# Generation and Propagation of a Hermite-Gaussian Correlated Schell-Model LG\({}_{0l}\) Beam
## 2 Propagation Equation of a HGCSMLG\({}_{0l}\) Beam  
For a PC LG\({}_{0l}\) beam, the cross-spectral density (CSD) function can be described as follows [45; 46],  
\[\mathbf{W}(\mathbf{r}_{1},\mathbf{r}_{2})=\left(\frac{2r_{1}r_{2}}{4\sigma_{0}^{2}}\right) ^{|l|}\exp\left(-\frac{\mathbf{r}_{1}^{2}+\mathbf{r}_{2}^{2}}{4\sigma_{0}^{2}}\right) \gamma(\mathbf{r}_{2}-\mathbf{r}_{1})\exp[-il(\varphi_{1}-\varphi_{2})], \tag{1}\]  
where \(\mathbf{r}_{1},\mathbf{r}_{2}\) are two arbitrary points in the source plane, \(\sigma_{0}\), \(\varphi\) and \(l\) are the beam width, azimuth angle, and the topological charge, respectively, \(\gamma(\mathbf{r}_{2}-\mathbf{r}_{1})\) denotes the correlation function. In order to explore the effects of a special correlation function on a PCV, the conventional Gaussian distribution function will be replaced by a Hermite-Gaussian distribution function, which can be expressed by:  
\[ \begin{split}\gamma(\mathbf{r}_{2}-\mathbf{r}_{1})&=\frac{ H_{2m}\left[(x_{2}-x_{1})/\sqrt{2}\delta_{0}\right]}{H_{2m}\left[0\right]} \exp\left[-\frac{(x_{2}-x_{1})^{2}}{2\delta_{0}^{2}}\right]\\ &\times\frac{H_{2m}\left[(y_{2}-y_{1})/\sqrt{2}\delta_{0}\right] }{H_{2m}\left[0\right]}\exp\left[-\frac{(y_{2}-y_{1})^{2}}{2\delta_{0}^{2}} \right],\end{split}  \tag{2}\]  
where \(\delta_{0}\) is the spatial coherence width, \(H_{2m}\), and \(H_{2n}\) are the Hermite polynomial with beam order \(2m\), and \(2n\), respectively. This new constructed beam is called the HGCSMLG\({}_{0l}\) beam. According to the generalized Huygens-Fresnel principle [26; 37], a HGCSMLG\({}_{0l}\) beam passing through an ABCD optical system can be dealt with the generalized Collins integral equation  
\[ \begin{split}&\mathbf{W}(\mathbf{\rho}_{1},\mathbf{\rho}_{2};z)=\frac{1}{ \left(\lambda B\right)^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}W(r_ {1},r_{2})\exp\Bigl{\{}-\frac{ik}{2B}\left[A\left(\mathbf{r}_{1}^{2}-\mathbf{r}_{2}^{2 }\right)\right.\\ &\left.-2\mathbf{r}_{1}\cdot\mathbf{\rho}_{1}+2\mathbf{r}_{2}\cdot\mathbf{\rho}_ {2}+D\left(\rho_{1}^{2}-\rho_{2}^{2}\right)\right]\Bigr{\}}d^{2}\mathbf{r}_{1}d^{ 2}\mathbf{r}_{2},\end{split}  \tag{3}\]  
where \(\mathbf{\rho}_{1}\) and \(\mathbf{\rho}_{2}\) are two arbitrary points in the output plane, \(k\) and A, B, D are the wave-number and the transfer matrix elements of an optical system. By substituting Equations (1) and (2) into Equation (3) and the integrating them, the CSD of the outgoing HGCSMLG\({}_{0l}\) beam can be acquired with the intermediate steps listed in Appendix A. Figure 4 shows the normalized focal plane intensity of an HGCSMLG\({}_{dl}\) beam, with \(l=2\) for different beam orders \(m\), \(n\) and the spatial coherence width \(\delta_{0}\). One can see that the beam orders \(m\), \(n\) and the spatial coherence width \(\delta_{0}\) determine the splitting array form of focal plane. Once the spatial coherence width \(\delta_{0}\) increases to a certain value, the far-field beamlets array distribution would disappear and degenerate into a semi-dark hollow beam profile. We can draw a conclusion that the  
Figure 1: Normalized intensity of a focused HGCSMLG\({}_{dl}\) beam at different distances with \(m=0\), \(n=1\) and increasing values of the TC. Figure 2: Normalized intensity of a focused HGCSMLG\({}_{dl}\) beam at different distances with \(m=0\), \(n=1\) and increasing values of the topological charge (TC). beam shape in the focal plane depends on the correlation function, as the initial coherence width is not large, and as the initial coherence width becomes large the focused beam profile is mainly determined by the vortex phase. As a result, coherence width should be suitably chosen to obtain the splitting array form of beam profile on propagation. Figures 5-7 displays the modulus of the DOC (\(|\mu(\boldsymbol{\rho},0;z)|\)) of a focused HGCSMLG\({}_{0l}\) beam at different distances with different m, n and the increasing values of the TC.