the entangled auto-focusing Airy beams in oceanic turbulence, revealing the co-effects between auto-focusing and OAM entanglement on mitigating turbulence-induced distortions. We constructed the OAM detection probability of the entangled bi-photon states by preparing the initial bi-photon in Bell-like states and derived the analytical spatial coherence radius of unstably stratified oceanic turbulence.

The OAM detection probability of the entangled bi-photon states was constructed, and the analytical spatial coherence radius of unstably stratified oceanic turbulence was derived. We disclosed that Airy beams would focus at a shorter distance, spread wider, and possess lower peak intensity in stronger oceanic turbulence. In the discussions, we found that beam parameters have reversed behaviors on the OAM probability after auto-focusing. We confirmed that auto-focusing would hinder the OAM conversion between the initial entangled OAM modes, therefore reducing the increment term on the OAM probability brought by entanglement. However, the increment brought by auto-focusing is much larger than that of the reduction. Through numerical validation, we have demonstrated a $118\%$ increment by using the entangled auto-focusing Airy beam compared to the general LG beam.

Our result suggests that using an entangled source with auto-focusing beams could enhance the detection probability. However, improvement of the signal OAM mode by modulating focal length may confine in short-distance communication. From our results, Airy beams with larger $\lambda$, smaller ${r_0}$, and w would obtain higher ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ before auto-focusing but smaller ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$ after auto-focusing. This reversion of the evolution laws after auto-focusing indicates the importance of matching the focal length to the practical communication link. It will be also interesting to modulate the auto-focusing distance according to the length of the practical communication link, the issue of which is out of the scope of this paper and will be addressed elsewhere. We believe that our results would provide significant insight into underwater OAM communication utilizing entangled auto-focusing resources.

Appendix

Here we illustrate the effects of grid numbers on the detection probability ${P_{\textrm{r}{\textrm{m}_\textrm{1}}}}(z )$, which denotes the sampling accuracy. We conduct the simulation for entangled Airy beams and five sampling distances are selected to include the variation of Airy beams before and after auto-focusing. Figure 12 illustrates the effects of different grid numbers on the calculation of OAM probability. Generally, the probabilities in different distances demonstrate great convergence when the grid number is larger than 300. Therefore, the grid number 512 used in this paper is large enough to ensure accuracy.

 

Fig. 12. OAM probability for entangled Airy beams versus gird number in different propagation distances.

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Funding

National Natural Science Foundation of China (61701196, 61871202).

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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