process. The essential point is that, the combined defocusing-refocusing system maps the inelastic scattering spectrum onto image plane $2$. The image is independent of the spectral composition $E+\delta E$ of the photons in the incident beam.

The spectral resolution $\Delta {\varepsilon}$ of the echo spectrometer is calculated from the condition, that the shift due to inelastic scattering $x_{{_{2}}}^{\prime}-x_{{_{2}}}=G_{{_{\mathrm {{R}}}}} 
{\varepsilon}$ is at least as large as the linear size $\Delta
x_{{_{2}}}$ of the echo signal : $$\Delta {\varepsilon}=    \frac{\Delta x_{{_{2}}}}{|G_{{_{\mathrm {{R}}}}}|}  \equiv  \frac{|A_{{_{\mathrm {{R}}}}}|\Delta x_{{_{1}}} }{|G_{{_{\mathrm {{R}}}}}|} \equiv  \frac{|A_{{_{\mathrm {{R}}}}} A_{{_{\mathrm {{D}}}}}| \Delta x_{{_{o}}}}{|G_{{_{\mathrm {{R}}}}}|}.
\label{resolution}$$ Here it is assumed that the spatial resolution of the detector is better than $\Delta x_{{_{2}}}$.

These results constitute the underlying principle of x-ray echo spectroscopy. Noteworthy, angular dispersion always results in an inclined intensity front, i.e., in dispersion both perpendicular to and along the beam propagation direction [@SL12]. Therefore, x-rays are defocused and refocused also in the time domain, as in spin-echo. As a result, inelastic scattering spectra can be also mapped by measuring time distributions in the detector, given a short-pulse source.

Perfect refocusing takes place if the linear dispersion of the combined system $G_{{_{\mathrm {{C}}}}}=A_{{_{\mathrm {{R}}}}}G_{{_{\mathrm {{D}}}}}+G_{{_{\mathrm {{R}}}}}$ vanishes, as in 