  Thick diffractive optical elements offer a promising way to achieve focusing
or imaging at a resolution approaching 1 nm for X-ray wavelengths shorter than
about 0.1 nm. Efficient focusing requires that these are fabricated with
structures that vary in period and orientation so that rays obey Bragg's law
over the entire lens aperture and give rise to constructive interference at the
focus. Here the analysis method of ray-tracing of thick diffractive optical
elements is applied to such lenses to optimise their designs and to investigate
their operating and manufacturing tolerances. Expressions are provided of the
fourth-order series expansions of the wavefront aberrations and transmissions
of both axi-symmetric lenses and pairs of crossed lenses that each focuses in
only one dimension like a cylindrical lens. We find that aplanatic zone-plate
designs, whereby aberrations are corrected over a large field of view, can be
achieved by axi-symmetric lenses but not the crossed lenses. We investigate the
performance of 1 nm-resolution lenses with focal lengths of about 1 mm and show
their fields of view are mainly limited by the acceptance angle of Bragg
diffraction, and that aberrations can limit the performance of lenses with
longer focal lengths. We apply the ray-tracing formalism for a tolerancing
analysis of imperfect lenses and examine some strategies for the correction of
their aberrations.
