in such a way that the location of the source image for photons with an energy $E+\delta E$ is displaced in the image plane by $G \delta E$ from the location of the image for photons with energy $E$. Here, $G$ is a linear dispersion rate of the system. As a result, although monochromatic x-rays are focused, the whole spectrum of x-rays is defocused, due to linear dispersion.

We will use the ray-transfer matrix technique [@KL66; @MK80-1; @Siegman] to propagate paraxial x-rays through such optical systems and to determine linear and angular sizes of the x-ray beam along the optical axis. A paraxial ray in any reference plane is characterized by its distance $x$ from the optical axis, by its angle $\xi$ with respect to that axis, and the deviation $\delta E$ of the photon energy from a nominal value $E$. The ray vector ${\mbox{\boldmath $r$}}_{{_{0}}}=(x_{{_{0}}},\xi_{{_{0}}},\delta E)$ at an input source plane is transformed to ${\mbox{\boldmath $r$}}_{{_{1}}}=(x_{{_{1}}},\xi_{{_{1}}},\delta
E)=\hat{O}{\mbox{\boldmath $r$}}_{{_{0}}}$ at the output reference plane (image plane), where $\hat{O}=\{ABG;CDF;001\}$ is a ray-transfer matrix of an optical element placed between the planes [^1]. Only elastic processes in the optical systems are taken into account, that is reflected by zero and unity terms 