femtoseconds and wavelength of the order of $1000$ nanometers, and $\frac{\tau[{\rm fs}]}{\lambda[{\rm nm}]}\approx 300$, this is justified. The time scale $\tau$ enters our calculation only as a measure of the interaction time (cf. below).

Let us emphasize that both invariants  vanish for a single Gaussian laser beam, modeled by one of the field configurations labeled by $\pm$ in [Eq. ]{}. However, nonzero invariants are attainable by superimposing multiple, e.g., two, Gaussian beams. Note that macroscopic, non-vanishing invariants could also be realized by a single laser beam if higher laser modes are utilized. However, in this case the focus area is increased in comparison to the ${\rm TEM}_{00}$ mode and correspondingly the available laser intensity diminished. Another option is to consider a single Gaussian beam in the limit of a substantial beam divergence $\theta\to\frac{\pi}{2}$ [@Monden:2011], such that wavefront curvature effects become dominant and cannot be neglected; cf. also [@Paredes:2014oxa]. Of course, under theses circumstances the laser beam does no longer correspond to a slight modification of a plane-wave like electromagnetic field configuration and both invariants  can assume nonzero values, facilitating stimulated photon emission from the vacuum.

At least one invariant can be rendered nonzero by superimposing the two counter-propagating laser beams 