Coherent Backscattering from Multiple Scattering Systems - KOPS ...

Dissertation
Coherent Backscattering from
Multiple Scattering Systems
Susanne Fiebig

Susanne Fiebig:
Coherent Backscattering from Multiple Scattering Systems
Dissertation
zur Erlangung des akademischen Grades ‘Doktor der Naturwissenschaften’ (Dr. rer. nat.) an
der Universität Konstanz, Mathematisch-Naturwissenschaftliche Sektion, Fachbereich Physik
Referenten: Prof. Dr. Georg Maret, PD Dr. Christof M. Aegerter
Tag der mündlichen Püfung: 8.9.2010
Diese Arbeit wurde an der Universität Konstanz am Lehrstuhl von Prof. Dr. G. Maret durchgeführt
und durch die Deutsche Forschungsgemeinschaft (DFG), das Internationale Graduiertenkolleg
(IRTG) ‘Soft Condensed Matter of Model Systems’ und das Center for Applied Photonics
(CAP) des Landes Baden-Württemberg und der Universität Konstanz finanziert. Vielen
Dank auch an Sigma-Aldrich und DuPont für die kostenlose Bereitstellung eines Großteils
der in dieser Arbeit verwendeten Proben.

Ein kurzer Überblick
Streuung ist ein Phänomen, auf das man auf dem Gebiet der Wellenenausbreitung überaus
häufig stößt. Insbesondere unsere Wahrnehmung der Umwelt ist ganz wesentlich durch Streuung
geprägt. Kaum eine Welle – ob nun Lichtwelle, akustische oder sogar seismische Welle –
erreicht uns auf geradem Weg. Auch für uns nicht direkt wahrnehmbare Wellen wie Radiooder
Mikrowellen oder auch die als Wellen beschreibbaren Elektronen unterliegen in nicht
unerheblichem Maße der Streuung.
Trotzdem hat die Physik besonders im Bereich der Vielfachstreuung noch viele offene Fragen
zu beantworten. Einige davon betreffen die so genannte kohärente Rückstreuung, ein
Phänomen, das durch Interferenz bestimmter vielfach gestreuter Wellen entsteht. Anhand
von elektromagnetischen Wellen im Spektralbereich des sichtbaren Lichts lassen sich diese Interferenzen
sehr präzise untersuchen, da hier nur Absorption als rivalisierender Effekt auftritt,
und die experimentelle Realisierung zudem nicht besonders kompliziert ist.
Die kohärente Rückstreuung lässt sich mit den Modell einer Zufallsbewegung oder Random
Walks der mit der vielfach gestreuten Welle assoziierten Teilchen durch das streuende Medium
beschreiben. In diesem Modell kann man sehr einfach verstehen, dass zu jedem Teilchenpfad
auch seine Umkehrung existiert, bei der ein anderes Teilchen den selben Pfad in
umgekehrter Richtung durchläuft, wenn beide Enden des Pfades von der einfallenden Welle
erreicht werden.
Interferenzen von aus unterschiedlichen Pfaden austretenden Wellen sind zufällig, da sie auf
den verschiedenen Random Walks unterschiedliche Phasenverschiebungen erfahren. Im Gegensatz
dazu hängt das Interferenzmuster der an den beiden Enden eines zeitumgekehrten
Pfades austretenden Wellen grundsätzlich nur vom Abstand der beiden Endpunkte und der
Richtung der einfallenden Welle ab.
Handelt es sich bei dem zeitumgekehrten Pfad um einen geschlossenen (Teil-)Pfad innerhalb
des Mediums, so führt konstruktive Interferenz am Pfadausgang zu einer erhöhten
Aufenthaltswahrscheinlichkeit der Welle an diesem Ort und damit zu einer Verlangsamung
der Wellenausbreitung. Bei makroskopischer Besetzung solcher Pfadringe kommt es zu einem
vollständigen Zusammenbruch der Wellenausbreitung und damit zum Übergang in eine
lokalisierende Phase. Nach ihrem Entdecker P. W. Anderson wird diese als ‘Anderson-
Lokalisierung’ bezeichnet.
Liegen die beiden Endpunkte des Pfades dagegen in einem gewissen Abstand voneinander
an der Oberfläche des Mediums, so ist nur die Interferenz in Rückstreurichtung, also
in Richtung entgegengesetzt zur einfallenden Welle, grundsätzlich konstruktiv. Dies führt
bei Überlagerung der Interferenzmuster einer großen Anzahl solcher Pfade zu einer konusförmigen
Intensitätsüberhöhung um den Faktor zwei, die als kohärenter Rückstreukonus
bezeichnet wird.
Die Breite dieses Konus ist umgekehrt proportional zu der Schrittlänge des Random Walk, der
mittleren freien Transportweglänge l ∗ . Diese wiederum bestimmt beispielsweise, ob in einem

Ein kurzer Überblick
bestimmten Medium ein Übergang zur Anderson-Lokalisierung möglich ist. Dieser Übergang
sollte nach dem so genannten Ioffe-Regel-Kriterium in etwa dann stattfinden, wenn l ∗ von
der Größenordnung der Wellenlänge des gestreuten Lichts ist. Für die Charakterisierung vielfachstreuender
Proben ist es daher wichtig, den Rückstreukonus in Experiment und Theorie
korrekt abzubilden.
Leider enthalten die bisher meist angewandten experimentellen und theoretischen Methoden
kleine aber signifikante Ungenauigkeiten, die besonders bei den breiten Konen ins Gewicht
fallen, die nach dem Ioffe-Regel Kriterium in der Nähe des Übergangs zur Anderson-
Lokalisierung auftreten sollten. Ein Ziel der vorliegenden Arbeit war deshalb eine Verbesserung
der experimentellen Methodik im Einklang mit einer genaueren theoretischen Beschreibung
des Rückstreukonus, die von E. Akkermans (Technion Israel Institute of Technology,
Haifa, Israel) und G. Montambaux (Université Paris-Sud, Orsay, France) erarbeitet wurde.
Ausgangspunkt war dabei die Feststellung, dass sowohl gemessener als auch theoretisch
berechneter Konus das fundamentale Prinzip der Energieerhaltung zu verletzen scheinen.
Da der Rückstreukonus ein Interferenzphänomen ist, sollte die im Vergleich zu einer inkohärenten
Addition der Vielfachstreuung verstärkte Intensität in Rückstreurichtung durch
eine Intensitätsabschwächung bei größeren Streuwinkeln ausgeglichen werden. Diese Abschwächung
ist jedoch bisher weder im Experiment noch in der Theorie beobachtet worden.
Im Fall der Theorie ist dies nicht weiter verwunderlich, da eine ganze Reihe von Annahmen
und Näherungen verwendet werden. Akkermans und Montambaux erweiterten daher die
allgemein übliche Theorie durch zwei zusätzliche Terme. Diese führen an den Flanken des
Rückstreukonus zu einer Abschwächung der gestreuten Intensität unter das Niveau der inkohärenten
Addition der Rückstreuung, die die Intensitätsüberhöhung des Konus ausgleicht.
Bei dieser neuen theoretischen Bescheibung des Rückstreukonus ist damit die Energie erhalten.
Experimentell wird die Winkelverteilung der gestreuten Intensität mit dem so genannten
Weitwinkel-Setup bestimmt, in dem eine große Anzahl von Photodioden in einem Bogen
von 180 ◦ um die Probe angeordnet sind. Um die exakte Form des Rückstreukonusbestimmen
zu können müssen die Dioden korrekt kalibriert werden. Dazu wird eine Referenzprobe mit
extrem schmalem Konus verwendet, der Unterschied in den Albedos von Probe und Referenz
wurde dabei bisher jedoch nicht berücksichtigt. Der Schlüssel zu einer präziseren Messung
des Rückstreukonus war daher die Bestimmung dieser Albedos. Damit lässt sich nun auch in
den experimentellen Daten eine Intensitätsabschwächung an den Konusflanken beobachten,
die mit der Vorhersage von Akkermans und Montambaux übereinstimmt.
Ein weiterer Fokus der vorliegenden Arbeit lag auf dem Neuaufbau des so genannten Kleinwinkel-Setups,
mit dem die Verteilung der rückgestreuten Intensität mittels einer CCD-Kamera
in einem sehr engen Bereich um die Rückstreurichtung gemessen wird. Das Ziel war eigentlich,
Anderson-Lokalisierung durch die durch sie verursachte Abrundung der Spitze des
Rückstreukonus nachzuweisen. Dafür wurden eine ältere Kamera durch ein hochauflösendes
Modell mit einem größeren CCD-Chip ersetzt. Es stellte sich jedoch heraus, dass zwischen
der Probe und der Kamera platzierte optischen Komponenten zu viel Störlicht verursachen,
so dass die notwendige Intensitätsauflösung trotzdem nicht erreicht wird.
Dafür kann mit dem verbesserten Aufbau die freie Transportweglänge von schwach streuenden
Materialien wie beispielsweise Teflon gemessen werden. Da die dabei gemessene Transii

Ein kurzer Überblick
portweglänge sowohl mit den Ergebnissen früherer Experimente als auch mit der theoretischen
Vorhersage im Rahmen der Messgenauigkeit übereinstimmt, kann die Zuverlässigkeit
der Ergebnisse des Kleinwinkel-Setups als bestätigt angesehen werden.
Eine weitere Anwendung für das Kleinwinkel-Setup ergab sich im Rahmen einer Zusammenarbeit
mit der Gruppe von M. Schröter (Max Planck Institut für Dynamik und Selbstorganisation,
Göttingen). Hier sollte die freie Transportweglänge von Licht in so genannten fluidisierten
Betten bestimmt werden. Da die streuenden Teilchen in diesem Experiment sehr groß sind und
eine gleichmäßig sphärische Form haben, ist die rückgestreute Intensitätsverteilung durch die
Ringstruktur der Einfachstreuung an Mie-Teilchen überlagert. Diese lässt sich jedoch theoretisch
berechnen und an die gemessenen Kurven anpassen, so dass sich der Rückstreukonus
aus den Daten extrahieren lässt.
Bei ersten Messungen führte die Breite dieses Konus zu einer Transportweglänge, die wesentlich
kleiner als der Teilchendurchmesser der Streuer ist. Dies widerspricht eklatant den Ergebnissen
ähnlicher Experimente, die von Transportweglängen in der Größenordnung mehrerer
Teilchendurchmesser berichten. Der Grund hierfür ist bisher nicht bekannt; die grundlegende
Auswerteprozedur für Rückstreudaten von fluidisierten Betten konnte jedoch erfolgreich
getestet werden.
Für zukünftige Experimente stehen damit nun zwei verbesserte Experimentaufbauten zur
Verfügung, mit denen kl ∗ über einen Bereich von mehr als drei Zehnerpotenzen hinweg gemessen
werden kann. Mögliche Anwendungen reichen von neuen, maßgeschneiderten Proben
mit kl ∗ am Übergang zur Anderson-Lokalisierung zu Schäumen oder biologischem Gewebe.
Einige dieser Experimente sind bereits für die nähere Zukunft geplant, und es steht zu hoffen,
dass sie einen weiteren Schritt hin zu einem vollen Verständnis der Vielfachstreuung bilden
werden.
Danksagungen
Viele haben auf die eine oder andere Art zu dieser Arbeit beigetragen. Besonders bedanken
möchte ich mich bei
Prof. Dr. G. Maret – für die interessante Aufgabenstellung, eine zuverlässige und geduldige
Finanzierung, und für eine in jeder Hinsicht entspannte Arbeitsumgebung
PD Dr. C. M. Aegerter – der nach einigen Jahren tapferer Betreuungsarbeit schließlich doch
die Flucht ergriff und nach Zürich auswanderte
W. Bührer – unter anderem für sämtliche Time of flight-Messungen; die komplette Dank-
Liste würde hier leider den Rahmen sprengen. . .
Prof. Dr. E. Akkermans und Prof. Dr. G. Montambaux – für die Ausarbeitung der Theorie
zu den Experimenten an stark streuenden Proben
der P10-Stammbelegschaft in Sektretariat, Chemielabor, Elektrotechnik und Werkstatt – für
die überaus professionelle Unterstützung bei Problemen von ‘Anträge stellen’ bis ‘Zusammenlöten’
iii

Ein kurzer Überblick
meinen Büro- und Laborkollegen – die jetzt vermutlich mehr über meine schlechten Angewohnheiten
wissen als mir lieb sein kann
allen P10lern (samt den Exilanten auf Z10) – für alle Hilfsbereitschaft und die entspannte
und freundschaftliche Zusammenarbeit (und die dazugehörigen Kaffeepausen)
meiner Familie – für die uneingeschränkte Unterstützung in allen Lebenslagen
יהוה – für alles . . .
iv

Contents
Ein kurzer Überblick
i
Danksagungen
iii
Contents
vi
1 Introduction 1
2 Theory 3
2.1 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Single scattering – Mie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Random walk and diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 The influence of boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Photon flux from a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 On polarization and interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 The theory of coherent backscattering . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Setups 25
3.1 Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Wide Angle Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Small Angle Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Time Of Flight Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Samples 35
4.1 Sample characterization techniques . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 The samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Contents
5 Experiments 43
5.1 Conservation of energy in coherent backscattering . . . . . . . . . . . . . . . . . 43
5.2 The coherent backscattering cone in high resolution . . . . . . . . . . . . . . . . 56
6 Summary 69
Bibliography 74
Figures and Tables 75
MATLAB codes 79
Angular intensity distribution of single scattering . . . . . . . . . . . . . . . . . . . . . . 79
Evaluation of the wide angle data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Evaluation of the small angle data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
vi

1 Introduction
There have been many spectacular findings in the field of multiple scattering of light in random
media, from the theoretical work of P. W. Anderson in the 1950s [13] and its application
to electromagnetic waves [14, 28] to the discovery of the coherent backscattering cone some
30 years ago [31, 52, 57] and the resent find of the onset of Anderson localization [5, 6, 48].
This work however focusses on the equally important improvements of the experimental techniques
for the investigation of multiple scattering phenomenons.
Of particular interest were experiments on the coherent backscattering cone, an interference
effect that causes a twofold intensity enhancement in the direction opposite to the incoming
wave. Although it can be observed not only on visible light in the laboratory, but also
in space [24], on microwaves [18], seismic waves [34], sound waves [15], or the de Broglie
waves of electrons in metals [29], the laboratory experiments with visible light have two major
advantages over other systems: There are no rivaling effects like interaction between the scattered
particles or binding in deep minima of the random potential, except for absorption, and
the necessary technical effort is comparatively low. Multiple scattering with electromagnetic
waves in the visible range is therefore used as model system to experimentally investigate
multiple scattering of waves [14, 28].
From the shape of the coherent backscattering cone one can read all kinds of information
about the scattering process in the medium. The cone width for example is a measure for the
transport mean free path l ∗ and therefore for the turbidity of the medium. The shape of the
conetip gives evidence of absorption and in some cases even the transition to a localizing state.
The experiments however require highly sensitive setups and refined theoretical descriptions
to precisely depict the angular distribution of the backscattered light and to correctly fit the
scattering parameters.
This thesis reports the work on two experimental setups, one to record the backscattered
radiation over a wide angular range, the other for detailed measurement of the intensity
distribution close to backscattering direction. The theoretical basis is laid in sec. 2, where
the mathematical description of coherent backscattering is developed starting from the wave
equation. Along the way, some insight is also given in phenomena like single scattering at
Mie particles and Anderson localization. Sec. 3 gives technical information about the setups
used for the scattering experiments. These are not only the two backscattering setups this
work focusses on, but also a time of flight setup which is used to record the time-dependent
scattering in transmission. The next section presents the samples used in the experiments
alongside with some additional characterizing methods. Finally, sec. 5 reports the revision
process of the two setups. For the wide angle setup the evaluation procedure was refined,
and an improved theory of coherent backscattering was developed, while the small angle
setup was tested in experiments on strongly scattering titanium dioxide samples, on teflon as
an example of a weakly scattering material, and on water-fluidized beds of glass beads. The
MATLAB [2] codes used for the evaluations can be found in the appendix.

2
1 Introduction

2 Theory
In scattering theory, the mathematical and physical framework for studying and understanding
scattering events, the interaction of scattering particles with electromagnetic waves is described
as the solution of a particular partial differential equation, the so-called wave equation.
For many systems, like for example scattering of light at a single spherical particle, one can
solve this equation exactly. Multiple scattering samples however, with millions and billions of
randomly distributed scattering particles, have to be described by an approximate, collective
solution, as the exact solution can be obtained neither analytically nor numerically.
2.1 The wave equation
The scattering system we will consider in the following consists of randomly distributed scatterers
with dielectric permittivity ɛ scat in a surrounding medium with ɛ surr . We assume both
media to be non-magnetic (i.e. permeabilities µ scat = µ surr = 1) which corresponds to the usual
experimental conditions and does not bring any structural changes into the calculations.
The vector wave equations for a multiply scattered electromagnetic wave can be derived from
Maxwell’s equations as
∇ 2 ⃗ E(⃗r) +
ω 2
c 2 0
ɛ(⃗r)⃗ E(⃗r) = 0 and ∇
2 H(⃗r) ⃗ +
ω 2
ɛ(⃗r)⃗ H(⃗r) = 0
c 2 0
where ⃗ E(⃗r) and ⃗ H(⃗r) are the electric and magnetic field components, ɛ(⃗r) is either ɛscat or
ɛ surr , ω is the light frequency, and c 0 is the vacuum speed of light.
Instead of directly solving the above vector equations it is however convenient to find a solution
for the scalar wave equation
∇ 2 Ψ(⃗r) + ω2 ɛ(⃗r)Ψ(⃗r) = 0 (2.1)
c 2 0
As it will be demonstrated in sec 2.2, one can construct two vector harmonics M ⃗ = ⃗∇ × (⃗c Ψ)
and N ⃗ =
⃗∇× M ⃗
k
from this solution using a suitable pilot vector ⃗c. M ⃗ and N ⃗ are orthogonal
solutions of the vector wave equation, so the complete solution for the electric field is the
linear combination ⃗ E = AM ⃗ + BN, ⃗ and the magnetic field H ⃗ can be calculated from ⃗ E using
Maxwell’s equations.

2 Theory
The scalar wave equation 2.1 can be written in form of the inhomogenous differential equation
∇ 2 Ψ(⃗r) + k 2 Ψ(⃗r) = −V(⃗r)Ψ(⃗r)
where k 2 [ ]
= ω2 ɛ
c 2 surr , and V(⃗r) = ω2
0
c ɛ(⃗r) − ɛsurr is the scattering potential. a Its solution at a
2
0
certain point⃗r is given by
∫
Ψ(⃗r) = Ψ in (⃗r) +
G 0 (⃗r,⃗r 1 )V(⃗r 1 )Ψ(⃗r 1 ) d⃗r 1 (2.2)
where Ψ in is the part of the incoming wave that has not been scattered before. G 0 is termed the
bare Green’s function and describes the propagation of the electromagnetic field in a medium
without scatterers. It is defined by
∇ 2 G 0 (⃗r,⃗r 1 ) + k 2 G 0 (⃗r,⃗r 1 ) = −δ(⃗r,⃗r 1 )
and is given by
G 0 (⃗r,⃗r 1 ) = e−ik|⃗r−⃗r 1|
4π|⃗r −⃗r 1 |
By applying eqn. 2.2 recursively, the wave function can be expanded into a perturbation series
∫
Ψ(⃗r) = Ψ in (⃗r) +
G 0 (⃗r,⃗r 1 )V(⃗r 1 )Ψ in (⃗r 1 ) d⃗r 1 +
∫∫
+ G 0 (⃗r,⃗r 1 )V(⃗r 1 )G 0 (⃗r 1 ,⃗r 2 )V(⃗r 2 )Ψ in (⃗r 2 ) d⃗r 1 d⃗r 2 + · · · (2.3)
It would be convenient to split off the incoming wave Ψ in like Ψ(⃗r) = ∫ G(⃗r,⃗r ′ )Ψ in (⃗r ′ ) d⃗r ′ .
This introduces the total Green’s function G, which describes the electromagnetic field at a
certain position⃗r due to a disturbance at another point⃗r ′ . It has a perturbation series
∫
G(⃗r,⃗r ′ ) = G 0 (⃗r,⃗r ′ ) + G 0 (⃗r,⃗r a )V(⃗r a )G 0 (⃗r a ,⃗r ′ ) d⃗r a +
∫∫
+ G 0 (⃗r,⃗r a )V(⃗r a )G 0 (⃗r a ,⃗r b )V(⃗r b )G 0 (⃗r b ,⃗r ′ ) d⃗r a d⃗r b + · · ·
and the formal definition
∇ 2 G(⃗r,⃗r ′ ) + ω2 ɛ(⃗r)G(⃗r,⃗r ′ ) = −δ(⃗r,⃗r ′ )
c 2 0
[a] Elsewhere [16, 56], the scattering potential is defined as V(⃗r) = ω2 [ɛ
c 2 surr − ɛ(⃗r)] (or similar), so that the wave
0
equation becomes ∇ 2 Ψ(⃗r) + k 2 Ψ(⃗r) = V(⃗r)Ψ(⃗r). This definition makes the perturbation series eqn. 2.3 look less
intuitive as half of the integrals seem to be subtracted.
4

2.2 Single scattering – Mie theory
Figure 2.1: Coordinate system of single scattering. The coordinate system (θ, φ) of
Mie scattering at a spherical particle with radius a is defined by the wave vector ⃗ kin
and the electric field vector ⃗ Ein of the incoming light wave. The wave vectors of the
incoming and outgoing waves⃗ kin and⃗ kout span the scattering plane.
The scattered light intensity at a certain point⃗r must then be
∫∫
I(⃗r) ∝
G(⃗r,⃗r 1 )G ∗ (⃗r,⃗r 2 )Ψ in (⃗r 1 )Ψ ∗ in(⃗r 2 ) d⃗r 1 d⃗r 2
For a dilute system with pointlike scatterers the perturbation series of eqn. 2.3 immediately
justifies treating multiple scattering of electromagnetic waves as random walks of photons
with different numbers of scattering events: The photons travel in free space (described by
Green’s functions G 0 , which have the form of spherical waves) until they hit a particle and are
scattered into the surrounding space, where they again propagate freely, hit another particle,
and so on.
The random walk picture can still be upheld if the size of the particles is not negligible.
However, in this case one needs to consider how the particles distribute the incoming intensity
into their surrounding to describe the random walk properly. If a spherical particle can be
considered as an acceptable approximation for the actual particle shape, one can use the
solution given by G. Mie [42] and others. In the next section, we will follow the approach of
[16] to derive the distribution of the scattered light around a single particle.
2.2 Single scattering – Mie theory
The problem of an electromagnetic wave scattered by a spherical particle clearly has a spherical
symmetry. Therefore it is convenient to treat the problem in a polar coordinate system
with the scattering particle of radius a located in the origin, and wave vector and polarization
of the incident light defining the angular coordinates θ = 0 and φ = 0. The wave equation in
polar coordinates
1
r 2
(
∂
r 2 ∂Ψ )
+ 1
∂r ∂r r 2 sin θ
(
∂
sin θ ∂Ψ )
+
∂θ ∂θ
1
r 2 sin 2 θ
∂ 2 Ψ
∂φ 2 + k2 Ψ = 0
5

2 Theory
can be separated into three independent differential equations using the ansatz Ψ(r, θ, φ) =
R(r) · Θ(θ) · Φ(φ):
1
R
(
d
dr
sin θ
Θ
r 2 dR )
+ k 2 r 2 = α
dr
(
sin θ dΘ )
= β − α sin 2 θ
dθ
d
dθ
1 d 2 Φ
Φ dφ 2 = −β
Setting β = m 2 and α = n(n + 1) where m = 0, 1, 2, . . . and n = m, m + 1, . . . we obtain the
complete solution Ψ for the wave equation, and from this the vector harmonics ⃗ M = ⃗∇ × (⃗rΨ)
and ⃗ N =
⃗∇× ⃗ M
k
.
The solution of the radial part of the wave equation R(r) is given by a linear combination of
the spherical Bessel functions j n (ρ) = √ π
2ρ J n+1/2(ρ) and y n = √ π
2ρ Y n+1/2(ρ) where J ν (ρ) and
Y ν (ρ) are Bessel functions of first and second kind, and ρ = kr. For the outgoing scattered
wave the appropriate linear combination is given by one of the spherical Bessel functions of
the third kind or spherical Hankel functions, h (1)
n (ρ) = j n (ρ) + iy n (ρ).
The zenith part of the wave equation Θ(θ) is solved by associated Legendre functions Pn m (cos θ).
For the following calculations it is convenient to define the angle-dependent functions π n =
Pn
1
sin θ and τ n = dP1 n
dθ .
It can be shown that the solution for the scattered electric field is given by
⃗ Escat =
∞
∑ i n 2n + 1
)
E 0
(ia nNn ⃗ − b nMn ⃗
n(n + 1)
n=1
where the applicable vector harmonics are given by
⃗M n = cos φ π n (cos θ) h (1)
n (ρ) ê θ − sin φ τ n (cos θ) h (1)
n (ρ) ê φ
⃗N n = cos φ n(n + 1) sin θ π n (cos θ) h(1) n (ρ)
ρ
ê r + cos φ τ n (cos θ) [ρh(1) n (ρ)] ′
ρ
ê θ −
− sin φ π n (cos θ) [ρh(1) n (ρ)] ′
ρ
ê φ
and the coefficients a n and b n are
a n = m ψ n(mx) ψ ′ n(x) − ψ n (x) ψ ′ n(mx)
m ψ n (mx) ξ ′ n(x) − ξ n (x) ψ ′ n(mx)
; b n = ψ n(mx) ψ ′ n(x) − m ψ n (x) ψ ′ n(mx)
ψ n (mx) ξ ′ n(x) − m ξ n (x) ψ ′ n(mx)
6

2.2 Single scattering – Mie theory
Figure 2.2: Vibration ellipse. The tip of the electric field vector of a light wave traces
out an ellipse with semimajor axis r 1 , semiminor axis r 2 , azimuth γ, and ellipticity
η = arctan(r 2 /r 1 ).
with the size parameter x = ka, the relative refractive index m = k scat
k
, and the Riccati-Bessel
functions ψ n (z) = z j n (z) and ξ n (z) = z h (1)
n (z). The prime indicates differentiation with
respect to the argument in parentheses.
⃗M has no radial component, and in the far-field limit with large ρ = kr the radial component
of N ⃗ is negligible compared to the transverse component, as the spherical Hankel function is
asymptotically given by h (1)
n (kr) ≈ − (−i)n e ikr
ikr
. The scattered electromagnetic field is therefore
basically transverse, and
E θ ≈ E 0 ·
E φ ≈ −E 0 ·
e ikr
−ikr · cos φ · S 2(cos θ) with S 2 =
e ikr
−ikr · sin φ · S 1(cos θ) with S 1 =
n c
∑
n=1
n c
∑
n=1
2n + 1
n(n + 1) (a nτ n + b n π n )
2n + 1
n(n + 1) (a nπ n + b n τ n )
Both sums converge, so that the series can be terminated after n c steps without causing major
inaccuracies. A good assumption is n c ≈ x.
We now focus on a certain scattering plane, which is defined by the wave vectors of the
incoming and emerging waves. Intensity and polarization of the scattered light are then
merely a function of the scattering angle θ and the polarization of the incoming light with
respect to the scattering plane.
In a non-absorbing medium, both incoming and scattered electromagnetic waves can be characterized
by a Stokes vector [30]
⎛ ⎞ ⎛
⎞
I
1
⃗p = ⎜Q
⎟
⎝U⎠ = ⎜cos(2η) cos(2γ)
⎟
⎝cos(2η) sin(2γ) ⎠
V
sin(2η)
7

2 Theory
45
ellipticity / azimuth [deg]
30
15
0
−15
−30
azimuth ellipticity
−45
−180 −135 −90 −45 0 45 90 135 180
scattering angle [deg]
Figure 2.3: Polarization of a scattered wave. The polarization of the scattered wave –
which can be described by azimuthal angle γ and ellipticity η – fluctuates strongly as
a function of the scattering angle theta. The example shows the scattering of a linearly
polarized wave with wavelength λ = 575 nm and azimuth γ = 45 ◦ on a spherical
particle with diameter 2a = 300 nm and refractive index n = 2.7.
The ellipticity of the vibration ellipse (fig. 2.2) is then given by | tan(2η)| = U Q
, the azimuthal
V
angle by tan(2γ) = √Q . 2 +U 2
The transformation between the Stokes vectors of the incident and the scattered light waves is
described by so-called Mueller matrices [43]. The Mueller matrix of a spherical particle is
⎛
⎞
S 11 S 12 0 0
S(cos θ) = 1
⎜S 21 S 22 0 0
⎟
k 2 r 2 ⎝ 0 0 S 33 S 34
⎠
0 0 S 43 S 44
where
S 11 = S 22 = 1 2 (S 2S2 ∗ + S 1S1 ∗)
S 12 = S 21 = 1 2 (S 2S2 ∗ − S 1S1 ∗)
S 33 = S 44 = 1 2 (S∗ 2 S 1 + S 2 S1 ∗)
S 34 = −S 43 = i 2 (S∗ 2 S 1 − S 2 S1 ∗)
As shown by the example in figs. 2.3 and 2.4, in the Mie scattering regime with wavelength
λ ≈ a the polarization of the scattered light and the scattered intensity itself are very unevenly
distributed around the scattering particle. Generally, one can recognize a transition from the
isotropic intensity distribution of Raleigh scattering in the regime λ ≪ a to a significantly
enhanced scattering in forward direction in the Mie regime.
The total scattering cross section
C scat = 2π ∞
k ∑ 2 (2n + 1) ( |a n | 2 + |b n | 2)
n=1
which is a measure of the scattering efficiency, also fluctuates strongly when the particle
diameter becomes of the order of the wavelength (fig. 2.4). For nearly monodisperse scatterers
these so-called Mie resonances can be used to create especially strongly scattering samples.
8

2.3 Random walk and diffusion
scattering cross section [m 2 ]
10 −12
10 −14
90 1
120
60
150
0.5
30
180 0
210
330
10 −10 particle radius [m]
10 −7 10 −6
240
270
300
Figure 2.4: Scattering cross section and scattering anisotropy. Scattering of a light
wave with wavelength λ = 575 nm on spherical particles with refractive index n = 2.7.
The scattering cross section (left) fluctuates strongly when the particle radius becomes
of the order of the wavelength. The polar plot of the relative intensity distribution
(right) shows a transition from isotropic Rayleigh scattering to strongly anisotropic Mie
scattering with the incident light wave impinging from the left on particles of radius
a = 30 nm (dark blue), a = 300 nm (blue), and a = 3 µm (light blue). The calculations
were performed with [3].
Strong polydispersity washes out the most prominent features, but the scattering cross section
still depends on the ratio between particle radius and wavelength of the scattered light.
With the help of the scattering cross section one can establish an anisotropy parameter, which
describes the anisotropy of the intensity distribution: b
〈cos θ〉 = 1 ∫
dC scat
· · cos θ dΩ =
C scat 4π dΩ
where dC scat
dΩ
= 4π2 a
x 2 C scat
[∑
n
n(n + 2)
n + 1
R{a na ∗ n+1 + b n bn+1} ∗ + ∑
n
]
2n + 1
n(n + 1) R{a nbn}
∗
denotes the differential scattering cross section of the particle.c
2.3 Random walk and diffusion
As it was pointed out in sec. 2.1, the movements of the photons through a multiple scattering
sample can be modeled by random walks of photons from one scattering site to the next. To
[b] R{z} = real part of the imaginary number z
[c] The differential scattering cross section is denoted only symbolically as dC scat
dΩ . It should not be interpreted as
the derivative of a function of Ω.
9

2 Theory
Figure 2.5: Random walk. The vector⃗r denotes the displacement of a photon after M
steps ∆ri ⃗ , where θ i is the angle between two consecutive steps ∆ri−1 ⃗ and ∆ri ⃗ .
characterize the spreading of a cloud of photons that started off at a certain point and a certain
time that we set⃗r = 0 and t = 0 we use the mean square displacement
〈r 2 (t)〉 = 1 N
( )
N M(t) 2
∑ ∑
⃗∆r m,n
n=1 m=1
where M(t) is the number of steps ⃗ ∆r the photons have traveled after a certain time t, and N
is the number of photon paths considered (fig. 2.5). For large M(t) this is [38]
〈r 2 (t)〉 ≈ M(t) 〈 ⃗ ∆r
2
〉 + 2 〈 ⃗ ∆r〉
2
M(t) 〈cos θ〉
1 − 〈cos θ〉
where θ is the angle between two photon steps, and 〈cos θ〉 expresses the anisotropy of the
scattering.
For an exponential step length distribution p(∆r) = 1 ∆r
l
e− l we obtain 〈∆r〉 = l and 〈∆r 2 〉 = 2l 2 ,
and
〈r 2 (t)〉 = 2 M(t) l ·
l
1 − 〈cos θ〉 ≡ 2 M∗ (t) l ∗2 ≡ 2s(t)l ∗ (2.4)
Hence the mean square displacement can be expressed with the help of three different length
scales.
The first length scale is the scattering mean free path l, which characterizes the distribution of
the physical step lengths from one scattering site to the next.
The angular distribution of single scattering given by Mie theory correlates the directions of
two successive photon steps. In the mean square displacement this correlation is expressed
by an additional factor (1 − 〈cos θ〉) −1 . The transport mean free path l ∗ l
= and the
1−〈cos θ〉
effective number of photon steps M ∗ = M(t)(1 − 〈cos θ〉) incorporate this anisotropy factor. l ∗
is therefore the distance after which the photons have lost the memory of their initial direction
10

2.3 Random walk and diffusion
of propagation. The difference between transport mean free path and scattering mean free
path is the larger the more anisotropic the scattering is. For completely isotropic scattering
both mean free paths are equal.
The product M ∗ l ∗ gives the length s of the photon paths that contribute to the mean square
displacement. As there is no reason to assume that the speed of the photons will change
along their path, the length of a photon path is proportional to time. The size of the photon
cloud 〈r 2 (t)〉 at a certain time is therefore linearly proportional to both the length s of the
contributing photon paths and the time t the photons have spent traveling along these paths.
On top of everything, the mean square displacement of eqn. 2.4 is structurally equivalent to
the variance
〈r 2 (t)〉 = 6Dt (2.5)
of a gaussian distribution in three dimensions
ρ(⃗r, t) =
1
r2
√ e− 4Dt (2.6)
3
4πDt
where ρ is the density distribution of photons that started off at the origin of the coordinate
system at time t = 0 in an infinitely extended medium, and D is the diffusion coefficient. So
obviously multiple scattering in the limit of large M(t) can also be described as diffusion of
light energy through the medium with the diffusion equation ∂ρ
∂t − D∇2 ρ = δ(t)δ(⃗r), whose
solution is given by eqn. 2.6
Comparing eqns. 2.4 and 2.5 one can give the diffusion coefficient as
D = sl∗
3t = vl∗
3
(2.7)
where v is the velocity of energy transport. Using this, the photon density distribution can
also be given as a function of the path length s:
ρ(⃗r, s) =
√
3
4πsl ∗ 3
e − 4
r2
3 sl∗ (2.8)
However, the photon density distribution in a multiply scattering sample is not purely determined
by the scattering properties of the scattering particles, but also by energy losses due to
absorption in the material. According to Lambert-Beer’s law, absorption weakens the intensity
of a light wave exponentially along the traveled path, so that its effect can be included
in the photon density distribution by an additional factor e − la
s
or e − τ t , respectively. The absorption
length l a and the absorption time τ are inversely proportional to the number density
ρ abs of absorbing particles on the path and their absorption cross section σ abs : l a = 1
ρ abs σ abs
and
τ = t s l 1
a =
v ρ abs σ abs
.
11

2 Theory
Figure 2.6: Scattering in the presence of a single boundary. All photon paths A → B
that are lost because they run partially outside the samples can be described by their
image paths A → B ′ . The (virtual) scatterer P is the point where a photon path leaves
the sample.
2.4 The influence of boundaries
Eqns. 2.6 and 2.8 describe the photon density distribution in infinitely extended samples. In
reality however, this distribution is strongly influenced by the borders of the system, where
photons are inserted and released, absorbed and reflected.
There are two different approaches to describe the influence of boundaries on a multiple
scattering system. The more intuitive one is the image point method, which uses symmetry
properties of the density distribution to describe the scattering in terms of the density distribution
in infinite space [38]. The other approach is in the framework of radiative transfer theory,
which describes a field of radiation by the intensity flux through area elements. This makes
it possible to compare the flux through an area element at the sample surface with an area
element inside the sample and derive theoretical descriptions for the various experimental
situations [58].
2.4.1 Image point method
To describe multiple scattering in a finite sample in terms of scattering in infinite space one
can make use of the symmetry properties of the density distribution: The photon density at
any point B in an infinite sample is equal to the density at its image point B ′ with respect
to any mirror plane that contains the starting point P of the photon cloud. Therefore we can
describe photon paths A → P → B by paths A → P → B ′ that end at the image point of B
with respect to P. If P is the endpoint of a photon step that leads the path out of the sample,
the path A → P → B is not possible in the presence of the boundary, and its contribution is
missing from the free space photon density in B.
12

2.4 The influence of boundaries
Figure 2.7: Scattering in the presence of two parallel boundaries. The presence of
two parallel boundaries with distance L creates two series of image points B
1 ′ , B′ 2 , B′ 3 , . . .
and B
1 ′′,
B′′ 2 , . . .. The effective sample thickness L′ = L + 2z 0 simplifies the series.
On the other hand, all photon paths that lead to B ′ contain at least one such point P, which
is located at an average distance z 0 from the boundary. The photon density that is missing
from the free space photon density in B due to the sample boundary is therefore the free
space photon density in B ′ , and the photon density distribution in the presence of a boundary
becomes
ρ(A → B, t) 1 surface =
= ρ(A → B, t) − ρ(A → B ′ , t)
= e− τ
t
√
(e ( − ⃗r ⊥,B −⃗r ⊥,A) 2 +(z B −z A ) 2
4Dt − e ( − ⃗r ⊥,B −⃗r ⊥,A) 2 +(−z B +2z 0 −z A ) 2
4Dt
3
4πDt
)
(2.9)
where we have also considered absorption by its τ-dependent exponential decay. The vector
⃗r ⊥ denotes the components of the position vector⃗r perpendicular to the z-axis and parallel to
the sample boundary. Note that z 0 and z B or respectively z A have different signs, as depicted
in fig. 2.6.
If a second surface opposite to the entrance surface is to be considered, things become a little
more complicated. In this case, the photon density in B is the free space photon density
distribution minus the photon densities at the two image points B ′ and B ′′ with respect to the
two mirror planes in front of the surfaces at z = z 0 and z = L + z 0 . These photon densities
again have to be calculated as the densities in the presence of the other surface, and so on.
13

2 Theory
Figure 2.8: Radiative transfer. Radiative transfer theory describes the photon flux
from volume element dV through area element dS without any intermediate scattering.
The result are the two series of image points sketched in fig. 2.7, which we can simplify by
using an effective sample thickness L ′ = L + 2z 0 :
ρ(A → B, t) 2 surfaces =
∞
∑
= ρ(A → B, t) − ρ(A → B ′ ∞
k , t) − ∑
k=1 k=1
ρ(A → B ′′
k , t)
= e− τ
t ∞
√ 3 ∑ e ( − ⃗r ⊥,B −⃗r ⊥,A) 2 +(2mL ′ +z B −z A ) 2
4Dt − e ( − ⃗r ⊥,B −⃗r ⊥,A) 2 +(2mL ′ −z B −z A ) 2
4Dt (2.10)
4πDt m=−∞
2.4.2 Radiative transfer theory
The image point method gives an intuitive picture of the photon density distribution, but fails
to explain the influence of internal reflections at the sample boundaries and to give a value
for the average penetration depth z 0 . This can be accomplished by radiative transfer theory,
where we follow the approach of [58].
Let us consider the flux of photons through a small area dS inside the sample, which for
simplicity we place at the origin and perpendicular to the z-axis, as sketched in fig. 2.8. The
number of photons scattered from a volume element dV directly through dS is given by
dS cos θ
the product of the number of photons ρ(⃗r)dV in dV, the fractional solid angle that
represents the cross section of dS for the photons coming from dV, the speed of photon
transport v, and the loss e − r
l ∗ due to scattering between dV and dS. The total flux in the
negative z-direction j − dS can then be obtained by integrating over the upper half space with
z > 0, the flux in positive direction j + dS by integration over the lower half space with z < 0:
4πr 2
∫
j ∓ dS =
z > < 0 ρ(⃗r) · v · e − r
l ∗ ·
dS cos θ
4πr 2
dV
14

2.5 Photon flux from a surface
The main contribution of the flux comes from the immediate neighborhood of dS, so that
the photon density can be replaced by its first-order Taylor expansion around the origin.
Evaluation of the integral then yields
j ∓ = ρ 0v
4 ± vl∗
6
( ) ∂ρ
∂z
0
where the photon density and its derivative have to be taken at the origin.
If dS is located at a boundary, there will be no flux from outside the sample, but internal
reflections will create an apparent flux from this direction, which is related to the outgoing
flux by the reflectivity R of the surface: j in = R · j out . This gives the boundary conditions
ρ − 2l∗
3
ρ + 2l∗
3
1 + R
1 − R · ∂ρ
∂z = 0 for an upper boundary, i.e. j + = R · j − and
1 + R
1 − R · ∂ρ
∂z = 0 for a lower boundary, i.e. j − = R · j +
The point where the photon density drops to zero is not located at the boundary but at a
distance 2l∗ 1+R
3 1−R · ∂ρ
∂z
in front of it. We can identify this distance with the average penetration
depth z 0 which we have used above for the image point method, as this also is the point where
the photon density vanishes. Assuming a constant gradient of the photon density distribution
close to the surface, one obtains |z 0 | = 2l∗ 1+R 1+R
3 1−R
≈ 0.67
1−R l∗ . Other solutions are mostly of the
order of |z 0 | ≈ 0.7 for the limit of non-reflective surfaces (see e.g. [53], [54]).
2.5 Photon flux from a surface
The experimentally accessible quantity in multiple scattering experiments is not the photon
density distribution inside the sample, but the light intensity emitted from a sample surface.
It is obvious that this intensity (or photon flux density) is proportional to the number of
photons, i.e. the photon density, at the surface. One can therefore easily calculate the expected
intensities for the various experimental situations from eqns. 2.9 and 2.10.
2.5.1 Backscattering geometry
Backscattering experiments are usually performed with thick samples, where the influence of
the rear and side surfaces on the photon density distribution can be neglected, and the sample
is well described as an infinite half-space.
In backscattering geometry, photon paths of all lengths contribute to the intensity at the surface,
so that the latter is not fully described by the solution of the diffusion equation, which is
valid only for long photon paths. It has been shown however [44] that correct results can be
15

2 Theory
Figure 2.9: Photon flux in backscattering geometry. The photon flux in an infinite
half-space is determined not only by the flux between the first and the last scatterer of
the photon path, A and B, but also by the probabilities f (z A ) and f (z B ) of the photon
to travel between the surface and A or B without being scattered.
obtained from the diffusion equation by assuming that the conversion between propagating
light outside the sample and diffusing light inside does not happen at the sample surface, but
at a certain depth inside the sample. The flux density at the sample surface z = 0 is then
j back (⃗r ⊥,A ,⃗r ⊥,B , t) ∝
∫ ∞ ∫ ∞
0 0
f (z A ) · f (z B ) · ρ(A → B, t) 1 surface dz A dz B
where ρ(A → B, t) 1 surface is given by eqn. 2.9. The two functions f (z A ) and f (z B ) represent the
probability distribution of the conversion depth inside the sample. Usually, one assumes nearnormal
incidence and an exponentially decaying conversion probability with decay length l ∗ ,
so that f (z A ) = e − z A
l ∗
and f (z B ) = e − z B
l ∗ cos θ , where the scattering angle θ is the inclination of
the outgoing light wave with respect to the sample surface. d
The photon flux distribution for steady-state experiments is obtained from above equation
by integrating over time. For later calculations the case of negligible absorption is especially
important: e
j back (⃗r ⊥,A ,⃗r ⊥,B , θ) ∝
∝
∫ ∞ ∫ ∞
0
0
e − z A
l∗ · e − z B
l ∗ cos θ
(
)
√ 1
− √ 1
(⃗r⊥,B −⃗r ⊥,A ) 2 +(z B −z A ) 2 (⃗r⊥,B −⃗r ⊥,A ) 2 +(z B +2z 0 +z A ) 2
dz A dz B (2.11)
[d] The definition of the scattering angle θ used for backscattering geometry is inconsistent with the definition
used in previous sections. While e.g. Mie theory denotes backscattering with θ = π, the theory of coherent
backscattering uses θ = 0 for the same angle.
[e] As z A and z B are now taken to be positive, z 0 would be negative (see sec. 2.4.1). We simplify the notation by
using |z 0 | and omitting the absolute value bars.
16

2.6 On polarization and interference
2.5.2 Transmission geometry
While the samples for backscattering experiments are usually thick, transmission experiments
require rather thin samples, which resemble an infinite slab of thickness L. Still, if the samples
are not too thin, transmission experiments can be fully described by diffusion as all photon
paths have at least the length of the sample thickness.
As a consequence, the exact depth of the conversion between plane wave and diffusive transport
plays only a subordinate role. One can therefore assume that the conversion happens at
a single depth l ∗ .
An important experimental quantity is the distribution of photon flight times (which is equivalent
to the path length distribution) of the transmitted light
∫
j trans (t) ∝
rear surface
δ (z A − l ∗ ) · δ ( z B − (L ′ −l ∗ ) ) · ρ(A → B, t) 2 surfaces d⃗r ⊥ =
= e− τ
t ∞
√
4πDt
∑ e − (2mL′ +(L ′ −l ∗ )−l ∗ ) 2
4Dt − e − (2mL′ −(L ′ −l ∗ )−l ∗ ) 2
4Dt
m=−∞
Using Poisson’s sum formula ∑ ∞ n=−∞ f (n) = ∑ ∞ m=−∞
∫ ∞
−∞ e−2πima f (a) da this can be turned
into [38]
j trans (t) ∝ 2e− τ
t ∞ (
L ∑ ′ e − n2 π 2
L ′2 Dt nπ
( nπ
)
sin
L
n=1
′ l∗) sin
L ′ (L′ − l ∗ )
Obviously, it is l∗
L
′ 0, so that the sine can be replaced by its argument. Likewise, it is
1. Here we can replace the sine by (−1)
n+1 nπ
l ∗ , yielding
L ′ −l ∗
L ′
j trans (t) ∝ −2e − t τ
∞
∑
n=1
( (−1) n e − n2 π 2
L ′2 Dt nπ
L ′
) 2 l
∗2
L ′ L ′ (2.12)
2.6 On polarization and interference
With the introduction of the models ‘random walk’ and ‘diffusion’ an important property
of the scattered light has been lost: Despite of the vectorial nature of electromagnetic waves
both models describe the scattering of scalar waves. To find a correct description of multiple
scattering of vector waves we have to go back to single scattering once more.
Fig. 2.3 shows that the polarization of light scattered by a spherical particle depends strongly
on the scattering angle and the polarization of the incoming wave with respect to the scattering
plane. The result is that after a few scattering events in a multiple scattering sample the
photons will have completely lost the memory of their initial polarization. Multiply scattered
light is therefore unpolarized.
17

2 Theory
300
250
200
150
100
50
x 10 4
5
4
3
2
100 200 300
1
Figure 2.10: Speckles. Random interferences of the photons emerging from a multiple
scattering medium result in a random distribution of high and low light intensities.
Still, interference is possible between equally oriented components of the light waves. The
interference pattern observed on a multiple scattering sample results from the coherent addition
of the corresponding components of the waves that emerge from the ends of the light
paths in the sample. To first order it is therefore the superposition of the interference patterns
of the photons on all pairs of light paths in the sample.
This implies of course that a certain pair of light paths is not only theoretically possible,
but actually has photons traveling on it. The interference pattern of a infinitely extended
incoming wave will therefore differ in some way from that of a spatially restricted incoming
wave. Likewise, the interference pattern of multiply scattered light with restricted spatiotemporal
coherence will be only a modified version of the interference pattern of light with
infinite temporal and spatial coherence length.
2.6.1 Speckles
Most of the light paths in the sample are completely unrelated, so that their interference results
in a random speckle pattern of high and low light intensities (fig. 2.10). The speckle pattern
is therefore a subtle image of the positions of the scatterers inside the sample, and is unique
for every sample and every lighting and imaging situation. It is also extremely sensitive to
motions of the scattering particles. Even sub-wavelength movements of the particles lead
to significant variations in the overall phaseshift of the photons and to fluctuations of the
speckles. Averaging over speckle fluctuations or a sample average lead to a detected light
intensity approximately proportional to the cosine of the scattering angle θ, as described by
Lambert’s well-known emission law [33].
18

2.6 On polarization and interference
Figure 2.11: Theorem of reciprocity. Amplitudes and phases of direct and reversed
paths are equal if the incident and detected light is completely polarized, and if the
incident polarization P in,direct of the direct path is identical to the detected polarization
P out,reversed of the reversed path and vice versa [38]. With a single light source, direct and
reverted paths can not be distinguished, so that P in,direct = P in,reversed and P out,direct =
P out,reversed . For linear polarization we denote P in ‖ P out as the parallel polarization
channel and P in ⊥ P out as the crossed polarization channel. The respective channels for
circular polarization are called the helicity conserving and the helicity breaking channel.
2.6.2 Coherent backscattering
However, for multiply scattered photons that run on a certain path S 1 → S 2 → · · · → S n
and on its time-inverted path S n → S n−1 → · · · → S 1 the theorem of reciprocity (fig. 2.11)
predicts special correlations: Photons in the parallel polarized or helicity conserving channel
are always in phase when they leave the sample. This phase coherence is independent of the
pathlength or the exact positions of the scatterers, and therefore also insensitive to particle
motions, which are of course slow compared to the speed of light. The interference pattern of
the direct and time-inverted photons therefore survives any average over the random speckle
pattern.
Equal phases for photons on direct and time-inverted paths means also that in direct backscattering
direction all interferences are constructive, regardless of the end-to-end distances of the
photon paths that determine the angular intensity distributions of the interference patterns.
The intensity scattered in this direction is therefore twice the amount expected for an incoherent
superposition of the light waves. This all-constructive interference decays – for an
incoming spatially and temporally infinitely extended plane wave over an angular scale of
(kl ∗ ) −1 [11] – as constructive and destructive interferences superimpose for angles deviating
from backscattering direction. The resulting cone-shaped intensity enhancement is referred to
as the coherent backscattering cone.
2.6.3 Anderson localization
The concept of interference on time-inverted photon paths works also if the path forms a
closed loop inside the sample. Here, the place of the interference is the point where the light
19

2 Theory
waves enter and leave the loop. As the light waves are always in phase at this point, constructive
interference leads to an enhanced photon density compared to the normal diffusive
behavior. In strongly scattering media the probability of photons running on closed loops is
enhanced, so that a macroscopically reduced diffusion can be observed. Eventually this leads
to a complete breakdown of photon transport and a transition to a localizing state, which is
called Anderson localization [13].
The critical amount of disorder for this phase transition can be expressed by the criterion
proposed by Ioffe and Regel [27], namely that kl ∗ ≈ 1. The width of the steady-state coherent
backscattering cone, which is inversely proportional to this quantity, is therefore an important
experimental measure for the disorder in the sample [12].
2.7 The theory of coherent backscattering
To develop a theory for the steady-state coherent backscattering cone, the easiest case to consider
is a uniform plane wave with infinite spatio-temporal coherence impinging perpendicularly
on the surface of a non-absorbing multiple scattering medium. In this case the photon
flux distribution has the simple form given in eqn. 2.11, and furthermore is translationally
invariant, i.e. j back (⃗r ⊥,A ,⃗r ⊥,B , θ) becomes j back (⃗r ⊥ , θ) with⃗r ⊥ =⃗r ⊥,B −⃗r ⊥,A .
We define the cooperon α c (θ) as the coherent and the diffuson α d (θ) as the incoherent addition
of the flux emerging from the time-inverted paths in the sample:
∫
α d (θ) =
∫
j back (⃗r ⊥ , θ) d⃗r ⊥
j back (⃗r ⊥ , θ = 0) d⃗r ⊥
and α c (θ) =
∫
j back (⃗r ⊥ , θ) · e i ⃗ k⃗r⊥
d⃗r ⊥
∫
(2.13)
j back (⃗r ⊥ , θ = 0) d⃗r ⊥
where⃗ k is the wave vector of the emitted light wave. The diffuson is sometimes also referred
to as the incoherent background of the backscattering cone.
If single and low order scattering can be blocked completely, the backscattered intensity measured
in an experiment is the sum of these two contributions. Otherwise, the height of the
cooperon compared to the diffuson is reduced, as single scattering contributes to the incoherent,
but not to the coherent addition of the photon flux [38].
Evaluating the above equations yields [10]
( )
α d (θ) = µ z0
l
+ µ
∗ µ+1
z 0
l
+ 1 ∗ 2
(2.14)
and
α c (θ) =
2 ( z 0
l ∗ + 1 2
1−e −2qz 0
ql ∗ + 2µ
µ+1
) ( ql ∗ + µ+1
2µ
) 2
(2.15)
20

2.7 The theory of coherent backscattering
diffuson / cooperon
2
1.8
1.6
1.4
1.2
1
0.8
0.6
diffuson α d
(θ)
cooperon α c
(θ)
α d
(θ) + α c
(θ)
0.4
0.2
0
−90 −60 −30 0 30 60 90
scattering angle [deg]
Figure 2.12: Diffuson and cooperon. The graph shows diffuson α d (θ) and cooperon
α c (θ) calculated with eqns. 2.14 and 2.15 for λ = 590 nm, kl ∗ = 5, and non-reflective
sample surface.
with µ = cos θ and q = k| sin θ|.
In exact backscattering direction θ = 0, diffuson and cooperon are both equal to one, so that
the backscattered intensity is enhanced by a factor of two (fig. 2.12). For larger angles the
cooperon drops rapidly to zero, and the backscattered intensity approaches the value of the
diffuson.
The initial slope of the coherent backscattering cone close to θ = 0 and therefore also the width
of the cone is strongly affected by internal reflections. For non-reflective sample surfaces
the inverse angular width is approximately 4 3 kl∗ [58]. Internal reflections – described by the
reflectivity R of the surface – narrow the cone to f
FWHM −1 =
(
1 + 1 + R ) 2
1 − R 3 kl∗ (2.16)
The numerator of the cooperon in eqn. 2.13 is a Fourier transform of the flux distribution [38]:
∫∫
∫
j back (⃗r ⊥ , t) · e iqr ⊥
dt d⃗r ⊥ =
[ ] ∫
FT j back (⃗r ⊥ , t) dt =
j(t) · e −q2 Dt dt (2.17)
The cooperon is therefore not only the superposition of interference patterns as a function of
the distance of first and last scatterer, but also a superposition of Gaussian distributions as a
[f] FWHM = full width at half maximum
21

2 Theory
cooperon
1
0.8
0.6
0.4
l abs
= 10 −5 m
l abs
= 3 ⋅ 10 −5 m
l abs
= 10 −4 m
no absorption
0.2
0
−90 −60 −30 0 30 60 90
scattering angle [deg]
Figure 2.13: Coherent backscattering with absorption. The graph shows the cooperon
α c (θ, l abs ) calculated with eqn. 2.18 for different absorption lengths l abs = 3Dτ/l ∗ with
λ = 590 nm, kl ∗ = 5, and non-reflective sample surface.
function of time or respectively the pathlength. The longest photon paths have the narrowest
Gaussians, while short paths have wide distributions.
An infinite series of Gaussians creates a triangular cusp at the tip of the coherent backscattering
cone. However, absorption as well as localization introduce a cutoff length for the photon
paths, so that the narrowest Gaussians are eliminated. This explains why one observes a
rounded conetip for absorbing or localizing samples (figs. 2.13, 3.4).
The effect of absorption can be handled mathematically by introducing a substitution q 2 →
q 2 + (Dτ) −1 in the cooperon [38]. With the correct normalization the cooperon for absorbing
samples becomes
α c (θ, τ) =
(
Dτ l ∗ +
(
l ∗ l ∗ + √ ) 2 (
Dτ 1−e
−2q abs z 0
(
1 − e − 2z 0
q abs l ∗ + 2µ
µ+1
)
) )
√ √Dτ ( )
Dτ q abs l ∗ + µ+1 2
(2.18)
2µ
with q abs = √ q 2 + (Dτ) −1 .
Anderson localization can be modeled by a transition from a constant diffusion coefficient D
to a time-dependent coefficient D(t) ∝ t −1 at the localization time t loc [45]. As the ‘cutoff’
mechanism is different, the resulting shape of the conetip also differs from that of a purely
absorbing sample (fig. 3.4).
22

2.7 The theory of coherent backscattering
cooperon [a.u.]
0.3
0.25
0.2
0.15
0.1
l abs
= 10 −5 m
l abs
= 3 ⋅ 10 −5 m
l abs
= 10 −4 m
no absorption
0.05
0
−90 −60 −30 0 30 60 90
scattering angle [deg]
Figure 2.14: Coherent backscattering with absorption – unnormalized cooperon. The
graph shows the unnormalized cooperon for different absorption lengths l abs = 3Dτ/l ∗
with λ = 590 nm, kl ∗ = 5, and non-reflective sample surface. Deviations from the
non-absorptive case at the cone flanks can be observed only for very short absorption
lengths, which are irrelevant for our experimental situations.
Both absorption and localization not only cause a rounding of the conetip, they also widen
the cooperon. However, in many experimental situations the normalization of the diffuson
and the cooperon by ∫ j(⃗r ⊥ , θ = 0) d⃗r ⊥ is unnecessary, as the experimental data are also not
normalized. Applying the above transformations only to the numerator of the cooperon in
eqn. 2.13 results in a lowered cone enhancement instead of a widened cooperon (fig. 2.14), so
that the cone flanks are unaffected by absorption or localization. In the measurement of kl ∗ ,
an imprecise rendition of the very tip of the backscattering cone – which is rather common for
narrow cones – is therefore no major source of errors, as the theory can be fitted to the flanks
of the cone.
23

24
2 Theory

3 Setups
3.1 Laser System
The key piece of the light scattering laboratory is the picosecond pulse laser system (fig. 3.1)
which serves as light source for all light scattering setups.
The system consists of a Rhodamin6G dye laser (699 from Coherent) which is pumped by the
514.5 nm-line of an Ar + -laser (Innova400 from Coherent) [49], a laser type which operates in
continuous wave (cw) mode. The lasing medium in the dye laser is a jet of Rhodamin6G, so
that the wavelength can be tuned roughly between 570 nm and 620 nm.
To be able to alternatively operate the system in pulsed mode, the Ar + -laser was modified
with a mode locker (from APE), which provides 100 ps-pulses at a repetition rate of 76 MHz.
The cavity lengths of pump and dye laser are matched to synchronize and overlap the pulses
coming in from the pump laser with those already circulating in the cavity of the dye laser. The
inversion in the dye jet is completely broken down by the front flank of the circulating pulse,
so that the pulse is shorted to approximately 12-15 ps. After about 30 cycles in the cavity of
the dye laser the now significantly amplified pulse can be released by a cavity dumper (from
APE).
3.2 Wide Angle Setup
The setup used for the study of the wide-ranged angular distribution of the backscattered
light consists of 256 photosensitive diodes attached to a semicircular arc with a diameter of
1.2 m (fig. 3.2). In its center the sample is located, facing the incoming cw laser beam which is
focussed through a tiny hole between the topmost diodes [22, 23, 47]. A simple motor rotates
Figure 3.1: Laser system. The Rhodamin6G dye laser and the Ar + -pumplaser were
modified to run also in pulsed mode. The acousto-optical modulator (AOM) in the
Ar + -laser serves as mode locker, the one in the dye laser as cavity dumper.

3 Setups
Figure 3.2: Wide angle setup. 256 photodiodes arranged in a semicircle around the
sample capture the the backscattered radiation over a range of nearly 180 ◦ .
the sample around an axis parallel to the incoming laser beam, providing a sample average
which eliminates the speckle pattern.
3.2.1 Optical setup
The intensity of the coherent backscattering cone varies rapidly around the backscattering direction,
while the variations at larger scattering angles are comparatively slow. Accordingly,
the angular resolution of the setup has to be rather high in the center, while at larger angles
the diodes can be set further apart. To meet these requirements, for angles θ < 9.75 ◦ eight
photodiode arrays with 16 photodiodes each (S5668 from Hamamatsu) are used, yielding an
angular resolution of 0.15 ◦ . For angles θ > 9.75 ◦ , single photodiodes (S4011 from Hamamatsu)
provide angular resolutions of 0.7 ◦ for 9.75 ◦ < θ < 19.55 ◦ , ∼ 1 ◦ for 19.55 ◦ < θ < 60 ◦
and ∼ 3 ◦ for 60 ◦ < θ < 85 ◦ . Still, it is not possible ro resolve the very tip of the cone at θ ≈ 0,
as this is the position where the laser beam passes through the diode arc. For this task the
small angle setup (see sec. 3.3) has to be used.
The desired angular resolution also implies some prerequisites for the optical setup: The
laser source has to be imaged completely on a single photodiode, without any light of a
certain scattering angle missing its diode or even illuminating a neighboring one. On the
other hand, the incoming laser beam is supposed to have a considerable width at the sample
surface to avoid finite size effects [32]. Therefore the incoming laser beam is first widened
and parallelized by a telescope, and then focussed on the point between the photodiodes,
diverging again on its way to the sample. Light scattered with a certain scattering angle is
then focussed again when it reaches the photodiodes.
26

3.2 Wide Angle Setup
1 .0
0 .8
h e lic ity c o n s e rv in g
h e lic ity b re a k in g
0 .6
c o o p e ro n
0 .4
0 .2
0 .0
-9 0 -6 0 -3 0 0 3 0 6 0 9 0
s c a tte rin g a n g le [d e g ]
Figure 3.3: Polarizer effectiveness. The circular polarizer foil is only about 97% effective,
so that each polarization channel contains 3% of the other channel. The resulting
small cone with an enhancement of 0.03 can easily be observed in the helicity breaking
channel. Due to some missing data points at θ ≈ 0 the enhancement of 0.97 in the
helicity conserving channel is more difficult to read from the graph.
3.2.2 Suppression of single scattering
The enhancement of the coherent backscattering cone, I m /(I m + I s ), depends on the intensities
of both singly (I s ) and multiply (I m ) scattered light, where single scattering results in a lowered
backscattering enhancement. To study the backscattering of multiply scattered light in detail,
single scattering has to be suppressed sufficiently.
According to the theorem of reciprocity (see sec. 2.6), interference of multiple scattered waves
is only possible for the parallel or respectively the helicity conserving polarization channel.
As explained in sec. 5.2.3, the sequence ‘polarizer – scatterer(s) – polarizer’ which is necessary
to select the channel with the backscattering cone simultaneously blocks single scattering if
circular polarizers are used. The wide angle setup is therefore equipped with a bendable
circular polarization foil (J53-333 from 3M), which is placed at the entrance of the setup and
in front of the diodes. This polarizer foil blocks about 97% of the single scattered intensity
(fig. 3.3).
3.2.3 Diode calibration
As each of the photodiodes is read by its own electronic circuit, it is necessary to calibrate the
diodes to make up for different gains of the processing electronics [22, 23, 47]. This is done by
27

3 Setups
measuring the backscattering of a sample with known scattering properties as a function of
the incoming laser power P, which is determined with a calibrated powermeter (FieldMaxII
from Coherent) from a reflection on a glass plate in the laser beam at the entrance of the
setup. A fit of the powers P(d θ ) as a function of the diode signals d θ with a polynomial then
yields a calibration function for each photodiode (see fig. 5.1).
As reference sample we use a block of teflon, the backscattering cone of which has a FWHM
of the order of 0.03 ◦ for visible light (see sec. 5.2.2). This is much narrower than the angular
resolution of the wide-angle setup, so that teflon can be considered to give a purely incoherent
signal proportional to P α d (θ).
3.3 Small Angle Setup
In sec. 2.7 the coherent backscattering cone was presented as a superposition of Gaussian
distributions j(s) · e − sin2 θ
3 k 2 sl ∗ , whose width is a function of the path length s of the timeinverted
photon paths. Absorption and localization both affect mainly long paths, which
contribute essentially at the very tip of the backscattering cone. Their reduced contribution
results in a rounding of the tip of the backscattering cone. With a high-resolving setup it
should therefore be possible to observe both phenomena in coherent backscattering. The
same setup could also be used to measure the transport mean free paths of samples with
extremely narrow backscattering cones and thus complement the wide angle setup.
3.3.1 Precision requirements
For an estimate of the required setup precision we assume for a moment that absorption
(or localization) results in an abrupt cutoff at path length s = L. The narrowest Gaussian
that contributes to the backscattering cone is therefore the one with standard deviation σ =
√
3/(2k 2 Ll ∗ ) = √ 1/(2k 2 Dτ). The angular width of the conetip rounding must be of the same
order of magnitude.
The absorption of a sample like the titania powder R700 (see sec. 4.2) with absorption time
τ = 2 ns and diffusion coefficient D = 15 m2 /s at wavelength λ = 590 nm will therefore require
to properly resolve an angular range of θ round ≈ ±0.02 ◦ . To observe localization, which sets
in after a localization length l a = 340 mm [48], a similar resolution is necessary.
Test calculations show that the intensity difference between a localizing and a non-localizing
sample is less than 0.1% of the maximum of the cooperon (fig. 3.4). As the detection must
be able to capture the maximal backscattered intensity at θ = 0, plus some external radiation
and electronic noise that can never be avoided completely, while still providing the necessary
intensity resolution, the digital range of the detection must be at least 2 14 , better 2 15 − 2 16 .
3.3.2 Optical setup
In the small angle setup a 4-megapixel 16-bit monochrome CCD camera (Alta U4000 from
Apogee) is placed opposite the sample (fig. 3.5). The camera can be cooled thermoelectrically
28

3.3 Small Angle Setup
1
0.998
0.996
plane
absorption
localization
0.994
cooperon
0.992
0.99
0.988
0.986
0.984
0.982
0.98
0 0.02 0.04 0.06 0.08 0.1
scattering angle (deg)
Figure 3.4: Conetip rounding from absorption and localization. The backscattering
cones were calculated using eqn. 2.17 with the simpler flux density distribution j(s) ∝
√ −3
4πDt given in [40], for which at small scattering angles θ the results are equivalent
to the ones obtained with eqn. 2.11. Absorption was incorporated by the substitution
q 2 → q 2 + (Dτ) −1 , localization was modeled by a transition from D = const. to D(t) ∝
t −1 at the localization time t loc . For the sample parameters of R700 and a localization
length of the order of τ, localization causes a shift at the tip of the backscattering cone
of less than 0.1% of the intensity maximum.
29

3 Setups
Figure 3.5: Small angle setup. A CCD camera captures the the backscattered radiation
around backscattering direction.
to 45 K below ambient temperature to reduce electronic noise. It turned out that the glass
window that covers the CCD chip needs to be wedge shaped to avoid interferences in the glass,
which cause a disturbing ripple structure on the CCD image. The camera offers exposure
times between 30 ms and 183 min; to improve the signal-to-noise ratio, the backscattering data
of our experiments are an average over a series of usually 100 pictures, taken with an exposure
time of 0.5 s.
The camera replaces an older 8-bit model with 580 × 740 pixels, which did not offer enough
angular and intensity resolution for the planned experiments. The exchange of the camera
improves the maximum intensity resolution by more than two orders of magnitude and the
maximum angular resolution or respectively the maximum angular range by a factor of three.
As the direct backscattering direction is blocked by the camera, a beamsplitter directs the
laser beam onto the sample. The remaining part of the beam that passes straight through the
beamsplitter must be dumped completely, as even weak backreflection will disturb the image
on the camera. For similar reasons, a pellicle beamsplitter instead of a glass beamsplitter has
to be used, as even anti-reflection coated surfaces of a glass plate would cause distinct ghost
images.
In contrast to the wide angle setup described in the previous section, where the sample is
illuminated by a slightly divergent light beam, the incoming laser beam in the small angle
setup is exactly parallel. Backscattered light with a certain scattering angle θ then forms a
cone-shaped shell with a thickness that is determined by the diameter of the sample area
illuminated by the incoming laser beam (fig. 3.6). The lens above the beamsplitter converges
30

3.3 Small Angle Setup
Figure 3.6: The optical setup. Backscattered light with a certain scattering angle θ
forms a cone-shaped shell, which is converged in the focal plane on a circle with radius
r by the lens with focal length f .
this ring of light on a circle in the focal plane, the radius of which is given by
r = f · tan θ (3.1)
where f is the focal length of the lens.
Like in the wide angle setup, single scattering is blocked by a circular polarizer. In the small
angle setup it is unnecessary to have a bendable polarizer foil, so a high-quality circular
polarizer from an industrial optics manufacturer (AUC circular polarizer from B+W) can be
used. These are available with larger diameters than usual polarizers offered by laboratory
suppliers and provide extinction ratios up to 4000:1 [1].
3.3.3 Sample average
Solid samples like teflon or the titania powders have to be moved during the measurement to
average over the speckle pattern. The method used in the wide angle setup, where the sample
is simply rotated, however turned out to be inconvenient as the high-resolving camera picked
up this rotational motion as a circular structure in the images. To have a larger ensemble to
average over in the small angle experiments, the sample is therefore pulled on Lissajous loops
by two motors with a frequency of several Hertz on each axis (fig. 3.7).
3.3.4 Data evaluation
For the evaluation of the backscattering data the exact position of the tip of the coherent
backscattering cone at θ = 0 has to be obtained from the CCD image. We do this by converting
31

3 Setups
Figure 3.7: Sample shaker. The two motor-driven eccentric wheels M (red) pull the
sample S back and forth between the levers and the two rubber bands (brown), so that
the samples moves on Lissajous loops.
the greyscale image into a binary image, based on a threshold value of 95% of the maximum
intensity in the picture. The center of the cone is then identical with the center of mass of
the white region. The angular intensity distribution of the backscattered light can then be
obtained either by a profile section through the image or by an azimuthal average.
The angular resolution of the setup can be tested by replacing the multiple scattering sample
with a mirror. The image on the CCD chip is then that of a single scattering angle. If the lens
in front of the camera were able to focus the light on a single pixel, the angular resolution of
the setup would be given by the pixel size, which is 7.4 µm for the camera in use. However, in
the present setup configuration the focus spot is 4 − 8 pixels wide, depending on the diameter
of the incoming laser beam, which is between 0.5 cm and 2 cm.
To account for the structure of the focus spot, the data are therefore fitted with theory curves
that are convoluted with the intensity profile of the spot. a The resulting angular resolution of
the measured backscattering data is then close to the maximal resolution of 0.00085 ◦ , which
is given by the pixel size of the CCD chip.
32

3.4 Time Of Flight Setup
Figure 3.8: Time of flight setup. The setup measures a histogram of time delays between
photons arriving at detector 1 (photomultiplier) behind the sample and the signal
of the reference detector 2 (photodiode).
3.4 Time Of Flight Setup
Apart from investigations of the path length distribution in transmission and the search for
Anderson localization [48], the time of flight setup is used to determine diffusion coefficients
and absorption times of the samples. These quantities are obtained by fitting the measured
distribution of the times photons need to travel through the sample with the theoretical prediction
of eqn. 2.12.
To obtain a distribution of photon flight times, the time of flight setup measures a histogram
of time delays between photons arriving at a detector behind the sample and the signal of a
reference detector, as shown schematically in fig. 3.8. The signal from a photon arriving at the
photomultiplier (H5784 from Hamamatsu) starts a time measurement in the single photon
detection card (SPC-140 from Becker & Hickl). The stop signal is provided by a photodiode
which receives a small part of the unscattered pulse and which is followed by a cable delay to
have the signal come in after the signal from the photomultiplier.
It is important to have the samples turbid and thick enough so that at most a single photon
of each pulse actually reaches the detector. This ensures that the probability to trigger a time
measurement is not biased towards photons on short paths.
The measured time delays are sorted into a histogram with 1024 time slots. The width of
this histogram has to be matched to the desired time range, which is of the order of 20-40 ns.
As with the resulting time resolution the shape of the unscattered laser pulse is not a delta
function, the data of a broadened pulse have to be deconvoluted with the systemic answer of
the unscattered pulse.
[a] Deconvoluting the data with the reflection profile would be more straightforward, but is not possible due to
mathematical difficulties.
33

34
3 Setups

4 Samples
4.1 Sample characterization techniques
The light scattering experiments performed with the setups introduced in sec. 3 are the main
source of data which characterize the multiple scattering samples. However, there are a few
additional sample properties which have to be obtained otherwise. These are mainly the
effective refractive index of the sample and the reflectivity of the sample surface, which are
necessary for example to calculate the average penetration depth z 0 . Also important are the
particle size and polydispersity of the samples and the filling fraction of the sample. Some
of these calculations are trivial and can be found in every physics textbook, but still are to be
mentioned briefly.
4.1.1 Particle size and polydispersity
Particle size and polydispersity of the the colloidal particles give first indications for the scattering
properties of the multiple scattering samples. Strongly scattering samples have particle
diameters of the order of the wavelength and low polydispersity. Especially strong scattering
is obtained when the scattering in the particles becomes resonant.
The commercial titanium dioxide samples have been characterized before by M. Störzer [47],
who used electron microscopy to determine size and polydispersity of the particles. The
device used was an XL Scanning Electron Microscope from Phillips, which provides a spatial
resolution of up to 50 nm. To avoid charge building in the sample the surface was covered with
a gold layer of approximately 10 nm thickness, which was brought onto the sample using a
gas discharge sputter technique (Scancoat SIX, Edwards). The distribution of the particle sizes
was obtained by measuring the diameters of 150-200 particles from the pictures (fig 4.1).
4.1.2 Filling fraction
Another important feature is the filling fraction of the scatterers, which not only characterizes
the sample for itself, but is also needed for calculating the effective refractive index of the
sample.
The filling fraction is defined as
f = V scatterers
V sample
=
m
ρ · r 2 πh

4 Samples
3 0
2 5
n u m b e r o f p a rtic le s
2 0
1 5
1 0
5
0
1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0
p a rtic le d ia m e te r [n m ]
Figure 4.1: Particle size and polydispersity. From the electron microscope image
(left) the distribution of the particle sizes of R700 (right) was obtained by measuring the
diameters of approximately 150 particles [47].
and can quite easily be calculated by measuring the weight m and the spatial dimensions r
and h of the sample, while the mass density ρ of the scattering material can be obtained from
numerous literature sources.
For strong scattering and low mean free paths, the sample should be as dense as possible. If
however the filling fraction is larger than a certain value (which depends on the mismatch of
the refractive indices of scatterers and surrounding medium) [20], scattering takes place at the
holes rather than the particles, so that there is no use increasing the volume fraction beyond
this value.
4.1.3 Effective refractive index
The refractive index mismatch between the particles and the surrounding medium determines
the effectiveness of the scattering. In contrast, the effective refractive index, which is an average
over the indices of the scatterers, n scat , and the medium in between, n surr , describes
the sample as a whole and is for example needed to calculate the reflectivity of the sample
surface.
Linear approach
The most simple and straightforward approach to calculate the effective refractive index n eff
of a sample is to establish an average medium with an averaged refractive index
36
n eff = f · n scat + (1− f ) · n surr

4.1 Sample characterization techniques
refractive index
3
2.5
2
1.5
linear
Garnett
1
0 0.2 0.4 0.6 0.8 1
filling fraction
Figure 4.2: Refractive index. The results of the theory of J. C. M. Garnett differs
from the linear approach especially for volume fractions around 50%. The refractive
indices in the graph were calculated for scatterers with refractive index n sc = 2.7 and
a surrounding medium with n m = 1. The maximal difference is found for f = 50.4%,
where the refractive index calculated with the linear approach is about 16% larger than
with Garnett’s theory [49].
This approach works well as long as the microscopic structure of the sample is much smaller
than the wavelength. However, it must fail as soon as the complicated interactions of the
inhomogenous material distribution start to matter.
Garnett effective refractive index
The theory of J. C. M. Garnett [21] calculates the average electrical field inside the medium
assuming that the fields in the scatterers and in the embedding matrix are related linearly. It
yields an effective dielectric constant
ɛ eff = ɛ surr ·
(
1 + 3 f · ɛscat−ɛ )
surr
ɛ scat +2ɛ surr
1 − f · ɛscat−ɛ surr
ɛ scat +2ɛ surr
(4.1)
for spherical scatterers. As the above assumption neglects effects like the Mie resonances, the
theory is valid only for Raleigh scatterers, or for samples with rather high polydispersity or
irregularly shaped particles, where resonant effects average out. As the latter is true for the
samples used in this work, Garnett’s theory will be used to calculate the refractive index.
4.1.4 Surface reflectivity
Reflections at the sample surfaces strongly influence the distribution of the photon density
inside [58]. The angle dependent reflectivity of the boundary between the sample and the
37

4 Samples
1
0.8
reflectivity
0.6
0.4
0.2
0
1 1.5 2 2.5 3
effective refractive index
Figure 4.3: Reflectivity. The graph shows the reflectivity of the sample-air interface as
a function of the effective refractive index of the sample. For the titania samples with
n eff of the order of 1.5 the reflectivity is roughly 0.5 to 0.6.
surrounding medium with refractive indices n eff and n surr can be estimated from Fresnel’s
equations as
R(η) = R ‖(η) + R ⊥ (η)
2
⎧
tan ⎪⎨
2 (θ − η)
= 2 tan 2 (θ + η) + sin2 (θ − η)
2 sin 2 (θ + η) for η ≤ arcsin n surr
n eff
⎪⎩
1 for η > arcsin n surr
n eff
were η and θ are the directions of the incident and the transmitted light beam.
reflectivity is then obtained from the boundary conditions at the surface [58]:
The total
R = 3C 2 + 2C 1
3C 2 − 2C 1 + 2
(4.2)
with
C 1 =
∫ π/2
0
R(η) sin η cos η dη , C 2 =
∫ 0
−π/2
R(η) sin η cos 2 η dη
4.2 The samples
For the experiments three kinds of multiple scattering samples were used: dense colloidal
samples of titanium dioxide, solid teflon as reference sample, and water-fluidized beds with
soda lime glass beads.
38

4.2 The samples
sample particle size [nm] polydispersity [%] TiO 2 content [%] n eff D [m 2 /s] τ [ns]
a-1 1.38 25 1.05
a-2 1.46 13 1.31
NIX-2 1.38 27 0.43
NIX-3 1.31 41 0.67
R700 (DuPont) 245 22 1.58(6) 15(1) 2.0(1)
R700 + gypsum ≈ 17 1.29 250 0.1
R900 (DuPont) 359 29 1.44 17 1.05
R902 (DuPont) 279 38 1.63 15 0.88
S-25 1.35 21 0.65
Ti-pure (Aldrich) 540 37 1.38 19(1) 6.2(3)
Table 4.1: Colloidal samples. Particle size and polydispersity of the particles were
determined from electron microscope images [47]. The effective refractive indices n eff
were calculated using eqn. 4.1, diffusion coefficient D and absorption time τ were measured
in time of flight experiments at laser wavelength λ = 590 nm. Note that the data
are of strongly varied quality. Errors are given only when it was possible to quantify
them.
4.2.1 Titanium dioxide
Titanium dioxide (TiO 2 ), or titania, is widely used as white pigment for paint, plastics or
papers, in cosmetics, medicines and food, and in many other applications. The reason are its
extreme ‘whiteness’ and opacity, which are due to its high refractive index (n = 2.7 in rutile
phase) and its low absorption. In this work we use it to examine the structure of the coherent
backscattering cone. This requires samples with kl ∗ close to unity, as in this regime the cones
become very wide and their features are easier to observe in the experiment.
We use both custom-made TiO 2 particles that were produced in cooperation with the chemistry
department of the University of Konstanz and commercially available titania powders
which were kindly provided for free by Sigma-Aldrich and DuPont. These contain more or
less spherical particles of rutile titania with diameters between 200 and 600 nm (tab. 4.1). As
conglomerates are rare, the powders can quite easily be compressed to volume fractions of the
order of 40%.
The pure titania samples have diffusion coefficients in the low two-digit range. As we also
intended to do measurements on samples with higher diffusion coefficients, we mixed titania
and ground blackboard chalk (gypsum) in a weight ratio of 1 to 5.
4.2.2 Teflon
Polytetrafluoroethylene (PTFE), best known under its DuPont brand name Teflon, is our reference
sample for backscattering experiments. It is a solid (thus avoiding the problem of
re-creating the sample with exactly the same properties for later measurements, which one
would have with a colloidal reference), but without any particular order (crystalline or otherwise),
has low absorption (though not as low as the TiO 2 samples), and a very large transport
mean free path and therefore an extremely narrow backscattering cone.
39

4 Samples
Figure 4.4: Titania and teflon samples. (Clockwise from left:) Teflon sample with 4 cm
diameter and 5 cm height, titania sample with 15 mm diameter and < 1 mm height in
sample container, titania powder. Even on the photo one can observe the significant
difference in the albedos or ”whiteness” of Teflon and titania.
With a diameter of 4 cm and a thickness of 5 cm our teflon sample is not exactly a slab,
which however would be the condition for eqn. 2.12 to be valid. The value of D = 27500 m2 /s
measured in time of flight experiments is therefore not correct. In time of flight experiments
on shorter teflon blocks the diffusion coefficient drops to D = 20000 m2 /s for samples with
thickness L ≈ 2 − 3 cm, and even to D = 16500 m2 /s for L ≈ 1 − 1.5 cm. With this last value
eqns. 2.7 and 2.16 propose a transport mean free path of about 220 µm and a cone width
of approximately 0.01 ◦ . This is very close to the results of the backscattering experiments
reported in sec. 5.2.2, which yield l ∗ = 180 µm, and correspondingly a diffusion coefficient
D = 13300 m2 /s, which however could not be measured in the time of flight experiments due
to the still insufficient size of the teflon samples. a For the absorption time τ the time of flight
data yield 3.3 ns, the refractive index of teflon is given in literature as n = 1.35 [4].
4.2.3 Fluidized beds
A fluidized bed is a mixture of a fluid and a granular medium that has fluid-like properties. As
a dispersion of particles in an upward flux of low-viscosity fluid (gas or liquid) it represents an
intermediate system between suspensions in viscous fluids and dry granular materials. It has
excellent heat and mass transfer characteristics, and the fluid-solid mixture is easy to remove
[a] For technical reasons – the samples have to fit into the time of flight setup – it is impossible to use teflon samples
with larger radii that can reasonably be described as an infinite slab. The adaption of the setup to accommodate
larger samples seems to be the only solution.
40

4.2 The samples
Figure 4.5: Fluidized bed. To fluidize the granular medium, water is pressed through
it from below. The sinter plates in the container below the fluidized bed make the water
flow smooth and laminar to establish a uniform fluidization.
and rejuvenate, which is why fluidized beds are used in oil and coal industrial processes
such as fluidized catalytic cracking and fluidized coal combustion, but also pharmaceutical
processes and bio-reactors [26].
The granular medium in a fluidized bed is fluidized only if the flow rate of the fluid exceeds
a certain minimum value. The drag force on the particles is then sufficient to balance their
net weight, and the grains become free to move. All fluidized beds tend to be unstable and
to form bubbles of fluid without particles. However, for viscous liquids or light and small
particles, a stable uniform fluidization regime can be observed over a range of flow rates just
above minimal fluidization [26].
The particles in the fluidized bed examined in this work are soda lime glass spheres from
Whitehouse Scientific, with diameters around 150 µm (fig. 4.6). We take the refractive index
of soda lime to be 1.52(1), the mass density is given by Whitehouse Scientific as 2.6 g /cm 3 [50].
As the transport mean free path is expected to be proportional to the distance between the
particles in the fluidized bed, the volume fraction was varied between 35% and 52% by using
different flow rates of the pump.
41

4 Samples
0.09
0.08
0.07
relative particle number
0.06
0.05
0.04
0.03
0.02
0.01
0
80 100 120 140 160 180 200
particle diameter [µm]
Figure 4.6: Particle size distribution in the fluidized bed. The size distribution was
measured at the University of Magdeburg on a Retsch Technology Camsizer.
42

5 Experiments
5.1 Conservation of energy in coherent backscattering
Conservation of energy is one of the most fundamental principles in physics. However, the
intensity enhancement of the coherent backscattering cone is one instance where it seems to
be violated at first glance:
The origin of the backscattering enhancement lies in the interference of waves propagating
along reciprocal paths. a This interference can only spatially re-distribute the light energy that
emerges from the sample surface; it can not destroy photons or create new ones. The total
amount of energy per unit time emerging from the sample must therefore be the same with
and without interference:
∫
half-space
∫
α d (θ) dΩ =
half-space
α d (θ) + α c (θ) dΩ
where diffuson α d (θ) and cooperon α c (θ) are the coherent and the incoherent addition of the
photon flux as defined in sec. 2.7. It follows for the coherent backscattering enhancement that
∫
half-space
α c (θ) dΩ = 0 (5.1)
Thus the intensity enhancement of the coherent backscattering cone at small angles should be
balanced by a corresponding intensity cutback to ensure conservation of energy.
Unfortunately, such an intensity cutback had never been observed experimentally, and the
theory of coherent backscattering as developed in sec. 2.7 does not predict an intensity cutback
either. As the principle of conservation of energy holds in any case, the only possible
conclusion is that both the experimental procedure and the theoretical description of the
backscattering cone are too inaccurate to render the cone correctly.
The question if the backscattering cone is depicted correctly by experiment and theory is not
just of purely academic interest. The accurate measurement and description of the cone is
important as the scaling of its width with the inverse product of the wave vector k of the
scattered light and the transport mean free path l ∗ is commonly used to characterize multiple
scattering materials. In particular in the study of Anderson localization of light [13] a reliable
[a] The interference nature of coherent backscattering can be proved for example by the influence of Faraday
rotation on the backscattering cone [35, 36, 37].

5 Experiments
0.35
0.3
0.25
linear
2nd order polynomial
3rd order polynomial
4th order polynomial
laser power [a.u.]
0.2
0.15
0.1
0.05
0
−0.05
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
diode signal
x 10 4
Figure 5.1: Calibration. The graph shows an example calibration function of a single
photodiode. The errors of both the power and the diode signals are hardly recognizable
in the graph and are therefore not drawn explicitly. Overall, 3rd order polynomials
seem to be most appropriate to describe the gain characteristics of the photodiodes and
their electronics.
knowledge of the parameter kl ∗ is needed to characterize the phase transition from diffusive
transport to a localizing state [7, 48].
Our recent progress in this field [48] and the closely related experiments on extremely wide
backscattering cones, where conservation of energy demands a sizeable correction to the
present-day theory, prompted us to attentively review our backscattering experiments and
the evaluation of the backscattering data. An improved theory was contributed by Akkermans
and Montambaux, who managed to eliminate certain improper assumptions from the
calculations.
5.1.1 The power scale of the wide angle setup
One difficulty of the coherent backscattering experiments is to distinguish between the inevitable
background of stray light and electronic dark count, the incoherent background
(which might be hidden not only by the backscattering cone, but also by the compensating
intensity cutback), and the enhancement of the backscattering cone (which does not necessarily
reach a factor of 2 due to incomplete blocking of single scattering). In other words, the
experiments require an independent intensity scale.
We figured that for experiments with our wide angle setup (see sec. 3.2) this independent scale
could be provided by a reference sample with a very high kl ∗ , as it is already used to calibrate
44

5.1 Conservation of energy in coherent backscattering
the photodiodes. For such samples, the angular distribution of the backscattered intensity is
known to be proportional to the diffuson α d (θ) given in eqn. 2.14, as any deviations from the
incoherent background (i.e. the backscattering cone and the energy cutback) are too narrow
or too small to be detected by the setup.
The backscattering of the reference is also proportional to the incoming laser power P and
the total fraction of the backscattered intensity, the albedo A. The latter is reduced by energy
losses in the sample, mostly due to absorption of photonic energy, but also because of photon
leakage at the side and the rear surfaces.
For the calibration of the photodiodes only a relative power scale is needed, as one is merely
interested in compensating different gains of the diodes and their electronics. Therefore the
exact value of the albedo is not important, as long as it is independent of the inserted light
power. If however the albedos are known for sample and reference, one can derive an absolute
power scale for the diode signals.
Herein lies the misapprehension of [22, 23, 47], where the reference sample was also used to
determine the intensity level of the diffuson. As the different albedos of sample and reference
were neglected, this approach confuses relative and absolute power scale and therefore yields
slightly wrong results.
The photodiodes are calibrated by a series of measurements of the diode signals d θ at different
incoming light powers P, so that a relation between the diode signals and the light power
scattered in a certain direction θ can be derived. Previously used were relations like d θ ∝
α d (θ) · P [19], d θ ∝ cos θ · P, or even simply d θ ∝ P [22, 23, 47]. The latter however distort both
diffuson and cooperon for angles θ > 0, so that they can be used only if the backscattering
cone is comparatively narrow.
The effect of the different albedos of sample and reference can be incorporated in the calibration
function by the albedo mismatch b ℵ = A reference /A sample :
d θ ∝ α d (θ) · ℵ · P (5.2)
The albedo mismatch rescales the emitted intensity of the reference, so that the backscattering
of reference and sample is directly comparable.
The precision of the experimental result is determined by the precision of the reference measurements.
The order of the fitted polynomial must therefore be chosen such that the errors
are not artificially reduced. Based on the observations on fig. 5.1 we use 3rd order polynomials
to fit the data and obtain the calibration functions for the photodiodes.
5.1.2 Calculating the albedo
The albedo A of a multiple scattering sample is defined as the ratio of the (back-)scattered
light power P out and the inserted power P in . The albedo of a sample without any losses would
[b] ℵ = hebr. aleph
45

5 Experiments
E sample
(t) / E half−space
(t)
1
0.8
0.6
0.4
0.2
0
free space distribution
half−space distribution
−9.5 −9 −8.5 −8 −7.5 −7 −6.5 −6
log 10
(t [sec])
Figure 5.2: Comparison of the albedos. The time-dependent albedos A(t) of an
absorbing cut-out of the free space (solid line) and respectively a half-space (dashed
line) at the position of the sample are similar enough to replace the latter by the former
in the calculation of the total albedo. Calculations for teflon sample with radius R =
20 mm, thickness L = 50 mm, diffusion coefficient D = 16500 m2 /s, absorption time
τ = 3.3 ns, transport mean free path l ∗ = 220 µm, refractive index n = 1.35.
be equal to one, meaning that the incoming light power P in is equal to the scattered power
P out of a lossless sample. The albedo can therefore also be defined as
A = P with losses
P lossless
=
∫ ∫
∫ surface ∫
surface
j with losses (⃗r ⊥ , t) d⃗r ⊥ dt
j lossless (⃗r ⊥ , t) d⃗r ⊥ dt
(5.3)
In our multiple scattering samples, most of the loss of light energy is due to absorption. If it
were the only reason for energy loss, the albedo could be easily calculated as
A(τ) = 1 − 2(l∗ +z 0 )
e− √
Dτ
(5.4)
2(l ∗ +z
√ 0 )
Dτ
with diffusion coefficient D, absorption time τ, and penetration depth z 0 .
However, as the samples are not infinitely large, one can not a priori exclude that losses at the
side and rear boundaries have significant influence on the photon distribution in the sample.
Unfortunately there is no expression for the photon density distribution in a 3-dimensionally
confined absorbing medium. Therefore we approximate the losses in a sample with finite
radius R and thickness L by comparing the energy inside the non-absorbing infinite half-
46

5.1 Conservation of energy in coherent backscattering
Calculation with free space photon density distribution:
0
t = 10 −11 s
0
t = 10 −10 s
0
t = 10 −9 s
1e8
z [mm]
10
20
30
10
20
30
10
20
30
1e6
1e4
40
40
40
1e2
50
−20 0 20
r [mm]
50
−20 0 20
r [mm]
50
−20 0 20
0
r [mm]
Calculation with photon density distribution of the infinite half-space:
0
t = 10 −11 s
0
t = 10 −10 s
0
t = 10 −9 s
10
10
10
1e6
z [mm]
20
30
20
30
20
30
1e4
40
40
40
1e2
50
−20 0 20
r [mm]
50
−20 0 20
r [mm]
50
−20 0 20
0
r [mm]
Figure 5.3: Calculated photon density distributions in the Teflon sample. The upper
row shows the density distributions calculated as a cutout of free space, the lower row
as a cutout of the infinite half-space. Note that the color axes are logarithmic. Assumed
sample parameters: radius R = 20 mm, thickness L = 50 mm, diffusion coefficient
D = 16500 m2 /s, absorption time τ = 3.3 ns, transport mean free path l ∗ = 220 µm,
refractive index n = 1.35.
47

5 Experiments
space with the energy inside an absorbing cut-out of the half-space at the position of the
sample:
j with losses (⃗r ⊥ , t) = A(t) · j lossless (⃗r ⊥ , t) = j lossless (⃗r ⊥ , t) · ∫
∫
finite sample
infinite half-space
ρ τ (⃗r, t) d⃗r
ρ τ→∞ (⃗r, t) d⃗r
This ignores the fact that the photon density distribution is altered by the sample boundaries,
but if the sample is large enough, only a few photons reach those boundaries anyway, so that
the presence of the boundaries is of nearly no consequence for the density distribution.
As solving eqns. 5.5 and 5.3 for the photon density distribution of the infinite half-space raises
considerable algebraic and numeric difficulties, we fall back to the density distribution in free
space, where the solution is easier to obtain. Fig. 5.2 shows that the albedos A(t) are similar
enough to make this assumption, although the actual photon density distributions look quite
different (fig. 5.3). The only difference in the albedos occurs when the photon cloud reaches
the boundaries, which for the teflon sample in the figure happens at photon travel times
around 10 to 100 ns. Even at this point the error is only a few percent, which presumably is
not larger than the error that is already made by ignoring the altered density distribution.
(5.5)
Evaluating eqn. 5.5 gives the albedo
∫ L
1
A(τ, L, R) =
π(l ∗ +z 0 )
0
(√
− K 0
K 0
(√
4(l ∗ +z 0 ) 2 +z 2
Dτ
z 2
Dτ
)
− 2K 0
(√
)
+ 2K 0
(√
) (√
R 2 +z 2
Dτ
+ K 0
R 2 +4(l ∗ +z 0 ) 2 +z 2
Dτ
)
− K 0
(√
)
2R 2 +z 2
Dτ
−
2R 2 +4(l ∗ +z 0 ) 2 +z 2
Dτ
)
dz (5.6)
where the K n (x) are modified Bessel functions of the second kind. The last step of integration
is performed numerically.
The transport mean free path l ∗ is the result of the evaluation of the backscattering data, for
which the albedo is an input parameter. Therefore the correct value for l ∗ is not yet known
when the albedo is calculated. Instead, we use l ∗ = 3D v
≈ 3n effD
c 0
(see sec. 2.3), which should at
least have the right order of magnitude.
The resulting albedos for the samples examined in the backscattering experiments are given in
tab. 5.1. One immediately observes the striking difference between the albedos of the titania
samples, which are in excess of 99%, and the albedo of teflon, which is about 90%. The
accuracy of the experiments therefore depends strongly on the accuracy of the teflon albedo.
It is however nearly impossible to give an error for the calculated albedos, as the validity of the
various assumptions is not known. Altogether, the error of the albedo mismatch is probably
of the order of one or two percent.
5.1.3 The correct theory of coherent backscattering
The first steps towards a theoretical description of coherent backscattering have already been
taken in sec. 2.1, where the intensity in a multiple scattering medium is presented as the
48

5.1 Conservation of energy in coherent backscattering
sample R [mm] L [mm] A (eqn. 5.4) A (eqn. 5.6)
a-1 7.5 1.35 0.994 0.994
a-2 7.5 1.03 0.995 0.995
NIX-2 7.5 1.26 0.990 0.990
NIX-3 7.5 1.75 0.992 0.992
R700 (DuPont) 7.5 1.0 0.995 0.995
R700 + gypsum 7.5 0.95 0.950 0.951
R900 (DuPont) 7.5 0.5 0.994 0.994
R902 (DuPont) 7.5 0.5 0.991 0.991
S-25 7.5 0.5 0.993 0.993
Teflon (D = 27500 m 2 /s) 20 50 0.901 0.890
Teflon (D = 16500 m 2 /s) 20 50 0.922 0.918
Teflon (D = 13300 m 2 /s) 20 50 0.930 0.927
Ti-pure (Aldrich) 7.5 1.8 0.998 0.998
Table 5.1: Albedos. The albedos for the samples with radius R and thickness L
were calculated using eqns. 5.4 and 5.6. For other sample parameters see tab. 4.1 and
sec. 4.2.2. The albedo of the teflon reference was calculated for the three different
diffusion coefficients mentioned in sec. 4.2.2.
solution of the wave equation by Green’s functions. This description would be exact, but
it can not be solved for a system of millions and billions of randomly distributed particles,
all of whose positions in addition would have to be known exactly. To turn it into a useful
mathematical form, one therefore has to find a collective description of the multiple scattering
medium by choosing exactly the right approximations.
To describe multiple scattering, one must consider the products of Green’s functions corresponding
to all possible scattering sequences. However, the coherent backscattering cone is
observed in the averaged intensity, where most of the pairs of Green’s functions vanish, as
their phase difference is large and random. In the so-called Drude-Boltzmann approximation,
where the average over the product of two Green’s functions can be approximated by the
product over the two averaged functions, the average intensity can therefore be written as
∫∫ 〈Ψin
I d (⃗r,⃗ r ′ ) ∝ (⃗r,⃗r 1 ) 〉 · 〈Ψ
in(⃗r,⃗r ∗ 1 ) 〉 · Γ(⃗r 1 ,⃗r 2 ) · 〈G(⃗r
2 ,⃗ r ′ ) 〉 · 〈G ∗ (⃗r 2 ,⃗ r ′ ) 〉 d⃗r 1 d⃗r 2
where we introduce the structure factor or vertex function Γ(⃗r 1 ,⃗r 2 ), which takes into account
all possible scattering sequences between the first and the last scattering site,⃗r 1 and⃗r 2 .
One contribution that is not taken into account by the diffuson approximation are timeinverted
photon paths, for which the phase difference also vanishes. It is given by
∫∫ 〈Ψin
I c (⃗r,⃗ r ′ ) ∝ (⃗r,⃗r 1 ) 〉 · 〈Ψ
in(⃗r,⃗r ∗ 2 ) 〉 · Γ(⃗r 1 ,⃗r 2 ) · 〈G(⃗r
2 ,⃗ r ′ ) 〉 · 〈G ∗ (⃗r 1 ,⃗ r ′ ) 〉 d⃗r 1 d⃗r 2
In the limit of dilute systems with approximately spherical scatterers, where the Green’s
functions are spherical waves and the transport mean free path is well described by the l ∗
derived in sec. 2.3, this contribution is equivalent to the cooperon derived in sec. 2.7. The
49

5 Experiments
Figure 5.4: Contributions to the cooperon. The diagram on the left depicts the interference
of two light waves running on time-inverted paths between ⃗r 1 and ⃗r 2 . The
diagram on the right illustrates the light paths that include an additional scatterer in⃗r 0 .
cooperon as defined above is often expressed in terms of the Hikami box H (A) as I c (⃗r) ∝
∫∫ H (A) (⃗r 1 ,⃗r 2 ) · Γ(⃗r 1 ,⃗r 2 ) d⃗r 1 d⃗r 2 [25]. H (A) combines the four amplitudes that cross at the entrance
of the loop that is formed by direct and time-inverted path and is therefore also called
a quantum crossing [10].
The total power the cooperon distributes into the half-space above the sample, ∫ I c (θ) dΩ, is
∝ 1/(kl ∗ ) 2 in leading order [9]. However, as Akkermans and Montambaux realized, there are
two other contributions to the multiple scattered power which are of the same order in 1/(kl ∗ )
and which therefore also must be taken into account. In contrast to the cooperon, where the
interference occurs between time-inverted amplitudes, these additional contributions contain
an additional scattering event in⃗r 0 (fig. 5.4).
The additional contributions are defined analogous to the cooperon H (A) by the two dressed
Hikami boxes H (B) and H (C) , which are complex conjugates:
H (B) (⃗r,⃗ r ′ ,⃗r 1 ,⃗r 2 ) =
∫ 〈Ψin
= (⃗r,⃗r 1 ) 〉 · 〈Ψ
in(⃗r,⃗r ∗ 0 ) 〉 · 〈G ∗ (⃗r 0 ,⃗r 2 ) 〉 · 〈G(⃗r
2 ,⃗ r ′ ) 〉 · 〈G ∗ (⃗r 1 ,⃗r 0 ) 〉 · 〈G ∗ (⃗r 0 ,⃗ r ′ ) 〉 d⃗r 0
Solving the integrals requires some mathematical ploys [9] and yields I (B+C)
c ≈ −a/(kl ∗ ) 2 ·
µ/(µ + 1), where µ = cos θ, and a is a fit parameter. Akkermans and Montambaux obtain
a = 1.15 by fitting an intermediate result to experimental data from samples with wide
backscattering cones [9, 19]. However, one can also use the fact that the cooperon together
with the additional contributions now satisfies conservation of energy [8, 9, 10], so that
∫
I (A)
c
+ I (B+C)
c d(⃗r − ⃗ r ′ ) = 0
50

5.1 Conservation of energy in coherent backscattering
1
kl* = 2
kl* = 4
kl* = 10
0.8
cooperon
0.6
0.4
0.2
0
−90 −60 −30 0 30 60 90
scattering angle [deg]
Figure 5.5: Old and new theory. The graph shows the cooperon calculated with
eqn. 2.15 (dashed lines) and including the additional contribution given in eqn. 5.7
(solid lines). The dotted lines give the enhancement of the cooperon for the new theory
(for the old theory the enhancement is always equal to 1). Parameters: wavelength
λ = 590 nm, diffusion coefficient D = 15 m2 /s, reflectivity R = 0.5.
The fit parameter a is therefore obtained by numerical integration of
a =
0
∫ π/2
0
∫ π/2
I (A)
c
1
(kl ∗ ) 2 ·
· sin θ dθ
µ · sin θ dθ
µ + 1
After normalization analogous to eqns. 2.13, the additional contribution to the cooperon finally
becomes
α (B+C)
c
8π · a
= −
3 k 2 l ∗ (l ∗ + 2z 0 ) ·
µ
µ + 1
(5.7)
does not contribute at a specific angular value but it is rather spread out over the
whole angular range, similar to the diffuson. In particular it reduces the enhancement of the
coherent backscattering cone, so that the total cooperon α (A)
c
α (B+C)
c
+ α (B+C)
c at the conetip at θ = 0
51

5 Experiments
is lower than 1 (fig. 5.5). At the wings of the backscattering cone an intensity cutback appears
that balances the intensity enhancement of the cone.
However, the effect of α (B+C)
c is significant only for small kl ∗ . For kl ∗ 10 its influence is negligible,
and the backscattering cone is well described by the cooperon α (A)
c given in eqn. 2.15.
In this regime the diffuson can be found from the large angle wings of the backscattering data
[55]. For small kl ∗ the diffuson is hidden by the intensity cutback, and the proceeding devised
above has to be used to extract the cooperon from the data.
Small kl ∗ however means strongly scattering dense samples, where the scattering particles are
in touching distance. In such systems, near-field effects become important for the scattering
process. Strictly speaking, the random walk model developed for dilute systems and the
deduction of the transport mean free path from the intensity distribution of Mie scattering
and the anisotropy parameter 〈cos θ〉 are no longer valid. The kl ∗ obtained by fitting with the
theory from Akkermans and Montambaux will therefore certainly have the correct order of
magnitude, but might deviate slightly from the real value.
5.1.4 The results of the test experiments
To test the theoretical prediction made above, the coherent backscattering cones of various
samples were measured with the wide angle setup. The diffusion coefficients lay in the range
between 13 m2 /s and 250 m2 /s (see tab. 4.1), so that we expected to observe extremely wide
backscattering cones with kl ∗ close to unity and a distinct intensity cutback as well as narrow
cones with high kl ∗ where no cutback is visible.
As the albedos of the titania samples are basically 1, the albedo mismatch of sample and
reference is mainly given by the albedo of the teflon block. The latter was calculated with
all three diffusion coefficients D = 27500 m2 /s, D = 16500 m2 /s, and D = 13300 m2 /s that can
be derived from time of flight and small angle backscattering experiments, as discussed in
sec. 4.2.2.
The data quality of the wide angle backscattering experiments varies strongly, as the wide
angle setup is extremely sensitive to parasitically scattered light. However, as stated above,
the backscattering cone is evidentially caused by interference, so that the total amount of
backscattered power must be the same both for the coherent and the incoherent addition of
∫ π/2
0
a c (θ) sin θ dθ dφ of the cooperon is
the backscattered intensity, and the integral E = ∫ 2π
0
necessarily zero. E can therefore be used as a criterion to judge the quality of the experimental
data.
Furthermore, it is also possible to substantiate the method to calculate the albedo that was
developed earlier. In fig. 5.6 one observes that the data are approximately centered at E = 0
if the albedo is calculated with D = 16500 m2 /s and D = 13300 m2 /s, but are shifted to E ≈
−0.1 sterad for an albedo calculated with D = 27500 m2 /s. Therefore the data on average
suggest conservation of energy for precisely those values of the diffusion coefficient which
are obtained from the small angle measurements (see sec. 5.2.2). This strongly indicates that
the values for the albedos are correct, although as proof further experiments on samples with
other albedos than teflon or titania are necessary.
52

5.1 Conservation of energy in coherent backscattering
7
6
5
#
4
3
2
1
0
−0.7−0.6−0.5−0.4−0.3−0.2−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
integrated cooperon E [sterad]
7
6
5
#
4
3
2
1
0
−0.7−0.6−0.5−0.4−0.3−0.2−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
integrated cooperon E [sterad]
7
6
5
#
4
3
2
1
0
−0.7−0.6−0.5−0.4−0.3−0.2−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
integrated cooperon E [sterad]
Figure 5.6: Distribution of backscattered energies. The distribution of backscattered
light energies E centers around E ≈ 0 if D = 13300 m2 /s (top) and D = 16500 m2 /s
(center) are used to calculate the albedo of teflon. For D = 27500 m2 /s (bottom) the
distribution is clearly shifted away from zero, so that conservation of energy would not
be obeyed in this case. Consequently, for all further evaluations the diffusion coefficient
obtained from the small angle experiments, D = 13300 m2 /s, is used.
53

5 Experiments
1
0.8
α c
(A)
α c
(A) + αc
(B+C)
R700 with ℵ=1
R700 with ℵ=0.92
cooperon
0.6
0.4
0.2
0
0 10 20 30 40 50 60 70 80 90
scattering angle [deg]
1
0.8
α c
(A)
α c
(A) + αc
(B+C)
NIX−2 with ℵ=1
NIX−2 with ℵ=0.93
cooperon
0.6
0.4
0.2
0
0 10 20 30 40 50 60 70 80 90
scattering angle [deg]
Figure 5.7: Example wide angle backscattering data. Top: R700, bottom: NIX-2. If the
different albedos of sample and reference are neglected (red), the data clearly violate
conservation of energy and do not fit with any theory. The correctly evaluated data
(orange) deviate significantly from α (A)
c , but fit very well with α c = α (A)
c + α (B+C)
c within
the region of confidence (which is only marginally larger than the data points and is
therefore not drawn explicitly). For R700 one obtains kl ∗ = 2.7(3) and l ∗ = 250(30) nm,
and kl ∗ = 6.4(3) and l ∗ = 600(30) nm for NIX-2. The integration over the cooperon
yields E = −0.024(5) sterad for R700 and E = −0.013(7) sterad for NIX-2 if the error of
the albedo is set to zero. E is zero within the margins of error if the error of the albedo
is 1%.
54

5.1 Conservation of energy in coherent backscattering
0.8
0.6
0.4
cooperon
1
0.5
0
theory
R700 data
0 30 60 90
scattering angle [deg]
a−1
a−2
NIX−2
NIX−3
R700
R700+gypsum
R900
R902
S−25
Ti−pure
approx. power ∫ α (A) dΩ
c
integrated cooperon E [sterad]
0.2
0
−0.2
−0.4
−0.6
cooperon
0.3
0.2
0.1
0
theory
R700 data
cooperon
0.4
0.2
0
theory
S−25 data
−0.2
0 30 60 90
scattering angle [deg]
−0.1
0 30 60 90
scattering angle [deg]
−0.8
0 2 4 6 8 10 12 14 16 18 20
kl*
Figure 5.8: Distribution of backscattered energies. The quality of the data varies
strongly. For the evaluation only data sets with E ≈ 0 can be used; data sets where
the backscattered power differs strongly from zero do not obey conservation of energy
and are distorted by experimental problems, as demonstrated by the insets. For the
errorbars an error of ±1% of the albedo mismatch was assumed in addition to the
measured variation of the data.
55

5 Experiments
As it seems to be the most well-founded value, all further evaluations were performed with
the diffusion coefficient obtained from the small angle experiments, D = 13300 m2 /s. On good
data sets with E ≈ 0 one can then observe the anticipated intensity cutback at the wings of the
backscattering cone which balances the intensity enhancement of the cone (fig. 5.7). Especially
for small kl ∗ the measured data therefore deviate considerably from α (A)
c , which earlier was
used to fit the backscattering data [22, 23, 47]. In contrast, the agreement with α (A)
c + α (B+C)
c is
nearly perfect, except for a slight drop in the measured data at scattering angles close to 90 ◦ ,
which however suspiciously looks as if it was caused by some technical problem in the setup.
These findings confirm that the improved theory developed by E. Akkermans and G. Montambaux
can indeed be applied to describe coherent backscattering, and conversely allow to
judge the experimental data not only by conservation of energy, but also by their agreement
with the theoretical curve. A combination of both criteria should hereby yield the best results,
as good agreement with the theory can also be found for E that deviate somewhat from zero,
and on the other hand even a few measurements with E ≈ 0 deviate from the theory (fig. 5.8).
5.2 The coherent backscattering cone in high resolution
The high resolution backscattering setup introduced in sec. 3.3 was developed as a possible
means to investigate Anderson localization. The photons that are trapped on closed loops
in the photon paths not only lead to a different path length distribution in time-resolved
transmission experiments, but also change the shape of the coherent backscattering cone. This
is because localization strongly reduces the number of photons emerging from long photon
paths, which leads to a rounding of the tip of the backscattering cone similar to that caused
by absorption (see sec. 2.7). At the same time the setup was built as a means to measure the
transport mean free path of samples with high kl ∗ and consequently extremely narrow cones,
which of course have other applications than the search for Anderson localization.
5.2.1 Tip rounding of the coherent backscattering cone
In time of flight experiments the titania powder R700 has shown an onset of localization
[5, 6, 48], so that it was our first choice to test if the small angle setup is applicable in the
search for Anderson localization. If the setup is indeed able to resolve the effect of localization,
the measured rounding of the cone tip will deviate from the theoretical prediction for pure
absorption given in eqn. 2.18. If on the other hand the rounding due to localization can not be
observed, this is a strong indication that the concept of the small angle setup is not suitable to
detect localization effects.
To evaluate the small angle backscattering data of R700, the procedure proposed in sec. 3.3
has to be altered a little: The backscattering cone of the titania powder is too wide to directly
find the backscattering direction – the white region in the binary image is too frayed as that
the center of mass could reliably identify the tip of the backscattering cone. Instead, the
backscattering direction is derived from a teflon reference measurement, where the conetip
can be found quite precisely (fig. 5.9). As the direction θ = 0 is determined only by the
56

5.2 The coherent backscattering cone in high resolution
2048
3.7
2048
x 10 4
4.4
3.6
4.2
1024
3.5
3.4
3.3
3.2
x 10 4 1 1024 2048
1024
4
3.8
3.6
3.1
3.4
1
1 1024 2048
3
1
3.2
Figure 5.9: CCD images of R700 and teflon. As the backscattering cone of R700 is very
wide, it is not possible to find the backscattering direction θ = 0 from the CCD image
(left). Instead, a teflon reference (right) has to be used, which gives the same position
for the conetip if the optical path is not readjusted between the measurements.
incoming laser beam and not by the sample, the backscattering directions of the titania sample
and the teflon reference are identical.
The resulting coherent backscattering cone of R700 is depicted in fig. 5.10. Although the
measured data do not contradict the theory for coherent backscattering, it is obvious that it
is impossible to derive more than the vague statement that the conetip is not triangular but
somehow rounded. The data are much too noisy to draw any further conclusions.
This disappointing result is however easy to explain. The small angle setup requires to have a
lot of optical components between the sample and the camera. As none of these are ideal, they
all scatter and give rise to the slightly inhomogenous and noisy background illumination that
can be observed on the CCD images. This noise can not even be removed by the azimuthal
average, as in the center the data are averaged over a comparatively low number of pixels.
Therefore even low-intensity speckles or other disturbances are enough to hide the effect we
are looking for.
Altogether, it must be stated that the small angle setup in its present form can provide no
information about Anderson localization. For investigations about this phenomenon other
experiments like time of flight measurements have proven to be a lot more suitable [5, 6, 48].
5.2.2 The transport mean free path of weakly scattering samples
In terms of intensity resolution, measuring the transport mean free path of weakly scattering
samples with large kl ∗ with the small angle setup is less challenging. The mean free path
of teflon, our standard reference sample, can be estimated from eqn. 2.7 to be something
like 220 µm, if the velocity of energy transport is given by the speed of light in a material
57

5 Experiments
3.65 x 104 scattering angle [deg]
3.6
theory (absorption)
data average
3.55
intensity [a.u.]
3.5
3.45
3.4
3.35
3.3
3.25
0 0.2 0.4 0.6 0.8 1
3.62 x 104 scattering angle [deg]
3.61
theory (absorption)
data average
3.6
intensity [a.u.]
3.59
3.58
3.57
3.56
3.55
3.54
0 0.05 0.1 0.15 0.2
Figure 5.10: Tip of the R700 cone. The azimuthal average of the backscattering data
(bottom: detail) was fitted with the theory for an absorbing R700 sample (D = 15 m2 /s,
τ = 2 ns, l ∗ = 250 nm, λ = 575 nm). The line width gives the approximate order of
magnitude of the intensity difference between a localizing and a non-localizing sample.
58

5.2 The coherent backscattering cone in high resolution
intensity [a.u.]
1
0.8
0.6
0.4
0.2
1 mm
2 mm
3 mm
4 mm
5 mm
6 mm
7 mm
9 mm
10 mm
11 mm
12 mm
13 mm
14 mm
15 mm
16 mm
0
0 0.02 0.04 0.06 0.08 0.1
scattering angle [deg]
normalized intensity
1
0.8
0.6
0.4
0.2
1 mm
2 mm
3 mm
4 mm
5 mm
6 mm
7 mm
9 mm
10 mm
11 mm
12 mm
13 mm
14 mm
15 mm
16 mm
0
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
scattering angle [pixels]
Figure 5.11: Focus accuracy. The position of the focussing lens was varied in a range
of 16 mm around its nominal position 50 cm in front of the CCD camera. The lens
is positioned correctly when the width of the teflon backscattering cone (top) and the
mirror spot (bottom) is minimal, i.e. between 7 and 10 mm. For higher numbers the lens
is too close to the camera, for lower numbers it too far away. (To make the measurements
easier to compare the data were scaled to have intensities between 0 and 1.)
59

5 Experiments
220
200
180
l* [µm]
160
140
120
100
2 4 6 8 10 12 14 16
lens position [mm]
Figure 5.12: Transport mean free path in dependence of the lens position. Between
7 and 10 mm the transport mean free path of teflon has a slight maximum of approximately
180 µm, which can be clearly derived from the trend of the data. (The orange
line was drawn as a guide to the eye.)
with the refractive index of teflon, n = 1.35, and the diffusion coefficient is D = 16500 m2 /s,
as suggested by the time of flight experiments (see sec. 4.2.2). With eqn. 2.16 one obtains
a FWHM of about 0.01 ◦ , which corresponds to 12 pixels on the CCD chip. [47, 49] c report
experimentally determined cone widths of about 0.03 ◦ , which would correspond to 35 pixels
in our setup.
This however is not that much wider than the intensity profile of the focus spot (see sec. 3.3.4),
which therefore will have a considerable influence on the shape of the intensity enhancement.
Consequently, the measured backscattering cone of teflon can be expected to be significantly
widened, and the convolution of the fitted theory with the focus spot profile will become
essential to yield correct results.
The accuracy of the setup is for the most part determined by the divergence of the incoming
laser beam (which should be as low as possible) and the correct position of the focussing lens
in front of the camera. The former seems to be comparatively uncritical, as the beam can easily
be adjusted to have a divergence lower than ±0.01 ◦ , which is the range where no significant
change in the cone shape seems to occur.
The appropriate distance between lens and camera is more difficult to determine. Fig. 5.12
shows that the fitted values for the transport mean free path can fluctuate strongly, even
if the lens–camera distance is set correctly. Hence it is reasonable to measure a series of
backscattering cones where the setting is deliberately varied slightly for each take. The correct
[c] In [49] there is obviously a typing error at this point.
60

5.2 The coherent backscattering cone in high resolution
1
0.8
l* = 175µm
l* = 180µm
l* = 185µm
l* = 220µm
teflon data
intensity [a.u.]
0.6
0.4
0.2
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
scattering angle [deg]
Figure 5.13: Coherent backscattering cone of teflon. The backscattering data can be
fitted with a cone (i.e. eqn. 2.18) with l ∗ around 180 µm. From eqn. 2.7 one would expect
l ∗ = 220 µm, which is only slightly narrower.
value of l ∗ can then be read from the trend of the data, even if these contain some outliers that
could otherwise be quite confusing.
The reason for this tendency to produce unreliable data is that the intensity profile of the
mirror reflex is rendered by extremely few data points. The convolution of mirror profile and
theoretical cone shape, which is used to fit the data, is therefore prone to slight variations
in the slope of the backscattering cone, which result in apparent variations of l ∗ . Although
one instinctively tries to make the focus spot as small as possible to enhance the angular
resolution, it is actually more preferable to go for larger focus spots, which are rendered more
accurately. If the information about the intensity profile of the focus spot is incorporated in
the data evaluation afterwards, this actually does not reduce the angular resolution of the
experiment.
For the teflon sample the data suggest l ∗ ≈ 180 µm, which is consistent with the results
obtained with an older version of the setup [19, 47]. It is also not much lower than the
220 µm that one calculates from eqn. 2.7 using the diffusion coefficient D = 16500 m2 /s. An
estimate with D = 27500 m2 /s on the other hand would give a transport mean free path of
about 370 µm, which agrees neither with the old nor the new experimental results. From
the measured transport mean free path one can derive a diffusion coefficient D = 13300 m2 /s,
which is probably the most correct of the three values, although it was not possible to confirm
it in time of flight experiments for technical reasons.
61

5 Experiments
Altogether the teflon backscattering measurements confirm that the newly revised small angle
setup yields reliable results for the transport mean free paths of weakly scattering samples.
The difficulty is rather to properly determine the diffusion coefficient that enters in eqn. 2.7
from which the expected value of l ∗ is calculated. The problem herein is a technical one, as
only samples of a certain size fit into the time of flight setup. If in the future the diffusion
coefficients of other weakly scattering samples are to be measured, an appropriate adaption
of the setup should be considered.
5.2.3 Coherent backscattering on fluidized beds
Parallel to the experiments on weakly scattering solid samples like teflon, another field of application
for the small angle setup arose from a cooperation with the group of Dr. M. Schröter
of the Max Planck Institute (MPI) for Dynamics and Self-Organization in Göttingen. Working
on statistical mechanics of granular media, their interest lies in the granular temperature of
fluidized beds, which is proportional to the average speed of the particles.
The group uses diffusing wave spectroscopy (DWS) to measure the typical timescale of the
movements of the particles in the bed. To obtain a speed, they also needed a length scale for
the transport. This scale is given by the transport mean free path l ∗ , which can for example
be measured in coherent backscattering experiments.
Setup modifications
The small angle setup (fig. 3.5) was originally designed to take up the backscattering from
the horizontal top surface of the sample. This is not possible for fluidized beds, which have
to be accessed from the side through the container walls (fig. 4.5), so that the setup has to be
modified accordingly. Here it is important to position the circular polarizer directly in front of
the container to avoid the polarization to be altered by additional optical components between
beamsplitter and bed. As the light now transmits the container walls before and after being
scattered in the fluidized bed, one has to take into account the refraction and reflection at the
acrylic glass surfaces.
In multiple scattering, the transition from the fluidized bed to the acrylic glass becomes noticeable
in the average penetration depth z 0 = 2/3 · (1 + R)/(1 − R) · l ∗ , where R is the reflectivity
of the sample–glass surface.
In addition, the scattering angle θ CCD measured at the CCD camera has to be corrected for
the refraction at the glass–air surface of the container (fig. 5.16) to be compared with the
theoretical description of the backscattering cone:
( )
nair
θ ms = arcsin · sin θ CCD ≈ n air
· θ CCD (5.8)
n glass n glass
Although the light passes a circular polarizer before and after scattering, the ring structure of
single scattering at Mie particles can also be found on the backscattering images (fig. 5.14).
62

5.2 The coherent backscattering cone in high resolution
1
3.7
3.6
x 10 4 0 0.2 0.4 0.6 0.8 1
15 x 107 scattering angle [deg]
before polarizer
after polarizer
1024
3.5
3.4
intensity [a.u.]
10
5
3.3
2048 3.2
1 1024 2048
0
Figure 5.14: Single and multiple scattering. The CCD image (left) of the backscattering
of a fluidized bed shows a pronounced ring structure caused by single scattering.
Calculations with Mie theory (see fig. 5.15) show, that for circularly polarized light the
central maximum is extinguished by a transition through a circular polarizer after the
scattering event (right; the graph was calculated using the MATLAB code from the appendix).
The backscattering image however still shows an intensity maximum at the
center of the ring structure, which therefore must be the coherent backscattering cone.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛
⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
S 11 + S 33
1 −1 0 0 1 0 0 0 S 11 S 12 0 0 1 0 0 0 1 −1 0 0 1
p = ⎜−S 11 − S 33
⎟
⎝ 0 ⎠ = 1 ⎜−1 1 0 0
⎟
2 ⎝ 0 0 0 0⎠ ·
⎜0 0 0 1
⎟
⎝0 0 1 0⎠ ·
⎜S 12 S 11 0 0
⎟
⎝ 0 0 S 33 S 34 ⎠ ·
⎜0 0 0 1
⎟
⎝0 0 1 0⎠ · 1
⎜−1 1 0 0
⎟
2 ⎝ 0 0 0 0⎠ ·
⎜−1
⎟
⎝ 0 ⎠
0
0 0 0 0 0 −1 0 0 0 0 −S 34 S 33 0 −1 0 0 0 0 0 0 0
p = LP · QWP · S · QWP · LP · p 0
p S = S · QWP · LP · p 0
⎛ ⎞ ⎛
⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
S 11 S 11 S 12 0 0 1 0 0 0 1 −1 0 0 1
p S = ⎜ S 12
⎟
⎝−S 34 ⎠ = ⎜S 12 S 11 0 0
⎟
⎝ 0 0 S 33 S 34 ⎠ ·
⎜0 0 0 1
⎟
⎝0 0 1 0⎠ · 1
⎜−1 1 0 0
⎟
2 ⎝ 0 0 0 0⎠ ·
⎜−1
⎟
⎝ 0 ⎠
−S 33 0 0 −S 34 S 33 0 −1 0 0 0 0 0 0 0
Figure 5.15: Calculation of the Stokes vector for the experiment. Linearly polarized
light with Stokes vector p 0 transmits a circular polarizer, which is represented by a
combination of a linear polarizer LP and a quarter-wave-plate QWP, is scattered at the
particle S, and again transmits the circular polarizer. The resulting Stokes vector p
can be compared with the Stokes vector p S directly after the scatterer. The scattered
intensities depicted in fig. 5.14 are given by the first elements of p and p S .
63

5 Experiments
Figure 5.16: Correction of the scattering angle. Left: Multiply scattered light from the
medium with effective refractive index n eff enters the container wall (refractive index
n glass ) with scattering angle θ ms . At the container–air interface this light is refracted. The
scattering angle measured at the CCD camera is therefore θ CCD . Right: Singly scattered
light from a particle with refractive index n particle in water is refracted at both surfaces
of the container walls.
The reason for this at first surprising observation is that the polarizer removes only the central
maximum of the backscattered intensity distribution around θ = 0, which for the large
particles in the fluidized bed is only about 0.2 ◦ wide.
The singly scattered light is refracted at both surfaces of the acrylic glass wall of the container
(fig. 5.16). The scattering angle at the CCD camera and the original scattering angle θ ss at the
particle are therefore related like
( )
nair
θ ss = arcsin · sin θ CCD ≈
n water
n air
n water
· θ CCD (5.9)
This means that the signal at the CCD camera contains two superimposed contributions,
which require different corrections to obtain the real scattering angles!
The contribution of single scattering
As explained before, the circular polarizer in the setup removes the central maximum of single
scattering, but leaves the rest of its intensity pattern more or less unaltered. If the refractive
indices of particles and water and the size distribution of the particles are known exactly, the
intensity distribution of single scattering can be calculated using Mie theory (fig. 5.15), and
can then be subtracted from the data.
The background illumination of the backscattering image however is rather non-uniform and
can therefore not be removed completely. This makes it an extremely delicate task to choose
the correct parameters for the Mie distribution to subtract (fig. 5.17). We adapt the Mie distribution
to the first order maximum and minimum of the azimuthally averaged data, where
the variance of the data is still comparatively low.
64

5.2 The coherent backscattering cone in high resolution
intensity [a.u.]
4000
3000
2000
1000
data profiles (f = 43%)
data average (f = 43%)
n water
= 1.334 , n particle
= 1.519 .
n water
= 1.334 , n particle
= 1.520
n water
= 1.334 , n particle
= 1.521
n water
= 1.334 , n particle
= 1.522
n water
= 1.334 , n particle
= 1.523
n water
= 1.333 , n particle
= 1.519
n water
= 1.333 , n particle
= 1.520
n water
= 1.333 , n particle
= 1.521
n water
= 1.333 , n particle
= 1.522
n water
= 1.333 , n particle
= 1.523
0
−1000
−2000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
scattering angle θ CCD
[deg]
Figure 5.17: Measured data and calculated Mie distributions. Due to non-uniform
background lighting the data vary strongly for angles larger than 0.3 ◦ and can not be
fitted properly. To get a reliable scaling, the Mie distributions were scaled to fit the first
order minimum and maximum of the azimuthally averaged data, whose heights are
marked in the graph. The first maximum of the calculated Mie distributions is shifted
towards smaller scattering angles, which shows that the particle size distribution used
for the calculations is slightly wrong.
65

5 Experiments
1
0.8
data − ( n water
= 1.334 , n particle
= 1.519 ) .
data − ( n water
= 1.334 , n particle
= 1.521 )
data − ( n water
= 1.334 , n particle
= 1.523 )
intensity [a.u.]
0.6
0.4
0.2
0
0 0.05 0.1 0.15 0.2
scattering angle θ ms
[deg]
Figure 5.18: Coherent backscattering cone of a fluidized bed. To obtain the coherent
backscattering cone, three Mie distributions (stretched along the angular axis to match
the slope of the first maximum of the data) with different zero levels were subtracted
from the measured intensity distribution depicted in fig. 5.17. The angular width of the
cone can be read from the graph to an accuracy of about 15%.
The graph also shows that the actual distribution of particle sizes in the sample differs slightly
from the one measured beforehand – either because the size distribution in the fluidized bed is
not homogenous or because the size distribution changed during the experiments – as the first
maxima of measured data and Mie theory do not fit. For the evaluation the Mie distribution
was therefore stretched to match the measured intensity distribution. This is acceptable as the
intensity characteristics are always the same at small angles; however, the zero level of the Mie
distribution varies with the input parameters, so that the width of the backscattering cone is
known only to an accuracy of about 15% (fig. 5.18).
The transport mean free path of a fluidized bed
The first backscattering experiments were performed on a fluidized bed of water and soda
lime glass spheres with particle diameters around 150 µm. The transport mean free path was
expected to be proportional to the distance between the particles in the fluidized bed. So the
backscattering was measured for different flux rates in the bed, resulting in particle volume
fractions between 39% and 58% (fig. 5.19). However, due to strong fluctuations in the data no
real trend in the width of the backscattering cone can be observed – which does not necessarily
mean that there is none. Here additional speckles from static particles stuck at the container
surface do have a rather disadvantageous impact.
66

5.2 The coherent backscattering cone in high resolution
3.25 x 104 scattering angle θ CCD
[deg]
3.2
3.15
3.1
f = 58%
f = 49%
f = 47%
f = 43%
f = 39%
intensity [a.u.]
3.05
3
2.95
2.9
2.85
2.8
0 0.05 0.1 0.15 0.2 0.25 0.3
Figure 5.19: Backscattering for various volume fractions. The transport mean free path
is expected to be proportional to the volume fraction of the fluidized bed. However, due
to strong fluctuations in the data no real trend in the width of the backscattering cone
can be observed.
For the fluidized bed with f = 43% one reads from fig. 5.18 and with eqn. 2.16 a transport
mean free path between 50 µm and 60 µm. This would mean that l ∗ is significantly shorter
than the diameter of the particles in the fluidized bed, and it stands in strong contrast to
reported mean free paths from similar experiments (e.g. [52, 17, 39, 41, 46, 51, 57]), which are
all of the order of several particle diameters.
As the present data are only first results and the experiments are still ‘work in progress’,
there are all kinds of possible explanations for this surprising result, of both theoretical and
experimental nature. These will have to be discussed and investigated if further experiments
on fluidized beds or similar systems are planned for the future.
67

68
5 Experiments

6 Summary
The focus of the work presented in this thesis lay on improvements of the experimental techniques
for the investigation of multiple scattering phenomena, and in particular the coherent
backscattering cone of visible light.
This interference effect had been found to apparently violate the principle of conservation of
energy: The total amount of backscattered energy can not exceed the energy which the light
source inserts into the sample. The intensity enhancement of the cone should therefore be balanced
by a corresponding intensity cutback, which however had never been observed neither
in experiment nor in theory. Inaccuracies in the latter two being the only possible explanation,
we reviewed our backscattering experiments and the evaluation of the backscattering
data, and also tested an improved theory which had been worked out by E. Akkermans and
G. Montambaux.
The two theoreticians extended the theoretical description of the backscattering cone by two
additional terms that contribute to the total backscattered energy in the same order as the
cone itself. They lead to a cutback of the scattered intensity below the level of the incoherent
addition of the backscattered intensity which balances the intensity enhancement of the cone.
In the new theoretical description of the backscattering cone the energy is therefore conserved.
The key to a precise measurement of the shape of the coherent backscattering cone was the
correct calibration of the photodiodes of the wide angle setup. For this a teflon reference
sample is used, the albedo of which differs from the titania albedo by about 10%. In earlier
experiments this difference had been neglected, which led to a misrepresentation of the
backscattering cone, so that conservation of energy seemed to be seriously violated. If however
the albedos of sample and reference are considered in the evaluation, one can observe
an intensity cutback at the wings of the cone, which balances the intensity enhancement in
backscattering direction, and which also agrees with the predictions made by Akkermans and
Montambaux.
A second part of this work concerned the remodeling of the small angle setup for the measurement
of the scattered intensity distribution in a small range around backscattering direction.
One goal was to detect Anderson localization by the rounding of the tip of the coherent
backscattering cone. For this an older CCD camera was replaced by a high resolving model,
which improves the maximum intensity resolution by more than two orders of magnitude and
the maximum angular resolution or respectively the maximum angular range by a factor of
three. However, it turned out that the optical components between sample and camera cause
too much extraneous light, so that the necessary intensity resolution can not be achieved.
Still, the improved setup can be used to measure the transport mean free path of weakly
scattering samples like for example teflon. For teflon one can measure a cone with FWHM ≈
0.03 ◦ , corresponding to a transport mean free path l ∗ = 180 µm and a diffusion coefficient

6 Summary
D = 13300 m2 /s. This agrees pretty well with results of measurements with the old setup and
with the results of the wide angle experiments, which yield correct results if the diffusion
coefficient obtained with the small angle setup is used for the evaluation. With these findings
the reliability of the results of the small angle setup is confirmed.
Another application for the small angle setup was to determine the transport mean free path
of fluidized beds. As the scattering particles in these experiments are rather large and have a
regularly spherical shape, the intensity distribution of the backscattered light is superimposed
by the ring structure of the single scattering on the Mie particles. This structure can however
be calculated theoretically and adapted to the measured curves, so that the backscattering
cone can be extracted from the data.
In first measurements the width of the backscattering cone led to a transport mean free path
that was significantly shorter than the particle diameter of the scatterers. This is in strong
contrast to the results of similar experiments, which are reported to be of the order of several
particle diameters. The reason for this is still under discussion; however, the basic procedure
for the evaluation of the backscattering data of fluidized beds was tested successfully.
For future experiments, the combination of the two revised backscattering setups offers the
unique possibility to measure kl ∗ over more than three decades, from samples at the transition
to Anderson localization at kl ∗ ≈ 1 to weakly scattering samples with extremely narrow
backscattering cones. Possible applications range from experiments on new, custom designed
particles with small kl ∗ to measurements on foams or biological tissue. Some of these experiments
are already planned for the near future, and will hopefully provide a step further
towards fully understanding multiple scattering.
70

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74

Figures and Tables
Figures
Backscattering of a fluidized bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Title
2.1 Coordinate system of single scattering . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Vibration ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Polarization of a scattered wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Scattering cross section and scattering anisotropy . . . . . . . . . . . . . . . . . 9
2.5 Random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Scattering in the presence of a single boundary . . . . . . . . . . . . . . . . . . 12
2.7 Scattering in the presence of two parallel boundaries . . . . . . . . . . . . . . . 13
2.8 Radiative transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.9 Photon flux in backscattering geometry . . . . . . . . . . . . . . . . . . . . . . . 16
2.10 Speckles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.11 Theorem of reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.12 Diffuson and cooperon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.13 Coherent backscattering with absorption . . . . . . . . . . . . . . . . . . . . . . 22
2.14 Coherent backscattering with absorption – unnormalized cooperon . . . . . . . 23
3.1 Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Wide angle setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Polarizer effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Conetip rounding from absorption and localization . . . . . . . . . . . . . . . . 29
3.5 Small angle setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 The optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.7 Sample shaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figures and Tables
3.8 Time of flight setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 Particle size and polydispersity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Refractive index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.4 Titania and teflon samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 Fluidized bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.6 Particle size distribution in the fluidized bed . . . . . . . . . . . . . . . . . . . . 42
5.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2 Comparison of the albedos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3 Calculated photon density distributions in the Teflon sample . . . . . . . . . . 47
5.4 Contributions to the cooperon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.5 Old and new theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.6 Distribution of backscattered energies. . . . . . . . . . . . . . . . . . . . . . . . . 53
5.7 Example wide angle backscattering data . . . . . . . . . . . . . . . . . . . . . . 54
5.8 Distribution of backscattered energies . . . . . . . . . . . . . . . . . . . . . . . . 55
5.9 CCD images of R700 and teflon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.10 Tip of the R700 cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.11 Focus accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.12 Transport mean free path in dependence of the lens position . . . . . . . . . . . 60
5.13 Coherent backscattering cone of teflon . . . . . . . . . . . . . . . . . . . . . . . . 61
5.14 Single and multiple scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.15 Calculation of the Stokes vector for the experiment . . . . . . . . . . . . . . . . 63
5.16 Correction of the scattering angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.17 Measured data and Mie distributions . . . . . . . . . . . . . . . . . . . . . . . . 65
5.18 Coherent backscattering cone of a fluidized bed . . . . . . . . . . . . . . . . . . 66
5.19 Backscattering for various volume fractions . . . . . . . . . . . . . . . . . . . . . 67
76

Figures and Tables
Tables
4.1 Colloidal samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1 Albedos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
77

78
Figures and Tables

MATLAB codes
Angular intensity distribution of single scattering
The following MATLAB script calculates the angular distribution of the singly scattered intensity
after the sequence ‘circular polarizer – spherical scatterer – circular polarizer’ in both
backscattering setups, as derived in secs. 2.2 and 5.2.3.
function mie_setup
refractive_particle = 1.52 ;
refractive_surround = 1.33 ;
wavelength = 575e-9 ;
diameter = 150e-6 ;
% refractive index of the scatterer
% refractive index of the surrounding medium
% wavelength of the scattered light
% particle diameter
a = diameter / 2 ;
m = refractive_particle / refractive_surround ;
k = 2 * pi * refractive_surround / wavelength ;
x = k * a ;
angle = 180 : 0.01 : 181 ;
theta = angle / 180 * pi ;
mu = cos(theta) ;
% or larger angular range for wide angle setup
S1 = 0 ;
S2 = 0 ;
N = 50 + 1.03 * x ;
N = fix (N) ;
% avoids numerical overflow in Bessel functions
for n = 1 : N
A = J(n,m*x) ;
B = J(n,x) ;
C = dJ(n,x) ;
D = dJ(n,m*x) ;
E = H(n,x) ;
F = dH(n,x) ;
an = ( m * A * C - B * D ) / ( m * A * F - E * D ) ;
bn = ( A * C - m * B * D ) / ( A * F - m * E * D ) ;

MATLAB codes
end
if n == 1
p1 = 0 ;
p = 1 ;
t = mu ; % because t = n * mu * p - (n+1) * p1 = 1 * mu * 1 - 2 * 0
else
p0 = p1 ;
p1 = p ;
p = (2*n-1) / (n-1) * mu .* p1 - n / (n-1) * p0 ;
t = n * mu .* p - (n+1) * p1 ;
end
S1 = S1 + (2*n+1) / (n*(n+1)) * (an*p + bn*t) ;
S2 = S2 + (2*n+1) / (n*(n+1)) * (an*t + bn*p) ;
S11 = 1/2 * ( S2 .* conj(S2) + S1 .* conj(S1) ) ;
S33 = 1/2 * ( S1 .* conj(S2) + S2 .* conj(S1) ) ;
I1 = S11 ;
I2 = 1/2 * ( S11 + S33 ) ;
% intensity after the scattering particle
% after second transmission of circular polarizer
plot (angle,I1,angle,I2)
xlabel (’scattering angle [deg]’)
ylabel (’intensity [a.u.]’)
legend (’after scatterer’,’after circular polarizer’)
end
function J = J (n,x)
J = sqrt(x*pi/2) * besselj(n+1/2,x) ;
end
function H = H (n,x)
H = sqrt(x*pi/2) * ( besselj(n+1/2,x) + i * bessely(n+1/2,x) ) ;
end
function dJ = dJ (n,x)
dJ = sqrt(pi/(2*x)) * ( (1+n) * besselj(n+1/2,x) - x * besselj(n+3/2,x) ) ;
end
function dH = dH (n,x)
dH = sqrt(pi/(2*x)) * ( (1+n) * ( besselj(n+1/2,x) + i * bessely(n+1/2,x) ) ...
- x * ( besselj(n+3/2,x) + i * bessely(n+3/2,x) ) ) ;
end
80

Evaluation of the wide angle data
Evaluation of the wide angle data
This MATLAB script calibrates and displays the data of the wide angle setup (secs. 3.2 and
5.1), calculates the integrated cooperon E, and draws a theory curve which can be fitted to the
data by varying the value for the transport mean free path l ∗ .
Each data set consists of a series of reference measurements at different incoming laser powers
and one measurement with the sample itself. The data are stored in ASCII files which contain
the diode signals and their errors. The diode positions (in degrees) are read from the file
‘diode angles.dat’. The filenames of the data files contain the average incoming laser powers
and their errors (in arbitrary units), which are measured by the power meter in the setup.
The albedo mismatch ℵ = A reference /A sample has to be calculated from eqn. 5.6, diffusion
coefficient D and absorption time τ are measured in time of flight experiments (sec. 3.4), and
the effective refractive index n eff is calculated using eqn. 4.1. With the variable ‘enhancement’
the height of the theoretical cone can be adapted to that of the measured one.
function wide_angle_evaluation
%% Initialize variables and load files
file_samp = ’C:\Examples\example_0.100+-0.001.dat’ ; % sample file
path_ref = ’C:\Examples\’ ;
% path reference files
name_ref = ’teflon_01_0.200+-0.001.dat’ ;
% arbitrary reference file
wavelength = 590e-9 ;
D = 15 ;
tau = 2e-9 ;
n_eff = 1.58 ;
l = 500e-9 ;
aleph = 0.890 ;
enhancement = 0.9 ;
% laser wavelength
% diffusion coefficient of the sample
% absorption time of the sample
% effective refractive index of the sample
% transport mean free path for theory fit
% albedo mismatch
% enhancement of the backscattering cone
% angular positions of the photodiodes
angles = load (’C:\Examples\diode_angles.dat’,’-ascii’) ;
theta = angles * pi / 180 ;
% sample file
data_samp = load (file_samp,’-ascii’) ;
parts = regexp (file_samp, ’_|+-|.dat’, ’split’) ;
P_samp = parts ( size(parts,2) - 2 ) ;
P_samp = str2double (P_samp:,1) ;
Perror_samp = parts ( size(parts,2) - 1 ) ;
Perror_samp = str2double (Perror_samp:,1) ;
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MATLAB codes
% reference files
index = regexp (name_ref, ’_’, ’start’) ;
index = index (length(index) - 1) ;
namebase = name_ref (1 : index) ;
list = dir ([path_ref,namebase,’*.dat’]) ;
data_ref = zeros (size(list,1),256,2) ;
P_ref = zeros (size(list,1),1) ;
for N = 1 : size(list,1)
filename = [path,list(N).name] ;
data_ref(N,:,:) = load (filename,’-ascii’) ;
parts = regexp (list(N).name, ’_|+-|.dat’, ’split’) ;
p_ref = parts ( size(parts,2) - 2 ) ;
P_ref(N) = str2double (p_ref:,1) ;
end
% other data
k = 2 * pi / wavelength ;
c = 3e8 ; % speed of light
l_approx = D * n_eff / c ;
n_surr = 1 ; % refractive index of surrounding medium = air
R = reflectivity (n_eff,n_surr) ;
%% Calibrate data
cone = zeros(256,1) ;
cone_delta = zeros(256,1) ;
for n = 1 : 256
x = data_ref(:,n) ;
y = P_ref * diffuson(theta(n),l_approx,R) ;
temp1 = sortrows ([x,y],1) ; % sort by x
x = temp1(:,1) ;
y = temp1(:,2) ;
warning off % many warnings because of defect photodiodes (causing NANs)!
[p,s,mu] = polyfit (x,y,3) ;
[cone(n),cone_delta(n)] = polyval (p,data_samp(n),s,mu) ;
warning on
end
%% Fold data
dummy = sortrows ([abs(theta), cone, cone_delta]) ;
theta = dummy (:,1) ;
cone = dummy (:,2) ;
cone_delta = dummy (:,3) ;
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Evaluation of the wide angle data
% remove NANs and data with huge errors from damaged photodiodes
m = 1 ;
for n = 1 : 256
if isnan (cone(n))
elseif cone_delta(n) > 0.1*P_samp
else
Cone(m) = cone(n) ;
Delta(m) = cone_delta(n) ;
Theta(m) = theta(n) ;
m = m + 1 ;
end
end
cone = Cone ;
cone_delta = Delta ;
theta = Theta ;
clear Theta Cone Delta
% fold data (i.e. -pi/2 < theta < pi/2 -> 0 < theta < pi/2)
m = 1 ;
n = 1 ;
while n < size(theta,2)
if theta(n+1) > theta(n)
Theta(m) = theta(n) ;
Theta(m+1) = theta(n+1) ;
Cone(m) = cone(n) ;
Cone(m+1) = cone(n+1) ;
Delta(m) = cone_delta(n) ;
Delta(m+1) = cone_delta(n+1) ;
n = n + 1 ;
else % theta(n) == theta(n+1)
Theta(m) = theta(n) ;
Cone(m) = (cone(n) + cone(n+1)) / 2 ;
Delta(m) = 0.5 * sqrt ( (cone_delta(n)).^2 + (cone_delta(n+1)).^2 ) ;
n = n + 2 ;
end
m = m + 1 ;
end
cone = Cone ;
cone_delta = Delta ;
theta = Theta ;
%% Derive experimental and theoretical cones
% Correct cone with albedo mismatch
corr_cone = aleph * cone ;
corr_cone_delta = aleph * cone_delta ;
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MATLAB codes
% Derive experimental diffuson and cooperon
exp_diffuson = diffuson (theta,l,R) * P_samp ;
exp_diffuson_delta = diffuson (theta,l,R) * Perror_samp ;
exp_cooperon = (corr_cone - exp_diffuson) / P_samp ;
exp_cooperon_delta = ...
sqrt ( (corr_cone_delta / P_samp).^2 ...
+ (exp_diffuson_delta / P_samp).^2 ...
+ ((corr_cone - exp_diffuson) / P_samp^2 * Perror_samp).^2 ) ;
% Calculate theory curve
theory_theta = 0 : 0.001*pi : pi/2 ;
theory_cooperon = enhancement * cooperon (theory_theta,k,l,R,D,tau) ;
%% Calculate energy
e_theta = [ 0 , theta , pi/2 ] ;
e_cooperon = [ 1 , exp_cooperon , 0 ] ;
e_cooperon_delta = [ 0 , exp_cooperon_delta , 0 ] ;
E = 2*pi ...
* sum ( ( e_theta(:,2:end) - e_theta(:,1:end-1) ) ...
* 1/2 .* ( e_cooperon(:,1:end-1) .* sin(e_theta(:,1:end-1)) ...
+ e_cooperon(:,2:end) .* sin(e_theta(:,2:end)) ) ) ;
E_delta = 2*pi ...
* sqrt ( sum ( ( 1/2 * ( e_theta(1,2:end) - e_theta(1,1:end-1) ) ...
.* e_cooperon_delta(1:end-1) ).^2 ...
.* sin (1/2 * ( e_theta(1,2:end) + e_theta(1,1:end-1) )) ...
+ ( 1/2 * ( e_theta(1,2:end) - e_theta(1,1:end-1) ) ...
.* e_cooperon_delta(2:end) ).^2 ...
.* sin (1/2 * ( e_theta(1,2:end) + e_theta(1,1:end-1) )) ) ) ;
disp ([’E = ’,num2str(E),’+-’,num2str(E_delta)])
%% Plot data
angles = theta / pi * 180 ;
theory_angles = theory_theta / pi * 180 ;
plot ( angles, exp_cooperon, ’b’, ...
theory_angles, theory_cooperon, ’r’, ...
angles, exp_cooperon+exp_cooperon_delta, ’:b’, ...
angles, exp_cooperon-exp_cooperon_delta, ’:b’ )
xlabel (’scattering angle [deg]’)
ylabel (’cooperon’)
legend (’backscattering data’,’theory’)
end
84

Evaluation of the wide angle data
function d = diffuson (theta,l,R)
z0 = 2/3 * l * (1 + R) / (1 - R) ;
mu = cos(theta) ;
d = ( mu .* (z0/l + mu./(mu+1)) ) ./ (z0/l + 1/2) ;
end
function c = cooperon (theta,k,l,R,D,tau)
z0 = 2/3 * l * (1 + R) / (1 - R) ;
mu = cos(theta) ;
q = k * sin(theta) ;
L_a = sqrt (D * tau) ;
q_a = sqrt (q.^2 + 1/(D*tau)) ;
c = ( l * (l + L_a)^2 * ( (1 - exp(-2*q_a*z0)) ./ ...
(q_a*l) + 2*mu ./ (mu+1) ) ) ...
./ ( L_a^2 * ( l + (1 - exp(-2*z0/L_a)) * L_a ) * ...
( l*q_a + (mu+1) ./ (2*mu) ).^2 ) ;
c_add = (8 * pi * a(k,l,R)) / (3 * k^2 * l * (l + 2 * z0)) * mu ./ (mu + 1) ;
c = c - c_add ;
end
function a = a (k,l,R)
theta = 0.00001*pi : 0.00001*pi : pi/2 ;
z0 = 2/3 * l * (1 + R) / (1 - R) ;
mu = cos(theta) ;
q = k * abs(sin(theta)) ;
E_A = trapz ( theta, ...
3/(8*pi) * 1./(q*l + (mu+1)./(2*mu)).^2 ...
.* ( (1 - exp(-2*q*z0))./(q*l) + (2*mu)./(mu+1) ) ...
.* sin(theta) ) ;
E_BC = trapz ( theta, ...
1/(k*l)^2 * mu./(mu+1) ...
.* sin(theta) ) ;
a = E_A / E_BC ;
end
85

MATLAB codes
function R = reflectivity (n1,n2)
if n1 > n2
theta_totalreflection = asin (n2/n1) ;
else
theta_totalreflection = pi/2 ;
end
theta1 = 0 : pi/100 : theta_totalreflection ;
theta2 = theta_totalreflection+pi/100 : pi/100 : pi/2 ;
theta = [ theta1, theta2 ] ;
R_parallel = ( n1 * cos(theta1) ...
- n2 * sqrt( 1 - (n1 / n2 * sin(theta1)).^2 ) ).^2 ./ ...
( n1 * cos(theta1) ...
+ n2 * sqrt( 1 - (n1 / n2 * sin(theta1)).^2 ) ).^2 ;
R_parallel = [ R_parallel , ones(size(theta2)) ] ;
R_perp = ( n1 * sqrt( 1 - (n1 / n2 * sin(theta1)).^2 ) ...
- n2 * cos(theta1) ).^2 ./ ...
( n1 * sqrt( 1 - (n1 / n2 * sin(theta1)).^2 ) ...
+ n2 * cos(theta1) ).^2 ;
R_perp = [ R_perp , ones(size(theta2)) ] ;
C1 = trapz ( theta , ...
(R_parallel + R_perp) / 2 .* sin(theta) .* cos(theta) ) ;
C2 = trapz ( -theta , ...
(R_parallel + R_perp) / 2 .* sin(-theta) .* (cos(-theta)).^2 ) ;
R = ( 3 * C2 + 2 * C1 ) / ( 3 * C2 - 2 * C1 + 2 ) ;
end
86

Evaluation of the small angle data
Evaluation of the small angle data
The following MATLAB code is a very simplified version of the evaluation program for the
small angle setup; it will not work with all data. The script finds the backscattering direction
θ = 0 in the CCD image and calculates either an azimuthal average or a radial profile section
of the backscattering cone.
The signals of the CCD chip and their errors (se = standard error) are stored in two uint16
binary files. Depending on the preprocessing the numbers can have big (ieee-be) or little
(ieee-le) endian.
function small_angle_evaluation
%% Open file
endian = ’ieee-be’ ;
% alternative: ’ieee-le’
fid = fopen (’C:\Examples\example.i16’, ’r’) ;
matrix = fread (fid, [2048,2048], ’uint16’, 0, endian) ;
fclose (fid) ;
fid = fopen (’C:\Examples\example_se.i16’, ’r’) ;
errormatrix = fread (fid, [2048,2048], ’uint16’, 0, endian) ;
fclose (fid) ;
%% Find peak
% the image has to be scaled to have intensities in [0,1] by dividing by the
% largest possible intensity (i.e. 65535)
dummy = matrix / 65535 ;
% turn matrix into black-and-white picture (conetip forms white blob,
% everything else is black)
n = 5 ;
dummy = medfilt2 (dummy, [n,n]) ;
maximum = max ( max (dummy) ) ;
level = 0.95 * maximum ;
bwimage = im2bw (dummy, level) ;
% smooth picture so that fluctuations do
% not disturb the search for the cone
% find center of the blob => scattering angle theta = 0
dummy = bwlabel (bwimage) ; % label connected components in binary image
regions = regionprops (dummy, ’Area’, ’Centroid’) ; % measure properties
% of image regions
[area,index] = max ( [ regions.Area ] ) ;
87

MATLAB codes
position = regions(index).Centroid ;
position = round (position) ;
% position = center of mass of blob area;
% position(1) = x, position(2) = y !!!
%% Calculate profile or average
mode_selector = ’average’ ;
% alternative: ’profile’
if strcmp(mode_selector, ’average’)
range = 1 ; % change value to average over a larger radial range
[intensity,error,pixels] = ...
calculate_average (matrix, errormatrix, position, range) ;
else
direction = 0 ; % change value to select other direction
range = 1 ; % change value to average over a larger radial range
[intensity,error,pixels] = ...
calculate_profile (matrix, errormatrix, position, direction, range) ;
end
%% Display result
plot ( pixels, intensity )
hold on
plot ( pixels, intensity-error, ’:’ )
plot ( pixels, intensity+error, ’:’ )
hold off
xlabel (’scattering angle [pixels]’)
ylabel (’intensity [a.u.]’)
end
function [intensity,error,pixels] = ...
calculate_average (matrix, errormatrix, position, range)
% create carthesian coordinate system
X = repmat(1:2048,2048,1) ;
Y = X’ ;
% shift origin of the coordinate system to ’position’
X = X - position(1) ;
Y = Y - position(2) ;
% turn carthesian coordinate system into polar coordinates
[theta,rho] = cart2pol (X,Y) ; % range of theta: [-pi,pi]
88

Evaluation of the small angle data
% reshape matrices into vectors
matrix = reshape (matrix,[],1) ;
errormatrix = reshape (errormatrix,[],1) ;
rho = reshape (rho,[],1) ;
% sort vectors for rho
sorted = sortrows ([rho,matrix,errormatrix], 1) ;
rho = sorted(:,1) ;
matrix = sorted(:,2) ;
errormatrix = sorted(:,3) ;
% define ranges for the radial average
rho = rho / range ;
rho = round (rho) ;
% radial = values of rho without repetitions, sorted in ascending order
[radial,first] = unique (rho, ’first’) ; % first = first occurrence of each
% unique value in rho
[radial,last] = unique (rho, ’last’) ; % last = last occurrence of each
% unique value in rho
% average of the values of matrix and errormatrix
weight = 1 ./ errormatrix.^2 ;
weighted_matrix = matrix .* weight ;
intensity = zeros (size(radial,1), 1) ;
error = zeros (size(radial,1), 1) ;
for k = 1 : size(radial,1)
intensity(k) = sum ( weighted_matrix(first(k):last(k)) ) ...
/ sum ( weight(first(k):last(k)) ) ;
error(k) = 1 / sqrt ( sum ( weight(first(k):last(k)) ) ) ...
* 1 / sqrt ( last(k)-first(k)+1 ) ;
end
% define scale
pixels = radial * range ;
end
function [intensity,error,pixels] = ...
calculate_profile (matrix,errormatrix,position,direction,range)
% calculate direction angle
angle = direction / 180 * pi ;
if angle < 0
angle = angle + 2 * pi ; % angles between 0 and 2*pi
89

MATLAB codes
end
% calculate profile endpoints at the image borders
if 0

Evaluation of the small angle data
% calculate intensity profile
part1 = improfile ( matrix, [endpoints(1),position(1)] , ...
[endpoints(3),position(2)] ) ;
part2 = improfile ( matrix, [position(1),endpoints(2)] , ...
[position(2),endpoints(4)] ) ;
part2 = part2(2:end) ; % first pixel is already in part1 !
intensity = [part1;part2] ;
% calculate error profile
part1 = improfile ( errormatrix, [endpoints(1),position(1)] , ...
[endpoints(3),position(2)] ) ;
part2 = improfile ( errormatrix, [position(1),endpoints(2)] , ...
[position(2),endpoints(4)] ) ;
part2 = part2(2:end) ; % first pixel is already in part1 !
error = [part1;part2] ;
% calculate scale
if range > 1
modulo1 = mod ( size(part1,1) - ceil(range/2) , range ) ;
modulo2 = mod ( size(part2,1) - floor(range/2) , range ) ;
intensity = intensity ( modulo1+1 : size(part1,1)+size(part2,1)-modulo2 ) ;
intensity = reshape (intensity, range, []) ;
error = error ( modulo1+1 : size(part1,1)+size(part2,1)-modulo2 ) ;
error = reshape (error, range, []) ;
weight = 1 ./ error.^2 ;
weighted_intensity = intensity .* weight ;
intensity = sum (weighted_intensity, 1) ./ sum (weight, 1) ;
intensity = intensity’ ;
error = 1 / sqrt(range) .* 1 ./ sqrt ( sum (weight, 1) ) ;
error = error’ ;
else
modulo1 = 0 ;
end
length = sqrt ( ( x_end - x_start )^2 + ( y_end - y_start )^2 ) ;
number = size ( intensity , 1 ) ;
pixels = 1 : length/number : length ;
pixels = pixels - pixels( ceil ( ( size(part1,1) - modulo1 ) / range ) ) ;
pixels = pixels’ ;
end
91