  Inverse Compton scattering (ICS) is a unique mechanism for producing fast
pulses - picosecond and below - of bright X- to gamma-rays. These nominally
narrow spectral bandwidth electromagnetic radiation pulses are efficiently
produced in the interaction between intense, well-focused electron and laser
beams. The spectral characteristics of such sources are affected by many
experimental parameters, such as the bandwidth of the laser, and the angles of
both the electrons and laser photons at collision. The laser field amplitude
induces harmonic generation and importantly, for the present work, nonlinear
red shifting, both of which dilute the spectral brightness of the radiation. As
the applications enabled by this source often depend sensitively on its
spectra, it is critical to resolve the details of the wavelength and angular
distribution obtained from ICS collisions. With this motivation, we present
here an experimental study that greatly improves on previous spectral
measurement methods based on X-ray K-edge filters, by implementing a
multi-layer bent-crystal X-ray spectrometer. In tandem with a collimating slit,
this method reveals a projection of the double-differential angular-wavelength
spectrum of the ICS radiation in a single shot. The measurements enabled by
this diagnostic illustrate the combined off-axis and nonlinear-field-induced
red shifting in the ICS emission process. They reveal in detail the strength of
the normalized laser vector potential, and provide a non-destructive measure of
the temporal and spatial electron-laser beam overlap.
