# Partially Coherent Flat-Topped Beam Generated by an Axicon
## 2 Analytical Expression of a MGSM Beam Focused by an Axicon Lens  
The cross-spectral density (CSD) function of a MGSM beam in the source plane is described as follows [31]:  
\[W(\rho_{1},\rho_{2})=G_{0}\exp\left(-\frac{\rho_{1}^{2}+\rho_{2}^{2}}{2\sigma^ {2}}\right)\mu(\rho_{1},\rho_{2}), \tag{1}\]  
where \(G_{0}\) is a normalized constant, \(\rho_{1},\rho_{2}\) are two arbitrary points in the source plane, \(\sigma\) is the beam waist width of the source, \(\mu(\rho_{1},\rho_{2})\) is the spectral DOC. The DOC of a MGSM beam is expressed by [31]:  
\[\mu(\rho_{1},\rho_{2})=\frac{1}{C_{0}}\sum\limits_{m=1}^{M}\frac{{(-1)}^{m-1}} {m}\left( \begin{array}{c}M\\ m\end{array} \right)\exp\left[-\frac{{(\rho_{2}-\rho_{1})}^{2}}{2m\delta^{2}} \right], \tag{2}\]  
where \(C_{0}=\sum\limits_{m=1}^{M}\frac{{(-1)}^{m-1}}{m}\left( \begin{array}{c}M\\ m\end{array} \right)\) is the normalization factor, \(M\) is the beam order and \(\delta\) is the initial correlation width of the beam. A MGSM beam would reduce to a GSM beam when \(M=1\) or a coherent Gauss beam when \(\delta=\infty\) and \(M=1\). Figure 1 shows the distribution of the DOC of a MGSM beam for different values of the beam order \(M\). Generally, it displays a more complicated form represented by a sum of positive and negative exponentials. Figure 1: Distribution of the degree of coherence (DOC) of a multi-Gaussian Schell-model (MGSM) beam in the source plane for different values of the beam order \(M\). According to the generalized Huygens-Fresnel principle [36; 37], a MGSM beam propagating through an axicon, as shown on Figure 2, can be dealt with the following generalized Collins integral formula:  
\[ \begin{array}{c}W(\textbf{r}_{1}\textbf{r}_{2})=\frac{1}{\lambda^{2}B^{2}} \iint W(\rho_{1},\rho_{2})T^{*}(\rho_{1})T(\rho_{2})\times\exp\Bigl{\{}\frac{- jk}{2}\Bigl{[}\Bigl{(}\frac{A^{*}}{B^{*}}\textbf{r}_{1}^{2}-\frac{A}{B}\rho_{2}^{2} \Bigr{)}\Bigr{]}\\ \qquad\qquad\qquad\qquad\qquad-2\Bigl{(}\frac{1}{B^{*}}\rho_{1}\textbf{r}_{1} -\frac{1}{B}\rho_{2}\textbf{r}_{2}\Bigr{)}+\Bigl{(}\frac{D^{*}}{B^{*}}\textbf {r}_{1}^{2}-\frac{D}{B}\textbf{r}_{2}^{2}\Bigr{)}\Bigr{]}\Bigr{\}}\textbf{d}^ {2}\rho_{1}\textbf{d}^{2}\rho_{2},\end{array}  \tag{3}\]  
where \(\textbf{r}_{1}\textbf{r}_{2}\) are two arbitrary points in the output plane, \(\lambda\) is the wavelength, \(A\), \(B\), \(C\) and \(D\) are the transfer matrix elements of an optical system, \(T\) is the transmittance function of the axicon lens, which can be expressed as follows:  
\[T(\rho)= \begin{cases}\exp[jk(n_{d}\Re-\beta\rho)]&\rho\leq R\\ 0&\rho>R^{{}^{\prime}}\end{cases} , \tag{4}\]  
with  
\[\beta=(n_{d}-1)\tan\frac{\pi-\alpha}{2}. \tag{5}\]  
Here \(n_{d}\) is the relative refractive index of the axicon to the medium around it, \(\Re\) is the maxima of its thickness, \(\alpha\) is the apex angle of the axicon lens. One can rewrite Equation (3) in a polar coordinate system with \(\textbf{\rho}=(\rho\cos\varphi,\rho\sin\varphi)\), \(\textbf{r}=(r\cos\theta,r\sin\theta)\), as:  
\[ \begin{array}{c}W(\textbf{r}_{1}\textbf{r}_{2})=\frac{1}{\lambda^{2}B^{2}} \frac{1}{C_{0}}\sum\limits_{m=1}^{M}\left(\begin{array}{c}M\\ m\end{array}\right)\frac{(-1)^{m-1}}{m}\exp\Bigl{[}\frac{-jk}{2}\Bigl{(}\frac {D^{*}}{B^{*}}\textbf{r}_{1}^{2}-\frac{D}{B}\textbf{r}_{2}^{2}\Bigr{)}\Bigr{]} \\ \qquad\qquad\qquad\times\int\limits_{0}^{+\infty}\int\limits_{0}^{\rho_{1}} \rho_{2}\exp\Bigl{[}-\frac{1}{2\Delta_{m}^{2}}\bigl{(}\rho_{2}^{2}+\rho_{1}^{ 2}\bigr{)}\Bigr{]}\exp[jkf(\rho_{1},\rho_{2})]\\ \qquad\qquad\qquad\qquad\times\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi} \exp\Bigl{[}\frac{\rho_{1}\rho_{2}\cos(\varphi_{2}-\varphi_{1})}{m\sigma^{2}} \Bigr{]}\exp\Bigl{\{}\Bigl{[}jk\Bigl{(}\frac{1}{B^{*}}\rho_{1}r_{1}\cos(\varphi _{1}-\theta_{1})\] \[-\frac{1}{B}\rho_{2}r_{2}\cos(\varphi_{2}-\theta_{2})\Bigr{)}\Bigr{]} \Bigr{\}}d\varphi_{1}d\varphi_{2}d\rho_{1}d\rho_{2},\end{array}