\quad.
\label{eq:lasfield-v}$$ The rules of gauge transformation relate time profiles and wave functions by : $$\begin{aligned}
  F(t)
  &=&
  \displaystyle \int^t_{-\infty} \mathrm dt'\,f(t') \, \sin(\omega_\mathrm{las}t')
  \quad,
\label{eq:gaugeF}
\\
  \varphi_i^{(v)}(\mathbf{r},t)  
  &=&
  \varphi_i^{\mbox{}}(\mathbf{r},t)  
  \exp{\left[\mathrm{i}\, E_0\, F(t)\, \mathbf{e}_\mathrm{pol}\cdot\mathbf{r}\right]}
  \quad.
\label{eq:gaugepsi}\end{aligned}$$ Both gauges are fully equivalent. Which one is to be preferred is a matter of the actual numerical scheme. Most observables are not even sensitive to gauge. An exception is the evaluation of photoelectron spectra where the phase of the wave function plays a role. In this case, one has to consider gauges carefully. This will be addressed in more detail in Sec. \[sec:observ\_theo\].

### Varying laser characteristics {#sec:las_charac}

As pointed out above, all laser parameters can be tuned in rather large ranges. The point is illustrated in Fig. \[fig:laser-regimes\] which displays typical regions of interest in the intensity-frequency plane.

![Schematic representation of various dynamical regimes as a function of laser intensity $I$ and photon frequency $\omega$. The dashed diagonal line represents frequency-intensity combinations with constant Keldysh parameter $\gamma=1$, see text for details. This line characterizes the transition from photon-dominated to field-dominated regime for an assumed IP of a few eV. The blocks to the right side indicate typical frequency ranges as labeled. The 