plane-wave type modulation of frequency $\Omega>0$; wavelength $\lambda=\frac{2\pi}{\Omega}$. Without loss of generality, the beam waist is assumed to be located at $z=0$, such that the Gaussian envelope $\sim{\rm exp}(-{\rm z}^2/\zeta^2)$ can be seen as mimicking the decrease of the field over the Rayleigh range $z_{\rm R}$ which is of the order of $\zeta$. Note that for real Gaussian beams (for ${\rm x}={\rm y}=0$) the field decrease over the Rayleigh range is described by a Lorentzian profile, which is harder to tackle analytically and thus, would result in less transparent and handy expressions for the vacuum emission probability. We argue that for our purposes the Gaussian profile captures all relevant features, and – when providing experimental estimates below – will actually identify $\zeta=z_{\rm R}$. Moreover, we neglect diffraction spreading and wavefront curvature effects about the beam waist, arguing that within the Rayleigh range they amount to subleading corrections. Outside the Rayleigh range the fields  rapidly drop to zero. High-intensity lasers deliver multicycle pulses of finite duration $\tau$. The pulse duration is also not accounted for explicitly here. Given that $\tau\gg\lambda$, which is typically fulfilled for near infrared high-intensity lasers (cf. also Tab. \[tab:lasers\], below) whose pulse duration is $\gtrsim$ tens of 