X-ray focusing properties of doubly bent crystals

The focusing properties of several bent crystal geometries, including the newly introduced Pestehe & Askari general point-focusing system [Pestehe & Askari Germi (2012), Opt. Soc. Am. A, 29, 68–77; Pestehe & Askari Germi (2012), J. Appl. Cryst. 45, 890–901], on an arbitrarily positioned detector plane are investigated and illustrated. The properties of the focal points and the generated images are theoretically related to the local, ℓ, and directional, γ, positions of the detector plane for a given position of a point source on the Rowland circle. A general relation is derived for the detector positioning to obtain a specially focused image. This formula for the polar position of the detector plane, given by the two ℓ and γ variables, enables the exact determination of the system astigmatism and the exact calculation of the sagittal and meridional image positions for the spectrometer under study. The astigmatisms of the above-mentioned bent crystal geometries have been studied, and their sagittal and meridional focal positions and characteristics have been obtained and are illustrated. It is also shown that there is a possibility of designing a spectrometer to focus rays from a linear source onto a point on the Rowland circle.