# Bi-Photon Entangled Airy Beams through Unstable Ocean Turbulence
## 4 Conclusions  
In this paper, we have proposed a theoretical model describing the OAM of bi-photon entangled Airy beams propagating through unstable stratifications of oceanic turbulence. We extend the theory of quantifying single photon OAM communication to obtain the relative detection probability in a bi-photon entangled case and express the bi-photon detection probability as two parts of a single photon detection probability. The analytical expression of the spatial coherence radius of a spherical wave in an unstably stratified oceanic turbulence is derived. Our results demonstrate that bi-photon entangled Airy beams obtain larger relative detection probability than non-entangled ones and are the largest for the maximally entangled case, which is caused by the modal transfer of the bi-photon. The order of Airy beams plays a key role in the detection. For low-order bi-photon entangled Airy beams, OAM entanglement effectively suppresses the crosstalk and improves the relative detection probability. The lowest-order modes of OAM carried by the bi-photon entangled Airy beams exhibit stronger resistance against oceanic turbulence. Consequently, up to a 62% improvement is achieved over a distance of 100 m compared to the same measurement for the single photon system. To detect more initial OAM modes at receivers, bi-photon entangled Airy beams with a larger wavelength \(\lambda\) and main ring radius \(r_{0}\) are preferable. Moreover, a larger main ring radius \(r_{0}\) is also found to help reduce the fluctuations in the relative detection probability in oceanic turbulence. The discussions with respect to oceanic turbulence reveal that bi-photon entangled Airy beams in the turbulence of temperature-dominated fluctuations are less affected. The performance of a UOC system is related to oceanic turbulence with a larger inner-scale \(\eta\), larger dissipation rate of turbulent kinetic energy per unit mass of fluid \(\epsilon\) and a smaller rate of the dissipation  
Figure 7: Relative detection probability of signal photon under (**a**) different outer scale turbulences \(L_{0}\) and inner-scale turbulences \(\eta\); (**b**) different dissipation rates of mean-squared temperature \(\chi_{T}\) and different dissipation rates of kinetic energy per unit mass of fluid \(\epsilon\) (\(z=50\) m).  
of mean-squared temperature \(\chi_{T}\). These findings may be useful in a UOC link based on the OAM of bi-photon entangled Airy beams.  
In this work, we study the probability distribution of the entangled Airy beams in weak-strength turbulence. One of our future works will be to disclose how entangled Airy beams evolve in all strength regions of turbulence. The recent report that states that the auto-focus of Airy beam is reported to help reduce OAM crosstalk [31] will motivate the direction of our future work to investigate the interaction effects between auto-focus and entanglement.  
Conceptualization, D.Y., Y.Z. and Z.H.; methodology, D.Y. and Y.Z.; software, D.Y., Z.Y. and W.W.; validation, D.Y., Z.H. and Y.Z.; formal analysis, D.Y. and Y.Z.; investigation, D.Y.; resources, D.Y.; data curation, D.Y. and Z.Y.; writing--original draft preparation, D.Y.; writing--review and editing, D.Y., Z.H. and Y.Z.; visualization, D.Y., Z.Y. and W.W.; supervision, Z.H. and Y.Z.; project administration, Y.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 61701196, 61871202) and the Key Laboratory Open Fund of Quantum Information of CAS (KQI201).  
Not applicable.  
The authors declare no conflict of interest.