# Bi-Photon Entangled Airy Beams through Unstable Ocean Turbulence
## 1 Introduction  
Underwater optical communication (UOC) based on orbital angular momentum (OAM) has a high capacity due to the arbitrary base-N quantum digits [1, 2, 3], while oceanic turbulence in the propagation would seriously distort the helical phase structure of OAM [4], giving rise to degradation of the UOC system. Spatial power spectrum as a statistical method can describe this kind of optical turbulence [5, 6, 7]. [PERSON] proposed an oceanic spectrum based on the common assumption that the eddy diffusivity of temperature is equal to that of salt, which implicitly assumes that seawater has a stable stratification [8]. Partly because of its relatively simple algebraic expression, [PERSON]'s spectrum has been widely used in theoretical calculations [9, 10, 11, 12]. Indeed, the eddy diffusivity of temperature and salinity are different from each other in most underwater regions [7, 13], and the spatial power spectrum of turbulent fluctuations of the seawater refraction index has been given by [PERSON] that includes an eddy diffusivity ratio [7]. However, [PERSON]'s theory of oceanic spectrum only worked for infinity outer scale and did not include the effects of outer scale oceanic turbulence. Recently, a spatial power spectrum of unstable stratification oceanic turbulence has been proposed as an explicit function of the outer scale factor [14]. By considering the effect of the outer scale turbulence, this spectrum agrees well with the experimental data in all wavenumber regions. On the other hand, motivated by the fact that OAM states of vortex beams are discrete, integer-valued and construct a theoretical infinite-dimensional Hilbert space [15], the entanglement of OAM has been demonstrated to carry more information in optical communication [16, 17]. It is possible to obtain entangled OAM states via spontaneous parametric down-conversion [18] and transport through turbulence [19, 20]. OAM entanglement not only presents an approach to enhance information capacity [21] but also promisesstrong robustness against decoherence effects [22] and higher security [23]. However, the transmission of OAM entanglement states in free space is challenging [20]. The entanglement of OAM would be destroyed by beam wandering, wavefront degradation and scintillation induced by turbulence [24]. There are many efforts toward OAM entanglement propagation through turbulence. By introducing the phase correlation length of Laguerre-Gaussian (LG) modes, the decay of OAM entanglement states upon propagation through atmospheric turbulence has been discussed [25]. It turns out that the entanglement of photonic OAM shows a universal exponential decay in weak turbulence. Moreover, the effects of non-Kolmogorov atmospheric turbulence on bi-photon entangled OAM states have been discussed by means of LG modes [26]. In the above studies, LG beams possessing a well-defined OAM serve as a common option for the light source. In fact, Airy beams have non-diffraction [27], autofocusing [4; 28] and self-healing [29; 30] properties, providing considerable potential as a bi-photon entangled UOC link. The propagation dynamic of Airy beams has been proven in simulations to mitigate the OAM degradation and reduce the crosstalk induced by turbulence [31; 32]. Recently, the spatially entangled Airy bi-photon has been generated experimentally and propagated in a vacuum [33]. To apply the entangled Airy source in the UOC link, the investigation of entangled Airy beams in oceanic turbulence is imminent. In this paper, we demonstrate the propagation of the OAM of bi-photon entangled Airy beams through unstable oceanic turbulence. The semi-classical method is extended into a bi-photon case, and the relative detection probability of bi-photon entangled Airy beams is developed. Based on the general integrals, the analytical expression of the spatial coherence radius of a spherical wave propagating in unstably stratified oceanic turbulence is derived.