  X-ray echo spectroscopy, a space-domain counterpart of neutron spin echo, is
a recently proposed inelastic x-ray scattering (IXS) technique. X-ray echo
spectroscopy relies on imaging IXS spectra, and does not require x-ray
monochromatization. Due to this, the echo-type IXS spectrometers are broadband,
and thus have a potential to simultaneously provide dramatically increased
signal strength, reduced measurement times, and higher resolution compared to
the traditional narrow-band scanning-type IXS spectrometers. The theory of
x-ray echo spectrometers presented in [1] is developed here further with a
focus on questions of practical importance, which could facilitate optical
design and assessment of the feasibility and performance of the echo
spectrometers. Among others, the following questions are addressed: spectral
resolution, refocusing condition, echo spectrometer tolerances, refocusing
condition adjustment, effective beam size on the sample, spectral window of
imaging and scanning range, impact of the secondary source size on the spectral
resolution, angular dispersive optics, focusing and collimating optics, and
detector's spatial resolution. Examples of optical designs and characteristics
of echo spectrometers with 1-meV and 0.1-meV resolutions are presented.
