Geometric optics of arrays of reflective surfaces

We present an analysis of the geometric optics of spherically curved arrays of reflective surfaces. In particular we consider optical devices in which reflective surfaces are arranged on a spherical interface and every ray reflects once from a reflector. The orientation of the reflective surfaces is not necessarily related in any way to the orientation of the interface. The analysis can be applied to any radiation that may specularly reflect from the reflectors. This may be reflection from stacks of mirrors or diffraction from the atomic planes. The principles are applied to x-ray optical systems such as capillary arrays and curved crystals. The calculations are used to find optimum configurations of reflective arrays for applications such as x-ray condensers and telescopes, to find the tolerances to which reflective arrays must be constructed, and to find the conditions in which primary aberrations are eliminated.