can be placed from the source at a large distance $l_{{_{1}}} \gg {b_{\cup_{{_{{D}}}}}}^2 l_{{_{2}}}$. In this case, the refocusing condition reads $$l_{{_{3}}}{{{\mathcal D}}_{\cup_{{_{{D}}}}}} \simeq  f_{{_{1}}} \frac{{{{\mathcal D}}_{\cup_{{_{{R}}}}}}}{{b_{\cup_{{_{{R}}}}}}}. 
\label{refocus23}$$ For the spectral resolution $\Delta {\varepsilon}$ of a hard x-ray echo spectrometer we obtain from Eqs. , , and : $$\Delta {\varepsilon}=   \frac{|{b_{\cup_{{_{{R}}}}}}|}{|{{{\mathcal D}}_{\cup_{{_{{R}}}}}}|} \frac{\Delta x_{{_{1}}}}{f_{{_{1}}}}. 
\label{resolution2}$$ As follows from Eq. , the spectral resolution of the echo spectrometer is determined solely by the parameters of the refocusing system, i.e., by the resolution of the hard x-ray spectrograph [@Shvydko15]. The parameters of the defocusing system determine only the size of the secondary monochromatic source on the sample $\Delta x_{{_{1}}}=|A_{{_{\mathrm {{D}}}}}|\Delta
x_{{_{0}}}$.

In the second step, we consider the particular optical designs of x-ray echo spectrometers with a very high spectral resolution $\Delta
{\varepsilon}\lesssim 0.1$ meV. For practical reasons, we will assume that the secondary [*monochromatic*]{} source size is $\Delta
x_{{_{1}}}\simeq 5~\mu$m, and the focal length is $f_{{_{1}}}\simeq 0.4$ m of the collimating element in the refocusing system. Then, Eq.  requires the ratio ${|{{{\mathcal D}}_{\cup_{{_{{R}}}}}}|} /{|{b_{\cup_{{_{{R}}}}}}|}\simeq 125~\mu$rad/meV for the dispersive element D$_{{_{\mathrm {{R}}}}}$. Assuming the distance $l_{{_{3}}}\simeq 2$ m from the focusing element to the sample in the defocusing system, and $l_{{_{1}}} \gg 