Paraxial diffusion-field retrieval. II. Fokker-Planck generalization of the transport-of-intensity equation

Abstract: The transport-of-intensity equation (TIE) has been very widely employed for phase retrieval. In particular, the TIE is an elliptic second-order partial differential equation which may be solved for the phase of a coherent paraxial field such as a monochromatic scalar optical beam, given the intensity and longitudinal intensity derivative in a plane perpendicular to the optical axis. We show how the coherent flow associated with the TIE may be augmented by a diffusive flow associated with a scalar or tensor diffusion field. Such diffusive flow can arise via scattering from unresolved spatially random microstructure within an illuminated sample, the blurring effects of an extended chaotic source that illuminates the sample, the resolution-reducing effect of shot noise in detected intensity images of the sample, and the sharpening effect (negative diffusion) associated with scattering from sharp sample edges. Augmenting the TIE's modeling of coherent flow with a diffuse-flow channel leads to a Fokker-Planck extension to this equation. Two different TIE augmentations are obtained, using three different derivations. The inverse problems of phase retrieval and diffusion-field retrieval are then studied, for both defocus-based imaging and mask-based imaging (structured-illumination imaging). When symmetric overfocus and underfocus images are employed for the purposes of phase retrieval, the diffusive term drops out and our Fokker-Planck formalism implies that any ensuing TIE-based phase-retrieval method needs no modification in light of our formalism. However, the same focal-series dataset -- which is typically an infocus image, a weakly overfocused image, and a weakly underfocused image -- may also be employed to access the additional channel of information associated with the Fokker-Planck diffusion field. Our formalism is applicable to visible light, x-rays, electrons, and neutrons.