## 3 Gauss-Maxwell Beams
### General Considerations on the Propagation Procedure
\begin{array}{cc} \mathbb{S}_{1}&\mathbb{S}_{2}\\ -\mathbb{S}_{2}&\mathbb{S}_{1}\\ \end{array} \right), \tag{33}\]  
with  
\[\mathbb{S}_{1}=\left( \begin{array}{ccc}0&0&0\\ 0&0&-i\\ 0&i&0\\ \end{array} \right),\qquad\mathbb{S}_{2}=\left( \begin{array}{ccc}0&0&i\\ 0&0&0\\ -i&0&0\\ \end{array} \right). \tag{34}\]  
The \(\mathbb{G}\)-matrices satisfy the properties:\[\left[\mathbb{G}_{a},\mathbb{G}_{b}\right] = 0, \tag{35}\] \[\sum_{a}\mathbb{G}_{a}\mathbb{G}_{a} = 0,\] (36) \[\mathbb{G}_{a}\mathbb{G}_{b}\mathbb{G}_{c} = 0, \tag{37}\]  
for \(a,b,c=x,y\), which can be proven taking into account the following matrix relations:  
\[\mathbb{S}_{1}^{2n} = \left( \begin{array}{ccc}0&0&0\\ 0&1&0\\ 0&0&1\end{array} \right)=\mathbb{S}_{1}^{2},\qquad\mathbb{S}_{1}^{2n-1}= \mathbb{S}_{1}, \tag{38}\] \[\mathbb{S}_{2}^{2n} = \left( \begin{array}{ccc}1&0&0\\ 0&0&0\\ 0&0&1\end{array} \right)=\mathbb{S}_{2}^{2},\qquad\mathbb{S}_{2}^{2n-1}= \mathbb{S}_{2},\] (39) \[\mathbb{S}_{2}^{2}-\mathbb{S}_{1}^{2}=\left( \begin{array}{ccc}1 &0&0\\ 0&-1&0\\ 0&0&0\end{array} \right),\] (40) \[\left[\mathbb{S}_{1},\mathbb{S}_{2}\right]_{+}=\left( \begin{array} []{ccc}0&-1&0\\ -1&0&0\\ 0&0&0\end{array} \right),\] (41) \[\left[\mathbb{S}_{1},\left[\mathbb{S}_{1},\mathbb{S}_{2}\right]_{+ }\right]_{+}=\mathbb{S}_{2},\] (42) \[\left[\mathbb{S}_{2},\left[\mathbb{S}_{2},\mathbb{S}_{1}\right]_{+ }\right]_{+}=\mathbb{S}_{1},\] (43) \[\mathbb{S}_{1}\mathbb{S}_{2}\mathbb{S}_{1}=\mathbb{S}_{2}\mathbb{ S}_{1}\mathbb{S}_{2}=0, \tag{44}\]  
with \(n\geq 1\) and where \(\left[\cdot,\cdot\right]_{+}\) denotes the anticommutator. Note that the action of \(\underline{\hat{\mathbb{U}}}\) on the vector field \(\mathbf{F}\) can be accomplished in two steps:  
1. First, there is a space translation, from \(\mathbf{r}_{\perp}\mathbb{I}\) to \(\mathbb{R}_{\perp}\), by an effective amount \(k^{-1}\mathbb{G}_{\perp}\), proportional to \(\lambda\). 2. Then, the beam undergoes a boost, accounted for by the action of the momentum operator \(\hat{\mathbf{p}}_{\perp}\mathbb{I}\), while it is freely propagating along the \(z\)-direction. The above two-step prescription allows us to determine in a relatively simple fashion the evolution (along \(z\)) of Gauss-Maxwell beams (or any linear combination of them) once the input amplitude, \(\mathbf{F}(\mathbf{r}_{\perp};0)\), is known. Monitoring this evolution with rays, which are in compliance with the paraxial form of Maxwell's equations, can be done now with the aid of the time-averaged Poynting vector, since  
\[\mathbf{S}=\langle\mathcal{E}(\mathbf{r},t)\times\mathcal{H}(\mathbf{r},t) \rangle_{T}=\frac{1}{2}\text{Re}\left\{\mathbf{E}\times\mathbf{H}^{*}\right\}, \tag{45}\]  
where \(\mathcal{E}(\mathbf{r},t)\) and \(\mathcal{H}(\mathbf{r},t)\) denote, respectively, the electric and magnetic monochromatic vector fields, solutions to the Maxwell equations, and \(\langle\quad\rangle_{T}\) is the average over the period of the radiation. Notice that this averaging allows us to also recast the time-averaged Poynting vector just in terms of the time-independent amplitudes \(\mathbf{E}(\mathbf{r})\) and \(\mathbf{H}(\mathbf{r})\), which arise from the generalization of \(\mathcal{E}(\mathbf{r},t)\) and \(\mathcal{H}(\mathbf{r},t)\), respectively, to the complex domain [2], but that here are directly determined from the output beam \(\mathbf{F}(\mathbf{r}_{\perp};z)\), at a distance \(z\) from the input (transverse) plane. In this case, the guidance equation is defined as  
\[\frac{d\mathbf{r}_{\perp}}{dz}=\frac{\mathbf{S}_{\perp}}{S_{z}}, \tag{46}\]  
where \(\mathbf{S}_{\perp}\) and \(S_{z}\) are, respectively, the transverse and longitudinal components of the time-averaged Poynting vector. As before, Equation (46) corresponds to a phase velocity, with its validity being determined by the fact that we are dealing with vacuum, where the phase velocity and the components of the time-averaged Poynting vector are proportional (in general, for nondispersive media, phase velocity and group velocity point in the same direction [2]).