  We study the Compton scattering of x-rays off electrons that are driven by a
relativistically intense short optical laser pulse. The frequency spectrum of
the laser-assisted Compton radiation shows a broad plateau in the vicinity of
the laser-free Compton line due to a nonlinear mixing between x-ray and laser
photons. Special emphasis is placed on how the shape of the short assisting
laser pulse affects the spectrum of the scattered x-rays. In particular, we
observe sharp peak structures in the plateau region, whose number and locations
are highly sensitive to the laser pulse shape. These structures are interpreted
as spectral caustics by using a semiclassical analysis of the laser-assisted
QED matrix element.
