## 2 Theory
\begin{split} W_{T}(\rho,\varphi_{1},\varphi_{2},z)& =W(\rho,\varphi_{1},\varphi_{2},z)\big{\langle}\exp\{i[\phi(\rho,\varphi_{1})-\phi(\rho,\varphi_{2})]\}\big{\rangle}\\ &=\langle E_{T}{}^{*}(\rho,\varphi_{1},z)E_{T}(\rho,\varphi_{2},z )\rangle\langle\exp\{i[\phi(\rho,\varphi_{1})-\phi(\rho,\varphi_{2})]\} \big{\rangle}\end{split}  \tag{7}\]  
Here, \(E_{T}(\rho,\varphi,z)\) is the electric field of the beam in the receiver plane. With the help of the quadratic approximation of the wave structure function, the atmospheric turbulence disturbance term in the above formula can be obtained as follows:  
\[\langle\exp\{i[\phi(\rho,\varphi_{1})-\phi(\rho,\varphi_{2})]\}\rangle=\exp \left\{-\frac{2\rho^{2}}{\rho_{0}^{2}}[1-\cos(\varphi_{1}-\varphi_{2})]\right\} \tag{8}\]  
Here, \(\rho_{0}\) is the spatial coherence length of the atmospheric turbulence, as follows [43]:  
\[\rho_{0}=\Big{(}0.545C_{n}^{2}k^{2}z\Big{)}^{-3/5}. \tag{9}\]  
Here, the parameter \(C_{n}^{2}\) is the turbulence structure constant, the value characterizing the strength of the atmospheric turbulence. The OAM mode carried by the vortex beam inevitably encounters atmospheric turbulence after propagating over a long distance; the phase and complex amplitude of non-uniformly correlated Bessel-Gaussian beams are disturbed by the refractive index fluctuations of the turbulent atmosphere; the phase becomes extremely complicated; and the OAM states cannot maintain their original status. In order to facilitate the analysis of the signal and crosstalk, Fourier transform theory was applied. The electric field in the receiver plane \(E_{T}(\rho,\varphi,z)\) can be written as a superposition of the plane waves with different OAM states; then, the complex-amplitude plane wave can be written as a coefficient and the term \(\exp(il\varphi)\), and the CSD function \(W_{T}(\rho,\varphi_{1},\varphi_{2},z)\) in the receiver plane can be rewritten as follows [13]:  
\[W_{T}(\rho,\varphi_{1},\varphi_{2},z)=\sum\limits_{l}\left\langle\left|a_{l}( \rho,z)\right|^{2}\right\rangle\exp[-il\left(\varphi_{1}-\varphi_{2}\right)] \tag{10}\]  
Here, \(a_{l}\) is the expansion coefficient with the new OAM quantum number \(l\). Then, the spatial distribution of intensity for a non-uniformly correlated high-order Bessel-Gaussian beam carrying the OAM modes can be determined as follows:  
\[\left\langle\left|a_{l}(\rho,z)\right|^{2}\right\rangle=\left(\frac{1}{2\pi} \right)\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi}W_{T}(\rho,\varphi_{1}, \varphi_{2},z)\exp[il(\varphi_{1}-\varphi_{2})]d\varphi_{1}d\varphi_{2} \tag{11}\]  
Therefore, the received intensity of the non-uniformly correlated high-order Bessel-Gaussian beam carrying the OAM modes can be obtained as follows [13]:  
\[I(z)=2\pi\int\limits_{0}^{D/2}\left\langle\left|a_{l}(\rho,z)\right|^{2} \right\rangle\!\rho d\rho \tag{12}\]  
Here, D is the diameter of the receiving aperture. Substituting Equations (2), (6), (7) and (11) into Equation (12), we determined the received intensity of the OAM modes of non-uniformly correlated Bessel-Gaussian beams:  
\[I(z)=\int p(v)I_{l}(\rho,z)dv \tag{13}\]  
where  
\[ \begin{split} I_{l}(\rho,z)&=\frac{\pi k^{2}}{2z^{2 }\left[\beta_{1}{}^{2}-\left(\beta_{2}-\beta_{3}\right)^{2}\right]}\exp\left\{ -\frac{a^{2}}{2w_{0}^{2}\left[\beta_{1}{}^{2}-\left(\beta_{2}-\beta_{3}\right) ^{2}\right]}\right\}\\ &\times\int\limits_{0}^{D/2}I_{l_{0}}\left[\frac{\alpha k\rho}{2 z(\beta_{1}+\beta_{2}-\beta_{3})}\right]I_{l_{0}}\left[\frac{\alpha k\rho}{2z( \beta_{1}-\beta_{2}+\beta_{3})}\right]I_{\Delta l}\left(\frac{2\rho^{2}}{r_{0 }^{6}}\right)\\ &\times\exp\left\{-\left[\frac{k^{2}}{2z^{2}w_{0}^{2}\left[\beta _{1}{}^{2}-\left(\beta_{2}-\beta_{3}\right)^{2}\right]}+\frac{2}{\rho_{0}^{2} }\right]\rho^{2}\right\}\rho d\rho\end{split}  \tag{14}\]  
and \(\Delta l=l_{0}-l\), \(\beta_{1}=1/w_{0}^{2}\), \(\beta_{2}=ik/2z\), \(\beta_{3}=ikv\).