Coma correction and extension of the focusing geometry of a soft-x-ray monochromator

I show analytically and numerically how to use a varied line-space grating to reduce coma aberration in a soft-x-ray synchrotron radiation monochromator: a plane grating in combination with one or more spherical mirrors arranged in front of the grating and rotated around a fixed axis outside the mirror surface. It seems that the variation extent of groove spacing is too small to be manufactured; thus, a holographic grating of equivalent imaging properties is proposed. Based on the analysis of a focusing equation, I found that, if the plane grating is replaced by a spherical one, the monochromator can also be made free of defocus.