# Partially Coherent Flat-Topped Beam Generated by an Axicon
## 2 Analytical Expression of a MGSM Beam Focused by an Axicon Lens
 \tag{6}\]  
where  
\[\frac{1}{\Delta_{m}^{2}}=\frac{1}{\sigma^{2}}+\frac{1}{m\delta^{2}}. \tag{7}\]  
And after tedious integration, the expression of the CSD in the output plane can be obtained as:  
\[f(\rho_{1},\rho_{2})=\left(\frac{A}{2B}\rho_{2}^{2}-\beta\rho_{2}\right)-\left( \frac{A^{*}}{2B^{*}}\rho_{1}^{2}-\beta\rho_{1}\right). \tag{8}\]  
Figure 2: Optical layout of a MGSM beam passing through an axicon lens. \[ \begin{array}{l}W(\mathbf{r_{1}}\mathbf{r_{2}})=\frac{k^{2}}{B^{2}C_{0}}\sum\limits \limits_{m=1}^{M}\sum\limits_{l=-\infty}^{\infty}e^{jl(\theta_{2}-\theta_{1})} \Bigg{(}\begin{array}{c}M\\ m\end{array}\right)\frac{(-1)^{n-1}}{m}\exp\left[\frac{-jk}{2}\Big{(}\frac{D^{* }}{B^{*}}\mathbf{r_{1}^{2}}-\frac{D}{B}\mathbf{r_{2}^{2}}\Big{)}\right]\\ \times\int\limits_{0}^{+\infty}\int\limits_{0}^{+\infty}\rho_{1}\rho_{2}\exp \left[-\frac{1}{2\Delta_{m}^{2}}\big{(}\rho_{2}^{2}+\rho_{1}^{2}\big{)}\right] \exp[jkf(\rho_{1},\rho_{2})]\\ \times C_{lm}(B,\rho_{1},\rho_{2},r_{1},r_{2})d\rho_{1}d\rho_{2},\end{array}  \tag{9}\]  
with  
\[C_{lm}(B,\rho_{1},\rho_{2},r_{1},r_{2})=j^{-l}I_{l}\left(j\frac{\rho_{1}\rho_ {2}}{m\delta^{2}}\right)I_{l}\Big{(}k\frac{\rho_{1}r_{1}}{B^{*}}\Big{)}I_{l} \Big{(}k\frac{\rho_{2}r_{2}}{B}\Big{)}. \tag{10}\]  
where \(J_{l}\) is the first kind Bessel function with order \(l\). In Equation (9), the phase factor \(\exp[jkf(\rho_{1},\rho_{2})]\) oscillates around zero intensively within the range of the optical frequencies. Applying the stationary phase method in [12, 13, 38], the integration result of Equation (9) can be acquired:  
\[ \begin{array}{l}W(\mathbf{r_{1}}\mathbf{r_{2}})=\frac{2\pi k}{B^{2}\sqrt{| ab|}C_{0}}\rho_{1s}\rho_{2s}\exp\left[\frac{-jk}{2}\Big{(}\frac{D^{*}}{B^{*}} \mathbf{r_{1}^{2}}-\frac{D}{B}\mathbf{r_{2}^{2}}\Big{)}\right]\exp[jkf(\rho_{ 1s},\rho_{2s})]\\ \times\sum\limits_{m=1}^{M}\sum\limits_{l=-\infty}^{\infty}\left(\begin{array} []{c}M\\ m\end{array}\right)\frac{(-1)^{n-1}}{m}\exp\left[-\frac{1}{2\Delta_{m}^{2}} \big{(}\rho_{2s}^{2}+\rho_{1s}^{2}\big{)}\right]\\ \times C_{lm}(B,\rho_{1s},\rho_{2s},r_{1},r_{2})\exp[jl(\theta_{2}-\theta_{1}) ],\end{array}  \tag{11}\]  
with  
\[a=\frac{\partial^{2}}{\partial r_{1}^{2}}f(\rho_{1},\rho_{2})=-\frac{A^{*}}{B ^{*}},\,b=\frac{\partial^{2}}{\partial r_{2}^{2}}f(\rho_{1},\rho_{2})=\frac{A }{B},\rho_{1s}=\frac{\beta B^{*}}{A^{*}},\quad\rho_{2s}=\frac{\beta B}{A}. \tag{12}\]  
We call the stationary phase points for \(\rho_{1s}\) and \(\rho_{2s}\) are roots of equation:  
\[\frac{\partial}{\partial\rho_{1}}f(\rho_{1},\rho_{2})=\frac{\partial}{ \partial\rho_{2}}f(\rho_{1},\rho_{2})=0. \tag{13}\]  
The transfer matrix for a MSSM beam passing through an axicon is given as [36]:  
\[\left( \begin{array}{cc}A&B\\ C&D\end{array} \right)=\left( \begin{array}{cc}1&z\\ 0&1\end{array} \right). \tag{14}\]  
Replacing the elements of the transfer matrix in Equation (11) with the elements in Equation (14), we can obtain the final analytical expression for the average intensity of a MSSM beam focused by an axicon:  
\[ \begin{array}{l}I(\mathbf{r_{2}}z)=\frac{2\pi k}{z^{2}\sqrt{|ab|}C_{0}}\rho_ {1s}\rho_{2s}\exp[jkf(\rho_{1s},\rho_{2s})]\\ \times\sum\limits_{m=1}^{M}\sum\limits_{l=-\infty}^{\infty}\left(\begin{array} []{c}M\\ m\end{array}\right)\frac{(-1)^{n-1}}{m}\exp\left[-\frac{1}{2\Delta_{m}^{2}} \big{(}\rho_{2s}^{2}+\rho_{1s}^{2}\big{)}\right]\\ \times C_{lm}(z,\rho_{1s},\rho_{2s},r,r).\end{array}  \tag{15}\]