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Thick diffractive optical elements offer a promising way to achieve focusingor imaging at a resolution approaching 1 nm for X-ray wavelengths shorter thanabout 0.1 nm. Efficient focusing requires that these are fabricated withstructures that vary in period and orientation so that rays obey Bragg's lawover the entire lens aperture and give rise to constructive interference at thefocus. Here the analysis method of ray-tracing of thick diffractive opticalelements is applied to such lenses to optimise their designs and to investigatetheir operating and manufacturing tolerances. Expressions are provided of thefourth-order series expansions of the wavefront aberrations and transmissionsof both axi-symmetric lenses and pairs of crossed lenses that each focuses inonly one dimension like a cylindrical lens. We find that aplanatic zone-platedesigns, whereby aberrations are corrected over a large field of view, can beachieved by axi-symmetric lenses but not the crossed lenses. We investigate theperformance of 1 nm-resolution lenses with focal lengths of about 1 mm and showtheir fields of view are mainly limited by the acceptance angle of Braggdiffraction, and that aberrations can limit the performance of lenses withlonger focal lengths. We apply the ray-tracing formalism for a tolerancinganalysis of imperfect lenses and examine some strategies for the correction oftheir aberrations.

A ray-trace analysis of X-ray multilayer Laue lenses for nanometer focusing