Varenna, June 23, 24, 26, 2009

Georg Maret Dept. of Physics, University of Konstanz, Germany

Lecture I: Introduction to light scattering and interference

http://hera.physik.uni-konstanz.de

It´s all scattered light

Single scattering

Reflection

Multiple scattering

Outline – today:Introduction to light scattering and interference

Static and dynamic single light scattering

Scattering and transport mean free path

Diffusion approximation

Photon random walks and speckles

Basics of multiple light scattering

Outline – tomorrow:(Anderson) localization of light

Strong multiple scattering, Anderson Localization

Coherent backscattering and weak localization

Principle & physical pictureRecent experiments

CB, TOF, T(L), absorption

Quantitative analysis

Work in progress, outlook

Outline, Friday: Diffusing Wave Spectroscopy

Probing different motions:

Dynamic multiple scattering, DWS principle

Some medical applications

Brownian, shear, oscillatory,

Dynamic long range speckle correlations

Optical analog of Universal Conductance Fluctuations

Static and dynamic single light scattering

Sun

λ-4

Static single scattering of light

Rayleigh scattering,

Rayleigh scattering from one point particle

Scattered far field

Scattered intensity

Rayleigh scattering from two point particles1

2

incoherent average

Rayleigh scattering from N point particlesi

j

Incoherent intensity speckle

False color intensity

Rayleigh–Debye–Gans scattering from one particle with finite radius R

i

j

Elements inside particle

R

Form factor:

P(q) ~ Is(q) = I(q)

q

r

R

Example: homogeneous dielectric sphere

Example: arbitrary particles (Guinier’s expression)

Rm

Np volume elements

Scattering particle with arbitrary shape

RCM

form factor of an ensemble of arbitrarily oriented particles

Average over orientations

qssθ

Beyond Rayleigh–Debye–Gansapproximation: finite index particles

i

j

x Phase shift of internal wave

RDG criterion:

Mie-scattering: Incident wave, internal wave, reflected waveboundary conditions, solve Helmholtz wave equation

Example: water dropletSize parameter

Polarized depolarized

Example: bigger water dropletSize parameter

Example: Polystyrene sphere in water

The glory

Big water drops

Rayleigh scattering (λ >> R) is essentially isotropic

Mie and RDG scattering (λ < R) is mostly forward scattering

No flip+ - flip

+ -No flip

flip

Circ.pol.

General case:

dense & isotropic ensembles of particles:

Scattering volume

Fluctuations of ε

Scattering length of particle j

Interparticle interferenceOR

For all particles identical, , , and normalized to E02/R2

Structure factor

rewrite

Pair correlation function

jk

Example: Hard sphere S(q), Percus Yevick

Average over isotropic distribution of viz.

Total scattering cross section

Total transport cross section

Isolated Mie particleMie resonances

Typical correlation time of Is(q,t)

Moving particles: Dynamic light scattering

Time dependent phase

i

j

Normalized time autocorrelation function

For small τ:

For large τ:

Time autocorrelation function of scattered intensity I(q,t):

Time averaged intensity for uncorrelated particles

Time averaged intensity correlation function

Self motion

Autocorrelation function of scattered field

Siegert relation

Example: Diffusing (Brownian) particles with radius R0

Diffusion constant D0 = kBT/6πηR0

Distribution of time dependent displacements

ΔR(τ) = R(τ) – R(0)

Sphere radius R0

Particle sizing by DLS

Diffusing (Brownian) particles with hydrodynamic interactions

flow field

Beenakker, Mazur, Physica 126A, 349 (1984)

Basics of multiple light scattering

Light transport in random media

Obergabelhorn 4063m

Sun

Diffuse reflection, (unpolarized)

„white“

e.g.cloud, snow, milk...

Multiple scattering of light

Turbid medium

Light beamDirectbeam

multiply scattered (diffuse) light

Multiple scattering of electrons

dirty metal

e-

Drude conductance

e- hνinspiration

L >> l*

diffusingintensity

l*

ρσ∗

scattering mean free path

transport mean free path

bulkboundary boundary

incidentintensity I0

Transmitted direct beam

Diffuse transmission

decay length of unscattered beam

decay length of direction of intensity

“dilute” limit

Rayleigh scattering:

Rayleigh Debye Gans or Mie scattering:

Example: Polystyrene spheres in water

Example: Titania particles (rutile) in air

Hard sphere S(q)

Many ways to describe light transport in random media

EM wave equation:

Potential

+ +

Average Green´s function for field amplitude (Dyson)Average Green´s function for intensity (Bethe Salpeter)Diagrammatic expansionRadiation transfer theory………

Photon Random Walk

Diffusion equation

Infinite medium ( )

Semi-infinite half space (Method of images)

Semi-infinite slabL

Multiple images

Absorption (abs.length la)

(Ohm’s law)

(Beer Lambert´s law)

e.g. Watson et.al. PRL 58, 945 (1987).

Time of flight distributions

escape absorptionL

Outline:(Anderson) localization of light

Strong multiple scattering, Anderson Localization

coherent backscattering and weak localization

Principle & physical pictureRecent experiments

CB, TOF, T(L), absorption

Quantitative analysis

Work in progress, outlook

Coherent wave transport, interferences

Ei

Ej

Average intensity

Laser

Configurational average

Speckle

L

L

Speckle statistics

Angular size

Coherent backscattering and weak localization

Coherent backscattering

Interference between reversed path .......

.... is constructive in backscattering

..... whatever the path´s configuration! is destructive off backscattering, depending on phase shift (r)

r

double slit

One interference effect survivesconfigurational average!

2 1.5 1

I = ( + )

I = 4 E = 2 I

2

2

koh inkoh

E1 E2E1 E2

~(

kl*

)-1

|E |1 = =E|E |2

Ang

le

- any elastically scattered wave- any (disordered) medium- time reversal symmetry of wave propagation

CB occurs for:

Our first cones

P.E Wolf, G.M. July – August 85

M.P. van Albada, A.Lagendijk,

Very narrow cones for kl* >>1

BaSO4 -powder

Physics Today Dec. 1988

sample

θLaser

CCD

f

4o

Colloidal suspension

D.S. Wiersma, M.P. van Albada, B.A. van Tiggelen, A. Lagendijk, Phys.Rev.Lett. 74, 4193 (1995)

wider cones

Angular shape of CB-ConeMany contributions from different and

E. Akkermans et.al. J.Phys.France 49, 77 (1988)

Contributions for all at fixed s

cone shape with absorption

Cut off

0

Energy conservation in CB

S.Fiebig et.al.: EPL, 81, 64004 (2008)

(2π)−1

λ

cosθ

..and destroyscoherentbackscattering

F.Erbacher et al. Europhys Lett, 21, 551 (1993) R.Lenke et al. Eur.Phys.J E 17, 171 (2000)

A.A. Golubentsev Sov, Phys. JETP, 59, 26 1984

F.C. MacKintosh, S.John Phys.Rev.B 37, 1884 (1988)

Faraday effect brakes reciprocity of light propagation

Exotic cone shapes

R.Lenke et.al. EPL 52, 620 (2000)

Fit to Akkermans et.al. (1988)

Zhu et.al. PRA 44, 3948 (1991)

Internal reflections change the distribution of light at thesurface

n

CB and weak localization

lesstransmission(λ/l*)2

kl* >> 1

L L

Correction to Ohm’s law

2π/λ

λ

Strong multiple scattering & Anderson-Localization

metal – insulator transition

“Stopped” light

kl* ~ 1

critical regime

R.Lenke et.al. EPJB 26, 235 (2002)

Τ(t)

Time dependent diffusion coefficient D(t)

Simulations

critical regime

strong localization

Berkovits, KavehJ Phys C 2, 307 (1990).

Anderson Phil.Mag. 1985

long time tail

Scaling theory of localization

strong localization

classical, but absorption

D.Wiersma et.al. Nature 390, 671 (1997) F.Scheffold et.al. Nature 398, 206 (1999)

Distinction between localization and absorption ??

absorption + classical diffusionD = D0

J.M. Drake, A.Z. Genack Phys.Rev.Lett. 63, 259, 1989

Small but constant diffusion constant

Mie resonances cause long dwell times

Small D0 explained by low effective speed due to resonant scattering

M.v.Albada, B.v.Tiggelen, A.Lagendijk A.Tip, Phys.Rev.Lett. 66, 3132 (1991)

Measure D(t), l*, la, v independently on samples with small kl*

Time of flight T(t)

CB cone widthFWHM = 0.95 (kl*)-1

D0 = v l*/3

T(L)

• High refractive index (2.8 for titania in rutile structure)

• Particle size 220 – 540 nm (i.e. comparable to λ)

• Polydispersity not too high (15-20%)

• Low absorption ( la ~ 0.3 – 2.6 m)

• TiO2 powders – as commercially available fromDupont and Aldrich

1 μm

Strongly scattering samples: TiO2

S. Eiden, J.Widoniak

Very wide angle CB setup

• 256 photodiodes attached to an 180o arc• angles up to 65° in both directions• one shot measurements

P.Gross et.al., Rev.Sci.Instr., 78, 033105 (2007)

(kl*)-1

Large CB-cones: – indication of low kl*

Time of flight experiment, setup

•rep.rate ~ 1 MHz

For high kl*, perfect agreement with classical diffusion theory (Do = const).

kl* = 6.3

10 m !> 107 scattering events

Time dependent diffusion coefficient D(t) for lower kl*

Sample R700

kl* = 2.5

deviations fromclassicaldiffusion

M.Störzer et.al. Phys.Rev.Lett. 96, 063904 (2006)

Systematic dependence of the deviations on kl*

classical

No stratification or layering within sample

No fluorescence

Quantitative analysis

Fits of long time tails with power law D(t) ~ t -a

a ~ 1/3

kl* = 2.5 kl* = 4.3

tloc

a ~ 1

C.M.Aegerter et.al., Europhys.Lett. 75, 562 (2006)

Systematic dependence of Lloc on kl*

Lloc = (D0 tloc)1/2

Localization length Lloc = (D0 tloc)1/2

kl*c ~ 4.2

Critical exponent ν = 0.45 (10)

The localization t-exponent a

classicalcriticallocalized

Absorption

No systematic dependence of measured la on kl*

kl* = 25

no absorption, L-1

expected from absorption values as extracted from dynamic T(t)

directly measured T(L)

L/l*

(classical)

T(L)

kl* = 2.5

no absorption

with measured absorption (la)

with la and Lloc as measured from T(t)

direct T(L) data

T(L) decays by 12 orders over 2.5 mm !!

No adjustable parameter

localizing

Directly measured transport velocity, from D0 = v l*/3Mie – resonances at λ/2n ~ d, 2d …

“no” reduction for themost localizing sample !M.Störzer et.al.

Phys.Rev.E 73, 065602(R), 2006

BvT

ECPA

Outlook- More strongly localizing samples ?

- Nature of the transition from weak to strong localization ?

- Speckle statistics ?

- Applications ? Ultrahigh reflectance coatings ?

Lasing paints, random laser…?

………?

- Localizing other types of waves !Microwaves (in quasi 1D)

Sound waves

Cold atoms

……..?

Even better candidates for smaller kl* ?

Hollow TiO2 spheres

ECPA

S.Eiden et.al. J.Coll.Int.Sci 2002

R.Tweer PhD-Thesis KN 2002

Outline, Friday: Diffusing Wave Spectroscopy

Probing different motions:

Dynamic multiple scattering, DWS principle

Some medical applications

Brownian, shear, oscillatory

Dynamic long range speckle correlations

Optical analog of Universal Conductance Fluctuations

Dynamic multiple scattering, DWS principle

Diffusing Wave Spectroscopy(DWS)

Time dependent phase φ(t)

Ione speckle spot

Motion of scatterers

(t)

t

t

Theory of DWS of scatterers undergoing Brownian motion:

Do

s/l*

<τ0-1 > ≈ D0k0

2

φ0(τ)

k0

p(s): average weight distribution for different path lengths s

<φ02(τ)> = <q2 ΔR2(τ)> s/l*

Like QELS single scattering

ΔR

= <q2> <ΔR2(τ)> s/l*

~ k02 6D0τ

A random walk of the phase φ0(τ)

q and R are uncorrelated

Average relaxation of correlations for paths of lengths s

Contributions from cross terms i = k vanish on average since paths are uncorrelated

We take together all paths of length s-

Short time scale = long paths Very sensitive to displacements << λ

G.M., P.E.Wolf Z.Physik (1987)

Siegert relation

many applications of DWSParticle sizing, brownian motion

Flow, flow visualization

Acoustic modulation

DWS imaging

Shape fluctuations of vesicles, cells...

Dynamics of foams

Visco-elasticity (DWS echo)

Motions in complex media (sand jets)

Aging in colloidal glasses

Rotational motions

Particle sizing in colloidalsuspensions

o

G.M., P.E.Wolf 1990

Particle diameter

Brownian scatterers with hydrodynamic interactions

form factor !!no interactions

with interactions

Corrections due to interactions:

Short time self diffusion coefficient of colloidswith hydrodyn.interaction

S. Fraden, GM, Phys. Rev. Lett. 65, 515 (1990)

Beenakker Mazur

Γ

increasing Γ

D.Bicout, G.M. Physica A (1994)

Ω

White stuffTiO2 particles in H2O, l*=90μm

L=0.9mm

0

27

43

137

Γ (1/s)

DWS under shear motion<δφ0

2> = ( k Γ l* )2 t2 = ( t/τS) 2

measure shear rates

Acoustic speckle modulation

W.Leutz, G.M., Physica B, 1995

Ultrasound transducer(2MHz)

measure sound amplitudes

Multiple scattering „imaging“

Dynamic contrast,

(but turbidity and absorption match)

The idea:

x

x

absorber

shadow

Brownian motion

Shear flow, sound wavesDWS imaging:

(Boas, Yodh)

y

x flow

Laserto detector

DWS imaging

displacement y (l*) - 30 - 20 - 10 0 10 20 30

g

0.08

0.04

0.00

Δg

capillary

M.Heckmeier, G.M. Europhys.Lett. (1996) JOSA A (1997)

Capillary position

Brownian motion Same colloidalsuspension

x

y

y

x flow

Laserto detector

Intensity distribution imaging

speckle washedout here

M.Heckmeier, G.M. Opt.Comm. (1998)

teflon

Some medical applications

NIR window

Deep tissue optical imaging

Near-infrared imaging NIR Laser

zmax

Laser source Detector

x

z

y“banana”

NIR absorption imaging: perfusion mapping

O2 saturation increases

cortical activation

NIR absorption increases

stimulus

M. Wolf et al., Neuroimage, 16, 707 (2002)

Tomographic perfusion mapping in neonates

Hb concentration HbO concentration

increased PaCO2and PaO2

increased PaO2 at baseline PaCO2

J. Hebden et al., Phys. Med. Biol. 49, 1117 (2004)

32-channel time-of-flight setup (University College London):

1.5cm spatial resolution

changes of- total blood volume- oxygen saturation- scattering

Imaging with dynamic contrast

M. Atlan, et.al. JOSA A 24, 2701 (2007)

F.Ramaz et al, Pour la Science, December 2005

Acoustic tagging breast imaging

Ultrasound

Near Infrared Diffusing Wave Spectroscopy probing Neural Activity

J.Li, F.Jaillon, G.Dietsche, T.Gisler, T.Elbert, B.Rockstroh, G.M.

banana

fNMR

evokedpattern

J. Li et al., J. Biomed. Opt. (2005)

contralateral:stimulation vs. baseline

ipsilateral vs. baseline

Motor cortex stimulation

from laserto detector90 s10 10 s×

stimulus

data acquisition

Dcort

Quantitative test of the 3 layer model

(2+1) layer phantom:

- agreement between theory and experimentwithout adjustable parameters.

F. Jaillon et al., Opt. Express, 14, 10181 (2006)

10mmρ =

15mmρ =

20mmρ =

cortex

skull

scalp

ρ

Origin of functional DWS signal?

depolymerization of cytoskeleton

increased vesicle mobility

Ca2+ release

action potentiala) a) vasodilation

increased cortical blood flow rate

fast response (ms) slow response (~ 5s)

Scenario 1: neural coupling Scenario 2: hemodynamic coupling

b) action potential

axonal swelling

b) vasodilation

increased shear rate in cortical tissue

t

t

t

t

bloodflow/volume

stimulus

gate

corticaldiffusioncoefficient

perfusiondominated

neurallydominated

time-resolved detection:

Pulsation-synchronized DWS

radial artery:forearm:

J. Li et al., Opt. Express, 14, 7841 (2006)

source-receiver distance: 23 mm

diastole

systole

(mainly venous capillaries)

Enhancing sensitivity: Multispeckle correlation

32-channel correlator

average DWS signals from independent speckles:

Time-resolved multi-speckle DWS

very strong pulsatile variations of DWS signalphase lag between NIRS and DWS signals

systole diastole

index finger tip:

DWS

NIRS

G. Dietsche et al., Appl. Opt. 46, 8506, 2007

Venous pulsation

DWS

NIRS

Medial cubital vein (elbow),16mm source-receiverdistance

DWS detects blood flow even when blood volume changes aretoo small to be detected.

R L

2.8 x 2 .8 mm pixel size 1 .4 x 1 .4 mm pixel size

2.8 x 2.8 mm pixel sizespatially filtered on

1.4 x 1.4 mm pixel size Angiogram (MRA)

Spatial Resolution

t - values15105

MRI

occipital cortex response:

stimulation:

50s flickeringat 8Hz

16 mm

30 mm

10-6 10-5 10-4

-0.04-0.020.000.02

0.0

0.2

0.4

0.6

0.8

1.0

diffe

renc

e

lag time [s]

field

aut

ocor

rela

tion

func

tionstimulation

baseline

F. Jaillon et al., Opt. Express, 15, 6643 (2007)visual cortex signals are small:2% in g(1)(τ )

DWS from deeper cortical areas: visual cortex

- marker-free- non-invasive- fast (26 ms resolution)- highly sensitive- cm resolution now- portable- cheap- method for measuring cortical blood flow velocity

- origin of fast signals ?- ultimate imaging resolution ?

NIR DWS Imaging

DWS principle used in other fields: e.g. geophysics, acoustics

C1 = <I.I>

C2 long range correlations

S.Feng, C.Kane, P.A.Lee, A.D.Stone, Phys. Rev. Lett. 61, 834 (1988)

Long range speckle intensity correlations

speckles

DWS

Very long range correlations C3

(Universal Conductance Fluctuations)

DetektorLaser

Number of (speckle) modes

Frequency-frequency correlations

J. de Boer et.al, Phys.Rev.B 45, 658 (1992)

0.001 0.01 0.1 10

5

10

15

20 w=11.6 μm w=17.6 μm w=32.1 μm

(1/g

)*C

2(t) x

105

t (ms)

Laser457.9nmL

IS

PM Correlator PC

Absorber

Sample

Time correlations C2

F. Scheffold et.al. Phys.Rev.B 56, 10942 (1997)

LL11=50=50--150150μμm, L=13m, L=13μμm, Lm, L22=100=100--300300μμm, m, PinholePinhole--diameter D=4diameter D=4--3030μμm, l*=1.35m, l*=1.35μμm, m, particle size d = 300 nmparticle size d = 300 nm

0.001 0.01 0.1 1 10 1000

3

6

9

12

15

C(t)

x 1

05

t (ms)

1/g2

C3

0.01 1 1001

10

100

1000

C3

C'2

F.Scheffold, G.M.Phys.Rev.Lett.81, 5800 (1998)

Connection to localization?

Thanks

Pierre-Etienne Wolf

Eric Akkermans

Roger Maynard

Sergey Skipetrov

Dominique Bicout

Willi Leutz

Michael Heckmeier

Frank Scheffold

Ralf Lenke

Ralf Tweer

Stephanie Eiden

Johanna Widoniak

Christof Aegerter

Martin Störzer

Peter Gross

Susanne Fiebig

Wolfgang Bührer

Thomas Gisler

Jun Li

Franck Jaillon

Gregor Dietsche

Markus Ninck

Brigitte Rockstroh

Thomas Elbert

D.J.Pine, D.A.Weitz, G.Maret, P.E.Wolf, E.Herbolzheimer and P.M.ChaikinDynamic Correlations of Multiply Scattered Light,In ”Scattering and Localization of Classical Waves in Random Media”,Ed. P.Sheng, World Scientific, Singapore, (1990)

G. MaretRecent Experiments on Multiple Scattering and Localization of LightIn: Mesoscopic Quantum Physics, Les Houches Lecture Notes in Theoretical Physics, E. Akkermans, G. Montambaux and L. Picard Eds., Elsevier Sci. Publ., pp. 147-179 (1995)

R. Lenke and G. MaretMultiple Scattering of Light: Coherent Backscattering and TransmissionIn: W. Brown Ed., Gordon and Breach Science Publishers, Reading U. K., 1-72 (2000)

C.M. Aegerter and G. MaretCoherent Backscattering and Anderson Localization of Light. Progress in Optics 52 (2009)

Reviews

http://hera.physik.uni-konstanz.de

NIRS: fast optical signal

visual stimulation:

checkerboard reversal

M. Wolf et al.,Neuroimage 2002, 17, 1868

motor stimulation:

too fast for perfusion changesactivation-induced turbidity changes?

maps over C3hemoglobinconcentration (slow)

fast signal

No effect of sample thickness (or illuminationintensity) on existence of long time tail

Sample R700

L

Deviations fromclassical diffusion@ small kl*

ToF

L = 1 – 3 mm

10m !

107

scatteringsevents !

6.3 = kl*

4.3

2.5

Classical limit (with absorption)

Absorption? Long time upturn D(t) slowed down

Photons in dielectric random media “beat” e-

+ very long coherence lengths of lasers

+ clean beams (low divergence, many hν/mode)

+ no hν-hν interaction

+ elastic scattering

+ weak absorption, very high order scattering

+ no “contacts” needed

+ no mass

- weak scattering

Very narrow cones forkl* >>1

BaSO4 -powder

Colloidal suspension

M. v.Albada, A.Lagendijk PRL 55, 2692 (1985)

P.E.Wolf, G.Maret PRL 55, 2696 (1985) The glory

sample

θLaser

CCD

f4o

No systematic dependence of measured la on kl*

Absorption length behaves „as it should“ as a function of volume fraction

Raw data are deconvoluted with input pulse

input Raw T(t)