$\omega_0|\Phi|\ll 1$ (see Eq. (\[Shift\])). This observation allows one to expand the four-vector potential in Eq. (\[M\_f\]) as[^5] $$\begin{aligned}
A^{\mu}(\phi+\Phi+\phi_j)\approx &A^{\mu}(\phi)-2E^{\mu}(\phi)\bigg(\Phi+\sum_{i=1}^j\delta\kappa_{i,X}s_i\bigg)\\
A^{\mu}(\phi+\Phi+\phi'_j)\approx &A^{\mu}(\phi)-2E^{\mu}(\phi)\bigg(\Phi+\sum_{i=1}^{j-1}\delta\kappa_{i,X}s_i+\delta\kappa_{j,X}s'_j\bigg),\end{aligned}$$ where $E^{\mu}(\phi)=-dA^{\mu}(\phi)/d\phi$ (note that $E^{\mu}(\phi)$ is not a four-vector). Analogously, one obtains $$\begin{aligned}
\label{Exp_1}
\hat{A}(\phi+\Phi+\phi_{j+1})-\hat{A}(\phi+\Phi+\phi_j)\approx &-2\hat{E}(\phi)\delta\kappa_{j+1,X}s_{j+1}\\
\label{Exp_2}
\begin{split}
\delta\pi_j^{\mu}(\phi+\Phi,s'_j)\approx &\delta \kappa_j^{\mu}+2eE^{\mu}(\phi)\bigg[\sum_{i=1}^{j-1}\delta\kappa_{i,X}s_i+\delta\kappa_{j,X}s'_j\\
&-\frac{1}{S}\sum_{l=1}^Ns_l\bigg(\sum_{i=1}^{l-1}\delta\kappa_{i,X}s_i+\frac{1}{2}\delta\kappa_{l,X}s_l\bigg)\bigg].
\end{split}\end{aligned}$$ Now, the fact that $F_{\phi}(\phi+\Phi,\{s\})\lesssim 1$ implies, as an order-of-magnitude estimate, that $[\delta\bm{\pi}_{j,\perp}(\phi+\Phi,s_j)]^2s_j\lesssim 1/N$. Thus, the above expansions, together with the fact that $|\bm{p}_{\perp}|\sim 1/\sqrt{S}$ (see Eq. (\[Sub\_perp\])), indicate that in the effective formation region of the process, the ratio between the terms in the pre-exponent proportional to the laser field and those which do not contain the laser field itself is less than unity. Therefore, terms containing higher powers of the external field are subdominant and, in conclusion, the probability of an exchange of a net number of photons much larger than unity is suppressed also for large values of $N$.

In order to make our analysis more concrete, we consider the particular case of a monochromatic, circularly polarized laser field. In this case, the vector potential is given by $\bm{A}(\phi)=-(E_0/\omega_0)[\cos(\omega_0\phi)\bm{a}_1+\sin(\omega_0\phi)\bm{a}_2]$. Starting again from the general expression in Eq. (\[M\_f\]) (see also Eq. (\[F\_perp\])), it is convenient to introduce the vectors $$\begin{aligned}
\bm{a}_{j,c}(s'_j)&=C_j(s'_j)\bm{a}_1+S_j(s'_j)\bm{a}_2\\
