## 2 Theoretical Model
### Adaptive Compensation  
Due to the wavefront distortion phenomenon of electromagnetic vortex waves being transmitted through atmospheric turbulence, both intensity and phase will be distorted. Usually, adaptive correction technology is used to compensate the wavefront distortion of the beam, so as to improve greatly and thereby recover the intensity and phase information of the beam. In this paper, we consider that the group of Laguerre-Gaussian beams carrying OAM mode is used to carry information data, as it has been used in numerous applications after the lab generation was performed in Reference [21]. The field distribution of the Laguerre-Gaussian beam is expressed as  
\[ \begin{array}{l}u(r,\theta,z)=\sqrt{\frac{2p!}{\pi(p+|l|)!}}\frac{1}{w(z)} \Big{[}\frac{\sqrt{2}}{w(z)}\Big{]}^{|l|}L_{p}^{l}\Big{[}\frac{2p^{2}}{w^{2}(z )}\Big{]}\exp\Big{[}\frac{-r^{2}}{w^{2}(z)}\Big{]}\\ \times\exp\left[\frac{-ikp^{2z}}{2\left(2^{2}+z_{R}^{2}\right)}\right]\exp \Big{[}i(2p+|l|+1)\tan^{-1}\frac{z}{z_{R}}\Big{]}\exp(-il\theta)\end{array}  \tag{9}\]  
where the function \(w(z)\) is the diffraction limited spot size of the fundamental Gaussian beam, \(z\) is the propagation distance, \(z_{R}\) is Rayleigh range, the symbols \(p\) and \(l\) are the radial and angular coordinate, respectively, in the \(z\) plane. The amplitude at the transmitting end can be expressed as  
\[u(r,\theta,z=0)=\sqrt{\frac{2p!}{\pi(p+|l|)!}}\frac{1}{w_{0}}\Bigg{(}\frac{ \sqrt{2}r}{w_{0}}\Bigg{)}^{|l|}L_{p}^{l}\Bigg{(}\frac{2r^{2}}{w_{0}^{2}}\Bigg{)} \exp\left(\frac{-r^{2}}{w_{0}^{2}}\right)\exp(-il\theta) \tag{10}\]  
By transmitting a certain distance in atmospheric turbulence, the electric field on the observed plane is obtained by the diffraction integral  
\[u(r,\theta,z)=-\frac{i}{\lambda z}\exp(ikz)\int\int u(r,\theta,z=0)\exp[\psi( r,\theta,z)]\exp\left[\frac{ik}{2z}(r-\rho)^{2}\right]\rho d\rho d\varphi, \tag{11}\]  
where the function \(\exp[\psi(r,\theta,z)]\) is the phase change introduced by atmospheric turbulence. The spatial inhomogeneity caused by atmospheric turbulence will change the structure function, which will lead to a change of OAM mode. In order to reduce the influence of atmospheric turbulence on electromagnetic waves, an adaptive phase compensation method was introduced. At the receiving end, the correlation coefficient of intensity affected by turbulence is as follows:  
\[\left\{ \begin{array}{l}c_{n1}=\int\int u(r,\phi)u^{*}(r,\phi)\exp(-il\phi) \exp[-il(\phi+\Delta\phi)]rdrd\phi\\ c_{n2}=\int\int u(r,\phi)u^{*}(r,\phi)\exp(-il\phi)\exp[-il(\phi-\Delta\phi)]rdrd \phi\end{array} \right., \tag{12}\]where the symbol \(\Delta\phi\) is the phase disturbance for the wavefront caused by atmospheric turbulence. The phase of the Laguerre-Gaussian beam can be expressed as the function \(P=\exp(-il\phi)\). Therefore, the output phase can be calculated by  
\[P_{n+1}=P_{n}+m\Delta P_{n}(c_{n1}-c_{n2}), \tag{13}\]  
where \(n\) represents the number of iterations, and \(m\) is the gain coefficient. The symbol \(P_{n}\) represents the phase within finite steps iterative computations, then the reconstructed phase can be obtained by Equation (12). The significance of this algorithm is to set the total number of iterations, in order to recover the best phase of the system and minimize the residual difference of phases after multiple iterations, so that we could achieve the best performance of the adaptive compensation system without a wavefront sensor. \[m_{n}=\frac{P_{n|n-1}}{\left|P_{n|n-1}-P_{n|n}\right|} \tag{14}\]  
The symbol \(P_{n|n-1}\) represents the phase ratio within \(n\) and \((n-1)\) iteration calculation.