Localization in fractal and multifractal media

TL;DR

This paper studies wave propagation in one-dimensional media with fractal or multifractal density profiles, revealing conditions under which localization effects do not prevent wave transmission, with implications for various scientific and technological fields.

Contribution

It demonstrates that in weakly disordered fractal and multifractal media, wave localization depends on the fractal dimension, providing a universal criterion for localization in such systems.

Findings

Localization does not occur if fractal dimension D < 3/2.

The model confirms the scaling theory of localization.

Results have practical applications in material design and wave control.

Abstract

The propagation of waves in highly inhomogeneous media is a problem of interest in multiple fields including seismology, acoustics and electromagnetism. It is also relevant for technological applications such as the design of sound absorbing materials or the fabrication of optically devices for multi-wavelength operation. A paradigmatic example of a highly inhomogeneous media is one in which the density or stiffness has fractal or multifractal properties. We investigate wave propagation in one dimensional media with these features. We have found that, for weak disorder, localization effects do not arrest wave propagation provided that the box fractal dimension D of the density profile is D < 3/2. This result holds for both fractal and multifractal media providing thus a simple universal characterization for the existence of localization in these systems. Moreover we show that our model verifies the scaling theory of localization and discuss practical applications of our results.