Specific properties of X-ray compound refractive lens. Theoretical analysis

We present the diffraction theory for the x-ray compound refractive lens (XCRL) operation as an imaging device in the paraxial approximation. We obtain the analytical expression for the image propagator in the case of parabolic XCRL that allows us to explain the peculiarities of imaging and focusing with the XCRL observed previously in the experiments. We propose the enhanced thin lens formula for the relatively long XCRL with the longitudinal size L taking into account the linear corrections in L/F where F is the focal length in the thin lens approximation. A relatively small aperture of XCRL due to absorption of x rays limits the resolution and, in addition, leads to phase effects visualizing the local phase gradient of the radiation wave field produced by transparent objects. This opens novel technique of imaging for purely phase objects, which is different from the in-line phase contrast imaging techniques.