A Lévy flight of light

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1 A Lévy flight of light Diederik Wiersma European Lab. For Non-linear Spectroscopy (LENS) INFM-CNR, Univ. of Florence

2 Micro and nano photonics group European Lab. For Non-linear Spectroscopy (LENS) INFM-CNR, Univ. of Florence Pierre Barthelemy Jacopo Bertolotti Francesca Intonti Rajesh Kumar Lorenzo Pattelli Francesco Riboli Silvia Vignolini Radha Vivekananthan Kevin Vynck Stefano Cavalieri Marcello Colocci Stefano Lepri (CNR) Roberto Livi Roberto Righini Matteo Burresi Paola Costantino Stefano Gottardo Yanjun Liu Sushil Mujumdar Shunsuke Murai Riccardo Sapienza Costanza Toninelli

3 Scattering single scattering phase is maintained multiple scattering interference

4 Interference effects Speckle pattern in transmitted light

5 Schrödinger s equation ) ( ) ( ) ( r E r r V m h Ψ = Ψ + Maxwell s equations ) ( ) ( ) ( r E c r E c r ε ω ε ω = Analogies electrons - photons

6 Transport processes Light transport Electron transport Anderson localization Coherent backscattering (weak localization) Optical Bloch oscillations / Zener tunneling Universal conductance fluctuations Ohm s law

7 Coherent backscattering θ θ A B white paint gysum fog/clouds... coherent backscattering cone A - B close: A - B distant: 0 0 θ θ sum W λ/

8 Coherent backscattering Strongy scattering powders: Phys. Rev. Lett. 74, 4193 (1995) - TiO2 powder - Barium Sulphate

9 Transmission

10 Ohm s law Electrons L Resistance L Photons Resistance L L Diffusive systems linear thickness dependence

11 Interference effects Speckle pattern in transmitted light

12 Vortices in optical speckle Amplitude Phase 10 μm 0 Max 0 2π

13 Phase vortices

14 Vortex repulsion

15 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

16 Lévy flights

17 Google search: Levy flights

18 Processes with Lévy statistics Stock market fluctuations Animal foraging Human travel Turbulent flow

19 Stable distributions Linear combination of elements remains in distribution Gaussian is limiting case with finite mean Rest has heavy tail asymptotic behaviour: Pz ( ) 1 1+ z α + 1 0< α < 2 α = 1: Cauchy distribution Infinite variance and (for α 1) also infinite mean

20 Lévy α-stable distributions

21 Lévy walks for light waves

22 Disorder: photonic glass Nature Photonics 2, 429 (2008) With group C. Lopez, Madrid

23 Gaussian random walk Δx Δx from distribution with finite average and variance 2 σ Central limit theorem 2 σ D t Gaussian distribution Diffusion process: x 2 = D t with 1 D = 3 v

24 Lévy walk Δx Δx from stable distribution Generalized central limit theorem Generalized diffusion process: x 2 = D γ t with: γ = 3 α 1 α < 2 for γ =1: Normal diffusion γ >1: Super diffusion

25 How to make materials with non- Gaussian disorder?

26 Lévy walk for light Control step size = n 1 σ σ n cross section density Fractal particle size distibution? does not work due to Rayleigh (Mie) scattering Engineer local particle density!

27 Lévy glass TiO2 nanoparticles + Glass Spheres + Sodium Silicate (liquid glass) Glass and glass: index matched Glass Spheres: introduce the density fluctuations Diameter distribution: Pd ( ) 1 d 2 + α (range between 5 and 650 μm)

28 Lévy glass Glass Spheres determine density fluctuations TiO2 particles Lévy flight from multiple scattering on TiO2 particles Nature 453, 498 (May 22, 2008)

29 Sample design Diameter distribution voids: Pd ( ) 1 d 2 + α Step length distribution: Pz ( ) 1 1+ z α + 1 Super diffusion: x 2 = D γ t with: γ = 3 α 1 α < 2 for

30 How to sample this distribution? Diameter distribution voids: Pd ( ) 1 d 2 + α d between 0 and 100 μm in n steps d between 1 and 100 μm in n steps

31 Logarithmic discrete sampling E-3 P(z) 1E-4 1E-5 Discrete sampling Levy Walk α=1.1 1E-6 1E z

32 Experimental observations

33 Super diffusion Enhanced diffusive transmission Transmission α=2 : Diffusive transport α=0,948: Levy transport Generalised Ohm s law: T 1 = 1 + al α / Thickness (μm)

Density 6 4 2 2 Levy Case: enhanced fluctuations

34 Transmission profiles Diffusive case: small fluctuations Diffusive Levy Probability Density Levy Case: enhanced fluctuations R/R average I/I average

35 Enhanced spreading Diffusive Case: Almost-gaussian profile Lévy Case: Cusped spatial profile Experiments Levy Diffusive Transmission Distance (mm)

36 Dynamic properties 1E7 Ti-Sapphire OPO Levy Diffusive Intensity Delay Line Superdiffusive part BBO crystal Diffusive Tail Photomultiply Time

37 Monte Carlo simulations 30

38 Monte Carlo simulations Free space Lévy flight with step length: Pz ( ) 1 1+ z α + 0< α < 2 1

39 Brownian motion vs. Lévy flight Lévy flight Brownian motion α = steps

40 Monte Carlo simulations Fixed geometry (quenched disorder)

41 Placing spheres in a space

42 Placing spheres in a space

43 Monte Carlo simulations Random walk with quenched disorder

44 Quenched vs annealed disorder 1E Quenched disorder Annealed disorder <x 2 >(a.u.) Time (a.u.)

45 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..

46 Micro and nano photonics group European Lab. For Non-linear Spectroscopy (LENS) INFM-CNR, Univ. of Florence Pierre Barthelemy Jacopo Bertolotti Francesca Intonti Rajesh Kumar Lorenzo Pattelli Francesco Riboli Silvia Vignolini Radha Vivekananthan Kevin Vynck Stefano Cavalieri Marcello Colocci Stefano Lepri (CNR) Roberto Livi Roberto Righini Matteo Burresi Paola Costantino Stefano Gottardo Yanjun Liu Sushil Mujumdar Shunsuke Murai Riccardo Sapienza Costanza Toninelli

Optics of complex micro structures

Optics of complex micro structures dielectric materials λ L disordered partially ordered ordered random multiple scattering liquid crystals quasi crystals (Fibonacci) photonic crystals Assembly of photonic