## Appendix A  
In this appendix, the analysis process of the BCP matrix of the signal beam is derived first. By substituting Equations (5) and (7) by Equation (6), \(W_{aij}(\mathbf{\rho}_{1}.\mathbf{\rho}_{2}.\mathbf{z})\) can be simplified as:  
\[ \begin{array}{ll}W_{aij}(\mathbf{\rho}_{1},\mathbf{\rho}_{2},z)=&\frac{A_{i}A_{j}B_{ij }}{\lambda^{2}z^{2}}\iint\exp\left[-\left(\frac{T_{1}^{2}}{4\epsilon_{ii}^{2}}+ \frac{T_{2}^{2}}{4\epsilon_{ii}^{2}}\right)\right]\times\exp\left(-\frac{\mathbf{ \
u}_{1}-\mathbf{\
u}_{2}^{2}}{2\epsilon_{ii}^{2}}\right)\\ &\times\exp\left\{-\frac{\mathbf{\kappa}}{2}\left(\left(\mathbf{\rho}_{1}-\mathbf{\tau}_{1 }\right)^{2}-\left(\mathbf{\rho}_{2}-\mathbf{\tau}_{2}\right)^{2}\right]\right\}\\ &\times\exp\left[-\frac{M_{1}}{2}(\rho_{1}-\rho_{2})^{2}-\frac{M_{2}}{2}(\mathbf{ \tau}_{1}-\mathbf{\tau}_{2})(\rho_{1}-\mathbf{\rho}_{2})-\frac{M_{3}}{2}(\mathbf{\tau}_{1} -\mathbf{\tau}_{2})^{2}\right]\!d^{2}\mathbf{\tau}_{1}d^{2}\mathbf{\tau}_{2}\end{array}  \tag{10}\]  
We assume that \(\mathbf{r_{c}}=(\mathbf{r}_{1}+\mathbf{r}_{2})/2\), \(\mathbf{r_{d}}=\mathbf{r}_{1}-\mathbf{r}_{2}\), \(p=\mathbf{\rho}_{1}-\mathbf{\rho}_{2}\), \(q=\mathbf{\rho}_{1}+\mathbf{\rho}_{2}\), for which, \(\mathbf{r_{1}}=\mathbf{r_{c}}+\mathbf{r_{d}}\mathbf{\rho}_{2}\), \(\mathbf{r_{2}}=\mathbf{r_{c}}-\mathbf{r_{d}}\mathbf{\rho}_{2}\). Equation (10) can be simplified as:  
\[ \begin{array}{ll}W_{aij}(\mathbf{\rho}_{1},\mathbf{\rho}_{2},z)=&\frac{A_{i}A_{j}B_{ ij}}{\lambda^{2}z^{2}}\times\iint d^{2}\mathbf{r}_{c}\exp\left[-\left(\frac{1}{4 \epsilon_{ii}^{2}}+\frac{1}{4\epsilon_{ii}^{2}}\right)r_{c}^{2}-\left(\frac{1 }{4\epsilon_{ii}^{2}}-\frac{1}{4\epsilon_{ii}^{2}}\right)r_{d}r_{c}\right]\\ &\times\exp\left(i\frac{\mathbf{\kappa}}{2}\mathbf{r}_{c}\right)\iint d^{2}\mathbf{r}_{d} \exp\left\{-\left[-\frac{1}{4\epsilon_{ii}^{2}}+\frac{1}{4\epsilon_{ii}^{2}} \right)-\frac{1}{2\epsilon_{ii}^{2}}-\frac{M_{1}}{2}\right\}r_{d}^{2}\right\} \\ &\exp\left(-\frac{M_{0}}{2}\mathbf{r}_{d}+i\frac{\mathbf{\kappa}}{2}\mathbf{r}_{d}\right) \exp\left(-\frac{M_{1}}{2}\mathbf{r}_{d}^{2}-i\frac{\mathbf{\kappa}}{2}\right)\end{array}  \tag{11}\]  
Using the identities written as formulas in Ref. [24]:  
\[ \begin{array}{ll}&\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\left[- \left(a\rho^{2}+2b\rho+c\right)\right]\exp\left[\pm i\left(p\rho^{2}+2q\rho+r \right)\right]d^{2}\rho\\ &=\frac{\pi}{\sqrt{a^{2}+p^{2}}}\exp\left[\frac{a\left(b^{2}-q^{2}\right)+2p \eta b}{a^{2}+p^{2}}-c\right]\\ &\times\exp\left\{\pm i\left[\arctan\frac{p}{a}-\frac{p\left(q^{2}-p^{2} \right)+2a\eta b}{a^{2}+p^{2}}+r\right]\right\}(a>0)\end{array}  \tag{12}\]  
With integral simplification of \(\mathbf{r_{c}}\), Equation (11) can be written as:  
\[ \begin{array}{ll}W_{aij}(\mathbf{\rho}_{1},\mathbf{\rho}_{2},z)=&\frac{A_{i}A_{j}B_{ ij}}{\lambda^{2}z^{2}\Lambda_{aij}}\exp\left(-\frac{k^{2}\rho^{2}}{4\epsilon^{2} \Lambda_{aij}}\right)\exp\left(-\frac{M_{1}\rho^{2}}{2}-i\frac{k\eta b}{2 \epsilon^{2}}\right)\\ &\iint d^{2}\mathbf{r}_{d}\exp\left\{-\frac{\Lambda_{aij}}{4}+\frac{1}{2\epsilon _{ii}^{2}}+\frac{M_{3}}{2}-\frac{\Lambda_{aij}^{2}}{4\Lambda_{aij}}+\frac{k^{ 2}}{4\epsilon^{2}\Lambda_{aij}}\right]\!d^{2}\right\}\\ &\times\exp\left[\left(\frac{p\mathbf{\kappa}}{2^{2}\Lambda_{aij}}-\frac{M_{0}p}{2 }\right)\mathbf{r}_{d}\right]\times\exp\left\{i\left[\frac{\Lambda_{aij}}{2\Lambda_ {aij}}r_{d}^{2}+\left(\frac{\mathbf{\kappa}}{2\epsilon^{2}}-\frac{\Lambda_{aij} }{2\epsilon_{ii}}\right)r_{d}\right]\right\}\end{array}  \tag{13}\]  
where  
\[ \begin{array}{ll}\left\{\begin{array}{ll}\Delta_{aij}=\frac{1}{4\epsilon_{ii }^{2}}+\frac{1}{4\epsilon_{ii}^{2}}\\ \Lambda_{aij}=\frac{1}{4\epsilon_{ii}^{2}}-\frac{1}{4\epsilon_{ii}^{2}}\\ Q(z)_{aij}=\frac{\Lambda_{aij}}{4}+\frac{1}{2\delta_{ii}^{2}}+\frac{M_{3}}{2}- \frac{\Lambda_{aij}^{2}}{4\Lambda_{aij}}+\frac{k^{2}}{4\epsilon^{2}\Lambda_{aij }}\\ T(z)_{aij}=\frac{\Lambda_{aij}}{2\epsilon\Lambda_{aij}}\end{array}\right.\end{array}