## 2 Theory  
Cross-spectral density (CSD) was used to characterize the partially coherent beams, which can be described at the source plane (z = 0) as follows:  
\[W_{0}(r_{1},r_{2},\theta_{1},\theta_{2},0)=\langle E^{*}(r_{1},\theta_{1})E(r_{ 2},\theta_{2})\rangle \tag{1}\]  
Here, (\(r\), \(\theta\)) are the polar coordinates, \(E\) (\(r\), \(\theta\)) denotes the electric field of the beam, the angular brackets \(<\)-\(>\) indicate the ensemble average, and the asterisk denotes the complex conjugate. In general, mode representation is applied to express the CSD of the beam as follows:  
\[W_{0}(r_{1},r_{2},\theta_{1},\theta_{2},0)=\int p(v)H(r_{1},\theta_{1},v)H^{*} (r_{2},\theta_{2},v)dv \tag{2}\]  
Here, \(v\) is the corresponding vector in Fourier space of a position vector \(r\) in the polar coordinate system. The evolution properties of the OAM state for a non-uniformly correlated high-order Bessel-Gaussian beam can be derived from the functions of \(p(v)\) and \(H(r,\theta,v)\). Then, a non-negative function \(p(v)\) is expressed as  
\[p(v)=\frac{1}{\sqrt{\pi a}}\exp\left(-\frac{v^{2}}{a^{2}}\right) \tag{3}\]  
and the kernel with an arbitrary function for the equation \(H\) (\(r\), \(\theta\), \(v\)) is described as  
\[H(r,\theta,v)=J_{l_{0}}(\alpha r)\exp\left(-\frac{r^{2}}{w_{0}^{2}}\right)\exp \left(-ikvr^{2}\right)\exp\left(il_{0}\theta\right) \tag{4}\]where the parameter \(a=2/kr_{c}^{2}\) is a positive real constant with a coherence length of \(r_{c}\), \(k=2\pi/\lambda\) is a wavenumber with a wavelength of \(\lambda\), \(l_{0}\) is the topological charge of the OAM state, \(J_{l_{0}}\) is the \(l_{0}\)-th order Bessel function of the first kind, \(\alpha\) is a radial wave vector, and \(w_{0}\) is the waist width of the Gaussian beam. Substituting Equations (3) and (4) into Equation (2), the CSD of the non-uniformly correlated high-order Bessel-Gaussian beam at the source plane can be obtained as follows:  
\[ \begin{split}& W_{0}(r_{1},r_{2},\theta_{1},\theta_{2},0)\\ &=J_{l_{0}}(\alpha r_{1})J_{l_{0}}(\alpha r_{2})\exp\left(-\frac{ r_{1}^{2}+r_{2}^{2}}{w_{0}^{2}}\right)\exp\left[-\frac{\left(r_{2}^{2}-r_{1}^{2} \right)^{2}}{r_{c}^{4}}\right]\exp[-il_{0}(\theta_{1}-\theta_{2})]\end{split}  \tag{5}\]  
Here, \(r_{c}=\sqrt{2/ka}\) is the coherence length. According to the Huygens-Fresnel principle and paraxial approximation, the field in the receiver plane of the CSD function of non-uniformly correlated high-order Bessel-Gaussian beams in free space without a turbulent atmosphere can be obtained as follows:  
\[ \begin{split} W(\rho_{1},\rho_{2},\varphi_{1},\varphi_{2},z)= \frac{k^{2}}{4\pi^{2}z^{2}}\int\int\int\int W_{0}(r_{1},r_{2},\theta_{1}, \theta_{2},0)\\ \times\exp\left\{-\frac{ik}{2z}\left[r_{1}^{2}-r_{2}^{2}-2r_{1} \rho_{1}\cos(\theta_{1}-\varphi_{1})+2r_{2}\rho_{2}\cos(\theta_{2}-\varphi_{2 })\right]\right\}r_{1}r_{2}dr_{1}dr_{2}d\theta_{1}d\theta_{2}\end{split}  \tag{6}\]  
Here, (\(\rho\), \(\varphi\)) are the polar coordinates in the receiver plane. When a non-uniformly correlated high-order Bessel-Gaussian beam propagates though a turbulent atmosphere, the evolution properties of the OAM states of this beam are influenced by atmospheric turbulence, the purity of the OAM mode decreases, and the crosstalk increases. To conveniently analyze the purity and crosstalk of the OAM modes, the parameter \(\rho_{1}=\rho_{2}=\rho\) was set, as the purity, SNR, and BER are affected by the intensity distribution of the beam. When a non-uniformly correlated high-order Bessel-Gaussian beam carrying the OAM state propagates in a turbulent atmosphere, the cumulative effect induced by the turbulence is assumed to be a pure phase perturbation effect on this beam [6]. Then, the CSD function of the non-uniformly correlated high-order Bessel-Gaussian beams propagating under conditions of atmospheric turbulence can be obtained as follows [26]:  
\[