## 2 Wave Structure Function of Beam Waves  
MG beam can be divided into four classes according to their even or odd modes [24; 26; 27], and the analytical expressions of four MG classes are similar in form. In this paper, for the sake of convenience, we consider \(2l_{0}\)th-order even MG beam. By utilizing research conclusions in [27], the complex amplitude of the \(2l_{0}\)th-order even MG beam propagation in turbulence-free communication link is expressed as  
\[ \begin{array}{c}E_{2l_{0}}\left(r,\varphi,z\right)=\frac{1}{\sqrt{1+z_{0}^{2 }}}\text{exp}[\mathrm{i}kz+\mathrm{i}\frac{r^{2}}{w_{0}^{2}z_{g}}-\mathrm{i} \arctan\left(z_{g}\right)]\text{exp}\left(-\frac{k_{F}^{2}w_{0}^{2}z_{g}}{4 \left(z_{g}-\mathrm{i}\right)}-\frac{r^{2}}{w_{0}^{2}z_{g}\left(z_{g}-\mathrm{ i}\right)}\right)\\ \times\sum\limits_{p=0}^{\infty}A_{2p}^{\left(2l_{0}\right)}\left(q\right) \left(-1\right)^{p}\text{Re}\left(\mathrm{e}^{\mathrm{i}2p\varphi}\right) \mathrm{J}_{2p}\left(\mathrm{i}\frac{k_{F}r}{z_{g}-\mathrm{i}}\right)\end{array} , \tag{1}\]  
where \(\left(r,\varphi,z\right)\) is the polar coordinates, \(\text{Re}\left(\cdot\right)\) indicates the real part, \(\mathrm{i}=\sqrt{-1}\), \(z_{g}=2z/kw_{0}^{2}\) is the non-dimensional coordinate, \(z\) is the propagation distance, \(k=2\pi/\lambda\) is the wavenumber with wavelength \(\lambda\), \(w_{0}\) is the initial beam width, \(k_{r}=k\sin\beta\) is the transverse wavenumbers, \(\beta\) is the semi-angle of the conical wave that forms the MG beam, \(2l_{0}\) is the vortex topological charge, \(p\) denote the number of sub-beams, \(q\) is the ellipticity parameter (\(q=h^{2}k_{r}^{2}/4\) and \(h\) is the interfocal parameter), \(\mathrm{J}_{2p}\left(\cdot\right)\) is the Bessel function of the first kind, \(A_{2p}^{\left(2l_{0}\right)}\left(q\right)\) is an expansion coefficient of Mathieu functions, and the \(A_{2p}^{\left(2l_{0}\right)}\left(q\right)\) can be obtained by using the recurrence equations between the expansion coefficient of Mathieu functions (details are found in Ref. [28]). Particularly, when \(2l_{0}=0\) and \(q=0\), MG beam degenerates into the zeroth-order Bessel-Gaussian beam. Figure 1 depicts the transverse intensity distribution of a MG beam at \(z=0\) plane for different values of \(q\) and \(2l_{0}\). Obviously, the MG beam morphology are determined by both \(q\) and \(2l_{0}\). Similar to the zero-order Bessel Gaussian beam, zeroth-order MG beam present a circular symmetry when the parameter \(q\) tends to zero. And the MG beam morphology that is more complex and abundant than that of the Bessel-Gaussian beam. Rytov approximation is a good approximation to describe the law of light transmission in weakly turbulent media [29; 30]. So, using the Rytov approximation, the complex amplitude \(E_{2l_{0}}\left(r,\varphi,z\right)\) of MG beam propagating through weak oceanic turbulence at the receiver plane can be approximated as  
\[E\left(r,\varphi,z\right)\approx E_{2l_{0}}\left(r,\varphi,z\right)\exp\left[ \psi\left(r,\varphi,z\right)\right], \tag{2}\]  
where \(\psi\left(r,\varphi,z\right)\) is the random complex phase distortion caused by oceanic turbulence and \(\psi\left(r,\varphi,z\right)\) satisfies [29]  
\[\left\langle\exp\left[\psi\left(r,\varphi,z\right)+\psi^{*}\left(r^{\prime}, \varphi^{\prime},z\right)\right]\right\rangle=\exp\left[-D_{\psi}\left( \mathbf{r},\mathbf{r}^{\prime},z\right)/2\right]=\exp\left(-\frac{\left| \mathbf{r}-\mathbf{r}^{\prime}\right|^{2}}{\rho_{oc}^{2}}\right), \tag{3}\]  
where \(\left\langle\cdot\right\rangle\) denotes the ensemble average of oceanic turbulence, \(D_{\psi}\left(\mathbf{r},\mathbf{r}^{\prime},z\right)\) is the wave structure function, and \(\rho_{oc}\) is the spatial coherence radius of beam waves propagating in oceanic turbulence. Mathematically, MG beam is an infinite sum of Bessel-Gaussian beam of various orders, and the Bessel-Gaussian beam is a kind of beam formed by truncating plane waves with Gaussian aperture.