Mean path length invariance in wave-scattering beyond the diffusive regime

Diffusive random walks feature the surprising property that the average length of all possible random trajectories that enter and exit a finite domain is determined solely by the domain boundary. Changes in the diffusion constant or the mean-free path, that characterize the diffusion process, leave the mean path length unchanged. Here, we demonstrate experimentally that this result can be transferred to the scattering of waves, even when wave interference leads to marked deviations from a diffusion process. Using a versatile microwave setup, we establish the mean path length invariance for the crossover to Anderson localization and for the case of a band gap in a photonic crystal. We obtain these results on the mean path length solely based on a transmission matrix measurement through a procedure that turns out to be more robust to absorption and incomplete measurement in the localized regime as compared to an assessment based on the full scattering matrix. In the diffusive transport regime, the mean path length of waves has been shown to be independent of the microstructure of the medium. Here, this invariance is shown to persist into the ballistic and localized regimes, as well as into the regime of band-gap materials like photonic crystals.