An exact theory of imaging with a parabolic continuously refractive X-ray lens

A theory is developed of image formation with an X-ray lens that consists of a large number of elements. Each element has a biconcave parabolic profile and weakly refracts an X-ray beam. Since such a lens can have a relatively large length comparable to the focal length, the thin-lens approximation is inapplicable. An exact expression for the propagator of a continuously refractive lens is derived that describes the transfer of radiation through a refractive parabolic medium. We calculate the image propagator that describes the focusing of a parallel beam and the image transfer (the focusing of a microobject), as well as the Fourier transform of the transmission function for a microobject with a lens, is calculated. The effective aperture of an X-ray lens is completely determined by the absorption of radiation and does not depend on its geometrical cross-sectional sizes. If we write the complex refractive index as n=1− δ+iβ, then the beam diameter at the focus is approximately a factor of 0.8β/δ smaller than the diameter of the effective aperture, with the index depending only slightly on the wavelength. A continuously refractive lens has no aberrations in the sense that all of the rays that passed through the lens aperture are focused at a single point. The lens can focus radiation inside it and has the properties of a waveguide; i.e., it can reconstruct the beam structure for some lengths to within the absorption-caused distortions. Nonuniform X-ray absorption in the lens leads to the interesting visualization effect of transparent microobjects when their image is focused. In this case, the phase shift gradient produced by the microobject is imaged. We discuss the properties of the Fourier transform pertaining to the absorption of radiation in the lens.