= Solution
In the cold unmagnetized linear plasma model, the refractive-index relation is $n^2=1-\omega_p^2/\omega_L^2$, with
$$
\omega_p^2=\frac{n_ee^2}{\epsilon_0m_e}.
$$
The critical density occurs at $\omega_p=\omega_L=2\pi c/\lambda_L$. Thus
$$
\boxed{n_{\rm crit}=\frac{\epsilon_0m_e}{e^2}\left(\frac{2\pi c}{\lambda_L}\right)^2
\simeq1.12\times10^{27}\,\mathrm{m^{-3}}=1.12\times10^{21}\,\mathrm{cm^{-3}},}
$$
for $\lambda_L=1\,\mu\mathrm m$. The corresponding frequency is
$$
\boxed{\omega_p\simeq1.88\times10^{15}\,\mathrm{rad\,s^{-1}},\qquad
f_p=\omega_p/(2\pi)\simeq3.0\times10^{14}\,\mathrm{Hz}.}
$$
A propagating conventional laser wake generally uses an underdense plasma, rather than this cutoff density. Strong relativistic laser motion, collisions or a magnetized plasma modify this simple critical-density condition.
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