in the lowest row of the ray-transfer matrices.

![Graphical presentation of the echo spectroscopy principles. (a) Photons from a source with a linear size $\Delta
  x_{{_{\mathrm {0}}}}$ in the reference source plane $0$ are focused onto a spot $\Delta x_{{_{\mathrm {1}}}}$ in the intermediate image plane $1$ by a focusing-dispersing system $\hat{O}_{{_{\mathrm {{D}}}}}$. Each spectral component, indicated by different color, is focused at a different location due to dispersion in $\hat{O}_{{_{\mathrm {{D}}}}}$. All spectral components of the x-rays are refocused by a consecutive time-reversal focusing-dispersing system $\hat{O}_{{_{\mathrm {{R}}}}}$ onto the same spot $\Delta x_{{_{\mathrm {2}}}}$ (echo) in the image plane $2$. (b) Inelastic x-ray scattering with an energy transfer ${\varepsilon}$ (indicated by changed color) from a sample in the reference plane $1$ results in a lateral shift $G_{{_{\mathrm {{R}}}}} {\varepsilon}$ of the echo signal equal for all spectral components. Here $G_{{_{\mathrm {{R}}}}}$ is a linear dispersion rate in the time-reversal system $\hat{O}_{{_{\mathrm {{R}}}}}$. Spectral resolution of the x-ray echo spectrometer is $\Delta {\varepsilon}=\Delta
  x_{{_{\mathrm {2}}}}/G_{{_{\mathrm {{R}}}}}$. []{data-label="fig001"}](1.pdf){width="50.00000%"}

Focusing of the monochromatic spectral components requires that matrix element $B=0$. The ray-transfer matrix of any focusing-dispersing system in a general case therefore reads as $$\hat{O}  = 
 \{A \,0\,G;\,CDF;001\}
\label{matrix}$$ with $A$ and 