Instituto de Física, Facultad de Ciencias, Pontifica Universidad Católica de Valparaíso
Darío G. Peréz
An wavefront propagating in free space […] $U_0(\mathbf r, z) = \exp \!(ikz) V(\mathbf r, z)$
$$ \frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial U_0}{\partial r}\right) + \frac{\partial ^2U_0}{ \partial z^2} + k^2 U_0=0, \quad\quad\tag{1} $$
The paraxial approximation consists in considering the propagation distance z through the optical axis is larger than the transversal spread of the beam, ||r|| ≪ z.
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A centered Gaussian beam at z = 0 is defined by
$$ U_0(\mathbf{r},0)=A_0 \exp \left(-\frac{1}{2} \alpha _0 k r^2\right),\quad\tag{6} $$
where
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$$ \gamma = \begin{cases} 1. & \text{plane wave,} \\ \frac{|F_0 -z|}{|F_0-L|}, & \text{spherical wave with focus at }F_0\text{, and}\\ \frac{|L-F_0 -z|}{|F_0-L|},&\text{spherical wave with focus at }L-F_0.\end{cases} $$
Yet, as we will discuss in this manuscript, we need to consider a descentered beam. Therefore, instead of (6), we define
$$ U_0(\mathbf{r},0)=A_0 \exp \left(-\frac{1}{2} \alpha _0 k ||\mathbf{r}-\mathbf{r}_0||^2\right),\quad\tag{7} $$