# Bi-Photon Entangled Airy Beams through Unstable Ocean Turbulence
## 3 Numerical Results and Discussion  
In this section, the impacts of beam parameters and turbulence factors on the propagation of bi-photon entangled Airy beams in oceanic turbulence are numerically simulated. Unless otherwise specified, the basic parameters are given in Table 1.  
Figure 3a shows a plot of the relative detection probability and crosstalk probability as a function of the initial probability amplitude \(\mu\) at a fixed propagation distance. The plot detail of Figure 3a is presented in Appendix A. By taking different values of \(\mu\) in Equation (7), Figure 3a demonstrates the relationship between OAM entanglement and OAM probability. The maximum value of the relative detection probability and the minimum value of the crosstalk probability of the signal photon are both located at \(\mu=1/\sqrt{2}\), which correspond to the maximal entanglement. With the increment of entanglement, the noise induced by turbulence in the UOC system is suppressed, and the signal becomes more distinguishable for receivers [26]. To evaluate the impacts of bi-photon entanglement on the OAM propagation in unstable oceanic turbulence, \((p_{rm_{1}}-p_{m_{0}})/p_{m_{0}}\) is calculated to quantify the improvement of the bi-photon entanglement in comparison with a single photon under the same UOC link, where \(p_{m_{0}}\) denotes the single photon detection probability in Equation (8). In what follows, we discuss the maximally entangled case of \(p_{rm_{1}}\), which is calculated according to Equation (7). As shown in Figure 3b, the smaller the OAM quantum number \(m_{0}\) is, the more pronounced the performance of the UOC link based on bi-photon entanglement improves. The maximal improvement reaches 62%, especially in the case \(m_{0}=1\). However, this improvement decreases with the increment of the initial OAM quantum number, and Figure 3b shows only a 17% improvement in the relative detection probability of the UOC link when \(m_{0}=5\). It reveals that bi-photon entangled Airy beams with smaller \(m_{0}\) are more robust to oceanic turbulence. Concurrently, the generation of vortex beams with a small \(m_{0}\) is easier than that with a large \(m_{0}\) in practical applications [39]. Therefore, the entangled Airy bi-photon with a small OAM number is the preferable light source in a practical UOC link. For OAM-based systems, information is encoded in the entire Hilbert space, where the turbulence effects, such as the fluctuations of the refractive index of the water flow [40], would cause the quantum state of photons to be partially transferred into other orthogonal complements, resulting in the mode scattering of OAM, or generally, the crosstalk, which is a disadvantage compared to polarization-based systems. The relative detection probability of a single photon and bi-photon entangled Airy beams are plotted in Figure 4, where we give no consideration to the trivial case \(m_{0}=0\) according to Equation (1). For short-distance communication, as shown in Figure 4a,c, the signal still remains distinguishable for the crosstalk only existing on the immediately adjacent channels. As the propagation distance increases, for example, \(z=50\) m (as shown in Figure 4b,d), the crosstalk is conspicuous in all the observed channels. For large values of transmitted \(|m_{1}\rangle\), oceanic turbulence makes the OAM state of the beam more prone to spread and the channel more vulnerable. Figure 4b,d shows that the OAM distribution of bi-photon entanglement for low-order OAM states is effectively enhanced in comparison with that of a single photon, while the phenomenon