High Energy & High Luminosity $\gamma\gamma$ Colliders

With the best of modern standard lasers, high-energy $\gamma\gamma$ colliders from electron beams of E larger than 250 GeV are only possible at the expense of photon luminosity, i.e. 10 times lower than for photon colliders at c.m. energies below 0.5 TeV. For existing state-of-the art lasers, an optimistic upper energy limit for x=4.8 is an electron beam of less than 250 GeV. This Snowmass21 Contributed Paper shows how Free Electron Lasers (FEL) pave the way for High Energy & High Luminosity $\gamma\gamma$ colliders. We present and assess a conceptual design study of a FEL with wavelength of 2.4 $\mu$m and an x-factor in the range of 2 to 40, to maximize the luminosity of a $\gamma\gamma$ collider as second interaction region of 0.5 TeV to 10 TeV c.m. $e^+e^-$ colliders.