# Wave equation with time dependent electric field amplitude in paraxial approximation

I want to reproduce some results from a paper I read (Link to paper). In it, they show for a slowly varying field amplitude $E_0(x,y,z,t)$ Maxwells wave equation, which takes the following form:

$2ik(\frac{\tilde n}{c_0} \partial_t + \partial_z)E_0 = (\nabla^2_\perp +k^2(\tilde n^2-1))E_0$

Here $\tilde n^2-1$ is described as the change in permitivity, $\nabla_\perp = \partial^2_x+\partial^2_y$, and the wave vector $k$. The paper is not very precise about the different quantities, so I need to reproduce this equation by myself. I started with the wave equation

$(\nabla^2-\frac{n^2(\mathbf{r})}{c^2_0}\partial_t^2)E=0$

and used the ansatz $E=E_0 \exp(i(kz-\omega t))$ where $k=n_0\omega/c_0$ including some mean value $n_0$ for the refractive index. Plugging this into the wave equation and applying the paraxial approximation ($\partial^2_z E_0 \approx0$) reproduces the given result assuming $\tilde n = n(\mathbf r)/n_0$, except the term involving the time derivative ($\propto \partial_t$ in the very first equation). The second term ($\propto \partial^2_t$) from the wave equation leads me to

$\exp(-i(kz-\omega t))\frac{n^2(\mathbf{r})}{c^2_0}\partial_t^2E = \frac{n^2(\mathbf{r})}{c^2_0}[\partial_t^2-2i\omega \partial_t -\omega^2]E_0$

The last term ($\propto \omega^2$) gets mapped to the $\tilde n^2(\mathbf r) k^2E_0$ term in the equation to be reproduced. The first term I can get rid of since $E_0$ is slowly varying (? not 100% sure about it). The term in the middle ($\propto \omega$) is what bothers me the most. I would reproduce the term with the time derivative in the very firts equation if I would use $k=n(\mathbf r)\omega/c_0$ (note $n(\mathbf r)$ instead of $n_0$). But then the term $\propto \omega^2$ will not map any more...

Any hints appreciated.