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We study in detail the strong-field QED process of non-linear Comptonscattering in short intense plane wave laser pulses of circular polarization.Our main focus is placed on how the spectrum of the back-scattered laser lightdepends on the shape and duration of the initial short intense pulse. Althoughthis pulse shape dependence is very complicated and highly non-linear, and hasnever been addressed explicitly, our analysis reveals that all the dependenceon the laser pulse shape is contained in a class of three-parameter masterintegrals. Here we present completely analytical expressions for the non-linearCompton spectrum in terms of these master integrals. Moreover, we analyse theuniversal behaviour of the shape of the spectrum for very high harmonic lines.

Analytical results for non-linear Compton scattering in short intense laser pulses