Spherical aberration it is to use a single aspheric lens. "Positive" spherical aberration means peripheral rays are bent too much. "Negative" spherical aberration means peripheral rays are not bent enough. The effect is proportional to the fourth power of the diameter and inversely proportional to the third power of the focal length, so it is much more pronounced at short focal ratios, i.e., "fast" lenses. In lens systems, the effect can be minimized using special combinations of convex and concave lenses, as well as using aspheric lenses or aplanatic lenses. For simple designs one can sometimes calculate parameters that minimize spherical aberration.
Compound refractive lens the order of meters, by using many lenses in series. To minimize the absorption the lens must be made from a material such as aluminium, beryllium, or lithium. CRLs were first demonstrated in the mid-1990s by a group of scientists at the ESRF. They drilled holes in an aluminium block, and achieved focusing in two dimensions. For X-rays a concave lens focuses the X-rays because the index of refraction is slightly below unity. In a CRL of this type the walls between the cylindrical holes act as concave lenses for X-rays traveling perpendicular to the axis of the drilled cylinders.