the average number of photons absorbed from the laser by the electron is of the order of $U_p/\omega_0$, where $U_p=e^2E_0^2/m\omega_0^2$ is its ponderomotive energy and $\omega_0$ is the central laser photon energy. HHG has also been observed for free electrons driven by an intense laser beam, being named nonlinear Thomson or nonlinear Compton scattering, depending on if quantum effects are negligible or not [@Moore_1995; @Bula_1996]. In both nonlinear Thomson and Compton scattering, the typical electric field strength, at which nonlinear effects set on, is given by $E_{\text{rel}}=m\omega_0/|e|$. The corresponding intensity is of the order of $10^{18}\;\text{W/cm$^2$}$ at optical photon energies $\omega_0\approx 1\;\text{eV}$. An electron in a laser field with central laser photon energy $\omega_0$ and electric field strength of the order of $E_{\text{rel}}$ is accelerated to relativistic velocities already within one laser period and its dynamic becomes highly nonlinear with respect to the laser field amplitude [@Landau_b_2_1975]. On the other hand, quantum effects such as the recoil of the photons emitted by the laser-driven electron, strongly modify the emission process when the electric field strength of the laser in the initial rest frame of the incoming electron is of the order of the so-called critical field $E_{\text{cr}}=m^2/|e|$ of QED, corresponding 