\bm{a}_{j,s}(s'_j)&=-S_j(s'_j)\bm{a}_1+C_j(s'_j)\bm{a}_2,\end{aligned}$$ where $C_j(s'_j)=\cos(\omega_0(\Phi+\phi'_j))$ and $S_j(s'_j)=\sin(\omega_0(\Phi+\phi'_j))$. In this way, we obtain $$\delta\bm{\pi}_{j,\perp}(\phi+\Phi,s'_j)=\delta\bm{\kappa}_{j,\perp}-m\xi[\cos(\omega_0\phi)\delta\bm{a}_{j,c}(s'_j)+\sin(\omega_0\phi)\delta\bm{a}_{j,s}(s'_j)]$$ and the function $F_{\phi}(\phi+\Phi,\{s\})$ can be written as $$F_{\phi}(\phi+\Phi,\{s\})=F_0(\{s\})+F_c(\{s\})\cos(\omega_0\phi)+F_s(\{s\})\sin(\omega_0\phi),$$ where $$\begin{aligned}
F_0(\{s\})=&\sum_{j=1}^N\int_0^{s_j}ds'_j\bm{(}(\delta\bm{\kappa}_{j,\perp})^2+m^2\{1+\xi^2[(\delta C_j(s'_j))^2+(\delta S_j(s'_j))^2]\}-i\epsilon\bm{)},\\
F_{c/s}(\{s\})=&-2m\xi\sum_{j=1}^N\int_0^{s_j}ds'_j\delta\bm{\kappa}_{j,\perp}\cdot\delta\bm{a}_{j,c/s}(s'_j).\end{aligned}$$ Note that the integrals in $ds'_j$ in $F_0(\{s\})$ and $F_{c/s}(\{s\})$ can be easily taken in the present case, which is however not necessary here. The discussion below Eq. (\[F\_perp\]) indicates that in the effective integration region it is $F_0(\{s\}),|F_{c/s}(\{s\})|\lesssim 1$. We consider now the prototype integral in $\phi$ $$\label{I}
\mathcal{I}(\{s\})=\int d\phi\, \text{e}^{-i[K_{\phi}\phi+F_{\phi}(\phi+\Phi,\{s\})]},$$ which is present in the partial amplitude $M^{(1)}$. After introducing the quantities $F_A(\{s\})$ and $\varphi_0(\{s\})$ according to the definitions $$\begin{aligned}
F_c(\{s\})&=F_A(\{s\})\cos(\varphi_0(\{s\})),\\
F_s(\{s\})&=F_A(\{s\})\sin(\varphi_0(\{s\})),\end{aligned}$$ and after passing to the variable $\varphi=\omega_0\phi-\varphi_0(\{s\})$, we obtain $$\label{I_f}
\mathcal{I}(\{s\})=2\pi \text{e}^{-i[(K_{\phi}/\omega_0)\varphi_0(\{s\})+F_0 (\{s\})]}\sum_{n_l=-\infty}^{\infty}i^{-n_l}\delta(K_{\phi}-n_l\omega_0)J_{n_l}(F_A(\{s\})),$$ where we employed the identity $\exp(iz\cos\varphi)=\sum_{n=-\infty}^{\infty}i^nJ_n(z)\exp(in\varphi)$ in terms of the ordinary Bessel functions $J_n(z)$ of integer order $n$, valid for an arbitrary complex number $z$ [@Gradshteyn_b_2000]. Equation (\[I\_f\]) shows that $n_l$ indicates the net number of photons absorbed from (if $n_l<0$) or ceded to (if $n_l>0$) the laser field. The well-known property of ordinary Bessel functions $J_n(x)$ of a real (positive) argument of being much smaller than unity at $n\gg x$ and the fact that $F_A(\{s\})=\sqrt{F^2_c(\{s\})+F^2_s(\{s\})}\lesssim 1$ shows, at least for the terms in the pre-exponent independent 