of the hard x-ray echo spectrometers are introduced with a spectral resolution $\Delta
{\varepsilon}=0.1-0.02$ meV at photon energies $E\simeq 9.1$ keV and $E\simeq 4.6$ keV, comprised of defocusing and refocusing systems with multi-crystal inline dispersing elements featuring both large cumulative dispersion rates ${{{\mathcal D}}_{\cup_{{_{}}}}} \gtrsim 25-60~\mu$rad/meV, transmission bandwidths $\Delta E_{{_{\cup}}}\simeq 5-13$ meV, and a dynamical range $\Delta E_{{_{\cup}}}/\Delta {\varepsilon}\simeq
100-500$. Because of much greater dispersion rates which are feasible in the crystal systems, as compared to the diffraction gratings, the spectral resolving power in the hard x-ray regime can be as large as $\gtrsim 10^8-10^9$, i.e., more than three orders of magnitude higher than in the soft x-ray regime.

Theory
======

We start here by considering optical systems featuring a combination of focusing and energy dispersing capabilities. We assume that such systems can, first, focus [*monochromatic*]{} x-rays from a source of a linear size $\Delta x_{{_{0}}}$ in a source plane (reference plane $0$ perpendicular to the optical axis $z$ in Fig. \[fig001\]) onto an intermediate image plane (reference plane $1$ in Fig. \[fig001\]) with an image linear size $\Delta x_{{_{1}}}= |A| \Delta
x_{{_{0}}}$, where $A$ is a magnification factor of the optical system. In addition, the system can disperse photons 