Focusing of an X-ray spherical wave by a uniaxially bent crystal: Johann scheme and logarithmic spiral

Focusing of a spherical wave in the well-known Johann scheme is considered theoretically. An analytical formula for the size of the crystal bent surface reflecting radiation in the Johann scheme is derived. The intensity distribution near the focus is found. The spherical aberration of the diffracted beam is analyzed. Back scattering is shown to minimize the aberration. Spectral characteristics of an Johann spectrometer are discussed. The focusing of a spherical wave by a crystal bent into a logarithmic spiral is considered. It is shown that the Johann scheme is a specific case of a logarithmic spiral.