# Bi-Photon Entangled Airy Beams through Unstable Ocean Turbulence
## 2 Methods
Moreover, as \(a>1\), the intensity distribution of the Airy beam evolves into a Gaussian distribution [38], as shown in Figure 2d. Figure 2: Intensity distribution for Airy beams in a vacuum of different beam parameters. Compared with (**a**) \(\omega_{0}=1\) mm, \(r_{0}=1\) mm, \(a=0.05\), the different parameters in (**b–d**) are: (**b**) \(\omega_{0}=2\) mm, (**c**) \(r_{0}=2\) mm and (**d**) \(a=1.1\). We now proceed to discuss how oceanic turbulence influences the OAM of bi-photon entangled Airy beams. Based on the quadratic approximation of the wave structure function [5], the circular harmonic transformation of the rotational coherence function \(\Theta(x)\) becomes  
\[\Theta(r_{i},m_{i},z)=\frac{1}{2\pi}\int_{0}^{2\pi}\exp\left[-\frac{2r_{i}^{2}-2 r_{i}^{2}\text{cos}(\Delta\varphi_{i})}{\rho_{\text{oc}}^{2}}-\text{i}m_{i} \Delta\varphi_{i}\right]\text{d}\Delta\varphi_{i}, \tag{10}\]  
where \(\Delta\varphi_{i}=\varphi_{i}-\varphi_{i}^{\prime}\), and \(\rho_{\text{oc}}\) is the spatial coherence radius of a spherical wave propagating in the oceanic turbulence calculated as [3]  
\[\rho_{\text{oc}}^{-2}=\frac{1}{3}\pi^{2}k^{2}z\int_{0}^{\infty}x^{3}\Phi_{n}( \kappa)\text{d}\kappa. \tag{11}\]  
\(\Phi_{n}(\kappa)\) in Equation (11), a theoretical description of the statistical characteristics of a wave propagating through optical turbulence, denotes the power spectrum of refractive fluctuations [5; 8] and is related to the magnitude of the wave number vector \(\kappa\). In a the real ocean circumstance, the eddy diffusivity ratio of temperature to salt is not equal to unity. When unstable stratification exists, and outer scale effects cannot be ignored in oceanic turbulence, the spectrum is given by [14]  
\[\Phi_{n}(\kappa)=\frac{\epsilon^{-1/3}\beta\chi_{T}A^{2}[1+C_{1}( \kappa\eta)^{2/3}]}{4\pi(\kappa^{2}+\kappa_{0}^{2})^{11/6}} \tag{12a}\] \[\times\Bigg{\{}\exp\left[-\frac{(\kappa\eta)^{2}}{R_{T}^{2}} \right]+\frac{1}{\varpi^{2}\theta}\Bigg{[}-\frac{(\kappa\eta)^{2}}{R_{S}^{2}} \Bigg{]}-\frac{1+\theta}{\varpi\theta}\Bigg{[}-\frac{(\kappa\eta)^{2}}{R_{TS }^{2}}\Bigg{]}\Bigg{\}},\]  
\[R_{j}=\sqrt{3}Q^{-3/2}\Bigg{(}W_{j}-\frac{1}{3}+\frac{1}{9W_{j}}\Bigg{)}^{3/2} \quad, \tag{12b}\]  
\[p_{i}(m_{i},z)=\int_{0}^{D/2}\Bigl{|}R^{|m_{i}|}(r_{i},z)\Bigr{|}^{2}r_{i} \Theta(r_{i},m_{i},z)\mathrm{d}r_{i}, \tag{8}\]0  
where \(j=T,S,TS\) refers to temperature fluctuations, salinity fluctuations and coupling fluctuations, \(R_{j}\) and \(W_{j}\) are dimensionless constants that vary with different \"\(\prime\)\", \(\kappa_{0}=1/L_{0}\) is the spatial cutoff frequency, \(L_{0}\) and \(\eta\) are the outer- and inner-scale of turbulence, respectively, \(e\) is the rate of dissipation of kinetic energy per unit mass of fluid, \(\beta\) is the Obukhov-Corrsin constant, \(\chi_{T}\) represents the dissipation rate of mean-squared temperature, \(A=2.6\times 10^{-4}\) L/deg, \(Q=2.5\) is a non-dimensional constant, \(Pr_{j}\) is the Prandtl number with \(Pr_{TS}=2 Pr_{T}Pr_{S}/(Pr_{T}+Pr_{S})\), \(C_{1}=2.35\) is a free parameter, \(\varpi\) defines the ratio of temperature and salinity contributions to the refractive index spectrum that varies from \(-5\) to \(0\) and \(\theta=K_{T}/K_{S}\) is the eddy diffusivity ratio where \(K_{T}\) and \(K_{T}\) denote the eddy thermal diffusivity and the diffusion of salinity, respectively. According to Ref. [7], we have the relationship of \(\varpi\) and \(\theta\) in the unstable stratification as  
\[\theta=\frac{|\varpi|}{R_{F}}\approx \begin{cases}\left(1-\sqrt{(|\varpi|-1|)/ |\varpi|}\right)^{-1}&|\varpi|\geq 1\\ 1.85|\varpi|-0.85&0.5\leq|\varpi|\leq 1^{\prime}\\ 0.15|\varpi|&|\varpi|\leq 0.5\end{cases}  \tag{13}\]  
with \(R_{F}\) is the eddy flux ratio. 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 Equation (11) as  
\[