Paraxial diffusion-field retrieval

Unresolved spatially random microstructure, in an illuminated sample, can lead to position-dependent blur when an image of that sample is formed. For a small propagation distance, between the exit surface of the sample and the entrance surface of a position-sensitive detector, the paraxial approximation implies that the blurring influence of the sample may be modeled using an anomalous-diffusion field. This diffusion field may have a scalar or tensor character, depending on whether the random microstructure has an autocorrelation function that is rotationally isotropic or anisotropic, respectively. Partial differential equations are written and then solved, in a closed-form manner, for several variants of the inverse problem of diffusion-field retrieval given suitable intensity images. Both uniform-illumination and structured-illumination schemes are considered. Links are made between the recovered diffusion field and certain statistical properties of the unresolved microstructure. The developed theory - which may be viewed as a crudely parallel form of small-angle scattering under the Guinier approximation - is applicable to a range of paraxial radiation and matter fields, such as visible light, x rays, neutrons, and electrons.