Geometrical principles of the monolithic X-ray magnifier

The principles of the monolithic X-ray magnifier based on two successive asymmetric non-coplanar diffractions from the same crystal are analyzed from the geometrical point of view. The conditions of the distortion-free two-dimensional pattern magnification are formulated by means of matrix formalism. The general solution of the problem of determination of the crystal surface planes satisfying the conditions of distortion-free magnification is derived in the closed form. The calculation is performed on the basis of the kinematical theory of diffraction. It is shown that the number of solutions is infinite for any given pair of diffraction vectors and magnification. General properties of the solution are illustrated by some examples. The effect of the refraction of X-rays on the pattern distortion is estimated. The results suggest that the refractive index correction has a negligible influence on the distortion of the magnified pattern. The results presented enable us to prepare monolithic X-ray magnifiers or demagnifiers that can be used in various fields of X-ray optics, especially in X-ray topography and X-ray lithography.