dispersing optic, comprised of collimating (C), dispersing (D$_{{_{1}}}$, D$_{{_{2}}}$), and wavelength-selecting (W) crystals in a ($\pi+$,$0-$,$0+$,$0-$) scattering configuration. With the crystal parameters provided in Table \[tab14\], the dispersing element D$_{{_{\mathrm {D}}}}$ features a spectral transmission function with a $\Delta E_{{_{\cup}}}=13$ meV bandwidth (b), a cumulative angular dispersion rate ${{{\mathcal D}}_{\cup_{{_{{D}}}}}} =
  -42 ~\mu$rad/meV, and a cumulative asymmetry factor ${b_{\cup_{{_{{D}}}}}} =
  1.4$. The sharp line in (b) presents the 0.1-meV design spectral resolution of the x-ray echo spectrometer. $E=4.5686$ keV. []{data-label="fig00399"}](cddw-pi+0-0+0-comb-E4o57keV.pdf){width="50.00000%"}

![Dispersing element D$_{{_{\mathrm {{R}}}}}$ (a) of the refocusing system $\hat{O}_{{_{\mathrm {{R}}}}}$ (Fig. 2) and its spectral transmittance function (b). Similar to D$_{{_{\mathrm {{D}}}}}$ in Fig. \[fig00399\], D$_{{_{\mathrm {{R}}}}}$ is an example of the in-line four-crystal CDDW-type dispersing optic, but, in a ($\pi+$,$\pi+$,$\pi-$,$0-$) scattering configuration. With the crystal parameters provided in Table \[tab14\], the dispersing element D$_{{_{\mathrm {D}}}}$ features a spectral transmission function with a $\Delta
  E_{{_{\cup}}}=9.9$ meV bandwidth (b), a cumulative angular dispersion rate ${{{\mathcal D}}_{\cup_{{_{{D}}}}}} = -64 ~\mu$rad/meV, a cumulative asymmetry factor ${b_{\cup_{{_{{D}}}}}} = 0.39$, and ${{{\mathcal D}}_{\cup_{{_{{D}}}}}}/{b_{\cup_{{_{{D}}}}}}
  = -161 ~\mu$rad/meV. The sharp line in (b) presents the 0.1-meV design spectral resolution of the x-ray echo spectrometer. $E=4.5686$ keV. []{data-label="fig0049"}](cddw-pi+pi+pi-0-comb-4o57keV.pdf){width="50.00000%"}

In-line four-crystal CDDW-type dispersing optics [@ShSS11; @SSS13; @SSS14] are 