Aberration-free imaging of inelastic scattering spectra with x-ray echo spectrometers

We study conditions for aberration-free imaging of inelastic x-ray scattering (IXS) spectra with x-ray echo spectrometers. Aberration-free imaging is essential for achieving instrumental functions with high resolution and high contrast. Computational ray tracing is applied to a thorough analysis of a 0.1-meV/0.07-nm$^{-1}$-resolution echo-type IXS spectrometer operating with 9-keV x-rays. We show that IXS spectra imaged by the x-ray echo spectrometer that uses lenses for the collimating and focusing optics are free of aberrations. When grazing-incidence mirrors (paraboloidal, parabolic Kirkpatrick-Baez, or parabolic Montel) are used instead of the lenses, the imaging system reveals some defocus aberration that depends on the inelastic energy transfer. However, the aberration-free images can be still recorded in a plane that is tilted with respect to the optical axis. This distortion can be thus fully compensated by inclining appropriately the x-ray imaging detector, which simultaneously improves its spatial resolution. A full simulation of imaging IXS spectra from a realistic sample demonstrates the excellent performance of the proposed designs.