## 3 Gauss-Maxwell Beams
### Linearly Polarized Gauss-Maxwell Beams
\tag{59}\]  
At this level of approximation, we already find the first-order corrections to the paraxial approximation that lead to usual Gaussian beams, with the electric and magnetic fields (57) and (58) being  
\[\mathbf{E}(\mathbf{r}_{\perp};z) = E_{0}\psi(\mathbf{r}_{\perp};z)\left(\hat{\mathbf{x}}-\frac{ \Delta x}{q_{z}}\;\hat{\mathbf{z}}\right)=E_{0}\psi(\mathbf{r}_{\perp};z)\left[ \hat{\mathbf{x}}+\frac{\Delta x}{2k\sigma_{0}\sigma_{z}}\;e^{-i(\sigma_{z}+ \pi/2)}\hat{\mathbf{z}}\right], \tag{60}\] \[\mathbf{H}(\mathbf{r}_{\perp};z) = H_{0}\psi(\mathbf{r}_{\perp};z)\left(\hat{\mathbf{y}}-\frac{ \Delta y}{q_{z}}\;\hat{\mathbf{z}}\right)=H_{0}\psi(\mathbf{r}_{\perp};z)\left[ \hat{\mathbf{y}}+\frac{\Delta y}{2k\sigma_{0}\sigma_{z}}\;e^{-i(\sigma_{z}+ \pi/2)}\hat{\mathbf{z}}\right], \tag{61}\]  
respectively (notice that the \(\pi/2\) phase factor is not an extra factor, but it arises from the expression for \(q_{z}\) introduced above, in Equation (50)). As it can be noticed, the two fields do not remain in-plane (i.e., contained within the same constant \(z\) plane), as one would expect from a usual scalar field guess, but they contain a small out-of-plane component, along the longitudinal (\(z\)) direction. From these expressions, it is now possible to obtain some information about the polarization state of the electromagnetic field in the near and far fields, that is, in terms of the value of \(z\) compared to that of \(z_{R}\). Without loss of generality, in this regard, let us consider the electric component (same holds for the magnetic one). The two cases of interest worth stressing are:  
* If \(z\ll z_{R}\), then \(\sigma_{z}\approx\sigma_{0}\) and \(\varphi_{z}\approx z/z_{R}\ll\pi/2\). So, in the near field regime, we have \[\mathbf{E}(\mathbf{r}_{\perp};z)\approx E_{0}\psi(\mathbf{r}_{\perp};z)\left( \hat{\mathbf{x}}+\frac{\Delta x}{z_{R}}\;e^{-i\pi/2}\hat{\mathbf{z}}\right),\] (62) and the field is elliptically polarized, with the polarization plane being the \(XZ\)-plane (perpendicular to the \(y\) transverse coordinate), since \[\frac{E_{z}}{E_{x}}=\frac{\Delta x}{z_{R}}\;e^{-i\pi/2}.\] (63) As it can be noticed, only at the center of the beam there is horizontal polarization (parallel to the \(x\) transverse coordinate); anywhere else, the \(z\)-component of the polarization increases linearly with the distance with respect to the center of the beam. Since this a regime where the expansion of the beam is still negligible, this polarization effect should be particularly relevant for distances (from the center) of the order of the beam waist, \(w_{0}\), where the fast decrease of the intensity might complicate its detection (particularly, if we consider typical values used in diffraction experiments). * On the other hand, for \(z\gg z_{R}\), in the far field regime, we have \(\sigma_{z}\approx z\sigma_{0}/z_{R}\) and \(\varphi_{z}\approx\pi/2\). Accordingly, the electric field component reads as \[\mathbf{E}(\mathbf{r}_{\perp};z)\approx E_{0}\psi(\mathbf{r}_{\perp};z)\left( \hat{\mathbf{x}}-\frac{\Delta x}{z}\;\hat{\mathbf{z}}\right),\] (64) that is, the field is still contained within the \(XZ\)-plane, but the elliptical polarization state has degenerated into linear, since \[\frac{E_{z}}{E_{x}}=-\frac{\Delta x}{z},\] (65)which becomes negligible very quickly as \(z\) increases, recovering the linear polarization along the \(x\) transverse coordinate at any distance from the center of the beam. Therefore, contrary to the case of a monochromatic plane wave, transversality is not ensured unless we consider the limit case \(\sigma_{0}\to\infty\), which is precisely the case near the maximum of the Gaussian, where the wavefront could be approximated by a nearly plane wave, as can be seen from Equations (60) and (61), when the \(z\)-component contribution is neglected. This model was considered in a previous work [8] to analyze the problem of interference, which is revisited and extended in next section for the case of interference between two Gauss-Maxwell beams.