 Solution

For linear polarization, $a_0=e{E}_L/(m_{e}c{\omega }_0)$ and $I={ϵ}_{0}c{E}_L^2/2$. Substitution gives $a_0\simeq 0.855{\lambda }_{\mu \mathrm{m}}\sqrt{I/(10^{18}\mathrm{W}\mathrm{c}{\mathrm{m}}^{-2})}\simeq 0.64$. Thus the specified peak electron quiver motion is at most mildly relativistic, not highly relativistic. Its quiver Lorentz factor is of order $\sqrt{1+a_0^2}\simeq 1.19$. The supplied 40 TW fixes a focal area through $P/I$, but does not increase $a_0$ once the intensity is specified. This exposes a limitation of assuming a strongly nonlinear, fully wavebreaking wake in the preceding estimates.