  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.
