# Bi-Photon Entangled Airy Beams through Unstable Ocean Turbulence
## 3 Numerical Results and Discussion
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Value** \\ \hline OAM number \(m_{0}\) & \(1\) \\ Wavelength \(\lambda\) & \(417\) nm \\ Exponential truncation parameter \(a\) & \(0.05\) \\ Arbitrary transverse scale \(\omega_{0}\) & \(1\) mm \\ Radius of the main ring \(r_{0}\) & \(1\) mm \\ Aperture diameter \(D\) & \(0.06\) m \\ Inner scale \(\eta\) & \(1\) mm \\ Outer scale \(L_{0}\) & \(10\) m \\ Obukhov-Corrsin constant \(\beta\) & \(0.72\) \\ Dissipation rate of kinetic energy per unit mass of fluid \(\epsilon\) & \(1.7\times 10^{-5}\) m\({}^{2}\)/s\({}^{3}\) \\ Dissipation rate of mean-squared temperature \(\chi_{T}\) & \(1.2\times 10^{-8}\) K\({}^{2}\)/s \\ Ratio of temperature and salinity contributions \(\varpi\) & \(-2\) \\ Prandtl number \(Pr_{s}\) & \(700\) \\ Prandtl number \(Pr_{T}\) & \(7\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values used in the simulations. gradually disappears as the OAM number increases. With the increase in the initial \(m_{0}\), the gap between \(m_{1}\) and \(m_{2}\) in Hilbert space becomes larger. Thus the mutual transfer is much harder than that for a smaller \(m_{0}\). For example, \(m_{0}=\pm 10\), the probability of \(|m_{2}\rangle\) transferred to \(|m_{1}\rangle\) will probably be negligible, while the transition probability is an essential component of the \(p_{rm_{1}}(z)\) in the case \(m_{0}=\pm 1\). This result also explains why the improvement in entanglement on the relative detection probability decreases with increases in OAM carried by an Airy beam in Figure 3b. In a practical UOC system, parameter selection of the light source attracts extensive interest. Figure 5a shows the evolution of the relative detection probability \(p_{rm_{1}}\) as the propagation distance increases for different wavelengths \(\lambda\). Here, the values of \(\lambda\) are selected from 417, 477 and 532 nm due to their small absorption loss under the sea. It can be seen from Figure 5a that \(p_{rm_{1}}\) increases with \(\lambda\) slightly. Airy beams with a shorter wavelength have a larger wave number that is more susceptible to turbulent fluctuations [41]. Moreover, we note that at propagation distance \(z=100\) m, the relative detection probabilities \(p_{rm_{1}}\) of  
Figure 4: Relative detection probability under oceanic turbulence of the single photon (**a**,**b**) and bi-photon entangled Airy beams (**c**,**d**) (for OAM, \(|m_{1}\rangle\) detections are plotted as the leading diagonal elements, while \(|m_{2}\rangle\) detections as the minor diagonal elements). With a propagation distance from \(z=10\) m (**a**,**c**), increasing to \(z=50\) m (**b**,**d**). Figure 3: (**a**) Relative detection probability and crosstalk probability of the signal photon versus initial probability amplitude \(\mu\) (\(z=50\) m, \(\Delta m_{i}=1\)); (**b**) \((p_{rm_{1}}-p_{m_{0}})/p_{m_{0}}\) versus OAM quantum numbers \(m_{0}\) (\(z=100\) m). different \(\lambda\) are approximately the same, which means that the communication wavelength could be equally optional at this distance. To investigate the effect of the main ring radius \(r_{0}\) on the OAM of bi-photon entangled Airy beams, we plot Figure 5b. As \(r_{0}\) increases from 1 to 5 mm, the detection probability \(p_{rm_{1}}\) rockets. The main ring radius \(r_{0}\) determines the energy characteristics of Airy beams larger than \(r_{0}\) corresponding to larger maximal intensity [42], thus elevating the detection probability. As we further increase \(r_{0}\) to 25 mm, the probability \(p_{rm_{1}}\) becomes saturated. It indicates that the main ring radius of bi-photon entangled Airy beams and the design of the UOC system should be appropriately matched. While a larger main ring radius does not contribute to the increment of the detection probability, fewer fluctuations of detection probability are observed for a larger \(r_{0}\) in \(z<100\) m. This effect is caused by the dispersion degree of the Airy beam being smaller so that the detection over different distances is more stable [43].