Abstract
The Fokker–Planck equation can be used in a partially-coherent imaging context to model the evolution of the intensity of a paraxial x-ray wave field with propagation. This forms a natural generalisation of the transport-of-intensity equation. The x-ray Fokker–Planck equation can simultaneously account for both propagation-based phase contrast, and the diffusive effects of sample-induced small-angle x-ray scattering, when forming an x-ray image of a thin sample. Two derivations are given for the Fokker–Planck equation associated with x-ray imaging, together with a Kramers–Moyal generalisation thereof. Both equations are underpinned by the concept of unresolved speckle due to unresolved sample micro-structure. These equations may be applied to the forward problem of modelling image formation in the presence of both coherent and diffusive energy transport. They may also be used to formulate associated inverse problems of retrieving the phase shifts due to a sample placed in an x-ray beam, together with the diffusive properties of the sample. The domain of applicability for the Fokker–Planck and Kramers–Moyal equations for paraxial imaging is at least as broad as that of the transport-of-intensity equation which they generalise, hence the technique is also expected to be useful for paraxial imaging using visible light, electrons and neutrons.
Highlights
The Fokker–Planck equation can be used in a partially-coherent imaging context to model the evolution of the intensity of a paraxial x-ray wave field with propagation
This is the key situation, of combined coherent and diffusive energy transport in a near-field paraxial imaging setting, that we wish to consider in the present paper
While we focus on the case of hard x rays, the methods considered here will be applicable to paraxial imaging using visible light, electrons, neutrons etc
Summary
The Fokker–Planck equation can be used in a partially-coherent imaging context to model the evolution of the intensity of a paraxial x-ray wave field with propagation. This is the key situation, of combined coherent and diffusive energy transport in a near-field paraxial imaging setting, that we wish to consider in the present paper. If there is unresolved spatially random micro-structure within the sample[11], as hinted in our opening paragraph, small-angle x-ray scatter[12] (SAXS) will be present This augments the previously mentioned coherent energy-flow vector at each point in space, with an ensemble (“SAXS fan”) of energy-flow vectors associated with diffusive energy transport. This equation can simultaneously model four effects, resulting from the illumination of a thin object by normally incident coherent x-ray plane waves:
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