Non-linear Compton scattering of ultra-relativistic electrons traversing high-intensity laser pulses generates also hard photons. These photon high-energy tails are considered for parameters in reach at the forthcoming experiments LUXE and E-320. We consider the invariant differential cross sections $d sigma / du$ between the IR and UV regions and analyze the impact of the laser polarization and find q-deformed exponential shapes. (The variable $u$ is the light-cone momentum-transfer from initial electron to final photon.) Optical laser pulses of various durations are compared with the monochromatic laser beam model which uncovers the laser intensity parameter in the range $xi = 1 cdots 10$. Some supplementary information is provided for the azimuthal final-electron/photon distributions and the photon energy-differential cross sections.
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