 Solution

Emittance reduction has little effect when radiation dominates every source width and divergence. At fixed $\beta =L=2$ m, the strongest condition is $\lambda L/(16{\pi }^2)\gg {ϵ}_x\beta$, or $\boxed{\lambda \gg 4.7\times 10^{-7}\mathrm{m}}$. The divergence condition only requires $\lambda \gg {ϵ}_{x}L/\beta =3$ nm. There is no sharp “negligible” cutoff without an allowed percentage. If the optics were reoptimized to match diffraction instead, the weaker comparison $\lambda /(4\pi )\gg {ϵ}_x$ would set the relevant scale; it is not the same fixed-beta problem.