## 2 Standard Scalar Gaussian Beams
### Gaussian Beam Propagation
The intensity or irradiance obtained from (12) is  
\[\psi(\mathbf{r}_{\perp};z)=\hat{\mathcal{U}}(\mathbf{\hat{r}}_{\perp},\mathbf{ \hat{p}}_{\perp})\psi(\mathbf{r}_{\perp};0), \tag{10}\]0  
The streamlines that describe the spatial propagation of this intensity along \(z\) can be obtained, according to Equation (7), from the equation of motion  
\[\psi(\mathbf{r}_{\perp};z)=\hat{\mathcal{U}}(\mathbf{\hat{r}}_{\perp},\mathbf{ \hat{p}}_{\perp})\psi(\mathbf{r}_{\perp};0), \tag{10}\]1  
which, particularized to each transverse coordinate, leads to  
\[\psi(\mathbf{r}_{\perp};z)=\hat{\mathcal{U}}(\mathbf{\hat{r}}_{\perp},\mathbf{ \hat{p}}_{\perp})\psi(\mathbf{r}_{\perp};0), \tag{10}\]2 \[\psi(\mathbf{r}_{\perp};z)=\hat{\mathcal{U}}(\mathbf{\hat{r}}_{\perp},\mathbf{ \hat{p}}_{\perp})\psi(\mathbf{r}_{\perp};0), \tag{10}\]3  
The integration of Equations (21) and (22) is straightforward, leading to \[\psi(\mathbf{r}_{\perp};z)=\hat{\mathcal{U}}(\mathbf{\hat{r}}_{\perp},\mathbf{ \hat{p}}_{\perp})\psi(\mathbf{r}_{\perp};0), \tag{10}\]4 \[\psi(\mathbf{r}_{\perp};z)=\hat{\mathcal{U}}(\mathbf{\hat{r}}_{\perp},\mathbf{ \hat{p}}_{\perp})\psi(\mathbf{r}_{\perp};0), \tag{10}\]5  
where \(x_{0}\) and \(y_{0}\) denote the position of the optical streamline at the input plane \(z=0\), while its position on the output plane \(z\) is given by \(x_{z}\) and \(y_{z}\). Notice that the solutions rendered by Equations (23) and (24) display radial symmetry with respect to the center of the Gaussian beam, \((x_{c},y_{c})\). Accordingly, if their initial positions distribute around this point in a circle of radius \(\rho_{0}=\|\mathbf{r}_{\perp,0}-\mathbf{r}_{\perp,c}\|\), they will distribute on a plane at a distance \(z\) around a circle of radius \(\rho_{z}=\|\mathbf{r}_{\perp,z}-\mathbf{r}_{\perp,c}\|\) that has increased its size in a factor \(\sigma_{z}/\sigma_{0}\) with respect to \(\rho_{0}\). This is better seen if, from the definitions of \(\rho_{0}\) and \(\rho_{z}\), the two equations are rewritten in a more compact form as:  
\[\psi(\mathbf{r}_{\perp};z)=\hat{\mathcal{U}}(\mathbf{\hat{r}}_{\perp},\mathbf{ \hat{p}}_{\perp})\psi(\mathbf{r}_{\perp};0), \tag{10}\]6  
Relatively close to the input plane, this increase is negligible and then, as \(z\) starts becoming important compared to the critical value \(z_{R}\), there is a quadratic dependence on \(z\), as expected in a typical Fresnel regime. In contrast, asymptotically, far from the input plane, the increase of the radius is linear with \(z\), in agreement with a propagation in a Fraunhofer regime. This can also be noticed from the expression describing the expansion rate of the beam,  
\[\psi(\mathbf{r}_{\perp};z)=\hat{\mathcal{U}}(\mathbf{\hat{r}}_{\perp},\mathbf{ \hat{p}}_{\perp})\psi(\mathbf{r}_{\perp};0), \tag{10}\]7  
which can be obtained from Equations (21) and (22) [or also directly from (25)] and describes the beam expansion in relation to the actual beam size. This expression displays a maximum for \(z=z_{R}\). Thus, before reaching the position of this critical plane, the expansion rate is first negligible (for \(z\lll z_{R}\)) and then starts increasing nearly linear with \(z\) (for \(z\ll z_{R}\)). This implies that the radius will undergo a sort of initial fast boost that will accelerate its increase and then the expansion of the beam. However, after surpassing the critical plane, the rate falls with \(z\), which will keep the radius increasing at a constant rate, and hence the divergence displayed by any ray with respect to the center of the input beam, consistent with the fact that far from the input plane (\(z\rightarrow\infty\)), the radial speed,  
\[\psi(\mathbf{r}_{\perp};z)=\hat{\mathcal{U}}(\mathbf{\hat{r}}_{\perp},\mathbf{ \hat{p}}_{\perp})\psi(\mathbf{r}_{\perp};0), \tag{10}\]8  
becomes a constant. This is in compliance with the linear increase of the cross section that we observe in Gaussian beams far from the source (e.g., a non-collimated laser beam).