## 4 Young-Type Interference with Gauss-Maxwell Beams
This is in contrast with the evolution of the rays along the \(y\)-direction, which is fully analytical and corresponds to the result that we have already found in Section 3.1 for unpolarized standard Gaussian beams and in Section 3.2 for linearly polarized Gauss-Maxwell beams. Yet in the far field it is possible to perform some guesses of interest from Equation (103) without even solving it. Thus, if we assume \(z\gg z_{R}\) (and \(x\gg x_{c}\)), this equation reads as  
\[\frac{dx}{dz}\approx\frac{x}{z}-\frac{z_{R}x_{c}}{2z^{2}}\left[\frac{\sin\left( \kappa_{z}x\right)}{1+\cos\left(\kappa_{z}x\right)}\right]. \tag{105}\]  
According to this equation, the transverse momentum along the \(x\)-direction increases linearly with \(x\) at any sufficiently distant output plane \(z\) (with respect to the input plane), from negative values to positive ones. This behavior is governed by the first term in the equation (the second term decays as \(z^{-2}\)), which is interrupted whenever the denominator of the second term cancels out, that is, \(\kappa_{z}x=(2n+1)\pi\). In this case, the second term becomes the leading one, which gives rise to some sudden variations of the transverse momentum (a sort of spiky behavior) at those points, positive for \(x<0\) and negative for \(x>0\). Regarding the rays that can be expected, if we neglect this second term, the equation describes straight lines; taking into account the second term, on the other hand, the rays will be grouped in swarms, each one lying within two consecutive \"spikes\". Although, as said above, this is just a guess, note that it coincides with both previous models [8; 12] and the experiment reported in Reference [17]. Now, the interference with two polarized beams can be extended to a general case where the polarization state associated with each Gaussian beam is arbitrary. This involves an interesting subtlety worth emphasizing: different polarizations give rise to a loss of mutual coherence, related to both the so-called Arago-Fresnel laws of interference with polarized light in optics [11] and, within the quantum realm, the well-known problem of the quantum (which-way) erasure [44]. To better appreciate this fact, notice that the electric and magnetic components for each field are now  
\[\mathbf{E}_{i}(\mathbf{r}_{\perp};z) = E_{0}\psi_{i}(\mathbf{r}_{\perp};z)\left[\alpha_{i}\hat{\mathbf{ x}}+\beta_{i}\hat{\mathbf{y}}+\frac{\left(\alpha_{i}\Delta x_{i}+\beta_{i} \Delta y_{i}\right)}{2k\sigma_{0}\sigma_{z}}\;e^{-i\left(\varphi_{z}+\pi/2 \right)}\hat{\mathbf{z}}\right], \tag{106}\] \[\mathbf{H}_{i}(\mathbf{r}_{\perp};z) = H_{0}\psi_{i}(\mathbf{r}_{\perp};z)\left[-\beta_{i}\hat{\mathbf{ x}}+\alpha_{i}\hat{\mathbf{y}}+\frac{\left(\beta_{i}\Delta x_{i}-\alpha_{i} \Delta y_{i}\right)}{2k\sigma_{0}\sigma_{z}}\;e^{-i\left(\varphi_{z}+\pi/2 \right)}\hat{\mathbf{z}}\right], \tag{107}\]  
where we can appreciate that coordinates and polarization coefficients are both intertwined, which does not allow us to recast the electromagnetic field as a factorizable function of coordinates and polarization states, that is, in general terms, something of the kind \(|\psi_{i}\rangle|P_{i}\rangle\), which would simplify the final solution.