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Ray-transfer matrices {#raytransfer}
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Ray-transfer matrices $\{A0G,CDF,001\}$ of the defocusing $\hat{O}_{{_{\mathrm {D}}}}$ and refocusing $\hat{O}_{{_{\mathrm {R}}}}$ systems of the x-ray echo spectrometers used in the paper are given in the last two rows of Table \[tab2\]. They are equivalent to the derived in Ref. [@Shvydko15] ray-transfer matrices of x-ray focusing monochromators and spectrographs. The matrices of the multi-element systems $\hat{O}_{{_{\mathrm {D}}}}$ and $\hat{O}_{{_{\mathrm {R}}}}$ are obtained by successive multiplication of the matrices of the constituent optical elements, which are given in the upper rows of Table \[tab2\].

In the first three rows, 1–3, matrices are shown for the basic optical elements, such as propagation in free space $\hat{{P}}(l)$, thin lens or focusing mirror $\hat{{L}}(f)$, and Bragg reflection from a crystal $\hat{{C}}(b,{s}{{\mathcal D}})$. Scattering geometries in Bragg diffraction from crystals are defined in Fig. \[fig005\]. In the following rows of Table 