x-ray regime without compromising the signal strength.

The origin of the proposed technique is in spin echo, a phenomenon discovered by Erwin Hahn in 1950 [@Hahn50]. Spin echo is the refocusing in the time domain of the defocused spin magnetization by time reversal. The spin echo technique is widely used in NMR. There is a photon echo analog in the optics. Neutron spin echo spectroscopy is an inelastic neutron scattering technique invented by Mezei, which uses time reversal of the neutron spin evolution to measure energy loss in an inelastic neutron scattering process [@Mezei80]. Photon polarization precession spectroscopy was proposed recently by Röhlsberger [@Roehle14] for studies of spin waves that exhibit similarities to the neutron spin echo. Fung et al. proposed a space-domain analog of the echo spectroscopy for resonant inelastic soft x-ray scattering applications [@Fung04]. Defocusing and refocusing of the spectral components is achieved by angular dispersion from curved diffraction gratings. This approach has been recently demonstrated by Lai et al. [@Lai14].

Here, we propose a hard x-ray version of the echo spectroscopy, which can be applied for non-resonant and resonant high-resolution inelastic x-ray scattering applications. Diffraction gratings are not practical in the hard x-ray regime. However, as was 