  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.
