Eq. . Refocusing can still take place with good accuracy if $|G_{{_{\mathrm {{C}}}}}|$ is sufficiently small: $$|G_{{_{\mathrm {{C}}}}}| \Delta E_{\cup}\ll \Delta x_{{_{2}}}.
\label{tolerances}$$ Here $\Delta E_{\cup}$ is the bandwidth of x-rays in image plane $2$. Tolerances on the echo spectrometer parameters and on the sample shape can be calculated with Eq. .

The above approach is general and applicable to any frequency domain. A particular version was proposed in the soft x-ray domain, for applications in resonant IXS spectroscopy at $L$-edges of $3d$ elements [@Fung04; @Lai14]. The dispersing elements in the soft x-ray and visible light domains are diffraction gratings. with diffraction gratings as dispersing elements [@Fung04; @Lai14].

Optical Design
==============

Diffraction gratings are not practical in the hard x-ray regime. Extension into the hard x-ray regime is therefore nontrivial. In this regard, as was demonstrated in [@Shvydko-SB; @SLK06], the angular dispersion in the hard x-ray regime can be achieved by Bragg diffraction from asymmetrically cut crystals, i.e., from crystals with the reflecting atomic planes not parallel to the entrance surface. This is a hard x-ray analog of the optical diffraction gratings or optical prisms. A large dispersion rate is a key for achieving high spectral resolution in angular-dispersive x-ray 