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

Abstract

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. Since the first demonstration (1) of the x-ray compound refractive lens (XCRL) for focusing a synchrotron radiation beam, the x-ray refractive optics is under extensive development. There are successful attempts to develop the refractive optics by means of various approaches. The XCRL with a parabolic profile of surfaces is more promising (2-5). Unlike visible optics, collecting XCRL has a concave shape and the material of the lens is always absorbing. The latter leads to a significant limitation of the effective aperture aγ which is smaller than the physical transverse size a of the XCRL. This property influences the XCRL operation as an imaging device. The XCRL has a rather large longitudinal size L, therefore the thin lens approximation must be verified. In most practical cases XCRL satisfies the condition L/F << 1, where F is the focal length in the thin lens approximation, so that the linear corrections in L/F are sufficient. In this work we present the diffraction theory for the parabolic planar (1D) XCRL in paraxial approximation. We solve the parabolic wave equation in terms of propagators. We calculate analytically the intermediate convolutions and obtain the image propagator which connects straightforwardly the wave field Ao(x) at the plane just after the object and the wave field Ai(x) at the image (detector) plane . Due to a relatively small effective aperture of the XCRL the image propagator stays the gauss function of finite width instead of the delta-function. This leads to developing phase or edge enhanced imaging effects, which make visible transparent objects at the image plane. We found out that these edge-enhanced images are associated with the local phase gradient of the wave field Ao(x). Moreover, these images are sensitive to the sign of the phase gradient. This opens quite a new technique of microimaging for purely phase objects, which is different from the traditional phase contrast microimaging techniques.