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Single-bounce hollow glass capillaries with ellipsoidal shapes have been used at the Cornell High Energy Synchrotron Source recently for various microbeam experiments, with focal spot sizes from 12 to 23 µm, divergences from 2 to 8 mrad, intensities up to 450 times the intensities of incident X-rays, and working distances up to 55 mm. Simple formulae are developed in this paper to explain capillary performance given the X-ray source size, capillary dimensions and slope errors. Capillary length is optimized for best focusing performance. Capillary fabrication accuracy is reported and capillary X-ray tests confirm the focusing properties expected from formulae. The application of capillaries to third-generation X-ray sources and future energy-recovery linac X-ray sources are discussed.

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