Lecture I: Introduction to light scattering and...
Transcript of Lecture I: Introduction to light scattering and...
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)