Bart A. van Tiggelen

Fifty years of Anderson localization

What began as a prediction about electron diffusion has spawned a rich variety of theories and experiments on the nature of the metal–insulator transition and the behavior of waves—from electromagnetic to seismic—in complex materials.

Correlation of random wavefields: An interdisciplinary review

This paper presents an interdisciplinary review of the correlation properties of random wavefields. We expose several important theoretical results of various fields, ranging from time reversal in acoustics to transport theory in condensed matter physics. Using numerical simulations, we introduce the correlation process in an intuitive manner.We establish a fruitful mapping between time reversal and correlation, which enables us to transpose many known results from acoustics to seismology. We show that the multiple-scattering formalism developed in condensed matter physics provides a rigorous basis to analyze the field correlations in disordered media. We discuss extensively the various factors controllingandaffectingtheretrievaloftheGreen’sfunction of a complex medium from the correlation of either noise or coda.Acousticimagingofcomplexsamplesinthelaboratory and seismic tomography of geologic structures give a glimpse of the promising wide range of applications of the correlationmethod.

Monte Carlo simulation of multiple scattering of elastic waves

We study multiple scattering of elastic waves with a Monte Carlo method. We take into account the mode conversions and the polarization of the S waves. Some important physical parameters relevant to the description of the polarization are recalled, such as the definition and properties of the elastic Stokes vector. We briefly derive the scattering and Mueller matrices, as well as the differential and total scattering cross sections for one spherical inclusion embedded in a homogeneous matrix. The results of the single-scattering problem are used as a building block for multiple scattering. A Monte Carlo method to simulate the propagation of full elastic waves is presented. We pay a special attention to the convergence toward the diffusive regime which exhibits the equilibration of the P and S energy densities. Our simulations show the shear energy to become very rapidly dominant in the coda and the S to P energy density ratio to tend to 10.4 for a Poisson solid, as predicted by the equipartition theorem. However, the typical timescale and length scale to reach equipartition heavily depend on the scattering parameter kpa, where kp is the P wave number and a is the sphere radius. For Rayleigh scattering (kpa ≪ 1) we find a smooth evolution of energy density with time and a slow convergence toward the equilibration, mainly because of the large difference between the P and S scattering mean free paths in this case. On the other hand, for Rayleigh-Gans scattering (kpa ∼ 1.2, 1.6) a peak of energy associated with the forward scattered waves is observed, followed by a slow decay according to the diffusion approximation. We find that after only a few mean free times, equipartition is reached in spite of the strong anisotropy of the scattering in this regime. As the scattering parameter kpa increases, we find that equipartition is again delayed because the transport mean free paths become quite large. We find that a large source-station distance favors a rapid equilibration. This effect is seen to be very pronounced for Rayleigh scatterers. When a source of P waves is considered, the equipartition time can be twice as long as compared with a shear source. The time evolution of the Ep/Es ratio could be used as a marker for the different scattering mechanisms.

Observation of Multifractality in Anderson Localization of Ultrasound

We report the experimental observation of strong multifractality in wave functions below the Anderson localization transition in open three-dimensional elastic networks. Our results confirm the recently predicted symmetry of the multifractal exponents. We have discovered that the result of multifractal analysis of real data depends on the excitation scheme used in the experiment.

Resonant point scatterers in multiple scattering of classical waves

Abstract Both for scalar and for vector waves it is shown in a simple manner how the limit of point scatterers can be achieved for spherical scattering objects in three dimensions. Applications for multiple scattering are discussed.

Generalized diffusion equation for multiple scattered elastic waves

Abstract We study multiple scattering of elastic waves using a generalized diffusion approximation of the radiative transfer equation applied to spherically symmetric scatterers. This generalized diffusion equation allows us to keep track of the two elastic wave types as well as the mode conversions. It describes the process towards equipartition and fills, as such, a gap between the radiative transfer equation and the conventional diffusion approximation. The effects of boundary conditions and dissipation on the energy partitioning are studied.

Quasi-two-dimensional transfer of elastic waves.

A theory for multiple scattering of elastic waves is presented in a heterogeneous plate bounded by two flat free surfaces, whose horizontal size is infinite and whose transverse size is smaller than the mean free path of the waves. We derive a time-dependent, quasi-two-dimensional radiative transfer equation (i.e., two horizontal dimensions with a finite number of vertical mode) that describes the coupling of the eigenmodes of the layer (surface Rayleigh waves, shear horizontal waves, and Lamb waves). The fundamentally different element is that the traction-free boundary conditions are treated on the level of the wave equation, whereas at the same time elastic transfer can be considered over macroscopic horizontal distances. Expressions are found that relate the small-scale fluctuations to the lifetime of the modes and to their mode-coupling rates. We discuss the diffusion approximation that simplifies the mathematics of this model significantly, and which should apply at large lapse times. Finally, this model facilitates a study of coherent backscattering near the plate surface for different sources and for different detection configurations.

Relation between time reversal focusing and coherent backscattering in multiple scattering media: A diagrammatic approach

In this paper, we revisit one-channel time reversal (TR) experiments through multiple scattering media in the framework of the multiple scattering theory. The hyperresolution and the self-averaging property are retrieved. The developed formalism leads to a deeper understanding of the role of the ladder and most-crossed diagrams in a TR experiment and also establishes the link between TR and coherent backscattering (CBS). Especially, we show that when the initial source and the time reversal point are at the same location, the time-reversed amplitude is twice higher. Surprisingly, this enhancement is due to the ladder diagrams and not to the most-crossed ones, contrary to CBS. These theoretical predictions are confirmed by experimental results. The experiments are performed with ultrasonic waves propagating through a random collection of parallel steel rods.

Anderson Mobility Gap Probed by Dynamic Coherent Backscattering

We use dynamic coherent backscattering to study one of the Anderson mobility gaps in the vibrational spectrum of strongly disordered three-dimensional mesoglasses. Comparison of experimental results with the self-consistent theory of localization allows us to estimate the localization (correlation) length as a function of frequency in a wide spectral range covering bands of diffuse transport and a mobility gap delimited by two mobility edges. The results are corroborated by transmission measurements on one of our samples.

Reciprocity and coherent backscattering of light

The aim of this paper is to summarize, qualitatively and very globally, our present understanding of interference in multiple light scattering in disordered media, thereby hoping to bridge part of a gap between mathematical and physical approaches. A special emphasis is laid on the role of reciprocity in multiple light scattering, its relation to Coherent Backscattering in particular, and how to manipulate it with a magnetic field.