a lévy flight of light - società italiana di...
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Diederik Wiersma
European Lab. For Non-linear Spectroscopy (LENS)INFM-CNR, Univ. of Florence
www.complexphotonics.org
A Lévy flight of light
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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
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Scattering
single scattering
multiple scattering
phase is maintained
interference
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Interference effects
Speckle pattern in transmitted light
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Schrödinger’s equation )()()(2
22
rErrVm
hΨ=Ψ⎥
⎦
⎤⎢⎣
⎡+∇−
Maxwell’s equations )()()(2
02
2
22 rE
crE
cr εωεω
=⎥⎦
⎤⎢⎣
⎡−∇−
Analogies electrons - photons
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Transport processes
• Anderson localization
• Coherent backscattering (weak localization)
• Optical Bloch oscillations / Zener tunneling
• Universal conductance fluctuations
• Ohm’s law
Light transport Electron transport
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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
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Coherent backscattering
Strongy scattering powders:
- TiO2 powder- Barium Sulphate
Phys. Rev. Lett. 74, 4193 (1995)
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Transmission
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Ohm’s law
ElectronsL
Resistance ∝ L
Resistance ∝ LPhotons
L
Diffusive systems linear thickness dependence
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Interference effects
Speckle pattern in transmitted light
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Vortices in optical speckle
10 μm
Amplitude Phase
00 Max 2π
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Phase vortices
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Vortex repulsion
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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
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Lévy flights
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Google search: Levy flights
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Processes with Lévy statistics
Stock market fluctuations
Turbulent flowHuman travel
Animal foraging
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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
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Lévy α-stable distributions
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Lévy walks for light waves
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Disorder: photonic glass
With group C. Lopez, MadridNature Photonics 2, 429 (2008)
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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
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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
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How to make materials with non-Gaussian disorder?
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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
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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)
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Lévy glass
• Glass Spheres determinedensity fluctuations TiO2particles
• Lévy flight from multiple scattering on TiO2 particles Nature 453, 498 (May 22, 2008)
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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:
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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
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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
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Experimental observations
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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:
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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
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-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
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-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
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Monte Carlo simulations
30
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Monte Carlo simulationsFree space Lévy flight with step length:
1
1( )1
P zzα++
∼ 0 2α< <
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Brownian motion vs. Lévy flight
Brownian motion
10000 steps
Lévy flight
1=α
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Monte Carlo simulationsFixed geometry (quenched disorder)
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Placing spheres in a space
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Placing spheres in a space
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Monte Carlo simulations
Random walk with quenched disorder
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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
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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..
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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