# Generation and Propagation of a Hermite-Gaussian Correlated Schell-Model LG\({}_{0l}\) Beam
## Appendix A  
The cross-spectral density (CSD) of the HGCSMLG\({}_{0l}\) beam in the source plane is defined as  
\[ \begin{split}\mathbf{W}(\mathbf{r}_{1},\mathbf{r}_{2})=\left(\frac{2\
u_{1} \
u_{2}}{4\
u_{0}^{2}}\right)^{|l|}\exp\left(-\frac{\
u_{1}^{2}+\
u_{2}^{2}}{4 \
u_{0}^{2}}\right)\exp[-il(\varphi_{1}-\varphi_{2})]\frac{H_{2m}[(x_{2}-x_{1} )/\sqrt{2}\
u_{0}]}{H_{2m}(0)}\\ \times\frac{H_{2m}[(y_{2}-y_{1})/\sqrt{2}\zeta_{0}]}{H_{2m}(0)} \exp\left[-\frac{(x_{2}-x_{1})^{2}}{2\
u_{0}^{2}}\right]\exp\left[-\frac{(y_{ 2}-y_{1})^{2}}{2\delta_{0}^{2}}\right],\end{split}  \tag{1}\]  
Using the following expansion  
\[H_{2n}(x)=(-1)^{n}2^{2n}n!L_{n}^{-1/2}\left(x^{2}\right) \tag{2}\]  
\[L_{n}^{\alpha}(x)=\sum_{m=0}^{n}(-1)^{m}\left( \begin{array}{c}n+\alpha\\ n-m\end{array} \right)\frac{x^{m}}{m! };L_{n}^{\alpha}(0)=\frac{\Gamma(n+\alpha+ 1)}{n!\Gamma(\alpha+1)} \tag{3}\]  
\[(x\pm iy)^{|l|}=\sum_{c1=0}^{|l|}\frac{|l|!(\pm i)^{c1}}{c1!(|l|-c1)!}x^{|l|-c1 }y^{c1} \tag{4}\]  
After some operation, Equation (1) can be expressed in the following alternative form:  
\[ \begin{split}& W(\mathbf{r}_{1},\mathbf{r}_{2})=\left(\frac{1}{2\
u_{0}^{ 2}}\right)^{|l|}\frac{m!\sqrt{\pi}}{\Gamma(m+1/2)}\frac{n!\sqrt{\pi}}{\Gamma(n +1/2)}\sum_{c1=0}^{|l|}\sum_{c2=0}^{|l|}(-1)^{c_{i}}\frac{|l|!(i)^{c1}}{c1!(|l| -c1)!}\frac{|l|!\
u^{2}}{c2!(|l|-c2)!}\\ &\times x_{1}^{|l|-c1}x_{2}^{|l|-c2}y_{1}c^{1}y_{2}c^{2}\exp\left[- \frac{\
u_{1}^{2}+\
u_{2}^{2}}{4\
u_{0}^{2}}\right]L_{m}^{-1/2}\left(\frac{(x_ {2}-x_{1})^{2}}{2\delta_{0}^{2}}\right)L_{n}^{-1/2}\left(\frac{(y_{2}-y_{1})^{ 2}}{2\delta_{0}^{2}}\right)\\ &\times\exp\left[-\frac{(x_{2}-x_{1})^{2}}{2\delta_{0}^{2}}\right] \exp\left[-\frac{(y_{2}-y_{1})^{2}}{2\delta_{0}^{2}}\right],(l\geq 0,c_{i}=c_{1} ;l<0,c_{i}=c_{2})\end{split}  \tag{5}\]With the help of the generalized Huygens-Fresnel integral, the propagation of a HGCSMLG\({}_{0l}\) beam through an ABCD optical system can be treated by the following generalized Collins integral formula:  
\[\begin{array}{c}\mathbf{W}(\mathbf{\rho}_{1},\mathbf{\rho}_{2};z)=\frac{1}{(\lambda B)^{2}} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}W(r_{1},r_{2})\exp\Bigl{\{}-  
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