1. Engineered-Disordered
Photonic Systems
Matteo Burresi
burresi@lens.unifi.it

2. 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.

3. Light and Disorder
Cloud, snow, fog
The whiteness of disorder….
… ops, absorption can color
disordered material

4. Bio-Disorder
Science (2007)
Luke, S. M. et al, Opt. Express 17, 14729
(2009).

5. Complex light transport
Speckle pattern from
disordered system
Diffraction from
ordered systems

6. How would you tackle the problem?
?
Statistically!
Probability of light to be transmitted at a
certain location

7. 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.

8. 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

9. Light as a Brownian particle
Under 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;

10. 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);

11. 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!
1skl
Optically dense systems;

12. 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)

13. 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.

14. 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

15. 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?

16. Lévy glass
An engineered heterogeneous disordered material
SEM image300 um
)1(
)(p
Sphere diameter distribution
 Power law distribution over 2 order of magnitude
 Fractal-like topology
P. Barthelemy et al., Nature (2008)
Burresi et al, PRL 108, 110604 (2012)

17. Lévy glass
A 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

18. The fractional propagator
Fractional 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….

19. 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.

20. 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.

21. 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

22. 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).

23. 2D correlated-disordered system
Conley G. et al, soon submitted (2013)

24. 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)

25. 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).

26. 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!