= -120 ~\mu$rad/meV. []{data-label="fig004"}](cddw-pi+pi+pi-0-comb-v2-art.pdf){width="50.00000%"}

For the spectrometer to feature a large throughput, the refocusing system $\hat{O}_{{_{\mathrm {{R}}}}}$ has to be capable of collecting x-ray photons in a large solid angle scattered from the sample. For this purpose, we propose using a hard x-ray focusing-dispersing system of a spectrograph-type considered in [@Shvydko15], and schematically shown in Fig \[fig002\]. A collimating focusing element $f_{{_{1}}}$ collects photons in a large solid angle and makes x-ray beams of each spectral component parallel. The collimated beams impinge upon the Bragg (multi)crystal dispersing element D$_{{_{\mathrm {{R}}}}}$ with the cumulative angular dispersion rate ${{{\mathcal D}}_{\cup_{{_{{R}}}}}}$, and the cumulative asymmetry factor ${b_{\cup_{{_{{R}}}}}}$. The focusing element $f_{{_{2}}}$ focuses x-rays in the vertical dispersion plane onto the detector placed in the image plane $2$. As shown in [@Shvydko15] (see also Appendix \[raytransfer\] and Table \[tab2\]) such a system is described by a ray-transfer matrix with the magnification $A_{{_{\mathrm {{R}}}}}$ and linear dispersion $G_{{_{\mathrm {{R}}}}}$ matrix elements given by $$A_{{_{\mathrm {{R}}}}}=-\frac{{b_{\cup_{{_{{R}}}}}}f_{{_{2}}}\!}{f_{{_{1}}}\!},\hspace{0.5cm}
G_{{_{\mathrm {{R}}}}}={{{\mathcal D}}_{\cup_{{_{{R}}}}}} f_{{_{2}}} .
\label{refocusing}$$ Using Eqs. , , and we obtain for the refocusing condition in the hard x-ray echo spectrometer $$\frac{l_{{_{3}}} l_{{_{1}}}}{l_{{_{1}}} + {b_{\cup_{{_{{D}}}}}}^2 l_{{_{2}}}} {{{\mathcal D}}_{\cup_{{_{{D}}}}}} = f_{{_{1}}} \frac{{{{\mathcal D}}_{\cup_{{_{{R}}}}}}}{{b_{\cup_{{_{{R}}}}}}}. 
\label{refocus2}$$ The dispersing element D$_{{_{\mathrm {{D}}}}}$ 