# Bi-Photon Entangled Airy Beams through Unstable Ocean Turbulence
## 2 Methods
\begin{split}& p_{m_{1}m_{2}}(z)=\int_{0}^{\infty}\int_{0}^{ \infty}\Bigl{|}R^{|m_{1}|}(r_{1},z)R^{|m_{2}|}(r_{2},z)\Bigr{|}^{2}\\ &\quad\times\Bigl{[}\bigl{|}|\mu|^{2}\Theta(r_{1},r_{2},m_{1}^{ -},m_{2}^{+},z)+|\
u|^{2}\Theta(r_{1},r_{2},m_{1}^{+},m_{2}^{-},z)\Bigr{]}r_{ 1}r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2},\end{split}  \tag{6}\]  
where \(m_{i}^{-}=m_{i}-m_{0}\), \(m_{i}^{+}=m_{i}+m_{0}\), and \(\Theta(x)\) represents the circular harmonic transformation of the rotational coherence function [36]. Given to Markov approximation, we consider the turbulent channels 1 and 2 are independent [25]. The relative version of the detection probability is calculated to study the influence of oceanic turbulence on the entangled bi-photon due to the existence of the launching probability at the source plane. By partially tracing the joint detection probability \(p_{m_{1}m_{2}}(z)\), and considering the probability of launching \(|m_{1}\rangle\), the relative detection probability \(p_{m_{1}}(z)\) of the signal photon reads  
\[p_{rm_{1}}(z)=\frac{\sum_{m_{2}}p_{m_{1}m_{2}}(z)}{|\mu|^{2}}=p_{1}(m_{1}^{-}, z)+\frac{|\
u|^{2}}{|\mu|^{2}}p_{1}(m_{1}^{+},z), \tag{7}\]  
where \(p_{i}(m_{i},z)\) is the detection probability of measuring the OAM of a single-photon Airy beam for a receiver with the aperture diameter \(D\)  
\[p_{i}(m_{i},z)=\int_{0}^{D/2}\Bigl{|}R^{|m_{i}|}(r_{i},z)\Bigr{|}^{2}r_{i} \Theta(r_{i},m_{i},z)\mathrm{d}r_{i}, \tag{8}\]  
for non-normalized wavefunctions, the normalization of Equation (8) must be performed via \(p_{i}(m_{i},z)/\sum_{m_{i}}p_{i}(m_{i},z)\). Without a loss in generality, we shall denote that the relative detection probability for faithful propagation is \(p_{rm_{1}}(z)\) (i.e., \(m_{1}=m_{0}\)) and the crosstalk probability that detects OAM modes \(m_{i}=\pm m_{0}+\Delta m_{i}\) is \(p_{\Delta m_{i}}(z)\). Moreover, Equation (8) is given under Rytov approximation [5], where the effects of the oceanic turbulence on the OAM beam are simplified as only phase disturbance. Thus the application of our model is theoretically limited to the weak fluctuation region. When launching photon i with probability 1 (\(\mu=1\)), the second term in Equation (7) is 0. In this case, Equation (7) is equivalent to Equation (8). Comparing the relative detection probability of a single photon in Equation (8), an extra item \(|\
u|^{2}p_{1}(m_{1}^{+},z)/|\mu|^{2}\) comes out in Equation (7) describing the relative detection probability of a bi-photon. Take OAM state \(|m_{1}\rangle\) as an example, \(p_{rm_{1}}(z)\geq p_{1}(m_{1},z)\) due to the detection probability always being a non-negative real number. Physically, the quantity \(|\
u|^{2}p_{1}(m_{1}^{+},z)/|\mu|^{2}\) represents the transition of an OAM state \(|m_{2}\rangle\) caused by oceanic turbulence, which results in the elevation of the relative probability \(p_{rm_{1}}(z)\). For an aperture-truncated Airy beam, \(R^{|m_{i}|}(r_{i},z)\) is given by [28]  
\[R^{|m_{i}|}(r_{i},z)=-\sqrt{2\pi}\frac{\mathrm{i}k}{z}\omega_{0}(r_{0}-\omega_ {0}a^{2})\mathrm{exp}\left(\mathrm{i}k\frac{r_{1}^{2}}{2z}+\frac{a^{3}}{3} \right)J_{m_{i}}\left(\frac{kr_{i}r_{0}}{z}\right), \tag{9}\]  
where \(k\) is the wave number and is related to the wavelength \(\lambda\) in general by \(k=2\pi/\lambda\), \(r_{0}\) and \(a\) are the radius of the main ring and the exponential truncation parameter, respectively, \(\omega_{0}\) is the arbitrary transverse scale [37] and \(J_{m}(x)\) is the first-kind Bessel function of the \(n^{th}\) order. To further understand the effects of the significant parameters of Airy modes on the beam structure, the transverse intensity distributions of Airy beams in a vacuum for different \(\omega_{0}\), \(r_{0}\) and \(a\) at the source plane (\(z=0\)) are plotted in Figure 2. The simulation is conducted by the software of Mathematica 13.0. It is clear to see that the beam size of the Airy beam increases with the increasing \(\omega_{0}\), and the dark spot at the center spreads with the increment of \(r_{0}\) simultaneously (Figure 2b,c).