 Solution

Let $\epsilon =\hslash \omega$ be the incident laser-photon energy and $\gamma =W/(m{c}^2)$. The maximum in the usual soft-laser geometry occurs for a head-on encounter followed by a forward outgoing photon along the electron beam. Boost into the initial electron rest frame: the incident energy becomes ${\epsilon }_{\ast }=\gamma (1+\beta )\epsilon$. Backscattering there gives the Compton scattering recoil relation ${\epsilon }_{\ast ,s}={\epsilon }_{\ast }/[1+2{\epsilon }_{\ast }/(m{c}^2)]$. Boosting this outgoing photon forward to the lab givesThe familiar $4{\gamma }^2\epsilon$ limit additionally requires $\gamma \gg 1$ and ${\epsilon }_{\ast }\ll m{c}^2$. The denominator is the recoil correction, not an optional factor at arbitrary incident energy.

For completeness, if the incidence direction is prescribed rather than optimized, let its unit vector be $\mathbf{n}$ and the outgoing direction ${\mathbf{n}}_s$. Squaring the residual electron four-momentum on its mass shell gives the more general kinematic relationThus an unspecified crossing angle must not silently be identified with head-on incidence; the boxed result is the maximum for the stated head-on soft-photon arrangement.

Numerically $W=2\mathrm{GeV}$ gives $\gamma =3913.89$, and a $1\mathrm{cm}$ photon has energy $1.23984\times 10^{-4}\mathrm{eV}$. The maximum scattered energy is $7.5970\mathrm{keV}$, givingThe rest-frame incident photon is only about one electronvolt, so the small recoil correction is consistent with the amplification estimate.