Superdiffusion of wavesin random media

Kevin Vynck

European Laboratory for Non-linear Spectroscopy (LENS)University of Florence & Istituto Nazionale di Ottica (CNR-INO)

Florence, Italy

Lévy walks & interferences

Group “Optics of Complex Systems”www.complexphotonics.org

Group leader:Diederik S. Wiersma

Researcher:Jacopo Bertolotti

Post-doctoral fellows:Matteo BurresiTomas SvenssonKevin Vynck

PhD students:Vivekananthan RadhalakshmiRomolo Savo

Former students:Pierre Barthelemy (now in TU Delft, NL)Lorenzo Pattelli

Collaborations

University of Florence (IT):Stefano LepriRoberto Livi

University of Parma (IT):Raffaella BurioniAlessandro VezzaniPierfrancesco Buonsante

University of Bologna (IT):Giampaolo CristadoroMirko Degli EspostiMarco Lenci

University of Twente (NL):Allard MoskThomas Huisman

The superdiffusive people

From Brownian motion to diffusion

x≈ℓs

Brownian random walk:Independent & small (finite variance) steps.⇒ Central Limit Theorem

Mean square displacement

Classical diffusion equation

Common disordered materials

Gallium Arsenide PowderScattering mean free path

Density

Scattering cross-section

Porous Gallium Phosphide

Aluminum beads

@ICMM, SP

@ Uni. Twente, NL @Uni. Manitoba, CA

Beyond Brownian motion

xLévy random walk:Strong fluctuations of x (diverging variance)⇒ Generalized Central Limit Theorem

-stable Lévy distribution

J. P. Nolan, Stable distributions – Modelsfor heavy-tailed data (Birkhauser, 2012).

Heavytail

Superdiffusive transport

Generalized diffusion process: with >1

Macroscopic displacement described by a fractional diffusion equation:

=1.510 000 steps

Animal foragingShlesinger and Klafter (1986).

Stock market fluctuationsMandelbrot (1963).

Human mobilityBrockmann et al., Nature 439, 462 (2006).

Lévy flights and walks

R. Klages, G. Radons, I. M. Sokolov,Anomalous transport (Wiley-VCH, 2008);

Metzler & Klafter, Phys. Rep. 339, 1 (2000).

Superdiffusion of photons

Lévy glasses: Transparent spheres, with diameters varying over orders of magnitude, embedded in a diffusive medium.

Barthelemy et al., Nature 453, 495 (2008);Bertolotti et al. , Adv. Funct. Mater. 20, 965 (2010).

~ 5

00

m

Barthelemy et al.,Phys Rev. E 82, 011101 (2010).

Fractional diffusion equation:

Laplacian operator spatially non-local Difficulty in defining boundary conditions

Discretized version of the operator:

Matrix of transition probabilities (includes long-range correlation)

Fractional diffusion

A. Zoia, A. Rosso, M. Kardar, Phys. Rev. E 76, 021116 (2007).

0 L

Superdiffusive propagator

Green's function for the intensity in a 1D finite-size medium (steady-state):

J. Bertolotti, K. Vynck, D. S. Wiersma, Phys. Rev. Lett. 105, 163902 (2010).

Interferences in random media

Intensity fluctuations due to interferences between multiply-scattered waves.

Akkermans & Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge, 2007).

Total complex amplitude of a multiply scattered wave given by the sum of thecomplex amplitude of all possible trajectories.

Speckle

Phase coherence

➢ For k+k'0, the intensity is twice higher than that predicted incoherently. Angular dependence of the reflected intensity Coherent backscattering

➢ For closed trajectories (r1=r

2), the coherent contribution survives for all k and k'.

Modification of the diffusive process Weak localization

B

Incoherent contribution Coherent contribution

Average over disorderTwo types of trajectories (for the amplitude) contribute to the backscattered intensity:

Superdiffusive coherent backscattering

Bertolotti et al., Phys. Rev. Lett. 105, 163902 (2010);Burresi et al., arXiv:1110.1447v1 (2011).

Weak interference effects can be calculated in the superdiffusion approximation from the Green's function retrieved semi-analytically.

Cone shape provides information on how waves propagate in the medium.

Concluding remarks

Important note:

➢ Structures exhibiting spatial non-locality (long-range correlation), fractality. Relation with graphs, networks, …

Examples of interesting problems:

➢ Lévy flights for other types of waves?

➢ Phase transition for Anderson localization: extended/localized in 2D systems?

➢ Atom-photon interaction (cavity QED): modification of spontaneous emission, LDOS...;

➢ Mesoscopic physics: intensity correlations, conductance fluctuations;

See recent studies by Eric Akkermans in Technion (Haifa, Israel)

Anomalous transport to investigate some unique propertiesof disordered (classical or quantum) systems