Highly efficient angularly resolving x-ray spectrometer optimized for absorption measurements with collimated sources

Highly collimated betatron radiation from a laser wakefield accelerator is a promising tool for spectroscopic measurements. Therefore there is a requirement to create spectrometers suited to the unique properties of such a source. We demonstrate a spectrometer which achieves an energy resolution of < 5 eV at 9 keV and is angularly resolving the x-ray emission allowing the reference and spectrum to be recorded at the same time. The single photon analysis is used to significantly reduce the background noise. Theoretical performance of various configurations of the spectrometer is calculated by a ray-tracing algorithm. The properties and performance of the spectrometer including the angular and spectral resolution are demonstrated experimentally on absorption above the K-edge of a Cu foil backlit by laser-produced betatron radiation x-ray beam.