X-ray focusing by the system of refractive lens(es) placed inside asymmetric channel-cut crystals

An X-ray one-dimensionally focusing system, a refracting-diffracting lens (RDL), composed of Bragg double-asymmetric-reflecting two-crystal plane parallel plates and a double-concave cylindrical parabolic lens placed in the gap between the plates is described. It is shown that the focal length of the RDL is equal to the focal distance of the separate lens multiplied by the square of the asymmetry factor. One can obtain RDLs with different focal lengths for certain applications. Using the point-source function of dynamic diffraction, as well as the Green function in a vacuum with parabolic approximation, an expression for the double-diffracted beam amplitude for an arbitrary incident wave is presented. Focusing of the plane incident wave and imaging of a point source are studied. The cases of non-absorptive and absorptive lenses are discussed. The intensity distribution in the focusing plane and on the focusing line, and its dependence on wavelength, deviation from the Bragg angle and magnification is studied. Geometrical optical considerations are also given. RDLs can be applied to focus radiation from both laboratory and synchrotron X-ray sources, for X-ray imaging of objects, and for obtaining high-intensity beams. RDLs can also be applied in X-ray astronomy.