## 2 Wave Structure Function of Beam Waves
\begin{array}{l}D_{\psi}\left(\mathbf{r},\mathbf{r},z\right)=2\pi^{2}k^{2}z \int_{0}^{\infty}\kappa^{3}\Phi\left(\kappa\right)\left\{\frac{1}{2\kappa} \sqrt{\frac{\pi k}{\Lambda z}}\left(\mathrm{e}^{\kappa\sqrt{\Lambda z/k}}-1 \right)\left[2r^{2}\left(1-\cos\left(\varphi-\varphi^{\prime}\right)\right) \right]\right.\\ \left.-\Theta\frac{\Gamma\left(1,0\right)-\Gamma\left(1,\Lambda z\kappa^{2}/k \right)}{\Lambda z\kappa^{2}/k}\left[2r^{2}\left(1-\cos\left(\varphi-\varphi^{ \prime}\right)\right)\right]+\Theta^{2}\frac{\Gamma\left(3/2,0\right)-\Gamma \left(3/2,\Lambda z\kappa^{2}/k\right)}{2\left(\Lambda z\kappa^{2}/k\right)^{ 3/2}}\right.\\ \left.\times\left[2r^{2}\left(1-\cos\left(\varphi-\varphi^{\prime}\right) \right)\right]\right.-\Lambda^{2}\left[2r^{2}\left(1+\cos\left(\varphi-\varphi^{ \prime}\right)\right)\right]\right.\frac{\Gamma\left(3/2,0\right)-\Gamma \left(3/2,\Lambda z\kappa^{2}/k\right)}{2\left(\Lambda z\kappa^{2}/k\right) ^{3/2}}\right\}\mathrm{d}\kappa\end{array} . \tag{7}\]  
Consider Gaussian beams (\(F_{0}=\infty\)) and \(300\) m long underwater optical communication link, we have following approximations \(\Lambda\approx\frac{2z}{kw_{0}^{2}}\), \(\Lambda^{2}\approx 0\), \(\mathrm{e}^{\kappa\sqrt{\Lambda z/k}}-1\approx\kappa\sqrt{\Lambda z/k}\), \(\tilde{\Theta}\approx 0\). Now, Equation (7) simplifies as  
\[D_{\psi}\left(\mathbf{r},\mathbf{r},z\right)=2\sqrt{\pi}\pi^{2}zk^{2}r^{2}\left[ 1-\cos\left(\varphi-\varphi^{\prime}\right)\right]\int_{0}^{\infty}\kappa^{3} \Phi\left(\kappa\right)\mathrm{d}\kappa. \tag{8}\]  
On substituting Equation (8) into Equation (3), the spatial coherence radius of beam waves \(\rho_{oc}\) can be expressed as  
\[\rho_{oc}=\left[\left(\sqrt{\pi}/2\right)\pi^{2}k^{2}z\int_{0}^{\infty}\kappa^{ 3}\Phi\left(\kappa\right)\mathrm{d}\kappa\right]^{-1/2}. \tag{9}\]  
As for an-isotropic oceanic turbulence, the power spectrum of refractive-index fluctuations is given by [32, 33, 34]  
\[\Phi\left(\kappa\right)=0.388\times 10^{-8}\chi t\varepsilon^{-1/3}\xi^{2}\kappa^{-11/3 }\left[1+2.35(\kappa\eta)^{2/3}\right]f\left(\kappa,\varpi\right)\text{,} \tag{10}\]  
where \(\kappa=\sqrt{\kappa_{x}^{2}+\zeta^{2}\kappa_{\rho}^{2}}\) is the spatial frequency of the refractive index fluctuations, \(\kappa_{\rho}=\sqrt{\kappa_{x}^{2}+\kappa_{\rho}^{2}}\), under the Markov approximation, \(\kappa\) can be further simplified as \(\kappa=\zeta x_{\rho}\), \(\zeta\) is the an-isotropicfactor of turbulence, \(\varpi\) is the contribution ratio of temperature and salinity fluctuations for turbulent ocean, \(\varepsilon\) is the dissipation rate of kinetic energy per unit mass of fluid, \(\chi_{t}\) is the dissipation rate of the mean-squared temperature, \(\eta\) is the inner scale of turbulence, \(f\left(\kappa,\varpi\right)=\left[\exp\left(-A_{T}\delta\right)+\varpi^{-2}\, \exp\left(-A_{S}\delta\right)-2\,\varpi^{-1}\exp\left(-A_{TS}\delta\right)\right]\), \(A_{T}=1.863\times 10^{-2}\), \(A_{S}=1.9\times 10^{-4}\), \(A_{TS}=9.41\times 10^{-3}\), \(\delta=8.284(\kappa\eta)^{4/3}+12.978(\kappa\eta)^{2}\). On substituting Equation (10) into Equation (9), after a long but straightforward calculation [35], the spatial coherence radius of beam waves in oceanic turbulence is given by  
\[\rho_{oc}=\left[3\times 10^{-8}k^{2}\pi^{2}\left(\sqrt{\pi}/2\right)z\chi_{t} \xi^{-2}\varpi^{-2}(\varepsilon\eta)^{-1/3}\,\,\left(0.876\varpi^{2}-2.247 \varpi+6.128\right)\right]^{-1/2}\,\,. \tag{11}\]