Measuring the fractal dimension of an optical random walk

Abstract

Random walks often grasp the essence of transport processes in complex systems, representing a model for a large variety of phenomena, from human travel, to molecular kinetics, to the propagation of light and sound in disordered media. Transport is generally driven by the topology of the system, which can range from a simply random distribution of scattering elements, to very rich self-similar structures like random fractals. In this context the fractal dimension of the random walk trajectory, $d_\mathrm{w}$, crucially determines the nature of the resulting transport process and provides information on the way the spatial evolution scales with time. In living cells and turbulent flow it has been possible to study anomalous dynamics showing $d_\mathrm{w}eq 2$. For wave transport phenomena, and light in disordered systems in particular, access to this fractal dimension has remained elusive up to today. Here, we report on the measurement of the fractal dimension $d_\mathrm{w}$ of the light trajectories in optically disordered media. We perform a series of ultrafast time-resolved transmission measurements on systems with different size and varying heterogeneity, and retrieve $d_\mathrm{w}$ by investigating how the lifetime scales with sample thickness. While, as expected, we find $d_\mathrm{w}=2$ for regular diffusive materials, disordered materials with an engineered self-similar (fractal) heterogeneity yield $d_\mathrm{w}<2$, indicating superdiffusive transport. This work provides a first experimental measurement of a fractal dimension describing anomalous wave propagation in disordered media, opening up a new possibility to investigate complex transport dynamics on fractal systems in a laboratory.