is instructive to also provide the total number of photons of frequency $\Omega$ emitted into directions outside the laser beam, to be denoted by $N^\Omega_{{\rm tot},>\theta}$ (cf. Fig. \[fig:Ntotal\]) and given by $$N^\Omega_{{\rm tot},>\theta}\approx 9.84\cdot10^{6}\left(\frac{W[J]}{\lambda[{\rm nm}]}\right)^3\!\left(\frac{1}{\tau[{\rm fs}]}\right)^2. \label{eq:res2}$$

As the laser field is polarized along $\vec{e}_{\rm x}$, it is particularly interesting to ask for the number of emitted photons with perpendicular polarization, fulfilling $\vec{e}_{\perp,\beta}\cdot\vec{e}_{\rm x}=0$ $\leftrightarrow$ $\beta=\arctan(\cot\varphi\cos\vartheta)$; cf. Eqs.  and above. Hence, to project out the emitted photons with polarization vector in the $\rm y$-$\rm z$ plane, the angle parameter $\beta$ has to be adjusted as a function of the emission direction parameterized by the angles $\varphi$ and $\vartheta$. With regard to Eqs.  and it is helpful to note that $\sin(2\arctan\chi)=\frac{2\chi}{1+\chi^2}$, while $\cos(2\varphi-2l\arctan\chi)=\frac{1-\chi^2}{1+\chi^2}\cos(2\varphi)+2l\frac{\chi}{1+\chi^2}\sin(2\varphi)$ for $l=\pm1$.

Defining $h_{\perp}^\omega(u,\varphi)\equiv h_{(1)}^\omega(u,\varphi)\big|_{\beta=\arctan(\cot\varphi\cos\vartheta)}$, the directional emission characteristics for photons with polarization vector perpendicular to the polarization direction of the laser $\vec{e}_{\rm x}$ are described by $h_{\perp}^\omega(\cos\vartheta,\varphi)\sin\vartheta$. We depict them in Fig. \[fig:Nperpx\].

The number of frequency $\Omega$ photons polarized perpendicular to $\vec{e}_x$ and emitted in the solid angle interval parameterized by $u_1\leq u\leq u_2$ and $\varphi_1\leq\varphi\leq\varphi_2$ is obtained by integration of $h_{\perp}^\omega(u,\varphi)$ (cf. Sec. \[sec:calculation\] above). Integrating over the full solid angle results in $N^\Omega_{\perp}$, 