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The use of refractive lenses for focusing x-ray beams has been the subject of publications since the early 1980s. Detailed calculations have been made for different shapes for the refractive lens: cylindrical, spherical, parabolic, and for a Fresnel-type refractive lens. The main drawback to the use of a single refractive lens to focus x-rays is that the index of refraction (n equals 1 - (delta) ) is very close to 1, which results in a lens with a very long focal length. Recently Snigerov and others have suggested and experimentally demonstrated, using cylindrical-shaped lenses, that this problem of long focal lengths can be overcome by using many lenses in series. Each lens refracts the photon through a small angle, but the sum of these sequential changes in direction can be moderately large. This increase in effective refraction angle reduces the focal length of the lens to a few meters or less and makes the multi-element lens a much more useful instrument for focusing x-rays. This paper, annualizes the expected performance of a lens consisting of a series of aligned hollow spheres in a beryllium substrate. The use of hollow spheres rather than hollow cylinders produces focusing of the x rays into a small focal spot in contrast to the single-directional focusing of the hollow cylinders, which produces a line focus. The use of beryllium as the substrate results in lower photo cross sections for both scattering and absorption relative to the value of the refractive index as compared to higher-Z materials and results in higher transmission values than for lenses with thin webs between the lens elements without distorting the surfaces of the neighbor lens element. This plus beryllium's low density, keep the absorption and scattering in the web at a minimum. The calculations suggest that one will be able to make Be lenses with short focal lengths (1 to 2 m) with useable transmissions (10 to 30%). Two multi-element lenses have been constructed: one with 20 1-mm-diameter hollow spheres in an aluminum substrate, and one with 50 hollow spheres, 1 mm in diameter, in a beryllium substrate. Some construction details and calculations of the expected performance, are given for these two multi-element lenses.
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