Bridging Three Orders of Magnitude: Multiple Scattered Waves Sense Fractal Microscopic Structures via Dispersion

Wave scattering provides profound insight into the structure of matter. We address the question of whether macroscopic waves can report back the fractality of microscopic scatterers despite several orders of magnitude difference in scale between wavelength and scatterer size. We show in theory, experiments, and simulations that the resulting coherent superposition of multiple reflections gives rise to power-law dispersion at the macroscopic scale if the scatterer distribution exhibits fractality. Since fractality is naturally present in any random medium, microstructure can thereby leave its fingerprint on the macroscopically quantifiable power-law exponent, paving the way to characterize non-invasively the fractality of vasculature.