Engineered-Disordered Photonic Systems

Matteo Burresiburresi@lens.unifi.it

Outline

• Introduction to light in disordered systems:Diffusion theory (where, when, what).

• Engineering disorder, Part 1 (mesoscopic optics):Random but not homogeneous systems.

• Engineering disorder, Part 2 (nanophotonics):Disordered but not random systems.

Light and Disorder

Cloud, snow, fog

The whiteness of disorder….

… ops, absorption can color disordered material

Bio-Disorder

Science (2007)

Luke, S. M. et al, Opt. Express 17, 14729 (2009).

Complex light transport

Speckle pattern from disordered

system

Diffraction from ordered systems

How would you tackle the problem?

?

Statistically!

Probability of light to be transmitted at a certain location

Discarding interference: The ‘walker’

The step length distribution, in close relation to Lambert-Beers law, is:

Akkermans, E. Physics of Electrons and Photons (2007).

Hulst, H. C. van de, Light Scattering by Small Particles. (1981).

Each scattering element is characterized by a scattering-cross section and a differential scattering-cross section:

For isotropic scattering, the scattering mean free path in a random and homogeneous system of n scatterers per unit volume is:

Monte Carlo Calculation of a fictitious random walker.

Light and Monte Carlo simulation: does it work?

Svensson et al, Optics Letters (2013)

YES!

It can also take into account anisotropic scattering in very thin random system

But is not a theory!

What does this look like?

Dunn, W. L. Exploring Monte Carlo Methods (2011)

Anisotropy factor

Transport mean free path

Transport cross section

Light as a Brownian particleUnder certain condition the energy density satisfies the diffusion equation:

Diffusion constant

Transport velocity

Dimension of the system

Two possible derivations:

Akkermans, E. Physics of Electrons and Photons (2007).

Mishenko, M. Multiple scattering of light by particles: radiative transfer and coherent backscattering (2006).

Radiative transfer approach:

•Conservation of energy flow inside a volume;

•Radiative transfer equation solved by expanding the solution in spherical

harmonics;

•The first order expansion gives diffusion equation;

Feynmann’s diagram approach:

•Calculating the Green’s function in vacuum from Maxwell equation;

•Iteratively summing up all scattered wave contribution;

•Important approximations;

Diffusion theory for light: does it work?

10/ tlL

1skl

Optically thick system (multiple scattering);

Optically dilute systems;

The system is HOMOGENEOUS and RANDOM;

Provided certain conditions are fulfilled:

R. Savo, Transport in complex heterogeneous photonic structures (2013)D. Contini et al, Appl. Opt., 36 45871(997).

Independent scattering (dilute systems);

Diffusion theory breaks down when interference effects are not negligible

Observation of Anderson Localization of light:Nature 390, 671 (1997) Nature Physics 4, 359 (2008)Nature Photonics 7, 48 (2012)

Sheng, P. Introduction to wave scattering, localization and mesoscopic phenomena (2006).

Halt of transport: Localization!

1sklOptically dense systems;

Questions?Let us summarize…..

In a HOMOGENEOUS diluted RANDOM system of isotropic scatterers Diffusion Theory describes properly light transport, completely disregarding the wave nature of light.The anisotropy (or finite size effect) of the single scatterer can be taken into account through Mie Theory.When diffusion theory breaks down Monte Carlo simulation can help.

Interference effects in highly dense material decrease the diffusion constant. A complete halt of transport is expected when the scattering mean free path is comparable with wavelength. (ELUSIVE OBSERVATION!)

Disorder Photonics Focus: Nature Photonics (2013)

Outline

• Introduction to light in disordered systems:Diffusion theory (where, when, what).

• Engineering disorder, Part 1 (mesoscopic optics):Random but not homogeneous systems.

• Engineering disorder, Part 2 (nanophotonics):Disordered but not random systems.

Homogeneous in nature? Realy?

Wu et al., Pharm. Sci. Tech. 9, 528-535 (2008)Svensson et al., Opt. Lett. 33, 80-82 (2008)

Cloudy Columns

A. Davis et al, ARM Science Team Meeting Proceedings (1999)

B. Davis & Marshak, Rep. Prog. Phys. 73, 026801 (2010)

Marble

Granular material

Paper

Central Limit Theorem supports Diffusion Theory for light

Svensson et al., PRE (2013)

For infinite heterogeneous disordered material Diffusion Theory (with some corrections) can still be applied.

Still a Random walk of light

?)( lp

The step length distribution is not exponential:

Holey random walk

And when the heterogeneity is of the same order of the system thickness?

Lévy glassAn engineered heterogeneous disordered material

SEM image300 um

)1()( pSphere diameter distribution

Power law distribution over 2 order of magnitudeFractal-like topology

P. Barthelemy et al., Nature (2008)Burresi et al, PRL 108, 110604 (2012)

Lévy glassA Lèvy walk for light

SEM image

300 um

)1()( llp

A controlled heavy tailed step length distribution

Consequences:Long jumps dominate transport (diverging variance);Superdiffusion (light spread faster than in the diffusive system);The exponent dictates the degree of superdiffusivity;Transport cannot be described by diffusion theory; Lévy walk

Diffusive walk

“Heavy” tailed distribution

The fractional propagatorFractional diffusion equation:

Laplacian operator spatially non-local → Difficulty in defining boundary conditions

Discretized version of the operator:

Matrix of transition probabilities(includes long jumps) Zoia, Rosso & Kardar, Phys. Rev. E 76, 021116 (2007);

Bertolotti et al, PRL 105, 163902 (2010)

Burresi et al, PRL 108, 110604 (2012)

The long steps make the propagator extremely cusped and heavy tailed….

Questions?Let us summarize…..

In an heterogeneous disordered system Diffusion Theory holds only for very thick materials by virtue of the Central Limit Theorem. For thickness comparable with the heterogeneities in which are present spatial correlations Diffusion Theory fails because the step length distribution is not exponential.The step length distribution depends on the type of heterogeneities. In particular, we study fractal-like engineered-disordered material (Levy Glass) in which the step length distribution has been imposed during fabrication.We find that the Fractional Diffusion Theory can properly describe transport in this systems.

Outline

• Introduction to light in disordered systems:Diffusion theory (where, when, what).

• Engineering disorder, Part 1 (mesoscopic optics):Random but not homogeneous systems.

• Engineering disorder, Part 2 (nanophotonics):Disordered but not random systems.

Random in nature? Realy?

Pair-correlation function

Yin et al, PNAS (2012)

Noh et al, Adv. Mat. (2010)

Garcia, Adv. Mat. (2009)

Donev, Science (2004)

Treacy, Science (2012)

Amorphous Silicon

Non-iridescent coloring

Correlations

When the correlation is strong and/or the correlation length is of the order of the wavelength, the phase lag between scattered wave does NOT average out.

Rojas-Ochoa et al, PRL (2004)

Structure factor

Modification of the scattering cross section due to the local distribution of the system

J. L. Yarnell et al, PR A (1973)

MacKintosh, F. C. & John, S. Phys. Rev. B (1989).

2D correlated-disordered system

Conley G. et al, soon submitted (2013)

Localization in correlated-disordered system

The introduction of correlations promote the formation of localized states.

The extent of the mode decreases as a function of the correlation.

Conley G. et al, soon submitted (2013)

Questions?Let us summarize…..

In a disordered DILUTED system in which are present spatial correlations Diffusion Theory can still be applied PROVIDED that corrections, which take into account the short-range ordered of the system, are applied to the scattering cross-section of the scatterers.When the density of scatterers is high, but not in the localization regime, this approach fails on the quantitative point of view: near-field effects (under investigation).Using the corrected scattering cross-section Monte Carlo simulation can still be employed.

When interference effects are not negligible, the introduction of correlation reduces the extent of the localized mode (under investigation).

Take-home message

We can manipulate disordered systems to control light transport;

By varying the KIND of disorder, which can also be found in nature, we can explore different regime of light transport;

This is important not only for fundamental physics but also for applications;

…and the fun just begun…. Now interference!