Diederik Wiersma

European Lab. For Non-linear Spectroscopy (LENS)INFM-CNR, Univ. of Florence

www.complexphotonics.org

A Lévy flight of light

Micro and nano photonics group

European Lab. For Non-linear Spectroscopy (LENS)INFM-CNR, Univ. of Florence

www.complexphotonics.org

Pierre BarthelemyJacopo BertolottiFrancesca IntontiRajesh KumarLorenzo PattelliFrancesco RiboliSilvia VignoliniRadha VivekananthanKevin Vynck

Stefano CavalieriMarcello ColocciStefano Lepri (CNR)Roberto LiviRoberto Righini

Matteo BurresiPaola CostantinoStefano GottardoYanjun LiuSushil MujumdarShunsuke MuraiRiccardo SapienzaCostanza Toninelli

Scattering

single scattering

multiple scattering

phase is maintained

interference

Interference effects

Speckle pattern in transmitted light

Schrödinger’s equation )()()(2

22

rErrVm

hΨ=Ψ⎥

⎦

⎤⎢⎣

⎡+∇−

Maxwell’s equations )()()(2

02

2

22 rE

crE

cr εωεω

=⎥⎦

⎤⎢⎣

⎡−∇−

Analogies electrons - photons

Transport processes

• Anderson localization

• Coherent backscattering (weak localization)

• Optical Bloch oscillations / Zener tunneling

• Universal conductance fluctuations

• Ohm’s law

Light transport Electron transport

Coherent backscattering

white paintgysumfog/clouds……...

θ

θ

A

B

θ

A - B close:

A - B distant:

0 θ

sum

0

W ∝ λ /

coherent backscattering cone

Coherent backscattering

Strongy scattering powders:

- TiO2 powder- Barium Sulphate

Phys. Rev. Lett. 74, 4193 (1995)

Transmission

Ohm’s law

ElectronsL

Resistance ∝ L

Resistance ∝ LPhotons

L

Diffusive systems linear thickness dependence

Interference effects

Speckle pattern in transmitted light

Vortices in optical speckle

10 μm

Amplitude Phase

00 Max 2π

Phase vortices

Vortex repulsion

Applications of light diffusion

• Medical diagnostics, imaging– caries in teeth– blood flow in tissue / brain functionality– optical mammography

• Diffusing Wave Spectroscopy (DWS)– dynamics of e.g. colloidal systems

• Visibility through fog in air/road traffic

• Random laser

Lévy flights

Google search: Levy flights

Processes with Lévy statistics

Stock market fluctuations

Turbulent flowHuman travel

Animal foraging

Stable distributions

• Linear combination of elements remains in distribution

• Infinite variance and (for ) also infinite mean1α ≤

1

1( )1

P zzα++

∼

• Rest has heavy tail asymptotic behaviour:

0 2α< <

1:α = Cauchy distribution

• Gaussian is limiting case with finite mean

Lévy α-stable distributions

Lévy walks for light waves

Disorder: photonic glass

With group C. Lopez, MadridNature Photonics 2, 429 (2008)

Gaussian random walk

Diffusion process: tDx ⋅=2 vD 31=with

xΔ

Central limit theorem

tD ⋅∝2σ Gaussian distribution

from distribution with finite average and variancexΔ 2σ

Lévy walk

xΔ

from stable distributionxΔ

Generalized central limit theorem

Generalized diffusion process: γtDx ⋅=2 αγ −= 3 21 <≤αwith: for

:1=γ :1>γNormal diffusion Super diffusion

How to make materials with non-Gaussian disorder?

Lévy walk for light

• Fractal particle size distibution?

• Engineer local particle density!

σ⋅=

n1• Control step size

cross section

density

σn

does not work due to Rayleigh (Mie) scattering

Lévy glass

TiO2 nanoparticles + Glass Spheres + Sodium Silicate (liquid glass)

Glass and glass: index matched

Glass Spheres: introduce the density fluctuations

Diameter distribution:2

1( )P dd α+≈

(range between 5 and 650 μm)

Lévy glass

• Glass Spheres determinedensity fluctuations TiO2particles

• Lévy flight from multiple scattering on TiO2 particles Nature 453, 498 (May 22, 2008)

Sample design

Diameter distribution voids:2

1( )P dd α+≈

γtDx ⋅=2 αγ −= 3 21 <≤αwith: for

1

1( )1

P zzα++

∼Step length distribution:

Super diffusion:

How to sample this distribution?

Diameter distribution voids:2

1( )P dd α+≈

between 0 and 100 μm ind stepsn

between 1 and 100 μm ind stepsn

Logarithmic discrete sampling

1 10 100 10001E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

P

(z)

z

Discrete sampling Levy Walk α=1.1

Experimental observations

Super diffusion

0 100 200 300 400 5000.0

0.2

0.4

0.6

0.8

1.0

α=2 : Diffusive transport α=0,948: Levy transport

Tran

smis

sion

Thickness (μm)

Enhanced diffusive transmission

/ 2

11

TaLα

=+

Generalised Ohm’s law:

Transmission profiles

0 1 2 3 4 5 60

2

4

0.0 0.5 1.0 1.50

2

4

6

8

10

Pro

babi

lity

Den

sity

R/Raverage

I/Iaverage

Diffusive Levy

Levy Case: enhanced fluctuations

Diffusive case: small fluctuations

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60.0

0.2

0.4

0.6

0.8

1.0

Levy Diffusive

Tran

smis

sion

Distance (mm)

Experiments

Lévy Case: Cusped spatial profileDiffusive Case: Almost-gaussian profile

Enhanced spreading

-200 0 200 400 600 800 1000

100000

1000000

1E7

Diffusive Tail

Inte

nsity

Time

Levy Diffusive

Superdiffusive part

Dynamic properties

Ti-Sapphire OPO

Delay Line

BBO crystal

Photomultiply

Monte Carlo simulations

30

Monte Carlo simulationsFree space Lévy flight with step length:

1

1( )1

P zzα++

∼ 0 2α< <

Brownian motion vs. Lévy flight

Brownian motion

10000 steps

Lévy flight

1=α

Monte Carlo simulationsFixed geometry (quenched disorder)

Placing spheres in a space

Placing spheres in a space

Monte Carlo simulations

Random walk with quenched disorder

Quenched vs annealed disorder

1 10 100 10001

10

100

1000

10000

100000

1000000

1E7

<x2 >(

a.u.

)

Time (a.u.)

Quenched disorder Annealed disorder

Summary

• Transport of light in random systems

• How to realize non-gaussian optical disorder: Levy glass

• Superdiffusion of light, possibility to study optical Levy flights

• Open questions:

- Unknown properties of Lévy flights (e.g. finite-size effects)

- Weak localization (coherent backscattering), strong localization, speckle correlations, etc..

Micro and nano photonics group

European Lab. For Non-linear Spectroscopy (LENS)INFM-CNR, Univ. of Florence

www.complexphotonics.org

Pierre BarthelemyJacopo BertolottiFrancesca IntontiRajesh KumarLorenzo PattelliFrancesco RiboliSilvia VignoliniRadha VivekananthanKevin Vynck

Stefano CavalieriMarcello ColocciStefano Lepri (CNR)Roberto LiviRoberto Righini

Matteo BurresiPaola CostantinoStefano GottardoYanjun LiuSushil MujumdarShunsuke MuraiRiccardo SapienzaCostanza Toninelli