## 4 Young-Type Interference with Gauss-Maxwell Beams
Following the procedure described in Section 3.2, we find that the electric and magnetic fields (within the paraxial approximation) associated with each partial wave (90) are  
\[\mathbf{E}_{i}(\mathbf{r}_{\perp};z) = E_{0}\psi_{i}(\mathbf{r}_{\perp};z)\left[\mathbf{\hat{x}}+\frac {\Delta x_{i}}{2k\sigma_{0}\sigma_{z}}\;e^{-i(\varphi_{z}+\pi/2)}\mathbf{\hat{ z}}\right], \tag{98}\] \[\psi_{i}(\mathbf{r}_{\perp};z)=\mathcal{N}\left(\frac{1}{2\pi\tilde{\sigma}_{ z,i}^{2}}\right)^{1/2}e^{-\|\mathbf{r}_{\perp}-\mathbf{r}_{\perp}^{(i)}\|^{2}/4 \sigma_{0,i}\tilde{\sigma}_{z,i}}, \tag{90}\]0  
where, again, we have assumed identical beams (\(E_{0,1}=E_{0,2}=E_{0}\) and \(H_{0,1}=H_{0,2}=H_{0}\)). The total electric and magnetic fields that result from the superposition of both beams read as  
\[\psi_{i}(\mathbf{r}_{\perp};z)=\mathcal{N}\left(\frac{1}{2\pi\tilde{\sigma}_{ z,i}^{2}}\right)^{1/2}e^{-\|\mathbf{r}_{\perp}-\mathbf{r}_{\perp}^{(i)}\|^{2}/4 \sigma_{0,i}\tilde{\sigma}_{z,i}}, \tag{90}\]1 \[\psi_{i}(\mathbf{r}_{\perp};z)=\mathcal{N}\left(\frac{1}{2\pi\tilde{\sigma}_{ z,i}^{2}}\right)^{1/2}e^{-\|\mathbf{r}_{\perp}-\mathbf{r}_{\perp}^{(i)}\|^{2}/4 \sigma_{0,i}\tilde{\sigma}_{z,i}}, \tag{90}\]2  
where \(\Delta x_{i}=x-x_{c,i}\) and \(\Delta\psi=\psi_{1}-\psi_{2}\). From these expressions, we now compute the time-averaged Poynting vector, which reads as  
\[\mathbf{S} = \frac{1}{2}\;E_{0}H_{0}\left(\begin{array}{c}\frac{\sin \varphi_{z}}{2k\sigma_{0}\sigma_{z}}\left[x|\psi|^{2}-x_{c}\left(|\psi_{1}|^{2} -|\psi_{2}|^{2}\right)\right]+\frac{\cos\varphi_{z}}{k\sigma_{0}\sigma_{z}}\; x_{c}\mathrm{Im}\left(\psi_{1}\psi_{2}^{s}\right)\\ \frac{\sin\varphi_{z}}{2k\sigma_{0}\sigma_{z}}\;y|\psi|^{2}\\ \left|\psi|^{2}\right.\\ = \frac{1}{2}\;E_{0}H_{0}\left( \begin{array}{c}\frac{z}{4k^{2} \sigma_{0}^{2}\sigma_{z}^{2}}\left[\Delta x_{1}|\psi_{1}|^{2}+\Delta x_{2}| \psi_{2}|^{2}+2x\mathrm{Re}\left(\psi_{1}\psi_{2}^{s}\right)\right]+\frac{z_{ R}}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}\;x_{c}\mathrm{Im}\left(\psi_{1}\psi_{2}^{s} \right)\\ \frac{z}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}\;y|\psi|^{2}\\ |\psi|^{2}\end{array} \right),\]  
As before, the ray equation is now obtained from the components of (103), since flow of the beam at each point is described by the direction of the time-averaged Poynting vector right on that point, which is also the direction along which the wave vector \(\mathbf{k}\) points [2, 41]. Accordingly, we obtain \[\frac{dx}{dz} = \frac{z}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}\left[\Delta x_{1}\;\frac{ |\psi_{1}|^{2}}{|\psi|^{2}}+\Delta x_{2}\;\frac{|\psi_{2}|^{2}}{|\psi|^{2}}+2x \frac{\text{Re}\left(\psi_{1}\psi_{2}^{*}\right)}{|\psi|^{2}}\right]+\frac{z_{ R}x_{c}}{2k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}\;\frac{\text{Im}\left(\psi_{1} \psi_{2}^{*}\right)}{|\psi|^{2}} \tag{103}\] \[= \frac{z}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}\left[\frac{\Delta x_ {1}\;e^{-(\Delta x_{1})^{2}/2\sigma_{z}^{2}}+\Delta x_{2}\;e^{-(\Delta x_{2})^ {2}/2\sigma_{z}^{2}}+2 xe^{-(x^{2}+x_{c}^{2})/2\sigma_{z}^{2}}\cos\left(\kappa_ {z}x\right)}{e^{-(\Delta x_{1})^{2}/2\sigma_{z}^{2}}+e^{-(\Delta x_{2})^{2}/2 \sigma_{z}^{2}}+2e^{-(x^{2}+x_{c}^{2})/2\sigma_{z}^{2}}\cos\left(\kappa_{z}x \right)}\right]\] \[-\frac{z_{R}x_{c}}{2k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}\left[ \frac{e^{-(x^{2}+x_{c}^{2})/2\sigma_{z}^{2}}\sin\left(\kappa_{z}x\right)}{e^{ -(\Delta x_{1})^{2}/2\sigma_{z}^{2}}+e^{-(\Delta x_{2})^{2}/2\sigma_{z}^{2}}+2 e^{-(x^{2}+x_{c}^{2})/2\sigma_{z}^{2}}\cos\left(\kappa_{z}x\right)}\right],\] \[\frac{dx}{dz} = \frac{z}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}\;y, \tag{104}\]  
where the last two terms in (103) depend on the effective spatial frequency \(\kappa_{z}\), as (92). It can be noticed from Equation (103) that when the wave function \(\psi\) is formed by only one beam, it reduces to the same expression we had for a single Gaussian beam. The presence of both beams, however, gives rise to a rather complex evolution, as described in Reference [42], which is highly nonlinear [43] and does not allow for analytical solutions.