$G$ elements defined above. The system blurs the polychromatic source image, because of linear dispersion, as mentioned earlier and graphically presented in Fig. \[fig001\](a). However, another focusing-dispersing system can be used to refocus the source onto reference plane 2. Indeed, propagation of x-rays through the defocusing system $\hat{O}_{{_{\mathrm {{D}}}}}$ and a second system, which we will refer to as a refocusing or time-reversal system $\hat{O}_{{_{\mathrm {{R}}}}}$ (see Fig. \[fig001\]) is given by a combined ray-transfer matrix $$\begin{gathered}
\hat{O}_{{_{\mathrm {{C}}}}}  = \hat{O}_{{_{\mathrm {{R}}}}} \hat{O}_{{_{\mathrm {{D}}}}} = \{A_{{_{\mathrm {{C}}}}}\,0\,G_{{_{\mathrm {{C}}}}};C_{{_{\mathrm {{C}}}}}D_{{_{\mathrm {{C}}}}}F_{{_{\mathrm {{C}}}}};001\}\\  = \left(\!\! \begin{array}{ccc} A_{{_{\mathrm {{R}}}}} A_{{_{\mathrm {{D}}}}}  & 0 &  A_{{_{\mathrm {{R}}}}}G_{{_{\mathrm {{D}}}}}+G_{{_{\mathrm {{R}}}}}  \\ C_{{_{\mathrm {{R}}}}}A_{{_{\mathrm {{D}}}}}\!+\!D_{{_{\mathrm {{R}}}}}C_{{_{\mathrm {{R}}}}}   & D_{{_{\mathrm {{R}}}}} D_{{_{\mathrm {{D}}}}}  & C_{{_{\mathrm {{R}}}}}G_{{_{\mathrm {{D}}}}}\!+\!D_{{_{\mathrm {{R}}}}}F_{{_{\mathrm {{D}}}}}\!+\!F_{{_{\mathrm {{R}}}}} \\ 0 & 0 & 1 \end{array}\!\! \right),
\label{comb}\end{gathered}$$ and by a ray vector ${\mbox{\boldmath $r$}}_{{_{2}}}=(x_{{_{2}}},\xi_{{_{2}}},\delta
E)=\hat{O}_{{_{\mathrm {{C}}}}}{\mbox{\boldmath $r$}}_{{_{0}}}$.

Here we arrive at a crucial point. If $$G_{{_{\mathrm {{C}}}}}=A_{{_{\mathrm {{R}}}}}G_{{_{\mathrm {{D}}}}}+G_{{_{\mathrm {{R}}}}}=0,
\label{refocus}$$ the linear dispersion at the exit of the combined system vanishes, because dispersion in the defocusing system is compensated (time reversed) by dispersion in the refocusing system. As a result, the combined system refocuses all photons independent of the photon energy to the same 