## 4 Young-Type Interference with Gauss-Maxwell Beams  
Now we are going to consider the case of a coherent superposition of two linearly polarized Gaussian beams at the input plane \(z=0\). This is the case, for instance, of the two diffracted beams produced by two slits (Young-type diffraction), which arise from the incidence of light of a conventional laser pointer linearly polarized. As mentioned above, the standard Gaussian beam description can be used to describe the eventual spatial distribution of radiation, but it does not provide us with any clue on polarization, in case we need it. That is, we would have a two-beam coherent superposition  
\[\psi(\mathbf{r}_{\perp};z)=\psi_{1}(\mathbf{r}_{\perp};z)+\psi_{2}(\mathbf{r} _{\perp};z), \tag{89}\]  
with  
\[\psi_{i}(\mathbf{r}_{\perp};z)=\mathcal{N}\left(\frac{1}{2\pi\tilde{\sigma}_{ z,i}^{2}}\right)^{1/2}e^{-\|\mathbf{r}_{\perp}-\mathbf{r}_{\perp}^{(i)}\|^{2}/4 \sigma_{0,i}\tilde{\sigma}_{z,i}}, \tag{90}\]  
denoting the diffracted Gaussian beams, with \(i=1,2\). For simplicity, although without loss of generality, we are going to consider that both beams are identical (\(\sigma_{0,i}=\sigma_{0}\)), they are symmetrically placed with respect to \(x=0\) (\(x_{0,1}=-x_{0,2}=x_{c}\)), and aligned along the \(x\)-axis (\(y_{0,1}=y_{0,2}=0\)). As for the prefactor \(\mathcal{N}\) in (90), it is just a \(z\)-independent global norm factor that amounts to approximately \(1/\sqrt{2}\) when the overlap of the two input beams is nearly zero, as it can readily be seen from its functional form,  
\[\mathcal{N}=\frac{1}{\sqrt{2}}\frac{1}{\sqrt{1+e^{-x_{c}^{2}/2\sigma_{0}^{2}}}}. \tag{91}\]  
Without any need for considering the effect of polarization (which is the same to say that both beams are equally polarized, as we have already seen above), the intensity of the radiation described by the scalar superposition field (89), except for constant factors, is going to be proportional to the density profile along the \(x\)-direction,  
\[|\psi|^{2}=|\psi_{1}|^{2}+|\psi_{2}|^{2}+2\text{Re}\left(\psi_{1}\psi_{2}^{*} \right)\sim e^{-(x-x_{c})^{2}/2\sigma_{z}^{2}}+e^{-(x+x_{c})^{2}/2\sigma_{z}^ {2}}+2e^{-(x^{2}+x_{c}^{2})/2\sigma_{z}^{2}}\cos\left(\kappa_{z}x\right), \tag{92}\]  
where  
\[\kappa_{z}=\frac{x_{c}z}{\sigma_{z}^{2}z_{R}}. \tag{93}\]  
Note that we have denoted (92) with the term density, because of its direct relation to the time-averaged electromagnetic energy density (except for a proportionality constant) [2]. Furthermore, it can readily be seen that, in the limit \(z\gg z_{R}\), where large \(x\) values are involved (\(x\gg x_{c}\)), the density (92) can be approximated to  
\[|\psi|^{2}\sim 4e^{-x_{R}^{2}x^{2}/2\sigma_{0}^{2}z^{2}}\cos^{2}\left(\frac{2 \pi x_{c}x}{\lambda z}\right), \tag{94}\]  
with  
\[\kappa_{z}\approx\frac{x_{c}z_{R}}{\sigma_{0}^{2}z}=\frac{4\pi x_{c}}{\lambda z}. \tag{95}\]  
The density (94) describes a Young-type fringe pattern (along the \(x\)-direction) modulated by a Gaussian envelop, arising from the Gauss shape of the input beams, and with maxima at positions  
\[x_{n}=\frac{\lambda z}{d}\ n, \tag{96}\]where \(d=2x_{c}\). Actually, if we assume that \(x_{n}/z=\sin\theta_{n}\), with \(\theta_{n}\) being the angular position of each maximum with respect to the coordinate origin at the input plane, the above condition becomes  
\[d\sin\theta_{n}=\lambda n, \tag{97}\]  
that is, the usual two-slit interference condition [2]. Let us now consider the case of polarized input beams, particularly, the case of horizontal polarization (in next section, we shall introduce the case of arbitrarily polarized superimposed beams).