Transport through modes in random media

Excitations in complex media are superpositions of eigenstates that are referred to as ‘levels’ for quantum systems and ‘modes’ for classical waves. Although the Hamiltonian of a complex system may not be known or solvable, Wigner conjectured that the statistics of energy level spacings would be the same as for the eigenvalues of large random matrices. This has explained key characteristics of neutron scattering spectra. Subsequently, Thouless and co-workers argued that the metal–insulator transition in disordered systems could be described by a single parameter, the ratio of the average width and spacing of electronic energy levels: when this dimensionless ratio falls below unity, conductivity is suppressed by Anderson localization of the electronic wavefunction. However, because of spectral congestion due to the overlap of modes, even for localized waves, a comprehensive modal description of wave propagation has not been realized. Here we show that the field speckle pattern of transmitted radiation—in this case, a microwave field transmitted through randomly packed alumina spheres—can be decomposed into a sum of the patterns of the individual modes of the medium and the central frequency and linewidth of each mode can be found. We find strong correlation between modal field speckle patterns, which leads to destructive interference between modes. This allows us to explain complexities of steady state and pulsed transmission of localized waves and to harmonize wave and particle descriptions of diffusion.