Statistical optics modeling of dark-field scattering in X-ray grating interferometers: Part 1. Theory

A grating-based Talbot-Lau X-ray interferometer provides three imaging modalities, namely attenuation, differential phase contrast, and dark field. Of these, dark-field imaging is uniquely capable of detecting and characterizing micron-scale fine structure in an object via small-angle scattering that reduces fringe visibility. Several empirical studies have been published showing the utility of this imaging modality for a wide range of applications. There also exists a more limited set of theoretical papers, based primarily on wave-optics formulations. In this two-part paper we present a comprehensive statistical optics model of the dark-field effect. In Part 1, we develop the theoretical underpinnings of the model with an emphasis on a scattering object comprising a random collection of microspheres, and in Part 2 [Opt. Express 29, 40917 (2021)10.1364/OE.447798], we provide a variety of example simulation results.