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.
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