Deriving a Comprehensive Formula for Meridional Focusing in Bragg Spectroscopy with Spherically Bent Crystals

Using spherically bent Bragg crystals for x-ray spectroscopy of hot plasmas has yielded measurements with high spectral resolution and high signal-to-noise ratio. However, optimizing the geometry of spectrometer configurations requires a deeper understanding of the meridional focus, since the minimum width of the envelope of meridional rays sets basic constraints on shielding. To our knowledge, no generalized equation for meridional focusing currently exists. We analyze reflection from a point source off of a spherical surface, where the angle of incidence for each ray is separated into components that vary along the crystal surface. Our geometrical analysis results in a general formula to describe meridional focusing, identifying parameters that affect the position and envelope of the meridional crossover region: crystal size, radius of curvature, Bragg angle, and source position. This formula builds intuition for distortions to the meridional focus and provides guidance for what parameters minimize the width of the meridional crossover.

This work was made possible by funding from the Department of Energy for the Summer Undergraduate Laboratory Internship (SULI) program. This work is supported by the US DOE Contract No. DE-AC02-09CH11466.