## 3 Gauss-Maxwell Beams
### Linearly Polarized Gauss-Maxwell Beams
\tag{53}\]  
According to the property (37), the only surviving terms in the second sum of (52) are those with \(m\leq 2\), hence such an expression can be further simplified to  
\[e^{\alpha_{z}\left(\Delta r_{\perp}^{2}\mathbb{I}+2k^{-1}\Delta r _{\perp}\cdot\mathbb{G}_{\perp}\right)} = \sum_{n=0}^{\infty}\frac{\alpha_{z}^{n}}{n!}\left[\Delta r_{ \perp}^{2n}\mathbb{I}+\left(\frac{2}{k}\right)n\Delta r_{\perp}^{2(n-1)}\left( \Delta r_{\perp}\cdot\mathbb{G}_{\perp}\right)\right. \tag{54}\] \[\left.\qquad\qquad+\left(\frac{4}{k^{2}}\right)n(n-1)\Delta r_{ \perp}^{2(n-2)}\left(\Delta r_{\perp}\cdot\mathbb{G}_{\perp}\right)^{2} \right],\]  
which can be recast in a more compact form as  
\[e^{\alpha_{z}\left(\Delta r_{\perp}^{2}\mathbb{I}+2k^{-1}\Delta r _{\perp}\cdot\mathbb{G}_{\perp}\right)} = \left(\sum_{n=0}^{\infty}\frac{\alpha_{z}^{n}}{n!}\Delta r_{\perp }^{2n}\right)\left[\mathbb{I}+\alpha_{z}\left(\frac{2}{k}\right)\left(\Delta r _{\perp}\cdot\mathbb{G}_{\perp}\right)+\alpha_{z}^{2}\left(\frac{4}{k^{2}} \right)\left(\Delta r_{\perp}\cdot\mathbb{G}_{\perp}\right)^{2}\right] \tag{55}\] \[= e^{\alpha_{z}\left\|\Delta r_{\perp}\right\|^{2}}\left[\mathbb{I} +\frac{i}{q_{z}}\left(\Delta r_{\perp}\cdot\mathbb{G}_{\perp}\right)-\frac{1}{ q_{z}^{2}}\left(\Delta r_{\perp}\cdot\mathbb{G}_{\perp}\right)^{2}\right],\]  
where the prefactor corresponds to the Gaussian function describing the beam (12), while the matrix terms provide us with the correct expression for the electric and magnetic fields, according to the rule (32). After evaluating the combined action of the \(\mathbb{G}\)-matrices and their products on the column vector (47), we obtain the expression for the output vector field \(\mathbf{F}\) at a distance \(z\),  
\[\mathbf{F}(\mathbf{r}_{\perp};z)=\sqrt{\frac{\epsilon_{0}}{2}}\;E_{0}\psi( \mathbf{r}_{\perp};z)\left( \begin{array}{c}1-[(\Delta x)^{2}-(\Delta y)^{2} ]/2q_{z}^{2}\\ -\Delta x\Delta y/q_{z}^{2}\\ -\Delta x/q_{z}\\ -\Delta x\Delta y/q_{z}^{2}\\ 1+[(\Delta x)^{2}-(\Delta y)^{2}]/2q_{z}^{2}\\ -\Delta y/q_{z}\end{array} \right), \tag{56}\]  
or, in a more conventional fashion, in terms of the separate electric and magnetic components, as  
\[\mathbf{E}(\mathbf{r}_{\perp};z) = E_{0}\psi(\mathbf{r}_{\perp};z)\left\{\left[1-\frac{(\Delta x)^ {2}-(\Delta y)^{2}}{2q_{z}^{2}}\right]\hat{\mathbf{x}}-\frac{\Delta x\Delta y }{q_{z}^{2}}\;\hat{\mathbf{y}}-\frac{\Delta x}{q_{z}}\;\hat{\mathbf{z}}\right\}, \tag{57}\] \[\mathbf{H}(\mathbf{r}_{\perp};z) = H_{0}\psi(\mathbf{r}_{\perp};z)\left\{-\frac{\Delta x\Delta y}{q_ {z}^{2}}\;\hat{\mathbf{x}}+\left[1+\frac{(\Delta x)^{2}-(\Delta y)^{2}}{2q_{z} ^{2}}\right]\hat{\mathbf{y}}-\frac{\Delta y}{q_{z}}\;\hat{\mathbf{z}}\right\}, \tag{58}\]  
which include up to second-order corrections in the transverse coordinates, with \(\hat{\mathbf{x}}\), \(\hat{\mathbf{y}}\) and \(\hat{\mathbf{z}}\) denoting the unit vectors along the three Cartesian directions. Next, we are going to analyze the physical implications carried by these contributions. Let us firstly start by neglecting the second-order contributions. Accordingly, the field vector (56) reads as  
\[\mathbf{F}(\mathbf{r}_{\perp};z)=\sqrt{\frac{\epsilon_{0}}{2}}\;E_{0}\psi( \mathbf{r}_{\perp};z)\left( \begin{array}{c}1\\ 0\\ -\Delta x/q_{z}\\ 0\\ 1\\ -\Delta y/q_{z}\end{array} \right).