Experimentally mapping the scattering phases and amplitudes of a finite object by optical mutual scattering

Mutual scattering arises when multiple waves intersect within a finite scattering object, resulting in cross-interference between the incident and scattered waves. By measuring mutual scattering, we determine the complex-valued scattering amplitude $f$ - both amplitude and phase - of the finite object, which holds information on its scattering properties by linking incident and outgoing waves from any arbitrary direction. Mutual scattering is present for any coherent wave - acoustic, electromagnetic, particle - and we here demonstrate the effect using optical experiments. We propose an experimental technique for characterization that utilizes mutual scattering and we present our results for four distinct finite objects: a polystyrene sphere (diameter $59\ \mu$m), a single black human hair (diameter $92\ \mu$m), a strip of pultruded carbon (edge length $140\ \mu$m), and a block of ZnO$_2$ (edge length $64\ \mu$m). Our measurements exhibit qualitative agreement with Mie scattering calculations where the model is applicable. Deviations from the model indicate the complexity of the objects, both in terms of their geometrical structure and scattering properties. Our results offer new insights into mutual scattering and have significant implications for future applications of sample characterization in fields such as metrology, microscopy, and nanofabrication.