[]{data-label="fig002"}](2-revised-more-in-and-elastic-prl.pdf){width="50.00000%"}

In the first step, we propose a principle optical scheme of a hard x-ray echo spectrometer (Fig. \[fig002\]) comprised of the defocusing $\hat{O}_{{_{\mathrm {{D}}}}}$ and refocusing $\hat{O}_{{_{\mathrm {{R}}}}}$ dispersing systems. The x-ray source is in reference plane $0$, the sample is in plane $1$, and the position-sensitive detector is in plane $2$. The defocusing system $\hat{O}_{{_{\mathrm {{D}}}}}$ is proposed here as a combination of a Bragg (multi)crystal dispersing element D$_{{_{\mathrm {{D}}}}}$ and a focusing element $f$. As has been shown in [@Shvydko15], such a system can be represented by a ray-transfer matrix with the magnification $A_{{_{\mathrm {{D}}}}}$ and linear dispersion $G_{{_{\mathrm {{D}}}}}$ matrix elements given by $$A_{{_{\mathrm {{D}}}}}\!=\! - \frac{1}{{b_{\cup_{{_{{D}}}}}}}\!\frac{l_{{_{3}}}}{l_{{_{12}}}},\hspace{0.20cm}
G_{{_{\mathrm {{D}}}}}\!=\!{{{\mathcal D}}_{\cup_{{_{{D}}}}}}  \frac{l_{{_{3}}} l_{{_{1}}}}{{b_{\cup_{{_{{D}}}}}}^2 l_{{_{12}}}},\hspace{0.20cm} l_{{_{12}}}\! =\!\frac{l_{{_{1}}}}{{b_{\cup_{{_{{D}}}}}}^2} + l_{{_{2}}}. 
\label{defocusing}$$ Here, $l_{{_{1}}}$, $l_{{_{2}}}$, and $l_{{_{3}}}$, are the distances between the x-ray source, the dispersing element D$_{{_{\mathrm {{D}}}}}$, the focusing element $f$ with focal length $f=(l_{{_{12}}}^{-1}+l_{{_{3}}}^{-1})^{-1}$, and the sample in the image plane $1$, respectively (Fig. \[fig002\]). The dispersing (multi)crystal system D$_{{_{\mathrm {{D}}}}}$ is characterized by the cumulative angular dispersion rate ${{{\mathcal D}}_{\cup_{{_{{D}}}}}}$, and cumulative asymmetry factor ${b_{\cup_{{_{{D}}}}}}$, which are defined in [@Shvydko15] (see also Appendix \[raytransfer\] and Table \[tab2\]).

![Dispersing element D$_{{_{\mathrm {{D}}}}}$ (a) of the defocusing system 