Exact revision of the elliptically bent mirror theory

One of the main hurdles for nanometer focusing by a bending mirror lies in the theoretical surface errors by its approximations used for the traditional theory. The impacts of approximations and analytical corrections have been discussed, and the elliptically bent mirror theory has been described during exact mathematical analysis without any approximations. These approximations are harmful for the focusing system with bigger grazing angle, bigger mirror length, and bigger numerical aperture. The properties of equal-moment and single-moment bent mirrors have been described and discussed. Because of its obvious advantages, a single-moment bending mirror has high potential ability for nanometer focusing.