## 2 Standard Scalar Gaussian Beams
### General Aspects for Monochromatic Scalar Fields
More specifically, if we consider a transverse section or plane of the full wave [consider, for instance, that such a wave describes a pulse with amplitude \(\psi(\mathbf{r})\)] within the \(XY\) plane for a given value \(z_{0}\) (in other words, the input plane \(z=z_{0}\), analogous to considering \(t_{0}\) in a time-propagation), we shall obtain its spatial redistribution or accommodation to the corresponding boundaries at subsequent planes, with \(z\) increasing. Consequently, if the pulse has an extension along the \(z\) direction, considering different \"slices\" of the pulse (i.e., different \(z_{0}\) planes), we may easily determine the shape of the pulse at a further distance by just combining all the resulting \"slices\" \(z_{f}\). Notice that this fact also allows to establish the validity limit for the approximation, which is going to remain correct provided dispersion along the \(z\) direction can be assumed to be negligible (i.e., as long as all \"slices\" travel with nearly the same speed). 2. On the other hand, such a relation enables a simple, direct link between wave optics and matter-wave optics, which arises from the formal relation established between [PERSON]'s equation and the Helmholtz equation in paraxial form, both being parabolic differential equationsdescribing the transport of the quantity \(\Psi\) (regardless of the nature of such a quantity, that is, whether it describes an electromagnetic amplitude or a probability amplitude) with an imaginary diffusion constant (this complex valuedness is precisely the fundamental trait that allows interference in the solution of these equations, but not in the heat equation, although it is also a differential equation of the parabolic class). This is also a rather convenient issue both analytically and computationally, because it explicitly shows that optical and matter-wave problems ruled by the same equation form have the same solution and, eventually, the same interpretations [33]. Taking into account point 2 above, Equation (5) can readily be recast in the form of a Schrodinger-type equation by replacing the coordinate \(z\) with the value indicated by the relation (6)--notice, however, that this gives rise to a two-dimensional Schrodinger equation, since the only space coordinates, after substitution, are \(x\) and \(y\). This direct analogy can be taken a step further to introduce a guidance equation in the Bohmian form, namely  
\[\frac{d\mathbf{r}_{\perp}}{dz}=\frac{\
abla_{\perp}S_{\phi}}{k}, \tag{7}\]  
with  
\[S_{\psi}=\frac{1}{2i}\ln\left(\frac{\psi}{\psi^{*}}\right) \tag{8}\]  
describing the space phase variations of the complex field amplitude \(\psi\). Since \(d\mathbf{r}_{\perp}^{\prime}/dz\approx\mathbf{k}_{\perp}/k\), this equation allows us to describe the transverse distribution of electromagnetic energy in terms of streamlines by providing the corresponding initial conditions and integrating it along \(z\)[32]. Further, observe that also in this case, the connection between Equation (7) and the Bohmian equation for matter waves is straightforward, since the latter can be directly obtained by considering the relation (6) and the change \(S_{\psi}\to S/\hbar\), from the phase of the amplitude \(\psi\) to the phase of the matter wave (with \(\hbar\) emphasizing the fact that \(S\) has units of action).