Note that the FWHM of the intensity $I(t)$, which is proportional to the square of the field $\mathbf E(t)$, is rather $T_{\rm pulse}/3$. The pulse maximum occurs at $t=T_\mathrm{pulse}/2$. Note that the sin$^2$ profile is written here for the laser field amplitude, which means that the time profile of the intensity time has a sin$^4$ shape. Thus far, we have discussed simple one-peak pulses. More flexibility is conceivable. The next important tool are dual pulses as used in pump-and-probe experiments in which the laser irradiation is performed in two steps. We shall illustrate such cases at several places below.

In practice, the effect of the laser field will be accounted for in our calculations as an external potential $U_{\rm ext}(\mathbf{r},t)$ which delivers a time-dependent perturbation. In the long wavelength limit, the electric field is homogeneous and delivers the potential : $$U_{\rm ext}(\mathbf{r},t)
  =
  -e\, E_0\, f(t)\, \sin(\omega_\mathrm{las}t)\, 
  \mathbf{e}_\mathrm{pol}\cdot\mathbf{r} 
  \quad,
\label{eq:lasfield-x}$$ where $f(t)$ is the time profile usually taken according to Eq. (\[eq:cos2\]). This is, in fact, the laser field in space gauge. Equivalently, one can use the velocity gauge for which the laser field is described by the interaction operator : $$\hat{U}_{\rm ext}^{(v)}
  =
  -\frac{e}{c} \, E_0 \, F(t)\, \mathbf{e}_\mathrm{pol}\cdot\hat{\mathbf{p}}
  