## 2 Standard Scalar Gaussian Beams
### Gaussian Beam Propagation  
Consider that at the input plane \(z_{0}=0\), the electromagnetic field is described by a beam with Gaussian amplitude (on the \(z_{0}\) plane),  
\[\psi(\mathbf{r}_{\perp};0)=\frac{1}{\sqrt{2\pi\sigma_{0}^{2}}}e^{-|\mathbf{r} _{\perp}-\mathbf{r}_{\perp,\epsilon}|^{2}/4\sigma_{0}^{2}}, \tag{9}\]  
which may represent a Gaussian mode released from an optical fiber or the light coming out from a laser pointer. The beam (9) is centered at \(\mathbf{r}_{\perp,\epsilon}=(x_{c},y_{c})\) and its width is related to its waist, \(w_{0}\), by the simple relation \(\sigma_{0}=w_{0}/2\). In this latter regard, we take here the conventional definition for the waist of a Gaussian beam, as its size at the point of its focus (here located at the input plane, \(z_{0}=0\)), which corresponds to the radius of the \(1/e^{2}\) irradiance contour at the plane (\(z_{0}\)) where the wavefront is flat. In order to provide some typical values, we can take them from the experiment reported in Reference [17]. In this experiment, a coherent superposition of two nearly Gaussian beams (in a good approximation [12]) is generated, with waists \(w_{0}=0.608\) mm, wavelength \(\lambda=943\) nm, and their centers separated a distance \(d=4.69\) mm. The propagation of the amplitude (9) along the \(z\)-axis is obtained by acting on it with the free-space propagator \(\hat{\mathcal{U}}\), that is,  
\[\psi(\mathbf{r}_{\perp};z)=\hat{\mathcal{U}}(\mathbf{\hat{r}}_{\perp},\mathbf{ \hat{p}}_{\perp})\psi(\mathbf{r}_{\perp};0), \tag{10}\]  
which is equivalent to considering the integral  
\[\psi(\mathbf{r}_{\perp};z)=\frac{1}{i\lambda z}\int\psi(\mathbf{r}_{\perp}^{ \prime},0)e^{ik\|\mathbf{r}_{\perp}-\mathbf{r}_{\perp}^{\prime}\|^{2}/2z}d \mathbf{r}_{\perp}^{\prime} \tag{11}\](for a derivation of this expression, see Appendix A). Accordingly, the substitution of the ansatz (9) into the integral (11) renders  
\[\psi(\mathbf{r}_{\perp};z)=A_{z}e^{-\|\mathbf{r}_{\perp}-\mathbf{r}_{\perp,\pm} \|^{2}/4\sigma_{0}\tilde{\sigma}_{z}}, \tag{12}\]  
where  
\[A_{z}=\frac{1}{\sqrt{2\pi\tilde{\sigma}_{z}^{2}}}=\frac{e^{-i\varphi_{z}}}{ \sqrt{2\pi\sigma_{z}^{2}}} \tag{13}\]  
is a complex-valued norm factor, \(\varphi_{z}\) is the well-known Gouy phase in optics (typical of Gaussian beams),  
\[\varphi_{z}=(\tan)^{-1}\left(\frac{z}{2k\sigma_{0}^{2}}\right), \tag{14}\]  
and  
\[\tilde{\sigma}_{z}=\sigma_{0}\left[1+\frac{iz}{2k\sigma_{0}^{2}}\right] \tag{15}\]  
is an also complex-valued spread factor. The dispersion of the beam at a distance \(z\) from the slits is given by the expression  
\[\sigma_{z}=|\tilde{\sigma}_{z}|=\sigma_{0}\sqrt{1+\left(\frac{z}{2k\sigma_{0 }^{2}}\right)^{2}}, \tag{16}\]  
related, by means of the simple relation \(\sigma_{z}=w_{z}/2\), to radius of the \(1/e^{2}\) contour when the wave has propagated a distance \(z\),  
\[w_{z}=w_{0}\sqrt{1+\left(\frac{2z}{kw_{0}^{2}}\right)^{2}}. \tag{17}\]  
As it can be noticed from (16), the expansion of the beam along the optical axis can be characterized in terms of the Rayleigh length,  
\[z_{R}=2k\sigma_{0}^{2}. \tag{18}\]  
which constitutes a critical or characteristic length that sets a separation between two dynamically different regimes regarding the beam expansion, namely the Fresnel or hyperbolic expansion regime and the Fraunhofer or linear expansion regime [34].