# Propagation Property of an Astigmatic sin-Gaussian Beam in a Strongly Nonlocal Nonlinear Media
## 2 Theoretical Formulation
Making use of the Gaussian integral formula  
\[\int_{-\infty}^{\infty}\exp\Bigl{(}-\alpha x^{2}+\beta x\Bigr{)}\ =\ \exp\biggl{(}\frac{\beta^{2}}{4\alpha}\biggr{)}\mathrm{Re}(\alpha)>0 \tag{7}\]  
One obtains  
\[E(x,y,z)\ =\ E_{0}\exp\biggl{[}-\frac{z_{R}^{2}}{B^{2}}\biggl{(}\frac{x_{w}^{2}} {1-i\alpha_{x}}+\frac{y_{w}^{2}}{1-i\alpha_{y}}\biggr{)}\biggr{]}\sinh\biggl{[} \frac{z_{R}}{B}\biggl{(}\frac{\beta_{x}x_{w}}{1-i\alpha_{x}}+\frac{\beta_{y}y_ {w}}{1-i\alpha_{y}}\biggr{)}\biggr{]} \tag{8}\]  
where  
\[E_{0}\ =\ \frac{z_{R}}{2B\sqrt{\bigl{(}1-i\alpha_{x}\bigr{)}\bigl{(}1-i \alpha_{y}\bigr{)}}}\exp\biggl{[}\frac{iDz_{R}}{B}\Bigl{(}x_{w}^{2}+y_{w}^{2} \Bigr{)}-\frac{\beta_{x}^{2}}{4\bigl{(}1-i\alpha_{x}\bigr{)}}-\frac{\beta_{y}^ {2}}{4\bigl{(}1-i\alpha_{y}\bigr{)}}\biggr{]} \tag{9}\]  
is an axial symmetry factor on a specified observation plane. Equation (8) is the main result obtained in this paper and is also the starting equation to discuss the evolution property of the propagating beam in the next section. Evolutions of Wave Front Dislocations and Intensity Patterns Occurring in an Astigmatic Sin-Gaussian Beam  
In this section, we mainly study the evolution of the intensity pattern as well as the wave front dislocation for a sin-Gaussian beam with astigmatism propagating in SNNM using the analytical formulae derived in previous section. According to Equation (8) and the relations of \(A\), \(B\), \(D\), \(\alpha_{x}\) and \(\alpha_{y}\) with the scaled propagation distance \(z_{s}\), one can generally prove  
\[E(x,y,z_{s})\ =\ E(x,y,z_{s}+1) \tag{10}\]  
Equation (10) reveals that the evolution of the field distribution with the scaled propagation distance \(z_{s}\) is periodic and the period of the field distribution evolution equals to 1 corresponding to the propagation distance \(\pi z_{p}\ =\ \pi z_{R}\sqrt{p_{Gc}/p_{0}}\)where \(p_{Gc}\ =\ 1/\bigl{(}\gamma^{2}z_{R}^{2}\bigr{)}\) is the critical power of Gaussian or higher-order Gaussian beams. Obviously, \(z_{p}\) = \(z_{R}\) corresponds to \(p_{0}\) = \(p_{Gc}\) while \(z_{p}\) < \(z_{R}\) corresponds to \(p_{0}\) > \(p_{Gc}\). It is known that a sin-Gaussian beam carries an edge dislocation along the straight line \(\beta_{x}x+\beta_{y}y\ =\ 0\) where \(\beta_{x}\) and \(\beta_{y}\) determines the orientation of the edge dislocation and its intensity pattern is a two-lobe configuration for sufficiently small \(\beta_{x}\) and \(\beta_{y}\). For \(R_{0x}\) = \(R_{0y}\) (i.e., \(\alpha_{x}\) = \(\alpha_{y}\)) corresponding to beams without astigmatism, Equation (8) turns to  
\[E(x,y,z)\ =\ E_{0}\exp\biggl{[}-\frac{z_{R}^{2}\bigl{(}x_{w}^{2}+y_{w}^{2} \bigr{)}}{B^{2}(1-i\alpha_{x})}\biggr{]}\sinh\biggl{[}\frac{z_{R}\bigl{(}\beta _{x}x_{w}+\beta_{y}y_{w}\bigr{)}}{B(1-i\alpha_{x})}\biggr{]} \tag{11}\]Regardless of astigmatism, for \(\beta_{x}\) or \(\beta_{y}\) being 0, to be specific, assuming \(\beta_{y}\) = 0, Equation (8) becomes  
\[E(x,y,z)\ =\ E_{0}\exp\left[-\frac{z_{R}^{2}}{B^{2}}\left(\frac{x_{w}^{2}}{1-ia_{x} }+\frac{y_{w}^{2}}{1-ia_{y}}\right)\right]\text{sinh}\left[\frac{z_{R}\beta_{x }x_{w}}{B(1-ia_{x})}\right] \tag{12}\]  
Therefore, Equations (11) and (12) clearly demonstrate that the initial edge dislocation remains unchanged for beams without astigmatism propagating in SNNM. Figure 1 shows typical evolutions of the intensity distributions of sin-Gaussian beams without astigmatism propagating in SNNM at various scaled propagation distance \(z_{s}\) for different values of \(R_{0x}\) (\(R_{0y}\)) and \(z_{p}\) (\(z_{R}\)) with sufficiently small \(\beta_{x}\) and \(\beta_{y}\). Note that only for \(z_{p}\) = \(z_{R}\) (i.e., \(p_{0}\) = \(p_{GC}\)) and \(R_{0x}\) = \(R_{0y}\) = \(\infty\), sin-Gaussian beam can roughly keep its structure and size unchanged during propagation in SNNM, seeing Row 2 in Figure 1. Otherwise, the pattern size varies with a period of \(\pi z_{p}\) although the intensity pattern continues to be a two-lobe structure during propagation.