## 3 Gauss-Maxwell Beams
### Arbitrarily Polarized Gauss-Maxwell Beams
\tag{74}\]  
As before, from this vector we readily obtain the electric and magnetic vector fields along the \(z\)-axis:  
\[\mathbf{E}(\mathbf{r}_{\perp};z) = E_{0}\psi(\mathbf{r}_{\perp};z)\left\{\left(\alpha-\frac{ \alpha[(\Delta x)^{2}-(\Delta y)^{2}]}{2q_{z}^{2}}-\frac{\beta\Delta x\Delta y }{q_{z}^{2}}\right)\mathbf{\hat{x}}\right. \tag{75}\] \[\left.\qquad+\left(\beta+\frac{\beta[(\Delta x)^{2}-(\Delta y)^{ 2}]}{2q_{z}^{2}}-\frac{\alpha\Delta x\Delta y}{q_{z}^{2}}\right)\mathbf{\hat{y }}-\frac{\alpha\Delta x+\beta\Delta y}{q_{z}}\mathbf{\hat{z}}\right\},\] \[\mathbf{H}(\mathbf{r}_{\perp};z) = H_{0}\psi(\mathbf{r}_{\perp};z)\left\{-\left[\beta-\frac{\beta [(\Delta x)^{2}-(\Delta y)^{2}]}{2q_{z}^{2}}+\frac{\alpha\Delta x\Delta y}{q_ {z}^{2}}\right]\mathbf{\hat{x}}\right.\] (76) \[\left.\qquad+\left[\alpha+\frac{\alpha[(\Delta x)^{2}-(\Delta y) ^{2}]}{2q_{z}^{2}}+\frac{\beta\Delta x\Delta y}{q_{z}^{2}}\right]\mathbf{\hat{ y}}+\frac{\beta\Delta x-\alpha\Delta y}{q_{z}}\mathbf{\hat{z}}\right\}.\]  
As seen above, dealing with these full expressions does not lead to any important difference with respect to retaining only up to the first order in the transverse displacement, \(\Delta\mathbf{r}_{\perp}\), and, in the present case, may lead to a rather complex expression for the time-averaged Poynting vector (which does not imply any significant advantage both at the conceptual level and at the methodological one). Therefore, let us thus keep terms at the lowest level in \(\Delta\mathbf{r}_{\perp}\), which reduces the six-component electromagnetic vector field to  
\[\mathbf{F}(\mathbf{r}_{\perp};z)=\sqrt{\frac{\epsilon_{0}}{2}}\;E_{0}\psi( \mathbf{r}_{\perp};z)\left( \begin{array}{c}\alpha\\ \beta\\ -(\alpha\Delta x+\beta\Delta y)/q_{z}\\ -\beta\\ \alpha\\ (\beta\Delta x-\alpha\Delta y)/q_{z}\end{array} \right), \tag{77}\]  
and the electric and magnetic fields to  
\[\mathbf{E}(\mathbf{r}_{\perp};z) = E_{0}\psi(\mathbf{r}_{\perp};z)\left[\alpha\mathbf{\hat{x}}+ \beta\mathbf{\hat{y}}+\frac{\alpha\Delta x+\beta\Delta y}{2k\sigma_{0}\sigma_ {z}}\;e^{-i(\sigma_{2}+\pi/2)}\mathbf{\hat{z}}\right], \tag{78}\] \[\mathbf{H}(\mathbf{r}_{\perp};z) = H_{0}\psi(\mathbf{r}_{\perp};z)\left[-\beta\mathbf{\hat{x}}+ \alpha\mathbf{\hat{y}}+\frac{\beta\Delta x-\alpha\Delta y}{2k\sigma_{0}\sigma_ {z}}\;e^{-i(\sigma_{2}-\pi/2)}\mathbf{\hat{z}}\right]. \tag{79}\]  
In order to determine the flux of associated rays, we first compute the time-averaged Poynting vector from the fields (78) and (79), which reads as \[\mathbf{S} = \frac{1}{2}\;E_{0}H_{0}|\psi|^{2}\left( \begin{array}{c}\frac{ \Delta x}{2k\sigma_{0}\sigma_{z}}\;\sin\varphi_{z}+\mathrm{Im}(\alpha\beta^{*}) \;\frac{\Delta y}{k\sigma_{0}\sigma_{z}}\;\cos\varphi_{z}\\ \\ \frac{z\Delta y}{2k\sigma_{0}\sigma_{z}}\;\sin\varphi_{z}-\mathrm{Im}(\alpha \beta^{*})\;\frac{\Delta x}{k\sigma_{0}\sigma_{z}^{2}}\;\cos\varphi_{z}\\ \\ 1\end{array} \right) \tag{80}\] \[= \frac{1}{2}\;E_{0}H_{0}|\psi|^{2}\left( \begin{array}{c}\frac{ z\Delta x}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}-\sin\theta\sin(\phi+\delta)\; \frac{z_{R}\Delta y}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}\\ \\ \frac{z\Delta y}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}+\sin\theta\sin(\phi+ \delta)\;\frac{z_{R}\Delta x}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}\\ \\ 1\end{array} \right),\]  
with  
\[\cos\varphi_{z}=\frac{z_{R}}{2k\sigma_{0}\sigma_{z}}. \tag{81}\]  
From (80), we obtain the following set of ray equations  
\[\frac{d\Delta x}{dz} = \frac{z\Delta x}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}-\sin\theta \sin(\phi+\delta)\;\frac{z_{R}\Delta y}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}, \tag{82}\] \[\frac{d\Delta y}{dz} = \frac{z\Delta y}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}+\sin\theta \sin(\phi+\delta)\;\frac{z_{R}\Delta x}{4k^{2}\sigma_{0}^{2}\sigma_{z}^{2}}, \tag{83}\]  
which are coupled by virtue of the polarization factor \(\sin\theta\sin(\phi+\delta)\).