laser beams is characterized by a single frequency scale $\Omega$. Each coupling to the background field configuration can be seen as coupling to a laser photon of frequency $\Omega$. The stimulated emission process involves three laser photons. Three laser photons can either give rise to a emitted photon of frequency $\Omega$ (two laser photons are scattered into one laser photon and one photon to be emitted) or merge to form a $3\Omega$ photon.

Hence, upon performing the integration over all possible values of ${\rm k}\in[0\ldots\infty)$ it is convenient to decompose the total number density $\rho_{(p)}(u,\varphi) \equiv \frac{1}{(2\pi)^3}\int_0^\infty{\rm dk}\,|{\rm k}\,S_{(p)}(u,\varphi)|^2$ of induced photons polarized in mode $p$ and emitted in $(u,\varphi)$ direction as $$\rho_{(p)}(u,\varphi)=\rho^\Omega_{(p)}(u,\varphi)+\rho^{3\Omega}_{(p)}(u,\varphi),$$ where $\rho^{\,\omega}_{(p)}(u,\varphi)$ refers to the number density of induced frequency $\omega$ photons.

These quantities are obtained straightforwardly from [Eq. ]{}, employing that $\delta^2({\rm k}-\omega)=\frac{\tau}{2\pi}\,\delta({\rm k}-\omega)$, with $\tau$ denoting the time scale of the interaction. Aiming at the number of photons per laser shot originating from the stimulated emission process, we identify this time scale with the laser pulse duration.

For the $p=1$ polarization mode they read $$\rho_{(1)}^{\omega}(u,\varphi)=\left(\frac{e{\cal E}}{m^2}\right)^6 \left(\frac{w}{2}m\right)^4\frac{\Omega \tau}{2\pi}\, h_{(1)}^{\omega}(u,\varphi)\,, \label{eq:rho1}$$ where $$\begin{gathered}
 h_{(1)}^{\Omega}(u,\varphi)=\frac{\alpha}{1749600\pi}(\Omega\zeta)^2\,{\rm e}^{-\frac{[\zeta^2+(\frac{w}{2})^2]\Omega^2}{6}}\,{\rm e}^{-\frac{[\zeta^2-(\frac{w}{2})^2]\Omega^2}{6}u^2}  \\
 \times\sum_{l=\pm1}\biggl\{
 \sum_{j=0,2} c_j\,{\rm e}^{+\tfrac{lj}{3}\zeta^2\Omega^2u}(1-u^2)\bigl[\cos(2\beta) + \cos(2\varphi)\bigr] \\
 +\sum_{j=1,3}c_j\,{\rm e}^{+\tfrac{lj}{3}\zeta^2\Omega^2u}(1-ul)^2[1+\cos(2\varphi-2l\beta)]  