# Propagation Property of an Astigmatic sin-Gaussian Beam in a Strongly Nonlocal Nonlinear Media
## 2 Theoretical Formulation  
In the Cartesian coordinate system, a sin-Gaussian beam with astigmatism in the source plane \(z=0\) takes the form as  
\[E(x,y,0)\ =\ \exp\left[-\frac{x^{2}+y^{2}}{w_{0}^{2}}-\frac{ik}{2}\left(\frac{x ^{2}}{R_{0x}}+\frac{y^{2}}{R_{0y}}\right)\right]\sin\left(\frac{\beta_{x}x+ \beta_{y}y}{w_{0}}\right) \tag{1}\]  
where \(\beta_{x}\) and \(\beta_{y}\) represent the parameters associated with the sin part, \(k=2\pi/\lambda\) and \(\lambda\) is the wavelength, \(w_{0}\) is the spot size of the beam, and \(R_{0x}\) and \(R_{0y}\) are respectively the curvatures along the \(x\) and \(y\) directions at \(z=0\). The astigmatism is represented by the curvature difference between \(x\) and \(y\) directions and can be easily realized in experiment [38; 39]. Because the nonlocal nonlinear Schrodinger equation governing the evolution dynamics of optical beams propagating in nonlocal nonlinear medium remains mathematically complicated, in this paper the medium is assumed to be SNNM and, then, Snyder and Mitchell model is appropriate. In fact, it has been found that the analytical solutions obtained from Snyder and [PERSON] model and the numerical simulations based on the nonlocal nonlinear Schrodinger equation are in good agreement in the case of strong nonlocality [12; 16; 18]. The relation between optical beams propagating in SNNM and in quadratic-index media or through optical fractional Fourier transform systems has also been determined [40; 41]. The propagation of a sin-Gaussian beam in SNNM is governed by the following equation:  
\[2 ik\frac{\partial}{\partial z}E(x,y,z)+\left(\frac{\partial^{2}}{\partial x^{ 2}}+\frac{\partial^{2}}{\partial y^{2}}\right)E(x,y,z)-k^{2}\gamma^{2}p_{0} \left(x^{2}+y^{2}\right)E(x,y,z)\ =\ 0 \tag{2}\]  
where \(\gamma\) is the material constant relating to the response function, and \(p_{0}\) is the input power in the source plane. The solution of Equation (2) can be expressed as [15; 21; 22]  
\[ \begin{array}{ll}E(x,y,z)&=\frac{i}{\lambda B}\exp\left[\frac{ikD}{2B}\left( x^{2}+y^{2}\right)\right]\\ &\iint dx_{0}dy_{0}E(x_{0},y_{0},0)\exp\left[\frac{ikA}{2B}\left(x_{0}^{2}+y_{0 }^{2}\right)-\frac{ik}{B}\left(xx_{0}+yy_{0}\right)\right]\end{array}  \tag{3}\]  
where the ABCD transfer matrix for SNNM has been obtained with the following:  
\[\left( \begin{array}{cc}A&B\\ C&D\end{array} \right)\ =\ \left( \begin{array}{cc}\cos(\pi z_{s})&-z_{p}\sin(\pi z _{s})\\ \sin(\pi z_{s})/z_{p}&\cos(\pi z_{s})\end{array} \right) \tag{4}\]  
with \(z_{p}\ =\ 1/\sqrt{\gamma^{2}p_{0}}\) and \(z_{s}\ =\ z/\left(\pi z_{p}\right)\) being the scaled propagation distance. Substituting Equation (1) into Equation (3) and performing some mathematical manipulations we have  
\[ \begin{array}{rcl}E(x,y,z)&=&\frac{1}{2\lambda B}\exp\biggl{[}\frac{iz_{R}D \left(\frac{x_{w}^{2}+y_{w}^{2}}{B}\right)}{B}\biggr{]}\iint dx_{0}dy_{0}\exp \biggl{[}-(1-i\alpha_{x})\frac{x_{0}^{2}}{w_{0}^{2}}-\bigl{(}1-i\alpha_{y} \bigr{)}\frac{y_{0}^{2}}{w_{0}^{2}}\biggr{]}\\ &&\times\biggl{\{}\exp\Bigl{[}i\Bigl{(}\beta_{x}-\frac{2z_{R}x_{w}}{B}\Bigr{)} \frac{x_{0}}{w_{0}}+i\Bigl{(}\beta_{y}-\frac{2z_{R}y_{w}}{B}\Bigr{)}\frac{y_{0 }}{w_{0}}\biggr{]}\\ &&-\exp\Bigl{[}-i\Bigl{(}\beta_{x}+\frac{2z_{R}x_{w}}{B}\Bigr{)}\frac{x_{0}}{ w_{0}}-i\Bigl{(}\beta_{y}+\frac{2z_{R}y_{w}}{B}\Bigr{)}\frac{y_{0}}{w_{0}}\Bigr{]} \biggr{\}}\end{array}  \tag{5}\]  
with  
\[\alpha_{u}\ =\ \frac{A}{B}\frac{kw_{0}^{2}}{2}-\frac{kw_{0}^{2}}{2R_{0u}}\ =\ \frac{Az_{R}}{B}-\frac{z_{R}}{R_{0u}} \tag{6}\]  
\(u_{w}\ =\ u/w_{0}\) (u = \(x\) and \(y\)) being scaled transversal coordinates and \(z_{R}\ =\ kw_{0}^{2}/2\ =\ \pi w_{0}^{2}/\lambda\) being the Rayleigh distance.