number of frequency $\omega$ photons originating from the stimulated emission process emitted in this solid angle interval ($\Delta u=u_1-u_2$, $\Delta\varphi=\varphi_2-\varphi_1$) is given by $$N^{\omega}(\Delta u,\Delta\varphi)=\left(\frac{e{\cal E}}{m^2}\right)^6 \left(\frac{w}{2}m\right)^4\frac{\Omega \tau}{2\pi}\,\int_{u_2}^{u_1}{\rm d}u\int_{\varphi_1}^{\varphi_2}{\rm d}\varphi\, h^{\omega}(u,\varphi)\,, \label{eq:Nsum}$$ with $\omega\in\{\Omega,3\Omega\}$. Obviously, the $\varphi$ integration in [Eq. ]{} is trivial. Also the $u$ integration can easily be performed analytically and the result be written in terms of exponential and error functions. As these results are rather lengthy and do not allow for any additional insights we do not represent them here.

Analogously, the number of emitted photons polarized in mode $p=1$ is obtained by $$N_{(p)}^{\omega}(\Delta u,\Delta\varphi)=\left(\frac{e{\cal E}}{m^2}\right)^6 \left(\frac{w}{2}m\right)^4\frac{\Omega \tau}{2\pi}\,\int_{u_2}^{u_1}{\rm d}u\int_{\varphi_1}^{\varphi_2}{\rm d}\varphi\, h_{(p)}^{\omega}(u,\varphi)\,. \label{eq:Np}$$ As before, the result for the $p=2$ mode follows upon substitution of $\beta\to\beta+\frac{\pi}{2}$. If the angle parameter $\beta$ is chosen independent of the values of $\varphi$ and $\vartheta$ both integrations can again be performed analytically as for [Eq. ]{}. However, note that the integrations over the solid angle interval can be significantly complicated if $\beta=\beta(\vartheta,\varphi)$ as is, e.g., necessary if we are interested in all photons polarized perpendicular to $\vec{e}_{\rm x}$; cf. Sec. \[seq:Ex+Res\] below.

To maximize the effect of stimulated photon emission, the laser field strength $\cal E$ is preferably rendered as large 