Diffraction tomography for incident Herglotz waves

Diffraction tomography (DT) is an inverse scattering technique used to reconstruct the spatial distribution of the material properties of a weakly scattering object. The object is exposed to radiation, typically light or ultrasound, and the scattered waves induced from different incident field angles are recorded. In conventional DT, the incident wave is assumed to be a monochromatic plane wave, an unrealistic simplification in practical imaging scenarios. In this article, we extend conventional DT by introducing the concept of customized illumination scenarios, with a pronounced emphasis on imaging with focused beams. More specifically, we consider incident Herglotz waves and extend the classical Fourier diffraction theorem to this setting. This yields a new two-step reconstruction process which we comprehensively evaluate through numerical experiments.