and just integrating over all directions outside the laser beam in $N^\Omega_{\perp,>\theta}$. Our explicit results are $$\left\{\begin{array}{c}
  \! N^\Omega_{\perp}\\
  \! N^{\Omega}_{\perp,>\theta}
 \end{array}\right\}
 \approx
 \left\{\begin{array}{c}
   1.35\cdot10^{4} \\
   1.18\cdot10^{4}
 \end{array}\right\}\left(\frac{W[J]}{\lambda[{\rm nm}]}\right)^3\!\left(\frac{1}{\tau[{\rm fs}]}\right)^2 . \label{eq:Nperp}$$ Note that these numbers are about a factor of $10^{-3}$ smaller than those for all polarizations given in the first line of [Eq. ]{} and [Eq. ]{}; cf. also Figs. \[fig:Ntotal\] and \[fig:Nperpx\]. In Tab. \[tab:lasers\] we list some explicit estimates for the numbers - of photons of frequency $\Omega$ originating from the stimulated emission process for various present and near future high-intensity laser facilities.

     Laser           $W$\[J\]           $\tau$\[fs\]      $\lambda$\[nm\]     $N^\Omega_{\rm tot}$   $N^\Omega_{{\rm tot},>\theta}$     $N^\Omega_\perp$      $N^\Omega_{\perp,>\theta}$
  ------------ ---------------------- ----------------- -------------------- ---------------------- -------------------------------- ----------------------- ----------------------------
    POLARIS            $150$                $150$              $1030$                $4.04$                      $1.35$                $1.86\cdot10^{-3}$        $1.62\cdot10^{-3}$   
     Vulcan            $500$                $500$              $1054$                $12.6$                      $4.22$                $5.77\cdot10^{-3}$         $5.04\cdot10^{-3}$
    Omega EP           $2000$              $10000$             $1054$                $2.01$                $6.75\cdot10^{-1}$          $9.20\cdot10^{-4}$         $8.08\cdot10^{-4}$
   ELI Prague          $1500$               $150$              $1054$           $3.77\cdot10^3$             $1.26\cdot10^3$                  $1.73$                     $1.51$
     ELI-NP       $2\,\times\,250$          $25$               $800$            $1.15\cdot10^4$             $3.86\cdot10^3$                  $5.28$                     $4.62$
     XCELS       $12\,\times\,400$          $25$               $910$            $6.90\cdot10^6$             $2.32\cdot10^6$              $3.17\cdot10^3$           $2.78\cdot10^3$

  : Numbers of induced photons resulting from the stimulated vacuum emission process for various present and near future high-intensity laser systems, characterized by their pulse energy $W$, pulse duration $\tau$ and wavelength $\lambda$. Apart from 