## 2 Theoretical Analysis
### Transmission from the Source to the Target Plane  
At z = 0, the general expression for the elements of the cross-spectral density (CSD) matrix of the isotropic EGSM beam in the space-frequency domain can be expressed as follows [4]:  
\[\mathrm{W}_{ij}(\mathbf{r}_{1},\mathbf{r}_{2})=\left\langle E_{i}(\mathbf{r}_{ 1})E_{j}^{*}(\mathbf{r}_{2})\right\rangle=A_{i}A_{j}B_{ij}\exp[-\frac{ \mathbf{r}_{1}^{2}}{4\sigma_{i}^{2}}-\frac{\mathbf{r}_{2}^{2}}{4\sigma_{j}^{2} }]\exp[-\frac{(\mathbf{r}_{2}-\mathbf{r}_{1})^{2}}{2{\delta_{ij}}^{2}}],\,(i= x,y;j=x,y). \tag{1}\]  
In this model, \(A_{i}\,,A_{j}\) are the spectral amplitudes of the electric field components; \(B_{ij}\) is the complex correlation coefficient between two electric field components; \(\sigma_{i}\,,\sigma_{j}\) are the beam radius along two orthogonal directions, respectively; \(\delta_{ij}\) denotes the transverse coherence radius of field components, respectively. Also, \(\delta_{ij}=\delta_{ji},B_{ij}=B_{ji}^{*}\) follow from the fact \(W_{ij}(\mathbf{r}_{1},\mathbf{r}_{2},\omega)=W_{ji}^{*}(\mathbf{r}_{2},\mathbf{ r}_{1},\omega)\), which is obtained from the definition of the correlation matrix W. For a uniformly polarized source, we set \(\sigma_{x}=\sigma_{y}=\sigma\)[60]. Under this condition, every point in the source plane has an identical degree of polarization. Finally, the constraints to ensure the source can be realized are obtained as follows [21]:  
\[\sqrt{\frac{\delta_{xx}^{2}+\delta_{yy}^{2}}{2}}\leq\delta_{xy}\leq\sqrt{ \frac{\delta_{xx}\delta_{yy}}{\left|B_{xy}\right|}}, \tag{2}\]  
\[\frac{1}{4\sigma^{2}}+\frac{1}{\delta_{ii}^{2}}\ll\frac{2\pi^{2}}{\lambda^{2}} \,(i=x,y). \tag{3}\]  
The factor \(B_{ij}\) obeys the following properties [40]:  
\[B_{ij}=1\text{ when }i=j;\,\left|B_{ij}\right|\leq 1\text{ when }i\
eq j \tag{4}\]  
In the source plane, the elements of the CSD matrix of the beam generated by an EGSM source can be expressed in the following form [17]  
\[W_{ij}(\overline{\mathbf{r}})=A_{i}A_{j}B_{ij}\exp[-\frac{ik_{-}}{2}\overline {\mathbf{r}}\mathbf{M}_{0 ij}^{-1}],\,(i=x,y;j=x,y), \tag{5}\]  
\[\mathbf{M}_{0 ij}^{-1}=\left\{ \begin{array}{cc}\left(-\frac{i}{2\omega^{2}}- \frac{i}{\delta_{0 ij}^{2}}\right)\mathbf{I}&\frac{i}{\delta_{0 ij}^{2}}\mathbf{ I}\\ \frac{i}{\delta_{0 ij}^{2}}\mathbf{I}&\left(-\frac{i}{2\omega^{2}}-\frac{i}{ \delta_{0 ij}^{2}}\right)\mathbf{I}\end{array} \right\}, \tag{6}\]  
where \(\overline{\mathbf{r}}^{T}\)=\((\mathbf{r}_{1},\mathbf{r}_{2})^{T}\), \(I\) is a \(2\times 2\) unit matrix and \(\mathbf{M}_{0 ij}^{-1}\) is a transpose symmetric matrix in the source plane. Following [54], the atmospheric turbulence is supposed to be homogeneous and isotropic. The transmission process of the beam can be expressed in the tensor form. From the source plane to the target plane, under the paraxial approximation, the atmospheric propagation obeys the following generalized Collins formula [52],  
\[ \begin{array}{ll}W_{ij}(\overline{\mathbf{t}})=&\frac{k^{2}}{4\pi\det( \overline{\mathbf{B}})^{2}}\iint\iint W_{ij}(\overline{\mathbf{r}})\exp[- \frac{ik}{2}\left(\overline{\mathbf{r}}^{T}\overline{\mathbf{B}}^{-1} \overline{\mathbf{A}}\overline{\mathbf{r}}-\overline{\mathbf{r}}^{T}\overline {\mathbf{B}}^{-1}\overline{\mathbf{t}}-\overline{\mathbf{t}}^{T}\overline{ \mathbf{B}}^{-1}\overline{\mathbf{r}}+\overline{\mathbf{t}}^{T}\overline{ \mathbf{D}}\overline{\mathbf{B}}^{-1}\overline{\mathbf{t}}}\right)]\\ &\times\exp[-\frac{ik}{2}\left(\overline{\mathbf{r}}^{T}\overline{\mathbf{P}} +\overline{\mathbf{r}}^{T}\overline{\mathbf{P}}\overline{\mathbf{t}}+ \overline{\mathbf{t}}^{T}\overline{\mathbf{P}}\overline{\mathbf{t}}\right)]d \overline{\mathbf{r}}.\end{array}  \tag{7}\]  
Substituting from Equations (5)-(7), we obtain the following expression for the elements of the CSD matrix of the EGSM beam at the target plane.