\[tab2\], ray-transfer matrices are shown for arrangements composed of several basic optical elements, such as successive multiple Bragg reflections from crystals $\hat{{C}}({b_{\cup_{{_{n}}}}},{{{\mathcal D}}_{\cup_{{_{n}}}}})$ and $\hat{{K}}({b_{\cup_{{_{n}}}}},{{{\mathcal D}}_{\cup_{{_{n}}}}},l)$, rows 4–5; and a focusing system $\hat{{F}}(l_{{_{2}}},f,l_{{_{1}}})$, row 6.

The matrices of the defocusing $\hat{O}_{{_{\mathrm {D}}}}$ and refocusing $\hat{O}_{{_{\mathrm {R}}}}$ systems presented in Table \[tab2\], rows 7 and 8, respectively, are calculated using the multi-crystal matrix $\hat{{C}}({b_{\cup_{{_{n}}}}},{{{\mathcal D}}_{\cup_{{_{n}}}}})$ from row 4, assuming zero free space between crystals in successive Bragg reflections. Generalization to a more realistic case of nonzero distances between the crystals requires the application of matrix $\hat{{K}}({b_{\cup_{{_{n}}}}},{{{\mathcal D}}_{\cup_{{_{n}}}}},l)$ from row 5.

We refer to Ref. [@Shvydko15] for details on the derivation of these matrices. Here, we provide only the final results, notations, and definitions.

![Definitions of scattering geometries in Bragg diffraction from a crystal: (a) $0+$, (b) $0-$, (c) $\pi+$, and (d) $\pi-$.[]{data-label="fig005"}](5.pdf){width="50.00000%"}

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  Optical system   Matrix notation                                                       Ray-transfer matrix                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Definitions and remarks

  \[-5pt\]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             

  \[-3.8mm\]       $\hat{{P}}(l)$                                                        $\left( \begin{array}{ccc} 1 & l  & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1  \end{array} \right)$                                                                                                                                                                                                                                                                                                                                                                                               

  \[-0.5mm\]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 

  \[-3.8mm\]       $\hat{{L}}(f)$                                                        $\left( \begin{array}{ccc} 1 & 0 & 0\\ -\frac{1}{f} & 1 & 0  \\ 0 & 0 & 1 \end{array} 