Focusing X-rays in two dimensions upon refraction in an inclined prism

In order to focus light in two dimensions a lens needs to be curved in two orthogonal directions. When the lens is symmetrically oriented around the optical axis this is usually achieved in an object with at least one rotationally symmetric surface. X-rays are refracted very little in such configuration as the refractive index differs very little from unity. However, the deflection angle can be increased at rather grazing angle of incidence. And in this condition X-rays can be refracted significantly and focused upon refraction at concave interfaces. It will be shown here that on passage through a slightly curved rectangular prism at very grazing angle of incidence an X-ray beam can be subject to two consecutive refraction processes at two orthogonal concave interfaces, which will lead to focusing. Such object thus functions like a lens, even though it has little similarity with a classical lens. A proof of principle experiment will be described, in which the focusing is observed. The optical properties of optimised lenses of this type will be very similar to those, which were discussed already for more classical stacks of concave lenses. This refers essentially to the absorption limited aperture and to the achievable spatial resolution. The latter could be smaller than 100 nm.