= Solution
For a sinusoidal laser field of peak amplitude $E_0$, an electron's peak nonrelativistic quiver speed is $v_q=eE_0/(m_e\omega)$. Choose the one-component electron thermal speed $v_T=\sqrt{k_BT/m_e}=1.33\times10^7\,\mathrm{m\,s^{-1}}$. The vacuum-equivalent laser intensity is $I=\epsilon_0cE_0^2/2$, so the onset $v_q\sim v_T$ occurs at
$$
I_{\rm th}\simeq\frac{\epsilon_0c}{2}\left(\frac{m_e\omega v_T}{e}\right)^2,\qquad \boxed{I_{\rm th}\simeq2.4\times10^{15}\,\mathrm{W\,cm^{-2}}.}
$$
Using $\sqrt{2k_BT/m_e}$ as the thermal-speed convention doubles this estimate; its order of magnitude is the physical content. Once the quiver is comparable to the thermal speed, a collision rate computed for an undriven thermal distribution is no longer adequate.
Relativistic motion begins when the normalized field $a_0=eE_0/(m_ec\omega)$ is of order one. Thus
$$
\boxed{I_{\rm rel}\simeq\frac{\epsilon_0c}{2}\left(\frac{m_ec\omega}{e}\right)^2\simeq1.2\times10^{18}\,\mathrm{W\,cm^{-2}}.}
$$
The ratio $I_{\rm rel}/I_{\rm th}=m_ec^2/(k_BT)\simeq511$ checks the separation of the two regimes. These estimates use peak field and cycle-averaged vacuum intensity. The small refractive-index and interface-transmission corrections in this underdense plasma change only the few-percent prefactor, below the accuracy intended here.
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