## 2 Standard Scalar Gaussian Beams  
### General Aspects for Monochromatic Scalar Fields  
Consider a general electromagnetic monochromatic scalar field in vacuum [2],  
\[\Xi(\mathbf{r},t)=\Psi(\mathbf{r})e^{-i\omega t}, \tag{1}\]  
with wavelength \(\lambda=c/\omega\). Within a generalized framework, the amplitude of this field, \(\Psi(\mathbf{r})\), can be specified as a complex-valued static (time-independent) field satisfying [PERSON]'s equation,  
\[\
abla^{2}\Psi+k^{2}\Psi=0, \tag{2}\]  
plus the corresponding boundary conditions. From this equation, the components of the wave vector \(\mathbf{k}=(k_{x},k_{y},k_{z})\) provide us with valuable information about how the field \(\Psi(\mathbf{r})\) changes spatially along each space direction, while the monochromaticity condition requires that the wave number, \(k=\|\mathbf{k}\|=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}=2\pi/\lambda\), remains constant. Because we are dealing with light, let us assume that there is a preferential direction for its space propagation. This direction is going to be referred to as the longitudinal or parallel (\(\|\)) direction, which eventually defines the system optical axis. Any other orthogonal direction is going to be denoted as the perpendicular or transverse (\(\perp\)) direction. Thus, in Cartesian coordinates, if we choose the longitudinal direction along the \(z\)-axis, the \(x\) and \(y\) axes specify the transverse (mutually orthogonal) directions. Consequently, the position vector can be recast as \(\mathbf{r}=(x,y,z)=(\mathbf{r}_{\perp},z)\) and the wave vector as \(\mathbf{k}=(\mathbf{k}_{\perp},k_{z})\), with the wave number being \(k=\sqrt{\|\mathbf{k}_{\perp}\|^{2}+k_{z}^{2}}\). If paraxial conditions are assumed, that is, space variations of the field are slower along the longitudinal direction than along the transverse ones, then \(k_{z}\gg k_{\perp}\) and hence \(k\approx k_{z}\). Accordingly, the scalar field \(\Psi(\mathbf{r})\) can be recast as a plane wave propagating along the \(z\)-direction, modulated by a spatially-dependent amplitude, that is,  
\[\Psi(\mathbf{r})\approx\psi(\mathbf{r})e^{ik_{z}z}\approx\psi(\mathbf{r})e^{ ikz}. \tag{3}\]  
The substitution of the ansatz (3) into the Helmholtz Equation (2) renders  
\[\
abla_{\perp}^{2}\psi+2 ik_{z}\frac{\partial\psi}{\partial z}+\frac{\partial^ {2}\psi}{\partial z^{2}}=0, \tag{4}\]  
where \(\
abla_{\perp}^{2}=\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}\) is the transverse Laplacian. Physically, paraxiality implies that the amplitude of the scalar field has to smoothly change along the longitudinal (\(z\)) direction. This allows us to neglect the second derivative in \(z\) in Equation (4), thus allowing us to simplify this equation, which reads as  
\[i\frac{\partial\psi}{\partial z}=-\frac{1}{2k}\;\
abla_{\perp}^{2}\psi. \tag{5}\]  
As it can readily be noticed, this equation is formally equivalent to the time-dependent Schrodinger equation for a free particle of mass \(m\) in two dimensions [32]. Actually, in this latter case, if it is assumed that propagation along \(z\) is classical, we find a relationship between this coordinate and time:  
\[z=\frac{p}{m}t=\frac{\hbar k}{m}t, \tag{6}\]  
where \(p=\hbar k\). For simplicity and convenience, we have chosen \(z_{0}=0\) in (6), although this is not necessary. It is worth highlighting that, in spite of its apparent simplicity, Equation (6) brings in two important consequences at a conceptual level:  
1. This relation enables a direct switch from propagation in time of an extended wave, namely the scalar field \(\Psi(\mathbf{r})\), to space diffusion along the longitudinal (axial) direction of a particular \"slice\" of such a wave.