## 2 Wave Structure Function of Beam Waves
Therefore, to accurately describe the effect of ocean turbulence on MG beam, the \(D_{\psi}\left(\mathbf{r},\mathbf{r}^{\prime},z\right)\) in Equation (3) should be calculated by the wave structure function of the beam wave. Suppose that light propagates in oceanic turbulence along horizontal path, so that the wave structure function is expressed in the form [29]  
\[ \begin{array}{l}D_{\psi}\left(\mathbf{r},\mathbf{r}^{\prime},z\right)=8\pi^{2} k^{2}z\int_{0}^{1}\int_{0}^{\infty}\kappa\Phi\left(\kappa\right)\exp\left(-\Lambda z \kappa^{2}\varsigma^{2}/k\right)\\ \times\left\{1-\mathrm{J}_{0}\left[\kappa\left|\left(1-\Theta\xi\right)\left( \mathbf{r}-\mathbf{r}^{\prime}\right)-\mathrm{i}\Lambda\xi\left(\mathbf{r}+ \mathbf{r}^{\prime}\right)\right|\right]\right\}\mathrm{d}k\mathrm{d}\xi\end{array} , \tag{4}\]  
here \(\Lambda=\frac{2z}{kw^{2}\left(z\right)}\), \(w\left(z\right)=w_{0}\sqrt{\left(1-z/F_{0}\right)^{2}+\frac{4z^{2}}{k^{2}w_{0 }^{2}}}\), and \(\tilde{\Theta}=1-\frac{\left(F_{0}-z\right)kw_{0}^{2}}{F_{0}k^{2}\left(z \right)}\), \(\Phi\left(\kappa\right)\) and \(\kappa\) represent the power spectrum of refractive-index fluctuations and the spatial frequency respectively. In general, the integral of Equation (4) cannot be calculated. But, in the case of the forward scattering approximation (or the near-axial transmission approximation), the zeroth-order Bessel function can be approximated as the first two terms of the expanded series  
\[ \begin{array}{l}\mathrm{J}_{0}\left[\kappa\left|\left(1-\Theta\xi\right) \left(\mathbf{r}_{1}-\mathbf{r}_{2}\right)-\mathrm{i}\Lambda\xi\left(\mathbf{ r}_{1}+\mathbf{r}_{2}\right)\right|\right]\approx\\ 1-\frac{\kappa^{2}}{4}\left[\left(1-\tilde{\Theta}\xi\right)^{2}\left|\mathbf{ r}_{1}-\mathbf{r}_{2}\right|^{2}-\mathrm{i}\Lambda\xi\left(1-\tilde{\Theta}\xi \right)\left(\mathbf{r}_{1}^{2}-\mathbf{r}_{2}^{2}\right)-\Lambda^{2}2^{2} \left|\mathbf{r}_{1}+\mathbf{r}_{2}\right|^{2}\right]\end{array} . \tag{5}\]  
When \(\left|\mathbf{r}\right|=\left|\mathbf{r}^{\prime}\right|=r\), make use of the relation \(\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{2}=2r^{2}\left(1-\cos\left( \varphi-\varphi^{\prime}\right)\right)\), \(\left|\mathbf{r}+\mathbf{r}^{\prime}\right|^{2}=2r^{2}\left(1+\cos\left( \varphi-\varphi^{\prime}\right)\right)\), \(\mathbf{r}^{2}-\mathbf{r}^{\prime 2}=0\). The Equation (5) is simplified as follows:  
\[ \begin{array}{l}D_{\psi}\left(\mathbf{r},\mathbf{r},z\right)=2\pi^{2}k^{2}z \int_{0}^{\infty}\kappa^{3}\Phi\left(\kappa\right)\left\{\int_{0}^{1}\left[ \left(1-2\tilde{\Theta}\xi+\tilde{\Theta}^{2}\xi^{2}\right)\left[2r^{2}\left( 1-\cos\left(\varphi-\varphi^{\prime}\right)\right)\right]\right.\\ \left.-\Lambda^{2}\xi^{2}\left[2r^{2}\left(1+\cos\left(\varphi-\varphi^{ \prime}\right)\right)\right]\right]\exp\left(-\Lambda z\kappa^{2}\xi^{2}/k \right)\mathrm{d}\xi\right\}\mathrm{d}\kappa\end{array} . \tag{6}\]  
Now we have the definite integral relations [31]\(\int_{0}^{1}\xi^{m}\exp\left(-\beta\xi^{n}\right)\mathrm{d}\xi=-\frac{\mathrm{ r}\left(\frac{m+1}{n},\varrho\right)}{\left(n\beta\frac{m+1}{n}\right)}+\frac{ \mathrm{r}\left(\frac{m+1}{n},\varrho\right)}{\left(n\beta\frac{m+1}{n}\right)}\) and \(\int_{0}^{1}\mathrm{e}^{-\beta\xi^{2}}\mathrm{d}\xi=\frac{1}{2}\sqrt{\pi/ \beta}\mathrm{e}\sqrt{\beta}-\frac{1}{2}\sqrt{\pi/\beta}\). Equation (6) therefore reduces to  
\[