  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.
