  Operation of an X-ray spectrometer based on a spherical variable line spacing
grating is analyzed using dedicated ray-tracing software allowing fast
optimization of the grating parameters and spectrometer geometry. The analysis
is illustrated with optical design of a model spectrometer to deliver a
resolving power above 20400 at photon energy of 930 eV (Cu L-edge). With this
energy taken as reference, the VLS coefficients are optimized to cancel the
lineshape asymmetry (mostly from the coma aberrations) as well as minimize the
symmetric aberration broadening at large grating illuminations, dramatically
increasing the aberration-limited vertical acceptance of the spectrometer. For
any energy away from the reference, we evaluate corrections to the entrance arm
and light incidence angle on the grating to maintain the exactly symmetric
lineshape. Furthermore, we evaluate operational modes when these corrections
are coordinated to maintain either energy independent focal curve inclination
or maximal aberration-limited spectrometer acceptance. The results are
supported by analytical evaluation of the coma term of the optical path
function. Our analysis gives thus a recipe to design a high-resolution
spherical VLS grating spectrometer operating with negligible aberrations at
large acceptance and over extended energy range.
