Optimizing and characterizing grating efficiency for a soft X-ray emission spectrometer

The efficiency of soft X-ray diffraction gratings is studied using measurements and calculations based on the differential method with the S-matrix propagation algorithm. New open-source software is introduced for efficiency modelling that accounts for arbitrary groove profiles, such as those based on atomic force microscopy (AFM) measurements; the software also exploits multi-core processors and high-performance computing resources for faster calculations. Insights from these calculations, including a new principle of optimal incidence angle, are used to design a soft X-ray emission spectrometer with high efficiency and high resolution for the REIXS beamline at the Canadian Light Source: a theoretical grating efficiency above 10% and resolving power E/ΔE > 2500 over the energy range from 100 eV to 1000 eV are achieved. The design also exploits an efficiency peak in the third diffraction order to provide a high-resolution mode offering E/ΔE > 14000 at 280 eV, and E/ΔE > 10000 at 710 eV, with theoretical grating efficiencies from 2% to 5%. The manufactured gratings are characterized using AFM measurements of the grooves and diffractometer measurements of the efficiency as a function of wavelength. The measured and theoretical efficiency spectra are compared, and the discrepancies are explained by accounting for real-world effects: groove geometry errors, oxidation and surface roughness. A curve-fitting process is used to invert the calculations to predict grating parameters that match the calculated and measured efficiency spectra; the predicted blaze angles are found to agree closely with the AFM estimates, and a method of characterizing grating parameters that are difficult or impossible to measure directly is suggested.