\quad \hat f^{31}_{(2)}(k)=\hat f^{31}_{(1)}(k)\big|_{\beta\to\beta+\frac{\pi}{2}} ,\end{gathered}$$ where we made use of the shorthand notation $u\equiv\cos\vartheta$.

According to Fermi’s golden rule, the number of induced photons with polarization $p\in\{1,2\}$ and momentum ${\rm k}$ in the interval ${\rm dk}$ emitted in the solid angle interval ${\rm d}u {\rm d}\varphi$ is obtained from the modulus squared of [Eq. ]{} as $\frac{1}{(2\pi)^3}\bigl|{\rm k}\,{\cal S}_{(p)}(\vec{k})\bigr|^2 {\rm dk}\,{\rm d}u\,{\rm d}\varphi$.

A straightforward but somewhat tedious calculation yields the following expressions for the modulus squared of ${\rm k}\,{\cal S}_{(p)}(\vec{k})$, $$\begin{gathered}
\bigl|{\rm k}\,{\cal S}_{(1)}(u,\varphi)\bigr|^2
 =\frac{\pi^2\,\alpha}{218700}\left(\frac{e{\cal E}}{m^2}\right)^6 \left(\frac{w}{2}m\right)^4\,{\rm k}^3\,\zeta^2\,{\rm e}^{-\frac{\zeta^2\Omega^2+(\frac{w}{2})^2{\rm k}^2}{6}}\,{\rm e}^{-\frac{[\zeta^2-(\frac{w}{2})^2]{\rm k}^2}{6}u^2}  \\
 \times\sum_{n=\pm1}\sum_{l=\pm1}\biggl\{\biggl[
 \sum_{j=0,2} c_j\,{\rm e}^{+\tfrac{lj}{3}\zeta^2{\rm k}\Omega u}(1-u^2)\bigl[\cos(2\beta) + \cos(2\varphi)\bigr] \\
 +\sum_{j=1,3}c_j\,{\rm e}^{+\tfrac{lj}{3}\zeta^2{\rm k}\Omega u}(1-unl)^2[1+\cos(2\varphi-2nl\beta)]  \\
 +{\rm e}^{+\tfrac{l}{3}\zeta^2{\rm k}\Omega u}\,8nl\Bigl[2u[1+\cos(2\varphi)\cos(2\beta)]-(1+u^2) \sin(2\varphi)\sin(2\beta)\Bigr]\biggr]\delta^2({\rm k}-n\Omega)  \\
+\biggl[(1-u^2)\bigl[\cos(2\beta) + \cos(2\varphi)\bigr]\\
+{\rm e}^{+\tfrac{l}{3}\zeta^2{\rm k}\Omega u}(1-unl)^2[1+\cos(2\varphi-2nl\beta)] \biggr]\,\delta^2({\rm k}-3n\Omega) 
     \biggr\}, \label{eq:S^2}\end{gathered}$$ and $\bigl|{\rm k}\,{\cal S}_{(2)}(u,\varphi)\bigr|^2=\bigl|{\rm k}\,{\cal S}_{(1)}(u,\varphi)\bigr|^2\big|_{\beta\to\beta+\frac{\pi}{2}}$, with $\alpha=\frac{e^2}{4\pi}$ and coefficients $$c_0=4+{\rm e}^{-\frac{4}{3}\zeta^2\Omega^2}, \quad c_1=4+4{\rm e}^{-\frac{2}{3}\zeta^2\Omega^2}\,, \quad c_2= 4{\rm e}^{-\frac{2}{3}\zeta^2\Omega^2}, \quad c_3={\rm e}^{-\frac{4}{3}\zeta^2\Omega^2}.$$ Evidently, only photons with the two distinct frequencies $\omega\in\{\Omega,3\Omega\}$ are induced. This is in agreement with elementary physical reasoning: In a Feynman diagrammatic expansion of the effective Lagrangian , the leading terms  taken into account by us actually amount to an effective four-photon interaction. Our electromagnetic background field configuration  modeling the counter-propagating 