the total numbers of frequency $\Omega$ photons emitted in all directions, $N^\Omega_{\rm tot}$, and in all directions outside the laser beam, $N^\Omega_{{\rm tot},>\theta}$, we provide the numbers of emitted photons polarized perpendicular to the initial laser field, $N^\Omega_\perp$ and $N^\Omega_{\perp,>\theta}$.[]{data-label="tab:lasers"}

However, let us emphasize that only those photons emitted in the $\rm y$-$\rm z$ plane ($\vartheta=\frac{\pi}{2}$) can be polarized in the same direction as the original laser beam. Only here, the polarization vectors which live in the tangent space of the unit sphere \[cf. [Eq. ]{}\] can point in the $\vec{e}_{\rm x}$ direction. With regard to the total number of emitted photons, these photons amount to a negligible fraction: This becomes particularly obvious when looking at the directional emission characteristics for the total number of photons depicted in Fig. \[fig:Ntotal\] (left). All photons that might have their polarization vector in the same direction as the original laser field lie on the intersection of the $\rm y$-$\rm z$ plane with the three dimensional emission characteristics. Clearly, their contribution to the integral  yielding the number of emitted photon number in three dimensions is negligible as it is to be multiplied with ${\rm d}\varphi\to0$ when performing the integration over any solid angle interval. 