 Solution

The laser driver should occupy at most roughly half a plasma period to excite the accelerating density response effectively. Interpreting pulse length as full significant spatial duration gives $\boxed{{\ell }_{\max }\sim {\lambda }_p/2\simeq 8.35\mu \mathrm{m},{\tau }_{\max }\sim 27.8\mathrm{fs}}$. If the source's $\sigma$ denotes the rms width of a Gaussian envelope instead, the resonant scale is commonly $k_p\sigma \sim 1$, or $\sigma \sim 2.66\mu$m and $8.9$ fs; full duration and rms width cannot be assigned the same numerical value without a pulse-shape convention. The robust condition is a sub-plasma-period pulse.