Polychromaticity in the combined propagation-based/analyser-based phase-contrast imaging

A theoretical analysis of the combined analyser-based/propagation-based x-ray phase-contrast imaging of weak objects is presented in the case of partially coherent illumination. It takes into account the finite size and divergence of a polychromatic x-ray source as well as the detector point-spread function. The proposed formalism describes the diffraction of x-rays in the monochromator and analyser crystals and free-space propagations between various elements of the imaging system. As in the case of a monochromatic plane wave incident on the object, the intensity in the image is expressed in terms of the amplitude and phase transfer functions. Analytical expressions for the new 'polychromatic' transfer functions are obtained and compared with the corresponding 'monochromatic' transfer functions by numerical simulations. The crucial role of the monochromator is elucidated in the formation of the analyser-based phase-contrast images with polychromatic illumination.