1. Superdiffusion of waves
in random media
Lévy walks & interferences
Kevin Vynck
European Laboratory for Non-linear Spectroscopy (LENS)
University of Florence & Istituto Nazionale di Ottica (CNR-INO)
Florence, Italy

2. The superdiffusive people
Group “Optics of Complex Systems” Collaborations
www.complexphotonics.org
University of Florence (IT):
Group leader: Stefano Lepri
Diederik S. Wiersma Roberto Livi
Researcher: University of Parma (IT):
Jacopo Bertolotti Raffaella Burioni
Alessandro Vezzani
Post-doctoral fellows: Pierfrancesco Buonsante
Matteo Burresi
Tomas Svensson University of Bologna (IT):
Kevin Vynck Giampaolo Cristadoro
Mirko Degli Esposti
PhD students: Marco Lenci
Vivekananthan Radhalakshmi
Romolo Savo University of Twente (NL):
Allard Mosk
Former students: Thomas Huisman
Pierre Barthelemy (now in TU Delft, NL)
Lorenzo Pattelli

3. From Brownian motion to diffusion
Brownian random walk:
x≈ℓs Independent & small (finite variance) steps.
⇒ Central Limit Theorem
Classical diffusion equation
Mean square displacement

4. Common disordered materials
Gallium Arsenide Powder
Scattering mean free path
Density
Scattering cross-section
@ICMM, SP
Aluminum beads
Porous Gallium Phosphide
@ Uni. Twente, NL @Uni. Manitoba, CA

5. Beyond Brownian motion
x
Lévy random walk:
Strong fluctuations of x (diverging variance)
⇒ Generalized Central Limit Theorem
Heavy
-stable Lévy distribution tail
J. P. Nolan, Stable distributions – Models
for heavy-tailed data (Birkhauser, 2012).

6. Superdiffusive transport
Macroscopic displacement described by a
fractional diffusion equation:
Generalized diffusion process: with >1
=1.5
10 000 steps

7. Lévy flights and walks
Animal foraging
Shlesinger and Klafter (1986).
Stock market fluctuations
Mandelbrot (1963).
R. Klages, G. Radons, I. M. Sokolov,
Anomalous transport (Wiley-VCH, 2008);
Human mobility
Metzler & Klafter, Phys. Rep. 339, 1 (2000).
Brockmann et al., Nature 439, 462 (2006).

8. Superdiffusion of photons
Lévy glasses: Transparent spheres, with diameters varying over orders
of magnitude, embedded in a diffusive medium.
~ 500 m
Barthelemy et al., Nature 453, 495 (2008); Barthelemy et al.,
Bertolotti et al. , Adv. Funct. Mater. 20, 965 (2010). Phys Rev. E 82, 011101 (2010).

9. Fractional diffusion
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)
0 L
A. Zoia, A. Rosso, M. Kardar, Phys. Rev. E 76, 021116 (2007).

10. 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).

11. Interferences in random media
Intensity fluctuations due to interferences between multiply-scattered waves.
Speckle
Total complex amplitude of a multiply scattered wave given by the sum of the
complex amplitude of all possible trajectories.
Akkermans & Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge, 2007).

12. Phase coherence
Average over disorder
Two types of trajectories (for the amplitude) contribute to the backscattered intensity:
Incoherent contribution Coherent contribution
➢ For k+k'0, the intensity is twice higher than that predicted incoherently. B
Angular dependence of the reflected intensity  Coherent backscattering
➢ For closed trajectories (r1=r2), the coherent contribution survives for all k and k'.
Modification of the diffusive process  Weak localization

13. Superdiffusive coherent backscattering
Weak interference effects can be calculated in the superdiffusion approximation from
the Green's function retrieved semi-analytically.
Bertolotti et al., Phys. Rev. Lett. 105, 163902 (2010);
Burresi et al., arXiv:1110.1447v1 (2011).
Cone shape provides information on how waves propagate in the medium.

14. 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 properties
of disordered (classical or quantum) systems