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

Wave scattering provides profound insight into the structure of matter. Typically, the ability to sense microstructure is determined by the ratio of scatterer size to probing wavelength. Here, we address the question of whether macroscopic waves can report back the presence and distribution of microscopic scatterers despite several orders of magnitude difference in scale between wavelength and scatterer size. In our analysis, monosized hard scatterers $5\mu \mathrm{m}$ in radius are immersed in lossless gelatin phantoms to investigate the effect of multiple reflections on the propagation of shear waves with millimeter wavelength. Steady-state monochromatic waves are imaged in situ via magnetic resonance imaging, enabling quantification of the phase velocity at a voxel size big enough to contain thousands of individual scatterers, but small enough to resolve the wavelength. 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 apparent fractality over an effective length scale that is comparable to the probing wavelength. Since apparent fractality is naturally present in any random medium, microstructure can thereby leave its fingerprint on the macroscopically quantifiable power-law exponent. Our results are generic to wave phenomena and carry great potential for sensing microstructure that exhibits intrinsic fractality, such as, for instance, vasculature.

Figure 1

Characterization of lag-time distribution (2). (a) From particle to lag-time distribution and (b) verification via simulation.

Figure 2

Characterization of gel samples and concept of shear-wave dispersion experiment. (a). Density dependence of fractal dimension. The inset shows a confocal microscopy image of a gel sample. (b) Phase of the main shear-wave displacement component and corresponding displacement in the through-slice direction as measured via MRE within a single slice of the gel sample. The wavelength λ ($\sim 3\mathrm{mm}$) is comparable to the effective correlation length ${\zeta }_f^{1D}$ whereby apparent fractality can be probed. The third image shows the derived phase velocity $c_p$ from which mean values of $c_p$ are calculated (yellow circle). Note, that $c_p$ is nearly constant as expected for a sample that appears homogeneous at the macroscopic imaging scale. Mean values of $c_p$ acquired at different vibration frequencies represent the basis for the subsequent analysis exploring the exponent of the frequency power law (4th image).

Figure 3

Geometrically induced dispersion due to fractal structures. (a) Frequency dependence of the phase velocity for various scatterer densities. (b) Density dependence of the power law exponent $y$ for the experiment, simulation, and theory. Experimental data for scatterers of different diameters are in good agreement with our theory. Error bars are only shown when errors are larger than 1%.