location, $x_{{_{2}}}$ in image plane $2$, to a spot with a linear size $$\Delta x_{{_{2}}}= |A_{{_{\mathrm {{R}}}}} A_{{_{\mathrm {{D}}}}}| \Delta x_{{_{0}}}\equiv  |A_{{_{\mathrm {{R}}}}} | \Delta x_{{_{1}}}, 
\label{magnification}$$ as shown schematically in Fig. \[fig001\](a). Such behavior is an analog of the echo phenomena. Here, however, it takes place in space, rather than in the time domain.

Now, what happens if a sample is placed into the intermediate image plane $1$, \[Fig. \[fig001\](b)\], which can scatter photons inelastically? In an inelastic scattering process, a photon with an arbitrary energy $E+\delta E$, changes its value to $E+\delta E+
{\varepsilon}$. Here ${\varepsilon}$ is an energy transfer in the inelastic scattering process. The ray vector ${\mbox{\boldmath $r$}}_{{_{1}}}=(x_{{_{1}}},\xi_{{_{1}}},\delta E)$ before scattering transforms to ${\mbox{\boldmath $r$}}_{{_{1}}}^{\prime}=(x_{{_{1}}},\xi_{{_{1}}}^{\prime},\delta
E+{\varepsilon})$ after inelastic scattering. Propagation of ${\mbox{\boldmath $r$}}_{{_{1}}}^{\prime}$ through the time-reversal system results in a ray vector ${\mbox{\boldmath $r$}}_{{_{2}}}^{\prime}=(x_{{_{2}}}^{\prime},\xi_{{_{2}}}^{\prime},\delta
E+{\varepsilon})= \hat{O}_{{_{\mathrm {{R}}}}}{\mbox{\boldmath $r$}}_{{_{1}}}^{\prime}$. Assuming that refocusing condition holds, we come to a decisive point: all photons independent of the incident photon energy $E+\delta E$ are refocused to the same location $$x_{{_{2}}}^{\prime}=x_{{_{2}}}+G_{{_{\mathrm {{R}}}}} {\varepsilon},\hspace{0.5cm}  x_{{_{2}}}=A_{{_{\mathrm {{R}}}}} A_{{_{\mathrm {{D}}}}} x_{{_{o}}},
\label{magnification2}$$ which is, however, shifted from $x_{{_{2}}}$ by $G_{{_{\mathrm {{R}}}}}
{\varepsilon}$, a value proportional to the energy transfer ${\varepsilon}$ in the inelastic scattering 