required dispersion rate ${|{{{\mathcal D}}_{\cup_{{_{{D}}}}}}|}\simeq 25~\mu$rad/meV, significant transmission bandwidth $\Delta E_{{_{\cup}}}\simeq 9$ meV, and is compact. The CDDW optic in the ($\pi+$,$\pi+$,$\pi-$,$0-$) configuration is better suited for the refocusing dispersing element D$_{{_{\mathrm {{R}}}}}$ (Fig. \[fig004\]). It provides the large ratio ${|{{{\mathcal D}}_{\cup_{{_{{R}}}}}}|} /{|{b_{\cup_{{_{{R}}}}}}|}\simeq 120~\mu$rad/meV required for the high spectral resolution \[see Eq. \] and substantial transmission bandwidth $\Delta E_{{_{\cup}}}\simeq
  5.5$ meV, although smaller than in the D$_{{_{\mathrm {{D}}}}}$-case.

The total beam size on the sample is $\simeq G_{{_{\mathrm {{D}}}}}\Delta E_{{_{\cup}}}$. For the spectrometer exemplified here, it is estimated to be $\simeq 275~\mu$m. Equation  together with Eqs.  and can be used to estimate tolerances on the admissible variations of the focal distances, sample displacement, sample surface imperfections, etc. As discussed in Appendix \[echotollerances\] the tolerances are in a millimeter range in the case of the 0.1-meV spectrometer.

Details on the optical designs and examples of the dispersing elements designed for use with a lower photon energy of 4.57 keV and larger $\Delta E_{{_{\cup}}}\simeq 13-10$ meV, can be found in Appendix \[cddwoptic\]. All these examples showcase the applicability of echo spectroscopy in a wide spectral range, and its feasibility both with synchrotron radiation and x-ray free electron laser sources.

The point is 