Coherent Backscattering from Multiple Scattering Systems - KOPS ...
Coherent Backscattering from Multiple Scattering Systems - KOPS ... Coherent Backscattering from Multiple Scattering Systems - KOPS ...
Dissertation Coherent Backscattering from Multiple Scattering Systems Susanne Fiebig
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Dissertation<br />
<strong>Coherent</strong> <strong>Backscattering</strong> <strong>from</strong><br />
<strong>Multiple</strong> <strong>Scattering</strong> <strong>Systems</strong><br />
Susanne Fiebig
Susanne Fiebig:<br />
<strong>Coherent</strong> <strong>Backscattering</strong> <strong>from</strong> <strong>Multiple</strong> <strong>Scattering</strong> <strong>Systems</strong><br />
Dissertation<br />
zur Erlangung des akademischen Grades ‘Doktor der Naturwissenschaften’ (Dr. rer. nat.) an<br />
der Universität Konstanz, Mathematisch-Naturwissenschaftliche Sektion, Fachbereich Physik<br />
Referenten: Prof. Dr. Georg Maret, PD Dr. Christof M. Aegerter<br />
Tag der mündlichen Püfung: 8.9.2010<br />
Diese Arbeit wurde an der Universität Konstanz am Lehrstuhl von Prof. Dr. G. Maret durchgeführt<br />
und durch die Deutsche Forschungsgemeinschaft (DFG), das Internationale Graduiertenkolleg<br />
(IRTG) ‘Soft Condensed Matter of Model <strong>Systems</strong>’ und das Center for Applied Photonics<br />
(CAP) des Landes Baden-Württemberg und der Universität Konstanz finanziert. Vielen<br />
Dank auch an Sigma-Aldrich und DuPont für die kostenlose Bereitstellung eines Großteils<br />
der in dieser Arbeit verwendeten Proben.
Ein kurzer Überblick<br />
Streuung ist ein Phänomen, auf das man auf dem Gebiet der Wellenenausbreitung überaus<br />
häufig stößt. Insbesondere unsere Wahrnehmung der Umwelt ist ganz wesentlich durch Streuung<br />
geprägt. Kaum eine Welle – ob nun Lichtwelle, akustische oder sogar seismische Welle –<br />
erreicht uns auf geradem Weg. Auch für uns nicht direkt wahrnehmbare Wellen wie Radiooder<br />
Mikrowellen oder auch die als Wellen beschreibbaren Elektronen unterliegen in nicht<br />
unerheblichem Maße der Streuung.<br />
Trotzdem hat die Physik besonders im Bereich der Vielfachstreuung noch viele offene Fragen<br />
zu beantworten. Einige davon betreffen die so genannte kohärente Rückstreuung, ein<br />
Phänomen, das durch Interferenz bestimmter vielfach gestreuter Wellen entsteht. Anhand<br />
von elektromagnetischen Wellen im Spektralbereich des sichtbaren Lichts lassen sich diese Interferenzen<br />
sehr präzise untersuchen, da hier nur Absorption als rivalisierender Effekt auftritt,<br />
und die experimentelle Realisierung zudem nicht besonders kompliziert ist.<br />
Die kohärente Rückstreuung lässt sich mit den Modell einer Zufallsbewegung oder Random<br />
Walks der mit der vielfach gestreuten Welle assoziierten Teilchen durch das streuende Medium<br />
beschreiben. In diesem Modell kann man sehr einfach verstehen, dass zu jedem Teilchenpfad<br />
auch seine Umkehrung existiert, bei der ein anderes Teilchen den selben Pfad in<br />
umgekehrter Richtung durchläuft, wenn beide Enden des Pfades von der einfallenden Welle<br />
erreicht werden.<br />
Interferenzen von aus unterschiedlichen Pfaden austretenden Wellen sind zufällig, da sie auf<br />
den verschiedenen Random Walks unterschiedliche Phasenverschiebungen erfahren. Im Gegensatz<br />
dazu hängt das Interferenzmuster der an den beiden Enden eines zeitumgekehrten<br />
Pfades austretenden Wellen grundsätzlich nur vom Abstand der beiden Endpunkte und der<br />
Richtung der einfallenden Welle ab.<br />
Handelt es sich bei dem zeitumgekehrten Pfad um einen geschlossenen (Teil-)Pfad innerhalb<br />
des Mediums, so führt konstruktive Interferenz am Pfadausgang zu einer erhöhten<br />
Aufenthaltswahrscheinlichkeit der Welle an diesem Ort und damit zu einer Verlangsamung<br />
der Wellenausbreitung. Bei makroskopischer Besetzung solcher Pfadringe kommt es zu einem<br />
vollständigen Zusammenbruch der Wellenausbreitung und damit zum Übergang in eine<br />
lokalisierende Phase. Nach ihrem Entdecker P. W. Anderson wird diese als ‘Anderson-<br />
Lokalisierung’ bezeichnet.<br />
Liegen die beiden Endpunkte des Pfades dagegen in einem gewissen Abstand voneinander<br />
an der Oberfläche des Mediums, so ist nur die Interferenz in Rückstreurichtung, also<br />
in Richtung entgegengesetzt zur einfallenden Welle, grundsätzlich konstruktiv. Dies führt<br />
bei Überlagerung der Interferenzmuster einer großen Anzahl solcher Pfade zu einer konusförmigen<br />
Intensitätsüberhöhung um den Faktor zwei, die als kohärenter Rückstreukonus<br />
bezeichnet wird.<br />
Die Breite dieses Konus ist umgekehrt proportional zu der Schrittlänge des Random Walk, der<br />
mittleren freien Transportweglänge l ∗ . Diese wiederum bestimmt beispielsweise, ob in einem
Ein kurzer Überblick<br />
bestimmten Medium ein Übergang zur Anderson-Lokalisierung möglich ist. Dieser Übergang<br />
sollte nach dem so genannten Ioffe-Regel-Kriterium in etwa dann stattfinden, wenn l ∗ von<br />
der Größenordnung der Wellenlänge des gestreuten Lichts ist. Für die Charakterisierung vielfachstreuender<br />
Proben ist es daher wichtig, den Rückstreukonus in Experiment und Theorie<br />
korrekt abzubilden.<br />
Leider enthalten die bisher meist angewandten experimentellen und theoretischen Methoden<br />
kleine aber signifikante Ungenauigkeiten, die besonders bei den breiten Konen ins Gewicht<br />
fallen, die nach dem Ioffe-Regel Kriterium in der Nähe des Übergangs zur Anderson-<br />
Lokalisierung auftreten sollten. Ein Ziel der vorliegenden Arbeit war deshalb eine Verbesserung<br />
der experimentellen Methodik im Einklang mit einer genaueren theoretischen Beschreibung<br />
des Rückstreukonus, die von E. Akkermans (Technion Israel Institute of Technology,<br />
Haifa, Israel) und G. Montambaux (Université Paris-Sud, Orsay, France) erarbeitet wurde.<br />
Ausgangspunkt war dabei die Feststellung, dass sowohl gemessener als auch theoretisch<br />
berechneter Konus das fundamentale Prinzip der Energieerhaltung zu verletzen scheinen.<br />
Da der Rückstreukonus ein Interferenzphänomen ist, sollte die im Vergleich zu einer inkohärenten<br />
Addition der Vielfachstreuung verstärkte Intensität in Rückstreurichtung durch<br />
eine Intensitätsabschwächung bei größeren Streuwinkeln ausgeglichen werden. Diese Abschwächung<br />
ist jedoch bisher weder im Experiment noch in der Theorie beobachtet worden.<br />
Im Fall der Theorie ist dies nicht weiter verwunderlich, da eine ganze Reihe von Annahmen<br />
und Näherungen verwendet werden. Akkermans und Montambaux erweiterten daher die<br />
allgemein übliche Theorie durch zwei zusätzliche Terme. Diese führen an den Flanken des<br />
Rückstreukonus zu einer Abschwächung der gestreuten Intensität unter das Niveau der inkohärenten<br />
Addition der Rückstreuung, die die Intensitätsüberhöhung des Konus ausgleicht.<br />
Bei dieser neuen theoretischen Bescheibung des Rückstreukonus ist damit die Energie erhalten.<br />
Experimentell wird die Winkelverteilung der gestreuten Intensität mit dem so genannten<br />
Weitwinkel-Setup bestimmt, in dem eine große Anzahl von Photodioden in einem Bogen<br />
von 180 ◦ um die Probe angeordnet sind. Um die exakte Form des Rückstreukonusbestimmen<br />
zu können müssen die Dioden korrekt kalibriert werden. Dazu wird eine Referenzprobe mit<br />
extrem schmalem Konus verwendet, der Unterschied in den Albedos von Probe und Referenz<br />
wurde dabei bisher jedoch nicht berücksichtigt. Der Schlüssel zu einer präziseren Messung<br />
des Rückstreukonus war daher die Bestimmung dieser Albedos. Damit lässt sich nun auch in<br />
den experimentellen Daten eine Intensitätsabschwächung an den Konusflanken beobachten,<br />
die mit der Vorhersage von Akkermans und Montambaux übereinstimmt.<br />
Ein weiterer Fokus der vorliegenden Arbeit lag auf dem Neuaufbau des so genannten Kleinwinkel-Setups,<br />
mit dem die Verteilung der rückgestreuten Intensität mittels einer CCD-Kamera<br />
in einem sehr engen Bereich um die Rückstreurichtung gemessen wird. Das Ziel war eigentlich,<br />
Anderson-Lokalisierung durch die durch sie verursachte Abrundung der Spitze des<br />
Rückstreukonus nachzuweisen. Dafür wurden eine ältere Kamera durch ein hochauflösendes<br />
Modell mit einem größeren CCD-Chip ersetzt. Es stellte sich jedoch heraus, dass zwischen<br />
der Probe und der Kamera platzierte optischen Komponenten zu viel Störlicht verursachen,<br />
so dass die notwendige Intensitätsauflösung trotzdem nicht erreicht wird.<br />
Dafür kann mit dem verbesserten Aufbau die freie Transportweglänge von schwach streuenden<br />
Materialien wie beispielsweise Teflon gemessen werden. Da die dabei gemessene Transii
Ein kurzer Überblick<br />
portweglänge sowohl mit den Ergebnissen früherer Experimente als auch mit der theoretischen<br />
Vorhersage im Rahmen der Messgenauigkeit übereinstimmt, kann die Zuverlässigkeit<br />
der Ergebnisse des Kleinwinkel-Setups als bestätigt angesehen werden.<br />
Eine weitere Anwendung für das Kleinwinkel-Setup ergab sich im Rahmen einer Zusammenarbeit<br />
mit der Gruppe von M. Schröter (Max Planck Institut für Dynamik und Selbstorganisation,<br />
Göttingen). Hier sollte die freie Transportweglänge von Licht in so genannten fluidisierten<br />
Betten bestimmt werden. Da die streuenden Teilchen in diesem Experiment sehr groß sind und<br />
eine gleichmäßig sphärische Form haben, ist die rückgestreute Intensitätsverteilung durch die<br />
Ringstruktur der Einfachstreuung an Mie-Teilchen überlagert. Diese lässt sich jedoch theoretisch<br />
berechnen und an die gemessenen Kurven anpassen, so dass sich der Rückstreukonus<br />
aus den Daten extrahieren lässt.<br />
Bei ersten Messungen führte die Breite dieses Konus zu einer Transportweglänge, die wesentlich<br />
kleiner als der Teilchendurchmesser der Streuer ist. Dies widerspricht eklatant den Ergebnissen<br />
ähnlicher Experimente, die von Transportweglängen in der Größenordnung mehrerer<br />
Teilchendurchmesser berichten. Der Grund hierfür ist bisher nicht bekannt; die grundlegende<br />
Auswerteprozedur für Rückstreudaten von fluidisierten Betten konnte jedoch erfolgreich<br />
getestet werden.<br />
Für zukünftige Experimente stehen damit nun zwei verbesserte Experimentaufbauten zur<br />
Verfügung, mit denen kl ∗ über einen Bereich von mehr als drei Zehnerpotenzen hinweg gemessen<br />
werden kann. Mögliche Anwendungen reichen von neuen, maßgeschneiderten Proben<br />
mit kl ∗ am Übergang zur Anderson-Lokalisierung zu Schäumen oder biologischem Gewebe.<br />
Einige dieser Experimente sind bereits für die nähere Zukunft geplant, und es steht zu hoffen,<br />
dass sie einen weiteren Schritt hin zu einem vollen Verständnis der Vielfachstreuung bilden<br />
werden.<br />
Danksagungen<br />
Viele haben auf die eine oder andere Art zu dieser Arbeit beigetragen. Besonders bedanken<br />
möchte ich mich bei<br />
Prof. Dr. G. Maret – für die interessante Aufgabenstellung, eine zuverlässige und geduldige<br />
Finanzierung, und für eine in jeder Hinsicht entspannte Arbeitsumgebung<br />
PD Dr. C. M. Aegerter – der nach einigen Jahren tapferer Betreuungsarbeit schließlich doch<br />
die Flucht ergriff und nach Zürich auswanderte<br />
W. Bührer – unter anderem für sämtliche Time of flight-Messungen; die komplette Dank-<br />
Liste würde hier leider den Rahmen sprengen. . .<br />
Prof. Dr. E. Akkermans und Prof. Dr. G. Montambaux – für die Ausarbeitung der Theorie<br />
zu den Experimenten an stark streuenden Proben<br />
der P10-Stammbelegschaft in Sektretariat, Chemielabor, Elektrotechnik und Werkstatt – für<br />
die überaus professionelle Unterstützung bei Problemen von ‘Anträge stellen’ bis ‘Zusammenlöten’<br />
iii
Ein kurzer Überblick<br />
meinen Büro- und Laborkollegen – die jetzt vermutlich mehr über meine schlechten Angewohnheiten<br />
wissen als mir lieb sein kann<br />
allen P10lern (samt den Exilanten auf Z10) – für alle Hilfsbereitschaft und die entspannte<br />
und freundschaftliche Zusammenarbeit (und die dazugehörigen Kaffeepausen)<br />
meiner Familie – für die uneingeschränkte Unterstützung in allen Lebenslagen<br />
יהוה – für alles . . .<br />
iv
Contents<br />
Ein kurzer Überblick<br />
i<br />
Danksagungen<br />
iii<br />
Contents<br />
vi<br />
1 Introduction 1<br />
2 Theory 3<br />
2.1 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3<br />
2.2 Single scattering – Mie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />
2.3 Random walk and diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9<br />
2.4 The influence of boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />
2.5 Photon flux <strong>from</strong> a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15<br />
2.6 On polarization and interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 17<br />
2.7 The theory of coherent backscattering . . . . . . . . . . . . . . . . . . . . . . . . 20<br />
3 Setups 25<br />
3.1 Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25<br />
3.2 Wide Angle Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25<br />
3.3 Small Angle Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28<br />
3.4 Time Of Flight Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33<br />
4 Samples 35<br />
4.1 Sample characterization techniques . . . . . . . . . . . . . . . . . . . . . . . . . 35<br />
4.2 The samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Contents<br />
5 Experiments 43<br />
5.1 Conservation of energy in coherent backscattering . . . . . . . . . . . . . . . . . 43<br />
5.2 The coherent backscattering cone in high resolution . . . . . . . . . . . . . . . . 56<br />
6 Summary 69<br />
Bibliography 74<br />
Figures and Tables 75<br />
MATLAB codes 79<br />
Angular intensity distribution of single scattering . . . . . . . . . . . . . . . . . . . . . . 79<br />
Evaluation of the wide angle data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81<br />
Evaluation of the small angle data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87<br />
vi
1 Introduction<br />
There have been many spectacular findings in the field of multiple scattering of light in random<br />
media, <strong>from</strong> the theoretical work of P. W. Anderson in the 1950s [13] and its application<br />
to electromagnetic waves [14, 28] to the discovery of the coherent backscattering cone some<br />
30 years ago [31, 52, 57] and the resent find of the onset of Anderson localization [5, 6, 48].<br />
This work however focusses on the equally important improvements of the experimental techniques<br />
for the investigation of multiple scattering phenomenons.<br />
Of particular interest were experiments on the coherent backscattering cone, an interference<br />
effect that causes a twofold intensity enhancement in the direction opposite to the incoming<br />
wave. Although it can be observed not only on visible light in the laboratory, but also<br />
in space [24], on microwaves [18], seismic waves [34], sound waves [15], or the de Broglie<br />
waves of electrons in metals [29], the laboratory experiments with visible light have two major<br />
advantages over other systems: There are no rivaling effects like interaction between the scattered<br />
particles or binding in deep minima of the random potential, except for absorption, and<br />
the necessary technical effort is comparatively low. <strong>Multiple</strong> scattering with electromagnetic<br />
waves in the visible range is therefore used as model system to experimentally investigate<br />
multiple scattering of waves [14, 28].<br />
From the shape of the coherent backscattering cone one can read all kinds of information<br />
about the scattering process in the medium. The cone width for example is a measure for the<br />
transport mean free path l ∗ and therefore for the turbidity of the medium. The shape of the<br />
conetip gives evidence of absorption and in some cases even the transition to a localizing state.<br />
The experiments however require highly sensitive setups and refined theoretical descriptions<br />
to precisely depict the angular distribution of the backscattered light and to correctly fit the<br />
scattering parameters.<br />
This thesis reports the work on two experimental setups, one to record the backscattered<br />
radiation over a wide angular range, the other for detailed measurement of the intensity<br />
distribution close to backscattering direction. The theoretical basis is laid in sec. 2, where<br />
the mathematical description of coherent backscattering is developed starting <strong>from</strong> the wave<br />
equation. Along the way, some insight is also given in phenomena like single scattering at<br />
Mie particles and Anderson localization. Sec. 3 gives technical information about the setups<br />
used for the scattering experiments. These are not only the two backscattering setups this<br />
work focusses on, but also a time of flight setup which is used to record the time-dependent<br />
scattering in transmission. The next section presents the samples used in the experiments<br />
alongside with some additional characterizing methods. Finally, sec. 5 reports the revision<br />
process of the two setups. For the wide angle setup the evaluation procedure was refined,<br />
and an improved theory of coherent backscattering was developed, while the small angle<br />
setup was tested in experiments on strongly scattering titanium dioxide samples, on teflon as<br />
an example of a weakly scattering material, and on water-fluidized beds of glass beads. The<br />
MATLAB [2] codes used for the evaluations can be found in the appendix.
2<br />
1 Introduction
2 Theory<br />
In scattering theory, the mathematical and physical framework for studying and understanding<br />
scattering events, the interaction of scattering particles with electromagnetic waves is described<br />
as the solution of a particular partial differential equation, the so-called wave equation.<br />
For many systems, like for example scattering of light at a single spherical particle, one can<br />
solve this equation exactly. <strong>Multiple</strong> scattering samples however, with millions and billions of<br />
randomly distributed scattering particles, have to be described by an approximate, collective<br />
solution, as the exact solution can be obtained neither analytically nor numerically.<br />
2.1 The wave equation<br />
The scattering system we will consider in the following consists of randomly distributed scatterers<br />
with dielectric permittivity ɛ scat in a surrounding medium with ɛ surr . We assume both<br />
media to be non-magnetic (i.e. permeabilities µ scat = µ surr = 1) which corresponds to the usual<br />
experimental conditions and does not bring any structural changes into the calculations.<br />
The vector wave equations for a multiply scattered electromagnetic wave can be derived <strong>from</strong><br />
Maxwell’s equations as<br />
∇ 2 ⃗ E(⃗r) +<br />
ω 2<br />
c 2 0<br />
ɛ(⃗r)⃗ E(⃗r) = 0 and ∇<br />
2 H(⃗r) ⃗ +<br />
ω 2<br />
ɛ(⃗r)⃗ H(⃗r) = 0<br />
c 2 0<br />
where ⃗ E(⃗r) and ⃗ H(⃗r) are the electric and magnetic field components, ɛ(⃗r) is either ɛscat or<br />
ɛ surr , ω is the light frequency, and c 0 is the vacuum speed of light.<br />
Instead of directly solving the above vector equations it is however convenient to find a solution<br />
for the scalar wave equation<br />
∇ 2 Ψ(⃗r) + ω2 ɛ(⃗r)Ψ(⃗r) = 0 (2.1)<br />
c 2 0<br />
As it will be demonstrated in sec 2.2, one can construct two vector harmonics M ⃗ = ⃗∇ × (⃗c Ψ)<br />
and N ⃗ =<br />
⃗∇× M ⃗<br />
k<br />
<strong>from</strong> this solution using a suitable pilot vector ⃗c. M ⃗ and N ⃗ are orthogonal<br />
solutions of the vector wave equation, so the complete solution for the electric field is the<br />
linear combination ⃗ E = AM ⃗ + BN, ⃗ and the magnetic field H ⃗ can be calculated <strong>from</strong> ⃗ E using<br />
Maxwell’s equations.
2 Theory<br />
The scalar wave equation 2.1 can be written in form of the inhomogenous differential equation<br />
∇ 2 Ψ(⃗r) + k 2 Ψ(⃗r) = −V(⃗r)Ψ(⃗r)<br />
where k 2 [ ]<br />
= ω2 ɛ<br />
c 2 surr , and V(⃗r) = ω2<br />
0<br />
c ɛ(⃗r) − ɛsurr is the scattering potential. a Its solution at a<br />
2<br />
0<br />
certain point⃗r is given by<br />
∫<br />
Ψ(⃗r) = Ψ in (⃗r) +<br />
G 0 (⃗r,⃗r 1 )V(⃗r 1 )Ψ(⃗r 1 ) d⃗r 1 (2.2)<br />
where Ψ in is the part of the incoming wave that has not been scattered before. G 0 is termed the<br />
bare Green’s function and describes the propagation of the electromagnetic field in a medium<br />
without scatterers. It is defined by<br />
∇ 2 G 0 (⃗r,⃗r 1 ) + k 2 G 0 (⃗r,⃗r 1 ) = −δ(⃗r,⃗r 1 )<br />
and is given by<br />
G 0 (⃗r,⃗r 1 ) = e−ik|⃗r−⃗r 1|<br />
4π|⃗r −⃗r 1 |<br />
By applying eqn. 2.2 recursively, the wave function can be expanded into a perturbation series<br />
∫<br />
Ψ(⃗r) = Ψ in (⃗r) +<br />
G 0 (⃗r,⃗r 1 )V(⃗r 1 )Ψ in (⃗r 1 ) d⃗r 1 +<br />
∫∫<br />
+ 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)<br />
It would be convenient to split off the incoming wave Ψ in like Ψ(⃗r) = ∫ G(⃗r,⃗r ′ )Ψ in (⃗r ′ ) d⃗r ′ .<br />
This introduces the total Green’s function G, which describes the electromagnetic field at a<br />
certain position⃗r due to a disturbance at another point⃗r ′ . It has a perturbation series<br />
∫<br />
G(⃗r,⃗r ′ ) = G 0 (⃗r,⃗r ′ ) + G 0 (⃗r,⃗r a )V(⃗r a )G 0 (⃗r a ,⃗r ′ ) d⃗r a +<br />
∫∫<br />
+ 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 + · · ·<br />
and the formal definition<br />
∇ 2 G(⃗r,⃗r ′ ) + ω2 ɛ(⃗r)G(⃗r,⃗r ′ ) = −δ(⃗r,⃗r ′ )<br />
c 2 0<br />
[a] Elsewhere [16, 56], the scattering potential is defined as V(⃗r) = ω2 [ɛ<br />
c 2 surr − ɛ(⃗r)] (or similar), so that the wave<br />
0<br />
equation becomes ∇ 2 Ψ(⃗r) + k 2 Ψ(⃗r) = V(⃗r)Ψ(⃗r). This definition makes the perturbation series eqn. 2.3 look less<br />
intuitive as half of the integrals seem to be subtracted.<br />
4
2.2 Single scattering – Mie theory<br />
Figure 2.1: Coordinate system of single scattering. The coordinate system (θ, φ) of<br />
Mie scattering at a spherical particle with radius a is defined by the wave vector ⃗ kin<br />
and the electric field vector ⃗ Ein of the incoming light wave. The wave vectors of the<br />
incoming and outgoing waves⃗ kin and⃗ kout span the scattering plane.<br />
The scattered light intensity at a certain point⃗r must then be<br />
∫∫<br />
I(⃗r) ∝<br />
G(⃗r,⃗r 1 )G ∗ (⃗r,⃗r 2 )Ψ in (⃗r 1 )Ψ ∗ in(⃗r 2 ) d⃗r 1 d⃗r 2<br />
For a dilute system with pointlike scatterers the perturbation series of eqn. 2.3 immediately<br />
justifies treating multiple scattering of electromagnetic waves as random walks of photons<br />
with different numbers of scattering events: The photons travel in free space (described by<br />
Green’s functions G 0 , which have the form of spherical waves) until they hit a particle and are<br />
scattered into the surrounding space, where they again propagate freely, hit another particle,<br />
and so on.<br />
The random walk picture can still be upheld if the size of the particles is not negligible.<br />
However, in this case one needs to consider how the particles distribute the incoming intensity<br />
into their surrounding to describe the random walk properly. If a spherical particle can be<br />
considered as an acceptable approximation for the actual particle shape, one can use the<br />
solution given by G. Mie [42] and others. In the next section, we will follow the approach of<br />
[16] to derive the distribution of the scattered light around a single particle.<br />
2.2 Single scattering – Mie theory<br />
The problem of an electromagnetic wave scattered by a spherical particle clearly has a spherical<br />
symmetry. Therefore it is convenient to treat the problem in a polar coordinate system<br />
with the scattering particle of radius a located in the origin, and wave vector and polarization<br />
of the incident light defining the angular coordinates θ = 0 and φ = 0. The wave equation in<br />
polar coordinates<br />
1<br />
r 2<br />
(<br />
∂<br />
r 2 ∂Ψ )<br />
+ 1<br />
∂r ∂r r 2 sin θ<br />
(<br />
∂<br />
sin θ ∂Ψ )<br />
+<br />
∂θ ∂θ<br />
1<br />
r 2 sin 2 θ<br />
∂ 2 Ψ<br />
∂φ 2 + k2 Ψ = 0<br />
5
2 Theory<br />
can be separated into three independent differential equations using the ansatz Ψ(r, θ, φ) =<br />
R(r) · Θ(θ) · Φ(φ):<br />
1<br />
R<br />
(<br />
d<br />
dr<br />
sin θ<br />
Θ<br />
r 2 dR )<br />
+ k 2 r 2 = α<br />
dr<br />
(<br />
sin θ dΘ )<br />
= β − α sin 2 θ<br />
dθ<br />
d<br />
dθ<br />
1 d 2 Φ<br />
Φ dφ 2 = −β<br />
Setting β = m 2 and α = n(n + 1) where m = 0, 1, 2, . . . and n = m, m + 1, . . . we obtain the<br />
complete solution Ψ for the wave equation, and <strong>from</strong> this the vector harmonics ⃗ M = ⃗∇ × (⃗rΨ)<br />
and ⃗ N =<br />
⃗∇× ⃗ M<br />
k<br />
.<br />
The solution of the radial part of the wave equation R(r) is given by a linear combination of<br />
the spherical Bessel functions j n (ρ) = √ π<br />
2ρ J n+1/2(ρ) and y n = √ π<br />
2ρ Y n+1/2(ρ) where J ν (ρ) and<br />
Y ν (ρ) are Bessel functions of first and second kind, and ρ = kr. For the outgoing scattered<br />
wave the appropriate linear combination is given by one of the spherical Bessel functions of<br />
the third kind or spherical Hankel functions, h (1)<br />
n (ρ) = j n (ρ) + iy n (ρ).<br />
The zenith part of the wave equation Θ(θ) is solved by associated Legendre functions Pn m (cos θ).<br />
For the following calculations it is convenient to define the angle-dependent functions π n =<br />
Pn<br />
1<br />
sin θ and τ n = dP1 n<br />
dθ .<br />
It can be shown that the solution for the scattered electric field is given by<br />
⃗ Escat =<br />
∞<br />
∑ i n 2n + 1<br />
)<br />
E 0<br />
(ia nNn ⃗ − b nMn ⃗<br />
n(n + 1)<br />
n=1<br />
where the applicable vector harmonics are given by<br />
⃗M n = cos φ π n (cos θ) h (1)<br />
n (ρ) ê θ − sin φ τ n (cos θ) h (1)<br />
n (ρ) ê φ<br />
⃗N n = cos φ n(n + 1) sin θ π n (cos θ) h(1) n (ρ)<br />
ρ<br />
ê r + cos φ τ n (cos θ) [ρh(1) n (ρ)] ′<br />
ρ<br />
ê θ −<br />
− sin φ π n (cos θ) [ρh(1) n (ρ)] ′<br />
ρ<br />
ê φ<br />
and the coefficients a n and b n are<br />
a n = m ψ n(mx) ψ ′ n(x) − ψ n (x) ψ ′ n(mx)<br />
m ψ n (mx) ξ ′ n(x) − ξ n (x) ψ ′ n(mx)<br />
; b n = ψ n(mx) ψ ′ n(x) − m ψ n (x) ψ ′ n(mx)<br />
ψ n (mx) ξ ′ n(x) − m ξ n (x) ψ ′ n(mx)<br />
6
2.2 Single scattering – Mie theory<br />
Figure 2.2: Vibration ellipse. The tip of the electric field vector of a light wave traces<br />
out an ellipse with semimajor axis r 1 , semiminor axis r 2 , azimuth γ, and ellipticity<br />
η = arctan(r 2 /r 1 ).<br />
with the size parameter x = ka, the relative refractive index m = k scat<br />
k<br />
, and the Riccati-Bessel<br />
functions ψ n (z) = z j n (z) and ξ n (z) = z h (1)<br />
n (z). The prime indicates differentiation with<br />
respect to the argument in parentheses.<br />
⃗M has no radial component, and in the far-field limit with large ρ = kr the radial component<br />
of N ⃗ is negligible compared to the transverse component, as the spherical Hankel function is<br />
asymptotically given by h (1)<br />
n (kr) ≈ − (−i)n e ikr<br />
ikr<br />
. The scattered electromagnetic field is therefore<br />
basically transverse, and<br />
E θ ≈ E 0 ·<br />
E φ ≈ −E 0 ·<br />
e ikr<br />
−ikr · cos φ · S 2(cos θ) with S 2 =<br />
e ikr<br />
−ikr · sin φ · S 1(cos θ) with S 1 =<br />
n c<br />
∑<br />
n=1<br />
n c<br />
∑<br />
n=1<br />
2n + 1<br />
n(n + 1) (a nτ n + b n π n )<br />
2n + 1<br />
n(n + 1) (a nπ n + b n τ n )<br />
Both sums converge, so that the series can be terminated after n c steps without causing major<br />
inaccuracies. A good assumption is n c ≈ x.<br />
We now focus on a certain scattering plane, which is defined by the wave vectors of the<br />
incoming and emerging waves. Intensity and polarization of the scattered light are then<br />
merely a function of the scattering angle θ and the polarization of the incoming light with<br />
respect to the scattering plane.<br />
In a non-absorbing medium, both incoming and scattered electromagnetic waves can be characterized<br />
by a Stokes vector [30]<br />
⎛ ⎞ ⎛<br />
⎞<br />
I<br />
1<br />
⃗p = ⎜Q<br />
⎟<br />
⎝U⎠ = ⎜cos(2η) cos(2γ)<br />
⎟<br />
⎝cos(2η) sin(2γ) ⎠<br />
V<br />
sin(2η)<br />
7
2 Theory<br />
45<br />
ellipticity / azimuth [deg]<br />
30<br />
15<br />
0<br />
−15<br />
−30<br />
azimuth ellipticity<br />
−45<br />
−180 −135 −90 −45 0 45 90 135 180<br />
scattering angle [deg]<br />
Figure 2.3: Polarization of a scattered wave. The polarization of the scattered wave –<br />
which can be described by azimuthal angle γ and ellipticity η – fluctuates strongly as<br />
a function of the scattering angle theta. The example shows the scattering of a linearly<br />
polarized wave with wavelength λ = 575 nm and azimuth γ = 45 ◦ on a spherical<br />
particle with diameter 2a = 300 nm and refractive index n = 2.7.<br />
The ellipticity of the vibration ellipse (fig. 2.2) is then given by | tan(2η)| = U Q<br />
, the azimuthal<br />
V<br />
angle by tan(2γ) = √Q . 2 +U 2<br />
The transformation between the Stokes vectors of the incident and the scattered light waves is<br />
described by so-called Mueller matrices [43]. The Mueller matrix of a spherical particle is<br />
⎛<br />
⎞<br />
S 11 S 12 0 0<br />
S(cos θ) = 1<br />
⎜S 21 S 22 0 0<br />
⎟<br />
k 2 r 2 ⎝ 0 0 S 33 S 34<br />
⎠<br />
0 0 S 43 S 44<br />
where<br />
S 11 = S 22 = 1 2 (S 2S2 ∗ + S 1S1 ∗)<br />
S 12 = S 21 = 1 2 (S 2S2 ∗ − S 1S1 ∗)<br />
S 33 = S 44 = 1 2 (S∗ 2 S 1 + S 2 S1 ∗)<br />
S 34 = −S 43 = i 2 (S∗ 2 S 1 − S 2 S1 ∗)<br />
As shown by the example in figs. 2.3 and 2.4, in the Mie scattering regime with wavelength<br />
λ ≈ a the polarization of the scattered light and the scattered intensity itself are very unevenly<br />
distributed around the scattering particle. Generally, one can recognize a transition <strong>from</strong> the<br />
isotropic intensity distribution of Raleigh scattering in the regime λ ≪ a to a significantly<br />
enhanced scattering in forward direction in the Mie regime.<br />
The total scattering cross section<br />
C scat = 2π ∞<br />
k ∑ 2 (2n + 1) ( |a n | 2 + |b n | 2)<br />
n=1<br />
which is a measure of the scattering efficiency, also fluctuates strongly when the particle<br />
diameter becomes of the order of the wavelength (fig. 2.4). For nearly monodisperse scatterers<br />
these so-called Mie resonances can be used to create especially strongly scattering samples.<br />
8
2.3 Random walk and diffusion<br />
scattering cross section [m 2 ]<br />
10 −12<br />
10 −14<br />
90 1<br />
120<br />
60<br />
150<br />
0.5<br />
30<br />
180 0<br />
210<br />
330<br />
10 −10 particle radius [m]<br />
10 −7 10 −6<br />
240<br />
270<br />
300<br />
Figure 2.4: <strong>Scattering</strong> cross section and scattering anisotropy. <strong>Scattering</strong> of a light<br />
wave with wavelength λ = 575 nm on spherical particles with refractive index n = 2.7.<br />
The scattering cross section (left) fluctuates strongly when the particle radius becomes<br />
of the order of the wavelength. The polar plot of the relative intensity distribution<br />
(right) shows a transition <strong>from</strong> isotropic Rayleigh scattering to strongly anisotropic Mie<br />
scattering with the incident light wave impinging <strong>from</strong> the left on particles of radius<br />
a = 30 nm (dark blue), a = 300 nm (blue), and a = 3 µm (light blue). The calculations<br />
were performed with [3].<br />
Strong polydispersity washes out the most prominent features, but the scattering cross section<br />
still depends on the ratio between particle radius and wavelength of the scattered light.<br />
With the help of the scattering cross section one can establish an anisotropy parameter, which<br />
describes the anisotropy of the intensity distribution: b<br />
〈cos θ〉 = 1 ∫<br />
dC scat<br />
· · cos θ dΩ =<br />
C scat 4π dΩ<br />
where dC scat<br />
dΩ<br />
= 4π2 a<br />
x 2 C scat<br />
[∑<br />
n<br />
n(n + 2)<br />
n + 1<br />
R{a na ∗ n+1 + b n bn+1} ∗ + ∑<br />
n<br />
]<br />
2n + 1<br />
n(n + 1) R{a nbn}<br />
∗<br />
denotes the differential scattering cross section of the particle.c<br />
2.3 Random walk and diffusion<br />
As it was pointed out in sec. 2.1, the movements of the photons through a multiple scattering<br />
sample can be modeled by random walks of photons <strong>from</strong> one scattering site to the next. To<br />
[b] R{z} = real part of the imaginary number z<br />
[c] The differential scattering cross section is denoted only symbolically as dC scat<br />
dΩ . It should not be interpreted as<br />
the derivative of a function of Ω.<br />
9
2 Theory<br />
Figure 2.5: Random walk. The vector⃗r denotes the displacement of a photon after M<br />
steps ∆ri ⃗ , where θ i is the angle between two consecutive steps ∆ri−1 ⃗ and ∆ri ⃗ .<br />
characterize the spreading of a cloud of photons that started off at a certain point and a certain<br />
time that we set⃗r = 0 and t = 0 we use the mean square displacement<br />
〈r 2 (t)〉 = 1 N<br />
( )<br />
N M(t) 2<br />
∑ ∑<br />
⃗∆r m,n<br />
n=1 m=1<br />
where M(t) is the number of steps ⃗ ∆r the photons have traveled after a certain time t, and N<br />
is the number of photon paths considered (fig. 2.5). For large M(t) this is [38]<br />
〈r 2 (t)〉 ≈ M(t) 〈 ⃗ ∆r<br />
2<br />
〉 + 2 〈 ⃗ ∆r〉<br />
2<br />
M(t) 〈cos θ〉<br />
1 − 〈cos θ〉<br />
where θ is the angle between two photon steps, and 〈cos θ〉 expresses the anisotropy of the<br />
scattering.<br />
For an exponential step length distribution p(∆r) = 1 ∆r<br />
l<br />
e− l we obtain 〈∆r〉 = l and 〈∆r 2 〉 = 2l 2 ,<br />
and<br />
〈r 2 (t)〉 = 2 M(t) l ·<br />
l<br />
1 − 〈cos θ〉 ≡ 2 M∗ (t) l ∗2 ≡ 2s(t)l ∗ (2.4)<br />
Hence the mean square displacement can be expressed with the help of three different length<br />
scales.<br />
The first length scale is the scattering mean free path l, which characterizes the distribution of<br />
the physical step lengths <strong>from</strong> one scattering site to the next.<br />
The angular distribution of single scattering given by Mie theory correlates the directions of<br />
two successive photon steps. In the mean square displacement this correlation is expressed<br />
by an additional factor (1 − 〈cos θ〉) −1 . The transport mean free path l ∗ l<br />
= and the<br />
1−〈cos θ〉<br />
effective number of photon steps M ∗ = M(t)(1 − 〈cos θ〉) incorporate this anisotropy factor. l ∗<br />
is therefore the distance after which the photons have lost the memory of their initial direction<br />
10
2.3 Random walk and diffusion<br />
of propagation. The difference between transport mean free path and scattering mean free<br />
path is the larger the more anisotropic the scattering is. For completely isotropic scattering<br />
both mean free paths are equal.<br />
The product M ∗ l ∗ gives the length s of the photon paths that contribute to the mean square<br />
displacement. As there is no reason to assume that the speed of the photons will change<br />
along their path, the length of a photon path is proportional to time. The size of the photon<br />
cloud 〈r 2 (t)〉 at a certain time is therefore linearly proportional to both the length s of the<br />
contributing photon paths and the time t the photons have spent traveling along these paths.<br />
On top of everything, the mean square displacement of eqn. 2.4 is structurally equivalent to<br />
the variance<br />
〈r 2 (t)〉 = 6Dt (2.5)<br />
of a gaussian distribution in three dimensions<br />
ρ(⃗r, t) =<br />
1<br />
r2<br />
√ e− 4Dt (2.6)<br />
3<br />
4πDt<br />
where ρ is the density distribution of photons that started off at the origin of the coordinate<br />
system at time t = 0 in an infinitely extended medium, and D is the diffusion coefficient. So<br />
obviously multiple scattering in the limit of large M(t) can also be described as diffusion of<br />
light energy through the medium with the diffusion equation ∂ρ<br />
∂t − D∇2 ρ = δ(t)δ(⃗r), whose<br />
solution is given by eqn. 2.6<br />
Comparing eqns. 2.4 and 2.5 one can give the diffusion coefficient as<br />
D = sl∗<br />
3t = vl∗<br />
3<br />
(2.7)<br />
where v is the velocity of energy transport. Using this, the photon density distribution can<br />
also be given as a function of the path length s:<br />
ρ(⃗r, s) =<br />
√<br />
3<br />
4πsl ∗ 3<br />
e − 4<br />
r2<br />
3 sl∗ (2.8)<br />
However, the photon density distribution in a multiply scattering sample is not purely determined<br />
by the scattering properties of the scattering particles, but also by energy losses due to<br />
absorption in the material. According to Lambert-Beer’s law, absorption weakens the intensity<br />
of a light wave exponentially along the traveled path, so that its effect can be included<br />
in the photon density distribution by an additional factor e − la<br />
s<br />
or e − τ t , respectively. The absorption<br />
length l a and the absorption time τ are inversely proportional to the number density<br />
ρ abs of absorbing particles on the path and their absorption cross section σ abs : l a = 1<br />
ρ abs σ abs<br />
and<br />
τ = t s l 1<br />
a =<br />
v ρ abs σ abs<br />
.<br />
11
2 Theory<br />
Figure 2.6: <strong>Scattering</strong> in the presence of a single boundary. All photon paths A → B<br />
that are lost because they run partially outside the samples can be described by their<br />
image paths A → B ′ . The (virtual) scatterer P is the point where a photon path leaves<br />
the sample.<br />
2.4 The influence of boundaries<br />
Eqns. 2.6 and 2.8 describe the photon density distribution in infinitely extended samples. In<br />
reality however, this distribution is strongly influenced by the borders of the system, where<br />
photons are inserted and released, absorbed and reflected.<br />
There are two different approaches to describe the influence of boundaries on a multiple<br />
scattering system. The more intuitive one is the image point method, which uses symmetry<br />
properties of the density distribution to describe the scattering in terms of the density distribution<br />
in infinite space [38]. The other approach is in the framework of radiative transfer theory,<br />
which describes a field of radiation by the intensity flux through area elements. This makes<br />
it possible to compare the flux through an area element at the sample surface with an area<br />
element inside the sample and derive theoretical descriptions for the various experimental<br />
situations [58].<br />
2.4.1 Image point method<br />
To describe multiple scattering in a finite sample in terms of scattering in infinite space one<br />
can make use of the symmetry properties of the density distribution: The photon density at<br />
any point B in an infinite sample is equal to the density at its image point B ′ with respect<br />
to any mirror plane that contains the starting point P of the photon cloud. Therefore we can<br />
describe photon paths A → P → B by paths A → P → B ′ that end at the image point of B<br />
with respect to P. If P is the endpoint of a photon step that leads the path out of the sample,<br />
the path A → P → B is not possible in the presence of the boundary, and its contribution is<br />
missing <strong>from</strong> the free space photon density in B.<br />
12
2.4 The influence of boundaries<br />
Figure 2.7: <strong>Scattering</strong> in the presence of two parallel boundaries. The presence of<br />
two parallel boundaries with distance L creates two series of image points B<br />
1 ′ , B′ 2 , B′ 3 , . . .<br />
and B<br />
1 ′′,<br />
B′′ 2 , . . .. The effective sample thickness L′ = L + 2z 0 simplifies the series.<br />
On the other hand, all photon paths that lead to B ′ contain at least one such point P, which<br />
is located at an average distance z 0 <strong>from</strong> the boundary. The photon density that is missing<br />
<strong>from</strong> the free space photon density in B due to the sample boundary is therefore the free<br />
space photon density in B ′ , and the photon density distribution in the presence of a boundary<br />
becomes<br />
ρ(A → B, t) 1 surface =<br />
= ρ(A → B, t) − ρ(A → B ′ , t)<br />
= e− τ<br />
t<br />
√<br />
(e ( − ⃗r ⊥,B −⃗r ⊥,A) 2 +(z B −z A ) 2<br />
4Dt − e ( − ⃗r ⊥,B −⃗r ⊥,A) 2 +(−z B +2z 0 −z A ) 2<br />
4Dt<br />
3<br />
4πDt<br />
)<br />
(2.9)<br />
where we have also considered absorption by its τ-dependent exponential decay. The vector<br />
⃗r ⊥ denotes the components of the position vector⃗r perpendicular to the z-axis and parallel to<br />
the sample boundary. Note that z 0 and z B or respectively z A have different signs, as depicted<br />
in fig. 2.6.<br />
If a second surface opposite to the entrance surface is to be considered, things become a little<br />
more complicated. In this case, the photon density in B is the free space photon density<br />
distribution minus the photon densities at the two image points B ′ and B ′′ with respect to the<br />
two mirror planes in front of the surfaces at z = z 0 and z = L + z 0 . These photon densities<br />
again have to be calculated as the densities in the presence of the other surface, and so on.<br />
13
2 Theory<br />
Figure 2.8: Radiative transfer. Radiative transfer theory describes the photon flux<br />
<strong>from</strong> volume element dV through area element dS without any intermediate scattering.<br />
The result are the two series of image points sketched in fig. 2.7, which we can simplify by<br />
using an effective sample thickness L ′ = L + 2z 0 :<br />
ρ(A → B, t) 2 surfaces =<br />
∞<br />
∑<br />
= ρ(A → B, t) − ρ(A → B ′ ∞<br />
k , t) − ∑<br />
k=1 k=1<br />
ρ(A → B ′′<br />
k , t)<br />
= e− τ<br />
t ∞<br />
√ 3 ∑ e ( − ⃗r ⊥,B −⃗r ⊥,A) 2 +(2mL ′ +z B −z A ) 2<br />
4Dt − e ( − ⃗r ⊥,B −⃗r ⊥,A) 2 +(2mL ′ −z B −z A ) 2<br />
4Dt (2.10)<br />
4πDt m=−∞<br />
2.4.2 Radiative transfer theory<br />
The image point method gives an intuitive picture of the photon density distribution, but fails<br />
to explain the influence of internal reflections at the sample boundaries and to give a value<br />
for the average penetration depth z 0 . This can be accomplished by radiative transfer theory,<br />
where we follow the approach of [58].<br />
Let us consider the flux of photons through a small area dS inside the sample, which for<br />
simplicity we place at the origin and perpendicular to the z-axis, as sketched in fig. 2.8. The<br />
number of photons scattered <strong>from</strong> a volume element dV directly through dS is given by<br />
dS cos θ<br />
the product of the number of photons ρ(⃗r)dV in dV, the fractional solid angle that<br />
represents the cross section of dS for the photons coming <strong>from</strong> dV, the speed of photon<br />
transport v, and the loss e − r<br />
l ∗ due to scattering between dV and dS. The total flux in the<br />
negative z-direction j − dS can then be obtained by integrating over the upper half space with<br />
z > 0, the flux in positive direction j + dS by integration over the lower half space with z < 0:<br />
4πr 2<br />
∫<br />
j ∓ dS =<br />
z > < 0 ρ(⃗r) · v · e − r<br />
l ∗ ·<br />
dS cos θ<br />
4πr 2<br />
dV<br />
14
2.5 Photon flux <strong>from</strong> a surface<br />
The main contribution of the flux comes <strong>from</strong> the immediate neighborhood of dS, so that<br />
the photon density can be replaced by its first-order Taylor expansion around the origin.<br />
Evaluation of the integral then yields<br />
j ∓ = ρ 0v<br />
4 ± vl∗<br />
6<br />
( ) ∂ρ<br />
∂z<br />
0<br />
where the photon density and its derivative have to be taken at the origin.<br />
If dS is located at a boundary, there will be no flux <strong>from</strong> outside the sample, but internal<br />
reflections will create an apparent flux <strong>from</strong> this direction, which is related to the outgoing<br />
flux by the reflectivity R of the surface: j in = R · j out . This gives the boundary conditions<br />
ρ − 2l∗<br />
3<br />
ρ + 2l∗<br />
3<br />
1 + R<br />
1 − R · ∂ρ<br />
∂z = 0 for an upper boundary, i.e. j + = R · j − and<br />
1 + R<br />
1 − R · ∂ρ<br />
∂z = 0 for a lower boundary, i.e. j − = R · j +<br />
The point where the photon density drops to zero is not located at the boundary but at a<br />
distance 2l∗ 1+R<br />
3 1−R · ∂ρ<br />
∂z<br />
in front of it. We can identify this distance with the average penetration<br />
depth z 0 which we have used above for the image point method, as this also is the point where<br />
the photon density vanishes. Assuming a constant gradient of the photon density distribution<br />
close to the surface, one obtains |z 0 | = 2l∗ 1+R 1+R<br />
3 1−R<br />
≈ 0.67<br />
1−R l∗ . Other solutions are mostly of the<br />
order of |z 0 | ≈ 0.7 for the limit of non-reflective surfaces (see e.g. [53], [54]).<br />
2.5 Photon flux <strong>from</strong> a surface<br />
The experimentally accessible quantity in multiple scattering experiments is not the photon<br />
density distribution inside the sample, but the light intensity emitted <strong>from</strong> a sample surface.<br />
It is obvious that this intensity (or photon flux density) is proportional to the number of<br />
photons, i.e. the photon density, at the surface. One can therefore easily calculate the expected<br />
intensities for the various experimental situations <strong>from</strong> eqns. 2.9 and 2.10.<br />
2.5.1 <strong>Backscattering</strong> geometry<br />
<strong>Backscattering</strong> experiments are usually performed with thick samples, where the influence of<br />
the rear and side surfaces on the photon density distribution can be neglected, and the sample<br />
is well described as an infinite half-space.<br />
In backscattering geometry, photon paths of all lengths contribute to the intensity at the surface,<br />
so that the latter is not fully described by the solution of the diffusion equation, which is<br />
valid only for long photon paths. It has been shown however [44] that correct results can be<br />
15
2 Theory<br />
Figure 2.9: Photon flux in backscattering geometry. The photon flux in an infinite<br />
half-space is determined not only by the flux between the first and the last scatterer of<br />
the photon path, A and B, but also by the probabilities f (z A ) and f (z B ) of the photon<br />
to travel between the surface and A or B without being scattered.<br />
obtained <strong>from</strong> the diffusion equation by assuming that the conversion between propagating<br />
light outside the sample and diffusing light inside does not happen at the sample surface, but<br />
at a certain depth inside the sample. The flux density at the sample surface z = 0 is then<br />
j back (⃗r ⊥,A ,⃗r ⊥,B , t) ∝<br />
∫ ∞ ∫ ∞<br />
0 0<br />
f (z A ) · f (z B ) · ρ(A → B, t) 1 surface dz A dz B<br />
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<br />
probability distribution of the conversion depth inside the sample. Usually, one assumes nearnormal<br />
incidence and an exponentially decaying conversion probability with decay length l ∗ ,<br />
so that f (z A ) = e − z A<br />
l ∗<br />
and f (z B ) = e − z B<br />
l ∗ cos θ , where the scattering angle θ is the inclination of<br />
the outgoing light wave with respect to the sample surface. d<br />
The photon flux distribution for steady-state experiments is obtained <strong>from</strong> above equation<br />
by integrating over time. For later calculations the case of negligible absorption is especially<br />
important: e<br />
j back (⃗r ⊥,A ,⃗r ⊥,B , θ) ∝<br />
∝<br />
∫ ∞ ∫ ∞<br />
0<br />
0<br />
e − z A<br />
l∗ · e − z B<br />
l ∗ cos θ<br />
(<br />
)<br />
√ 1<br />
− √ 1<br />
(⃗r⊥,B −⃗r ⊥,A ) 2 +(z B −z A ) 2 (⃗r⊥,B −⃗r ⊥,A ) 2 +(z B +2z 0 +z A ) 2<br />
dz A dz B (2.11)<br />
[d] The definition of the scattering angle θ used for backscattering geometry is inconsistent with the definition<br />
used in previous sections. While e.g. Mie theory denotes backscattering with θ = π, the theory of coherent<br />
backscattering uses θ = 0 for the same angle.<br />
[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<br />
using |z 0 | and omitting the absolute value bars.<br />
16
2.6 On polarization and interference<br />
2.5.2 Transmission geometry<br />
While the samples for backscattering experiments are usually thick, transmission experiments<br />
require rather thin samples, which resemble an infinite slab of thickness L. Still, if the samples<br />
are not too thin, transmission experiments can be fully described by diffusion as all photon<br />
paths have at least the length of the sample thickness.<br />
As a consequence, the exact depth of the conversion between plane wave and diffusive transport<br />
plays only a subordinate role. One can therefore assume that the conversion happens at<br />
a single depth l ∗ .<br />
An important experimental quantity is the distribution of photon flight times (which is equivalent<br />
to the path length distribution) of the transmitted light<br />
∫<br />
j trans (t) ∝<br />
rear surface<br />
δ (z A − l ∗ ) · δ ( z B − (L ′ −l ∗ ) ) · ρ(A → B, t) 2 surfaces d⃗r ⊥ =<br />
= e− τ<br />
t ∞<br />
√<br />
4πDt<br />
∑ e − (2mL′ +(L ′ −l ∗ )−l ∗ ) 2<br />
4Dt − e − (2mL′ −(L ′ −l ∗ )−l ∗ ) 2<br />
4Dt<br />
m=−∞<br />
Using Poisson’s sum formula ∑ ∞ n=−∞ f (n) = ∑ ∞ m=−∞<br />
∫ ∞<br />
−∞ e−2πima f (a) da this can be turned<br />
into [38]<br />
j trans (t) ∝ 2e− τ<br />
t ∞ (<br />
L ∑ ′ e − n2 π 2<br />
L ′2 Dt nπ<br />
( nπ<br />
)<br />
sin<br />
L<br />
n=1<br />
′ l∗) sin<br />
L ′ (L′ − l ∗ )<br />
Obviously, it is l∗<br />
L<br />
′ 0, so that the sine can be replaced by its argument. Likewise, it is<br />
1. Here we can replace the sine by (−1)<br />
n+1 nπ<br />
l ∗ , yielding<br />
L ′ −l ∗<br />
L ′<br />
j trans (t) ∝ −2e − t τ<br />
∞<br />
∑<br />
n=1<br />
( (−1) n e − n2 π 2<br />
L ′2 Dt nπ<br />
L ′<br />
) 2 l<br />
∗2<br />
L ′ L ′ (2.12)<br />
2.6 On polarization and interference<br />
With the introduction of the models ‘random walk’ and ‘diffusion’ an important property<br />
of the scattered light has been lost: Despite of the vectorial nature of electromagnetic waves<br />
both models describe the scattering of scalar waves. To find a correct description of multiple<br />
scattering of vector waves we have to go back to single scattering once more.<br />
Fig. 2.3 shows that the polarization of light scattered by a spherical particle depends strongly<br />
on the scattering angle and the polarization of the incoming wave with respect to the scattering<br />
plane. The result is that after a few scattering events in a multiple scattering sample the<br />
photons will have completely lost the memory of their initial polarization. Multiply scattered<br />
light is therefore unpolarized.<br />
17
2 Theory<br />
300<br />
250<br />
200<br />
150<br />
100<br />
50<br />
x 10 4<br />
5<br />
4<br />
3<br />
2<br />
100 200 300<br />
1<br />
Figure 2.10: Speckles. Random interferences of the photons emerging <strong>from</strong> a multiple<br />
scattering medium result in a random distribution of high and low light intensities.<br />
Still, interference is possible between equally oriented components of the light waves. The<br />
interference pattern observed on a multiple scattering sample results <strong>from</strong> the coherent addition<br />
of the corresponding components of the waves that emerge <strong>from</strong> the ends of the light<br />
paths in the sample. To first order it is therefore the superposition of the interference patterns<br />
of the photons on all pairs of light paths in the sample.<br />
This implies of course that a certain pair of light paths is not only theoretically possible,<br />
but actually has photons traveling on it. The interference pattern of a infinitely extended<br />
incoming wave will therefore differ in some way <strong>from</strong> that of a spatially restricted incoming<br />
wave. Likewise, the interference pattern of multiply scattered light with restricted spatiotemporal<br />
coherence will be only a modified version of the interference pattern of light with<br />
infinite temporal and spatial coherence length.<br />
2.6.1 Speckles<br />
Most of the light paths in the sample are completely unrelated, so that their interference results<br />
in a random speckle pattern of high and low light intensities (fig. 2.10). The speckle pattern<br />
is therefore a subtle image of the positions of the scatterers inside the sample, and is unique<br />
for every sample and every lighting and imaging situation. It is also extremely sensitive to<br />
motions of the scattering particles. Even sub-wavelength movements of the particles lead<br />
to significant variations in the overall phaseshift of the photons and to fluctuations of the<br />
speckles. Averaging over speckle fluctuations or a sample average lead to a detected light<br />
intensity approximately proportional to the cosine of the scattering angle θ, as described by<br />
Lambert’s well-known emission law [33].<br />
18
2.6 On polarization and interference<br />
Figure 2.11: Theorem of reciprocity. Amplitudes and phases of direct and reversed<br />
paths are equal if the incident and detected light is completely polarized, and if the<br />
incident polarization P in,direct of the direct path is identical to the detected polarization<br />
P out,reversed of the reversed path and vice versa [38]. With a single light source, direct and<br />
reverted paths can not be distinguished, so that P in,direct = P in,reversed and P out,direct =<br />
P out,reversed . For linear polarization we denote P in ‖ P out as the parallel polarization<br />
channel and P in ⊥ P out as the crossed polarization channel. The respective channels for<br />
circular polarization are called the helicity conserving and the helicity breaking channel.<br />
2.6.2 <strong>Coherent</strong> backscattering<br />
However, for multiply scattered photons that run on a certain path S 1 → S 2 → · · · → S n<br />
and on its time-inverted path S n → S n−1 → · · · → S 1 the theorem of reciprocity (fig. 2.11)<br />
predicts special correlations: Photons in the parallel polarized or helicity conserving channel<br />
are always in phase when they leave the sample. This phase coherence is independent of the<br />
pathlength or the exact positions of the scatterers, and therefore also insensitive to particle<br />
motions, which are of course slow compared to the speed of light. The interference pattern of<br />
the direct and time-inverted photons therefore survives any average over the random speckle<br />
pattern.<br />
Equal phases for photons on direct and time-inverted paths means also that in direct backscattering<br />
direction all interferences are constructive, regardless of the end-to-end distances of the<br />
photon paths that determine the angular intensity distributions of the interference patterns.<br />
The intensity scattered in this direction is therefore twice the amount expected for an incoherent<br />
superposition of the light waves. This all-constructive interference decays – for an<br />
incoming spatially and temporally infinitely extended plane wave over an angular scale of<br />
(kl ∗ ) −1 [11] – as constructive and destructive interferences superimpose for angles deviating<br />
<strong>from</strong> backscattering direction. The resulting cone-shaped intensity enhancement is referred to<br />
as the coherent backscattering cone.<br />
2.6.3 Anderson localization<br />
The concept of interference on time-inverted photon paths works also if the path forms a<br />
closed loop inside the sample. Here, the place of the interference is the point where the light<br />
19
2 Theory<br />
waves enter and leave the loop. As the light waves are always in phase at this point, constructive<br />
interference leads to an enhanced photon density compared to the normal diffusive<br />
behavior. In strongly scattering media the probability of photons running on closed loops is<br />
enhanced, so that a macroscopically reduced diffusion can be observed. Eventually this leads<br />
to a complete breakdown of photon transport and a transition to a localizing state, which is<br />
called Anderson localization [13].<br />
The critical amount of disorder for this phase transition can be expressed by the criterion<br />
proposed by Ioffe and Regel [27], namely that kl ∗ ≈ 1. The width of the steady-state coherent<br />
backscattering cone, which is inversely proportional to this quantity, is therefore an important<br />
experimental measure for the disorder in the sample [12].<br />
2.7 The theory of coherent backscattering<br />
To develop a theory for the steady-state coherent backscattering cone, the easiest case to consider<br />
is a uniform plane wave with infinite spatio-temporal coherence impinging perpendicularly<br />
on the surface of a non-absorbing multiple scattering medium. In this case the photon<br />
flux distribution has the simple form given in eqn. 2.11, and furthermore is translationally<br />
invariant, i.e. j back (⃗r ⊥,A ,⃗r ⊥,B , θ) becomes j back (⃗r ⊥ , θ) with⃗r ⊥ =⃗r ⊥,B −⃗r ⊥,A .<br />
We define the cooperon α c (θ) as the coherent and the diffuson α d (θ) as the incoherent addition<br />
of the flux emerging <strong>from</strong> the time-inverted paths in the sample:<br />
∫<br />
α d (θ) =<br />
∫<br />
j back (⃗r ⊥ , θ) d⃗r ⊥<br />
j back (⃗r ⊥ , θ = 0) d⃗r ⊥<br />
and α c (θ) =<br />
∫<br />
j back (⃗r ⊥ , θ) · e i ⃗ k⃗r⊥<br />
d⃗r ⊥<br />
∫<br />
(2.13)<br />
j back (⃗r ⊥ , θ = 0) d⃗r ⊥<br />
where⃗ k is the wave vector of the emitted light wave. The diffuson is sometimes also referred<br />
to as the incoherent background of the backscattering cone.<br />
If single and low order scattering can be blocked completely, the backscattered intensity measured<br />
in an experiment is the sum of these two contributions. Otherwise, the height of the<br />
cooperon compared to the diffuson is reduced, as single scattering contributes to the incoherent,<br />
but not to the coherent addition of the photon flux [38].<br />
Evaluating the above equations yields [10]<br />
( )<br />
α d (θ) = µ z0<br />
l<br />
+ µ<br />
∗ µ+1<br />
z 0<br />
l<br />
+ 1 ∗ 2<br />
(2.14)<br />
and<br />
α c (θ) =<br />
2 ( z 0<br />
l ∗ + 1 2<br />
1−e −2qz 0<br />
ql ∗ + 2µ<br />
µ+1<br />
) ( ql ∗ + µ+1<br />
2µ<br />
) 2<br />
(2.15)<br />
20
2.7 The theory of coherent backscattering<br />
diffuson / cooperon<br />
2<br />
1.8<br />
1.6<br />
1.4<br />
1.2<br />
1<br />
0.8<br />
0.6<br />
diffuson α d<br />
(θ)<br />
cooperon α c<br />
(θ)<br />
α d<br />
(θ) + α c<br />
(θ)<br />
0.4<br />
0.2<br />
0<br />
−90 −60 −30 0 30 60 90<br />
scattering angle [deg]<br />
Figure 2.12: Diffuson and cooperon. The graph shows diffuson α d (θ) and cooperon<br />
α c (θ) calculated with eqns. 2.14 and 2.15 for λ = 590 nm, kl ∗ = 5, and non-reflective<br />
sample surface.<br />
with µ = cos θ and q = k| sin θ|.<br />
In exact backscattering direction θ = 0, diffuson and cooperon are both equal to one, so that<br />
the backscattered intensity is enhanced by a factor of two (fig. 2.12). For larger angles the<br />
cooperon drops rapidly to zero, and the backscattered intensity approaches the value of the<br />
diffuson.<br />
The initial slope of the coherent backscattering cone close to θ = 0 and therefore also the width<br />
of the cone is strongly affected by internal reflections. For non-reflective sample surfaces<br />
the inverse angular width is approximately 4 3 kl∗ [58]. Internal reflections – described by the<br />
reflectivity R of the surface – narrow the cone to f<br />
FWHM −1 =<br />
(<br />
1 + 1 + R ) 2<br />
1 − R 3 kl∗ (2.16)<br />
The numerator of the cooperon in eqn. 2.13 is a Fourier transform of the flux distribution [38]:<br />
∫∫<br />
∫<br />
j back (⃗r ⊥ , t) · e iqr ⊥<br />
dt d⃗r ⊥ =<br />
[ ] ∫<br />
FT j back (⃗r ⊥ , t) dt =<br />
j(t) · e −q2 Dt dt (2.17)<br />
The cooperon is therefore not only the superposition of interference patterns as a function of<br />
the distance of first and last scatterer, but also a superposition of Gaussian distributions as a<br />
[f] FWHM = full width at half maximum<br />
21
2 Theory<br />
cooperon<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
l abs<br />
= 10 −5 m<br />
l abs<br />
= 3 ⋅ 10 −5 m<br />
l abs<br />
= 10 −4 m<br />
no absorption<br />
0.2<br />
0<br />
−90 −60 −30 0 30 60 90<br />
scattering angle [deg]<br />
Figure 2.13: <strong>Coherent</strong> backscattering with absorption. The graph shows the cooperon<br />
α c (θ, l abs ) calculated with eqn. 2.18 for different absorption lengths l abs = 3Dτ/l ∗ with<br />
λ = 590 nm, kl ∗ = 5, and non-reflective sample surface.<br />
function of time or respectively the pathlength. The longest photon paths have the narrowest<br />
Gaussians, while short paths have wide distributions.<br />
An infinite series of Gaussians creates a triangular cusp at the tip of the coherent backscattering<br />
cone. However, absorption as well as localization introduce a cutoff length for the photon<br />
paths, so that the narrowest Gaussians are eliminated. This explains why one observes a<br />
rounded conetip for absorbing or localizing samples (figs. 2.13, 3.4).<br />
The effect of absorption can be handled mathematically by introducing a substitution q 2 →<br />
q 2 + (Dτ) −1 in the cooperon [38]. With the correct normalization the cooperon for absorbing<br />
samples becomes<br />
α c (θ, τ) =<br />
(<br />
Dτ l ∗ +<br />
(<br />
l ∗ l ∗ + √ ) 2 (<br />
Dτ 1−e<br />
−2q abs z 0<br />
(<br />
1 − e − 2z 0<br />
q abs l ∗ + 2µ<br />
µ+1<br />
)<br />
) )<br />
√ √Dτ ( )<br />
Dτ q abs l ∗ + µ+1 2<br />
(2.18)<br />
2µ<br />
with q abs = √ q 2 + (Dτ) −1 .<br />
Anderson localization can be modeled by a transition <strong>from</strong> a constant diffusion coefficient D<br />
to a time-dependent coefficient D(t) ∝ t −1 at the localization time t loc [45]. As the ‘cutoff’<br />
mechanism is different, the resulting shape of the conetip also differs <strong>from</strong> that of a purely<br />
absorbing sample (fig. 3.4).<br />
22
2.7 The theory of coherent backscattering<br />
cooperon [a.u.]<br />
0.3<br />
0.25<br />
0.2<br />
0.15<br />
0.1<br />
l abs<br />
= 10 −5 m<br />
l abs<br />
= 3 ⋅ 10 −5 m<br />
l abs<br />
= 10 −4 m<br />
no absorption<br />
0.05<br />
0<br />
−90 −60 −30 0 30 60 90<br />
scattering angle [deg]<br />
Figure 2.14: <strong>Coherent</strong> backscattering with absorption – unnormalized cooperon. The<br />
graph shows the unnormalized cooperon for different absorption lengths l abs = 3Dτ/l ∗<br />
with λ = 590 nm, kl ∗ = 5, and non-reflective sample surface. Deviations <strong>from</strong> the<br />
non-absorptive case at the cone flanks can be observed only for very short absorption<br />
lengths, which are irrelevant for our experimental situations.<br />
Both absorption and localization not only cause a rounding of the conetip, they also widen<br />
the cooperon. However, in many experimental situations the normalization of the diffuson<br />
and the cooperon by ∫ j(⃗r ⊥ , θ = 0) d⃗r ⊥ is unnecessary, as the experimental data are also not<br />
normalized. Applying the above transformations only to the numerator of the cooperon in<br />
eqn. 2.13 results in a lowered cone enhancement instead of a widened cooperon (fig. 2.14), so<br />
that the cone flanks are unaffected by absorption or localization. In the measurement of kl ∗ ,<br />
an imprecise rendition of the very tip of the backscattering cone – which is rather common for<br />
narrow cones – is therefore no major source of errors, as the theory can be fitted to the flanks<br />
of the cone.<br />
23
24<br />
2 Theory
3 Setups<br />
3.1 Laser System<br />
The key piece of the light scattering laboratory is the picosecond pulse laser system (fig. 3.1)<br />
which serves as light source for all light scattering setups.<br />
The system consists of a Rhodamin6G dye laser (699 <strong>from</strong> <strong>Coherent</strong>) which is pumped by the<br />
514.5 nm-line of an Ar + -laser (Innova400 <strong>from</strong> <strong>Coherent</strong>) [49], a laser type which operates in<br />
continuous wave (cw) mode. The lasing medium in the dye laser is a jet of Rhodamin6G, so<br />
that the wavelength can be tuned roughly between 570 nm and 620 nm.<br />
To be able to alternatively operate the system in pulsed mode, the Ar + -laser was modified<br />
with a mode locker (<strong>from</strong> APE), which provides 100 ps-pulses at a repetition rate of 76 MHz.<br />
The cavity lengths of pump and dye laser are matched to synchronize and overlap the pulses<br />
coming in <strong>from</strong> the pump laser with those already circulating in the cavity of the dye laser. The<br />
inversion in the dye jet is completely broken down by the front flank of the circulating pulse,<br />
so that the pulse is shorted to approximately 12-15 ps. After about 30 cycles in the cavity of<br />
the dye laser the now significantly amplified pulse can be released by a cavity dumper (<strong>from</strong><br />
APE).<br />
3.2 Wide Angle Setup<br />
The setup used for the study of the wide-ranged angular distribution of the backscattered<br />
light consists of 256 photosensitive diodes attached to a semicircular arc with a diameter of<br />
1.2 m (fig. 3.2). In its center the sample is located, facing the incoming cw laser beam which is<br />
focussed through a tiny hole between the topmost diodes [22, 23, 47]. A simple motor rotates<br />
Figure 3.1: Laser system. The Rhodamin6G dye laser and the Ar + -pumplaser were<br />
modified to run also in pulsed mode. The acousto-optical modulator (AOM) in the<br />
Ar + -laser serves as mode locker, the one in the dye laser as cavity dumper.
3 Setups<br />
Figure 3.2: Wide angle setup. 256 photodiodes arranged in a semicircle around the<br />
sample capture the the backscattered radiation over a range of nearly 180 ◦ .<br />
the sample around an axis parallel to the incoming laser beam, providing a sample average<br />
which eliminates the speckle pattern.<br />
3.2.1 Optical setup<br />
The intensity of the coherent backscattering cone varies rapidly around the backscattering direction,<br />
while the variations at larger scattering angles are comparatively slow. Accordingly,<br />
the angular resolution of the setup has to be rather high in the center, while at larger angles<br />
the diodes can be set further apart. To meet these requirements, for angles θ < 9.75 ◦ eight<br />
photodiode arrays with 16 photodiodes each (S5668 <strong>from</strong> Hamamatsu) are used, yielding an<br />
angular resolution of 0.15 ◦ . For angles θ > 9.75 ◦ , single photodiodes (S4011 <strong>from</strong> Hamamatsu)<br />
provide angular resolutions of 0.7 ◦ for 9.75 ◦ < θ < 19.55 ◦ , ∼ 1 ◦ for 19.55 ◦ < θ < 60 ◦<br />
and ∼ 3 ◦ for 60 ◦ < θ < 85 ◦ . Still, it is not possible ro resolve the very tip of the cone at θ ≈ 0,<br />
as this is the position where the laser beam passes through the diode arc. For this task the<br />
small angle setup (see sec. 3.3) has to be used.<br />
The desired angular resolution also implies some prerequisites for the optical setup: The<br />
laser source has to be imaged completely on a single photodiode, without any light of a<br />
certain scattering angle missing its diode or even illuminating a neighboring one. On the<br />
other hand, the incoming laser beam is supposed to have a considerable width at the sample<br />
surface to avoid finite size effects [32]. Therefore the incoming laser beam is first widened<br />
and parallelized by a telescope, and then focussed on the point between the photodiodes,<br />
diverging again on its way to the sample. Light scattered with a certain scattering angle is<br />
then focussed again when it reaches the photodiodes.<br />
26
3.2 Wide Angle Setup<br />
1 .0<br />
0 .8<br />
h e lic ity c o n s e rv in g<br />
h e lic ity b re a k in g<br />
0 .6<br />
c o o p e ro n<br />
0 .4<br />
0 .2<br />
0 .0<br />
-9 0 -6 0 -3 0 0 3 0 6 0 9 0<br />
s c a tte rin g a n g le [d e g ]<br />
Figure 3.3: Polarizer effectiveness. The circular polarizer foil is only about 97% effective,<br />
so that each polarization channel contains 3% of the other channel. The resulting<br />
small cone with an enhancement of 0.03 can easily be observed in the helicity breaking<br />
channel. Due to some missing data points at θ ≈ 0 the enhancement of 0.97 in the<br />
helicity conserving channel is more difficult to read <strong>from</strong> the graph.<br />
3.2.2 Suppression of single scattering<br />
The enhancement of the coherent backscattering cone, I m /(I m + I s ), depends on the intensities<br />
of both singly (I s ) and multiply (I m ) scattered light, where single scattering results in a lowered<br />
backscattering enhancement. To study the backscattering of multiply scattered light in detail,<br />
single scattering has to be suppressed sufficiently.<br />
According to the theorem of reciprocity (see sec. 2.6), interference of multiple scattered waves<br />
is only possible for the parallel or respectively the helicity conserving polarization channel.<br />
As explained in sec. 5.2.3, the sequence ‘polarizer – scatterer(s) – polarizer’ which is necessary<br />
to select the channel with the backscattering cone simultaneously blocks single scattering if<br />
circular polarizers are used. The wide angle setup is therefore equipped with a bendable<br />
circular polarization foil (J53-333 <strong>from</strong> 3M), which is placed at the entrance of the setup and<br />
in front of the diodes. This polarizer foil blocks about 97% of the single scattered intensity<br />
(fig. 3.3).<br />
3.2.3 Diode calibration<br />
As each of the photodiodes is read by its own electronic circuit, it is necessary to calibrate the<br />
diodes to make up for different gains of the processing electronics [22, 23, 47]. This is done by<br />
27
3 Setups<br />
measuring the backscattering of a sample with known scattering properties as a function of<br />
the incoming laser power P, which is determined with a calibrated powermeter (FieldMaxII<br />
<strong>from</strong> <strong>Coherent</strong>) <strong>from</strong> a reflection on a glass plate in the laser beam at the entrance of the<br />
setup. A fit of the powers P(d θ ) as a function of the diode signals d θ with a polynomial then<br />
yields a calibration function for each photodiode (see fig. 5.1).<br />
As reference sample we use a block of teflon, the backscattering cone of which has a FWHM<br />
of the order of 0.03 ◦ for visible light (see sec. 5.2.2). This is much narrower than the angular<br />
resolution of the wide-angle setup, so that teflon can be considered to give a purely incoherent<br />
signal proportional to P α d (θ).<br />
3.3 Small Angle Setup<br />
In sec. 2.7 the coherent backscattering cone was presented as a superposition of Gaussian<br />
distributions j(s) · e − sin2 θ<br />
3 k 2 sl ∗ , whose width is a function of the path length s of the timeinverted<br />
photon paths. Absorption and localization both affect mainly long paths, which<br />
contribute essentially at the very tip of the backscattering cone. Their reduced contribution<br />
results in a rounding of the tip of the backscattering cone. With a high-resolving setup it<br />
should therefore be possible to observe both phenomena in coherent backscattering. The<br />
same setup could also be used to measure the transport mean free paths of samples with<br />
extremely narrow backscattering cones and thus complement the wide angle setup.<br />
3.3.1 Precision requirements<br />
For an estimate of the required setup precision we assume for a moment that absorption<br />
(or localization) results in an abrupt cutoff at path length s = L. The narrowest Gaussian<br />
that contributes to the backscattering cone is therefore the one with standard deviation σ =<br />
√<br />
3/(2k 2 Ll ∗ ) = √ 1/(2k 2 Dτ). The angular width of the conetip rounding must be of the same<br />
order of magnitude.<br />
The absorption of a sample like the titania powder R700 (see sec. 4.2) with absorption time<br />
τ = 2 ns and diffusion coefficient D = 15 m2 /s at wavelength λ = 590 nm will therefore require<br />
to properly resolve an angular range of θ round ≈ ±0.02 ◦ . To observe localization, which sets<br />
in after a localization length l a = 340 mm [48], a similar resolution is necessary.<br />
Test calculations show that the intensity difference between a localizing and a non-localizing<br />
sample is less than 0.1% of the maximum of the cooperon (fig. 3.4). As the detection must<br />
be able to capture the maximal backscattered intensity at θ = 0, plus some external radiation<br />
and electronic noise that can never be avoided completely, while still providing the necessary<br />
intensity resolution, the digital range of the detection must be at least 2 14 , better 2 15 − 2 16 .<br />
3.3.2 Optical setup<br />
In the small angle setup a 4-megapixel 16-bit monochrome CCD camera (Alta U4000 <strong>from</strong><br />
Apogee) is placed opposite the sample (fig. 3.5). The camera can be cooled thermoelectrically<br />
28
3.3 Small Angle Setup<br />
1<br />
0.998<br />
0.996<br />
plane<br />
absorption<br />
localization<br />
0.994<br />
cooperon<br />
0.992<br />
0.99<br />
0.988<br />
0.986<br />
0.984<br />
0.982<br />
0.98<br />
0 0.02 0.04 0.06 0.08 0.1<br />
scattering angle (deg)<br />
Figure 3.4: Conetip rounding <strong>from</strong> absorption and localization. The backscattering<br />
cones were calculated using eqn. 2.17 with the simpler flux density distribution j(s) ∝<br />
√ −3<br />
4πDt given in [40], for which at small scattering angles θ the results are equivalent<br />
to the ones obtained with eqn. 2.11. Absorption was incorporated by the substitution<br />
q 2 → q 2 + (Dτ) −1 , localization was modeled by a transition <strong>from</strong> D = const. to D(t) ∝<br />
t −1 at the localization time t loc . For the sample parameters of R700 and a localization<br />
length of the order of τ, localization causes a shift at the tip of the backscattering cone<br />
of less than 0.1% of the intensity maximum.<br />
29
3 Setups<br />
Figure 3.5: Small angle setup. A CCD camera captures the the backscattered radiation<br />
around backscattering direction.<br />
to 45 K below ambient temperature to reduce electronic noise. It turned out that the glass<br />
window that covers the CCD chip needs to be wedge shaped to avoid interferences in the glass,<br />
which cause a disturbing ripple structure on the CCD image. The camera offers exposure<br />
times between 30 ms and 183 min; to improve the signal-to-noise ratio, the backscattering data<br />
of our experiments are an average over a series of usually 100 pictures, taken with an exposure<br />
time of 0.5 s.<br />
The camera replaces an older 8-bit model with 580 × 740 pixels, which did not offer enough<br />
angular and intensity resolution for the planned experiments. The exchange of the camera<br />
improves the maximum intensity resolution by more than two orders of magnitude and the<br />
maximum angular resolution or respectively the maximum angular range by a factor of three.<br />
As the direct backscattering direction is blocked by the camera, a beamsplitter directs the<br />
laser beam onto the sample. The remaining part of the beam that passes straight through the<br />
beamsplitter must be dumped completely, as even weak backreflection will disturb the image<br />
on the camera. For similar reasons, a pellicle beamsplitter instead of a glass beamsplitter has<br />
to be used, as even anti-reflection coated surfaces of a glass plate would cause distinct ghost<br />
images.<br />
In contrast to the wide angle setup described in the previous section, where the sample is<br />
illuminated by a slightly divergent light beam, the incoming laser beam in the small angle<br />
setup is exactly parallel. Backscattered light with a certain scattering angle θ then forms a<br />
cone-shaped shell with a thickness that is determined by the diameter of the sample area<br />
illuminated by the incoming laser beam (fig. 3.6). The lens above the beamsplitter converges<br />
30
3.3 Small Angle Setup<br />
Figure 3.6: The optical setup. Backscattered light with a certain scattering angle θ<br />
forms a cone-shaped shell, which is converged in the focal plane on a circle with radius<br />
r by the lens with focal length f .<br />
this ring of light on a circle in the focal plane, the radius of which is given by<br />
r = f · tan θ (3.1)<br />
where f is the focal length of the lens.<br />
Like in the wide angle setup, single scattering is blocked by a circular polarizer. In the small<br />
angle setup it is unnecessary to have a bendable polarizer foil, so a high-quality circular<br />
polarizer <strong>from</strong> an industrial optics manufacturer (AUC circular polarizer <strong>from</strong> B+W) can be<br />
used. These are available with larger diameters than usual polarizers offered by laboratory<br />
suppliers and provide extinction ratios up to 4000:1 [1].<br />
3.3.3 Sample average<br />
Solid samples like teflon or the titania powders have to be moved during the measurement to<br />
average over the speckle pattern. The method used in the wide angle setup, where the sample<br />
is simply rotated, however turned out to be inconvenient as the high-resolving camera picked<br />
up this rotational motion as a circular structure in the images. To have a larger ensemble to<br />
average over in the small angle experiments, the sample is therefore pulled on Lissajous loops<br />
by two motors with a frequency of several Hertz on each axis (fig. 3.7).<br />
3.3.4 Data evaluation<br />
For the evaluation of the backscattering data the exact position of the tip of the coherent<br />
backscattering cone at θ = 0 has to be obtained <strong>from</strong> the CCD image. We do this by converting<br />
31
3 Setups<br />
Figure 3.7: Sample shaker. The two motor-driven eccentric wheels M (red) pull the<br />
sample S back and forth between the levers and the two rubber bands (brown), so that<br />
the samples moves on Lissajous loops.<br />
the greyscale image into a binary image, based on a threshold value of 95% of the maximum<br />
intensity in the picture. The center of the cone is then identical with the center of mass of<br />
the white region. The angular intensity distribution of the backscattered light can then be<br />
obtained either by a profile section through the image or by an azimuthal average.<br />
The angular resolution of the setup can be tested by replacing the multiple scattering sample<br />
with a mirror. The image on the CCD chip is then that of a single scattering angle. If the lens<br />
in front of the camera were able to focus the light on a single pixel, the angular resolution of<br />
the setup would be given by the pixel size, which is 7.4 µm for the camera in use. However, in<br />
the present setup configuration the focus spot is 4 − 8 pixels wide, depending on the diameter<br />
of the incoming laser beam, which is between 0.5 cm and 2 cm.<br />
To account for the structure of the focus spot, the data are therefore fitted with theory curves<br />
that are convoluted with the intensity profile of the spot. a The resulting angular resolution of<br />
the measured backscattering data is then close to the maximal resolution of 0.00085 ◦ , which<br />
is given by the pixel size of the CCD chip.<br />
32
3.4 Time Of Flight Setup<br />
Figure 3.8: Time of flight setup. The setup measures a histogram of time delays between<br />
photons arriving at detector 1 (photomultiplier) behind the sample and the signal<br />
of the reference detector 2 (photodiode).<br />
3.4 Time Of Flight Setup<br />
Apart <strong>from</strong> investigations of the path length distribution in transmission and the search for<br />
Anderson localization [48], the time of flight setup is used to determine diffusion coefficients<br />
and absorption times of the samples. These quantities are obtained by fitting the measured<br />
distribution of the times photons need to travel through the sample with the theoretical prediction<br />
of eqn. 2.12.<br />
To obtain a distribution of photon flight times, the time of flight setup measures a histogram<br />
of time delays between photons arriving at a detector behind the sample and the signal of a<br />
reference detector, as shown schematically in fig. 3.8. The signal <strong>from</strong> a photon arriving at the<br />
photomultiplier (H5784 <strong>from</strong> Hamamatsu) starts a time measurement in the single photon<br />
detection card (SPC-140 <strong>from</strong> Becker & Hickl). The stop signal is provided by a photodiode<br />
which receives a small part of the unscattered pulse and which is followed by a cable delay to<br />
have the signal come in after the signal <strong>from</strong> the photomultiplier.<br />
It is important to have the samples turbid and thick enough so that at most a single photon<br />
of each pulse actually reaches the detector. This ensures that the probability to trigger a time<br />
measurement is not biased towards photons on short paths.<br />
The measured time delays are sorted into a histogram with 1024 time slots. The width of<br />
this histogram has to be matched to the desired time range, which is of the order of 20-40 ns.<br />
As with the resulting time resolution the shape of the unscattered laser pulse is not a delta<br />
function, the data of a broadened pulse have to be deconvoluted with the systemic answer of<br />
the unscattered pulse.<br />
[a] Deconvoluting the data with the reflection profile would be more straightforward, but is not possible due to<br />
mathematical difficulties.<br />
33
34<br />
3 Setups
4 Samples<br />
4.1 Sample characterization techniques<br />
The light scattering experiments performed with the setups introduced in sec. 3 are the main<br />
source of data which characterize the multiple scattering samples. However, there are a few<br />
additional sample properties which have to be obtained otherwise. These are mainly the<br />
effective refractive index of the sample and the reflectivity of the sample surface, which are<br />
necessary for example to calculate the average penetration depth z 0 . Also important are the<br />
particle size and polydispersity of the samples and the filling fraction of the sample. Some<br />
of these calculations are trivial and can be found in every physics textbook, but still are to be<br />
mentioned briefly.<br />
4.1.1 Particle size and polydispersity<br />
Particle size and polydispersity of the the colloidal particles give first indications for the scattering<br />
properties of the multiple scattering samples. Strongly scattering samples have particle<br />
diameters of the order of the wavelength and low polydispersity. Especially strong scattering<br />
is obtained when the scattering in the particles becomes resonant.<br />
The commercial titanium dioxide samples have been characterized before by M. Störzer [47],<br />
who used electron microscopy to determine size and polydispersity of the particles. The<br />
device used was an XL Scanning Electron Microscope <strong>from</strong> Phillips, which provides a spatial<br />
resolution of up to 50 nm. To avoid charge building in the sample the surface was covered with<br />
a gold layer of approximately 10 nm thickness, which was brought onto the sample using a<br />
gas discharge sputter technique (Scancoat SIX, Edwards). The distribution of the particle sizes<br />
was obtained by measuring the diameters of 150-200 particles <strong>from</strong> the pictures (fig 4.1).<br />
4.1.2 Filling fraction<br />
Another important feature is the filling fraction of the scatterers, which not only characterizes<br />
the sample for itself, but is also needed for calculating the effective refractive index of the<br />
sample.<br />
The filling fraction is defined as<br />
f = V scatterers<br />
V sample<br />
=<br />
m<br />
ρ · r 2 πh
4 Samples<br />
3 0<br />
2 5<br />
n u m b e r o f p a rtic le s<br />
2 0<br />
1 5<br />
1 0<br />
5<br />
0<br />
1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0<br />
p a rtic le d ia m e te r [n m ]<br />
Figure 4.1: Particle size and polydispersity. From the electron microscope image<br />
(left) the distribution of the particle sizes of R700 (right) was obtained by measuring the<br />
diameters of approximately 150 particles [47].<br />
and can quite easily be calculated by measuring the weight m and the spatial dimensions r<br />
and h of the sample, while the mass density ρ of the scattering material can be obtained <strong>from</strong><br />
numerous literature sources.<br />
For strong scattering and low mean free paths, the sample should be as dense as possible. If<br />
however the filling fraction is larger than a certain value (which depends on the mismatch of<br />
the refractive indices of scatterers and surrounding medium) [20], scattering takes place at the<br />
holes rather than the particles, so that there is no use increasing the volume fraction beyond<br />
this value.<br />
4.1.3 Effective refractive index<br />
The refractive index mismatch between the particles and the surrounding medium determines<br />
the effectiveness of the scattering. In contrast, the effective refractive index, which is an average<br />
over the indices of the scatterers, n scat , and the medium in between, n surr , describes<br />
the sample as a whole and is for example needed to calculate the reflectivity of the sample<br />
surface.<br />
Linear approach<br />
The most simple and straightforward approach to calculate the effective refractive index n eff<br />
of a sample is to establish an average medium with an averaged refractive index<br />
36<br />
n eff = f · n scat + (1− f ) · n surr
4.1 Sample characterization techniques<br />
refractive index<br />
3<br />
2.5<br />
2<br />
1.5<br />
linear<br />
Garnett<br />
1<br />
0 0.2 0.4 0.6 0.8 1<br />
filling fraction<br />
Figure 4.2: Refractive index. The results of the theory of J. C. M. Garnett differs<br />
<strong>from</strong> the linear approach especially for volume fractions around 50%. The refractive<br />
indices in the graph were calculated for scatterers with refractive index n sc = 2.7 and<br />
a surrounding medium with n m = 1. The maximal difference is found for f = 50.4%,<br />
where the refractive index calculated with the linear approach is about 16% larger than<br />
with Garnett’s theory [49].<br />
This approach works well as long as the microscopic structure of the sample is much smaller<br />
than the wavelength. However, it must fail as soon as the complicated interactions of the<br />
inhomogenous material distribution start to matter.<br />
Garnett effective refractive index<br />
The theory of J. C. M. Garnett [21] calculates the average electrical field inside the medium<br />
assuming that the fields in the scatterers and in the embedding matrix are related linearly. It<br />
yields an effective dielectric constant<br />
ɛ eff = ɛ surr ·<br />
(<br />
1 + 3 f · ɛscat−ɛ )<br />
surr<br />
ɛ scat +2ɛ surr<br />
1 − f · ɛscat−ɛ surr<br />
ɛ scat +2ɛ surr<br />
(4.1)<br />
for spherical scatterers. As the above assumption neglects effects like the Mie resonances, the<br />
theory is valid only for Raleigh scatterers, or for samples with rather high polydispersity or<br />
irregularly shaped particles, where resonant effects average out. As the latter is true for the<br />
samples used in this work, Garnett’s theory will be used to calculate the refractive index.<br />
4.1.4 Surface reflectivity<br />
Reflections at the sample surfaces strongly influence the distribution of the photon density<br />
inside [58]. The angle dependent reflectivity of the boundary between the sample and the<br />
37
4 Samples<br />
1<br />
0.8<br />
reflectivity<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
1 1.5 2 2.5 3<br />
effective refractive index<br />
Figure 4.3: Reflectivity. The graph shows the reflectivity of the sample-air interface as<br />
a function of the effective refractive index of the sample. For the titania samples with<br />
n eff of the order of 1.5 the reflectivity is roughly 0.5 to 0.6.<br />
surrounding medium with refractive indices n eff and n surr can be estimated <strong>from</strong> Fresnel’s<br />
equations as<br />
R(η) = R ‖(η) + R ⊥ (η)<br />
2<br />
⎧<br />
tan ⎪⎨<br />
2 (θ − η)<br />
= 2 tan 2 (θ + η) + sin2 (θ − η)<br />
2 sin 2 (θ + η) for η ≤ arcsin n surr<br />
n eff<br />
⎪⎩<br />
1 for η > arcsin n surr<br />
n eff<br />
were η and θ are the directions of the incident and the transmitted light beam.<br />
reflectivity is then obtained <strong>from</strong> the boundary conditions at the surface [58]:<br />
The total<br />
R = 3C 2 + 2C 1<br />
3C 2 − 2C 1 + 2<br />
(4.2)<br />
with<br />
C 1 =<br />
∫ π/2<br />
0<br />
R(η) sin η cos η dη , C 2 =<br />
∫ 0<br />
−π/2<br />
R(η) sin η cos 2 η dη<br />
4.2 The samples<br />
For the experiments three kinds of multiple scattering samples were used: dense colloidal<br />
samples of titanium dioxide, solid teflon as reference sample, and water-fluidized beds with<br />
soda lime glass beads.<br />
38
4.2 The samples<br />
sample particle size [nm] polydispersity [%] TiO 2 content [%] n eff D [m 2 /s] τ [ns]<br />
a-1 1.38 25 1.05<br />
a-2 1.46 13 1.31<br />
NIX-2 1.38 27 0.43<br />
NIX-3 1.31 41 0.67<br />
R700 (DuPont) 245 22 1.58(6) 15(1) 2.0(1)<br />
R700 + gypsum ≈ 17 1.29 250 0.1<br />
R900 (DuPont) 359 29 1.44 17 1.05<br />
R902 (DuPont) 279 38 1.63 15 0.88<br />
S-25 1.35 21 0.65<br />
Ti-pure (Aldrich) 540 37 1.38 19(1) 6.2(3)<br />
Table 4.1: Colloidal samples. Particle size and polydispersity of the particles were<br />
determined <strong>from</strong> electron microscope images [47]. The effective refractive indices n eff<br />
were calculated using eqn. 4.1, diffusion coefficient D and absorption time τ were measured<br />
in time of flight experiments at laser wavelength λ = 590 nm. Note that the data<br />
are of strongly varied quality. Errors are given only when it was possible to quantify<br />
them.<br />
4.2.1 Titanium dioxide<br />
Titanium dioxide (TiO 2 ), or titania, is widely used as white pigment for paint, plastics or<br />
papers, in cosmetics, medicines and food, and in many other applications. The reason are its<br />
extreme ‘whiteness’ and opacity, which are due to its high refractive index (n = 2.7 in rutile<br />
phase) and its low absorption. In this work we use it to examine the structure of the coherent<br />
backscattering cone. This requires samples with kl ∗ close to unity, as in this regime the cones<br />
become very wide and their features are easier to observe in the experiment.<br />
We use both custom-made TiO 2 particles that were produced in cooperation with the chemistry<br />
department of the University of Konstanz and commercially available titania powders<br />
which were kindly provided for free by Sigma-Aldrich and DuPont. These contain more or<br />
less spherical particles of rutile titania with diameters between 200 and 600 nm (tab. 4.1). As<br />
conglomerates are rare, the powders can quite easily be compressed to volume fractions of the<br />
order of 40%.<br />
The pure titania samples have diffusion coefficients in the low two-digit range. As we also<br />
intended to do measurements on samples with higher diffusion coefficients, we mixed titania<br />
and ground blackboard chalk (gypsum) in a weight ratio of 1 to 5.<br />
4.2.2 Teflon<br />
Polytetrafluoroethylene (PTFE), best known under its DuPont brand name Teflon, is our reference<br />
sample for backscattering experiments. It is a solid (thus avoiding the problem of<br />
re-creating the sample with exactly the same properties for later measurements, which one<br />
would have with a colloidal reference), but without any particular order (crystalline or otherwise),<br />
has low absorption (though not as low as the TiO 2 samples), and a very large transport<br />
mean free path and therefore an extremely narrow backscattering cone.<br />
39
4 Samples<br />
Figure 4.4: Titania and teflon samples. (Clockwise <strong>from</strong> left:) Teflon sample with 4 cm<br />
diameter and 5 cm height, titania sample with 15 mm diameter and < 1 mm height in<br />
sample container, titania powder. Even on the photo one can observe the significant<br />
difference in the albedos or ”whiteness” of Teflon and titania.<br />
With a diameter of 4 cm and a thickness of 5 cm our teflon sample is not exactly a slab,<br />
which however would be the condition for eqn. 2.12 to be valid. The value of D = 27500 m2 /s<br />
measured in time of flight experiments is therefore not correct. In time of flight experiments<br />
on shorter teflon blocks the diffusion coefficient drops to D = 20000 m2 /s for samples with<br />
thickness L ≈ 2 − 3 cm, and even to D = 16500 m2 /s for L ≈ 1 − 1.5 cm. With this last value<br />
eqns. 2.7 and 2.16 propose a transport mean free path of about 220 µm and a cone width<br />
of approximately 0.01 ◦ . This is very close to the results of the backscattering experiments<br />
reported in sec. 5.2.2, which yield l ∗ = 180 µm, and correspondingly a diffusion coefficient<br />
D = 13300 m2 /s, which however could not be measured in the time of flight experiments due<br />
to the still insufficient size of the teflon samples. a For the absorption time τ the time of flight<br />
data yield 3.3 ns, the refractive index of teflon is given in literature as n = 1.35 [4].<br />
4.2.3 Fluidized beds<br />
A fluidized bed is a mixture of a fluid and a granular medium that has fluid-like properties. As<br />
a dispersion of particles in an upward flux of low-viscosity fluid (gas or liquid) it represents an<br />
intermediate system between suspensions in viscous fluids and dry granular materials. It has<br />
excellent heat and mass transfer characteristics, and the fluid-solid mixture is easy to remove<br />
[a] For technical reasons – the samples have to fit into the time of flight setup – it is impossible to use teflon samples<br />
with larger radii that can reasonably be described as an infinite slab. The adaption of the setup to accommodate<br />
larger samples seems to be the only solution.<br />
40
4.2 The samples<br />
Figure 4.5: Fluidized bed. To fluidize the granular medium, water is pressed through<br />
it <strong>from</strong> below. The sinter plates in the container below the fluidized bed make the water<br />
flow smooth and laminar to establish a uniform fluidization.<br />
and rejuvenate, which is why fluidized beds are used in oil and coal industrial processes<br />
such as fluidized catalytic cracking and fluidized coal combustion, but also pharmaceutical<br />
processes and bio-reactors [26].<br />
The granular medium in a fluidized bed is fluidized only if the flow rate of the fluid exceeds<br />
a certain minimum value. The drag force on the particles is then sufficient to balance their<br />
net weight, and the grains become free to move. All fluidized beds tend to be unstable and<br />
to form bubbles of fluid without particles. However, for viscous liquids or light and small<br />
particles, a stable uniform fluidization regime can be observed over a range of flow rates just<br />
above minimal fluidization [26].<br />
The particles in the fluidized bed examined in this work are soda lime glass spheres <strong>from</strong><br />
Whitehouse Scientific, with diameters around 150 µm (fig. 4.6). We take the refractive index<br />
of soda lime to be 1.52(1), the mass density is given by Whitehouse Scientific as 2.6 g /cm 3 [50].<br />
As the transport mean free path is expected to be proportional to the distance between the<br />
particles in the fluidized bed, the volume fraction was varied between 35% and 52% by using<br />
different flow rates of the pump.<br />
41
4 Samples<br />
0.09<br />
0.08<br />
0.07<br />
relative particle number<br />
0.06<br />
0.05<br />
0.04<br />
0.03<br />
0.02<br />
0.01<br />
0<br />
80 100 120 140 160 180 200<br />
particle diameter [µm]<br />
Figure 4.6: Particle size distribution in the fluidized bed. The size distribution was<br />
measured at the University of Magdeburg on a Retsch Technology Camsizer.<br />
42
5 Experiments<br />
5.1 Conservation of energy in coherent backscattering<br />
Conservation of energy is one of the most fundamental principles in physics. However, the<br />
intensity enhancement of the coherent backscattering cone is one instance where it seems to<br />
be violated at first glance:<br />
The origin of the backscattering enhancement lies in the interference of waves propagating<br />
along reciprocal paths. a This interference can only spatially re-distribute the light energy that<br />
emerges <strong>from</strong> the sample surface; it can not destroy photons or create new ones. The total<br />
amount of energy per unit time emerging <strong>from</strong> the sample must therefore be the same with<br />
and without interference:<br />
∫<br />
half-space<br />
∫<br />
α d (θ) dΩ =<br />
half-space<br />
α d (θ) + α c (θ) dΩ<br />
where diffuson α d (θ) and cooperon α c (θ) are the coherent and the incoherent addition of the<br />
photon flux as defined in sec. 2.7. It follows for the coherent backscattering enhancement that<br />
∫<br />
half-space<br />
α c (θ) dΩ = 0 (5.1)<br />
Thus the intensity enhancement of the coherent backscattering cone at small angles should be<br />
balanced by a corresponding intensity cutback to ensure conservation of energy.<br />
Unfortunately, such an intensity cutback had never been observed experimentally, and the<br />
theory of coherent backscattering as developed in sec. 2.7 does not predict an intensity cutback<br />
either. As the principle of conservation of energy holds in any case, the only possible<br />
conclusion is that both the experimental procedure and the theoretical description of the<br />
backscattering cone are too inaccurate to render the cone correctly.<br />
The question if the backscattering cone is depicted correctly by experiment and theory is not<br />
just of purely academic interest. The accurate measurement and description of the cone is<br />
important as the scaling of its width with the inverse product of the wave vector k of the<br />
scattered light and the transport mean free path l ∗ is commonly used to characterize multiple<br />
scattering materials. In particular in the study of Anderson localization of light [13] a reliable<br />
[a] The interference nature of coherent backscattering can be proved for example by the influence of Faraday<br />
rotation on the backscattering cone [35, 36, 37].
5 Experiments<br />
0.35<br />
0.3<br />
0.25<br />
linear<br />
2nd order polynomial<br />
3rd order polynomial<br />
4th order polynomial<br />
laser power [a.u.]<br />
0.2<br />
0.15<br />
0.1<br />
0.05<br />
0<br />
−0.05<br />
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5<br />
diode signal<br />
x 10 4<br />
Figure 5.1: Calibration. The graph shows an example calibration function of a single<br />
photodiode. The errors of both the power and the diode signals are hardly recognizable<br />
in the graph and are therefore not drawn explicitly. Overall, 3rd order polynomials<br />
seem to be most appropriate to describe the gain characteristics of the photodiodes and<br />
their electronics.<br />
knowledge of the parameter kl ∗ is needed to characterize the phase transition <strong>from</strong> diffusive<br />
transport to a localizing state [7, 48].<br />
Our recent progress in this field [48] and the closely related experiments on extremely wide<br />
backscattering cones, where conservation of energy demands a sizeable correction to the<br />
present-day theory, prompted us to attentively review our backscattering experiments and<br />
the evaluation of the backscattering data. An improved theory was contributed by Akkermans<br />
and Montambaux, who managed to eliminate certain improper assumptions <strong>from</strong> the<br />
calculations.<br />
5.1.1 The power scale of the wide angle setup<br />
One difficulty of the coherent backscattering experiments is to distinguish between the inevitable<br />
background of stray light and electronic dark count, the incoherent background<br />
(which might be hidden not only by the backscattering cone, but also by the compensating<br />
intensity cutback), and the enhancement of the backscattering cone (which does not necessarily<br />
reach a factor of 2 due to incomplete blocking of single scattering). In other words, the<br />
experiments require an independent intensity scale.<br />
We figured that for experiments with our wide angle setup (see sec. 3.2) this independent scale<br />
could be provided by a reference sample with a very high kl ∗ , as it is already used to calibrate<br />
44
5.1 Conservation of energy in coherent backscattering<br />
the photodiodes. For such samples, the angular distribution of the backscattered intensity is<br />
known to be proportional to the diffuson α d (θ) given in eqn. 2.14, as any deviations <strong>from</strong> the<br />
incoherent background (i.e. the backscattering cone and the energy cutback) are too narrow<br />
or too small to be detected by the setup.<br />
The backscattering of the reference is also proportional to the incoming laser power P and<br />
the total fraction of the backscattered intensity, the albedo A. The latter is reduced by energy<br />
losses in the sample, mostly due to absorption of photonic energy, but also because of photon<br />
leakage at the side and the rear surfaces.<br />
For the calibration of the photodiodes only a relative power scale is needed, as one is merely<br />
interested in compensating different gains of the diodes and their electronics. Therefore the<br />
exact value of the albedo is not important, as long as it is independent of the inserted light<br />
power. If however the albedos are known for sample and reference, one can derive an absolute<br />
power scale for the diode signals.<br />
Herein lies the misapprehension of [22, 23, 47], where the reference sample was also used to<br />
determine the intensity level of the diffuson. As the different albedos of sample and reference<br />
were neglected, this approach confuses relative and absolute power scale and therefore yields<br />
slightly wrong results.<br />
The photodiodes are calibrated by a series of measurements of the diode signals d θ at different<br />
incoming light powers P, so that a relation between the diode signals and the light power<br />
scattered in a certain direction θ can be derived. Previously used were relations like d θ ∝<br />
α d (θ) · P [19], d θ ∝ cos θ · P, or even simply d θ ∝ P [22, 23, 47]. The latter however distort both<br />
diffuson and cooperon for angles θ > 0, so that they can be used only if the backscattering<br />
cone is comparatively narrow.<br />
The effect of the different albedos of sample and reference can be incorporated in the calibration<br />
function by the albedo mismatch b ℵ = A reference /A sample :<br />
d θ ∝ α d (θ) · ℵ · P (5.2)<br />
The albedo mismatch rescales the emitted intensity of the reference, so that the backscattering<br />
of reference and sample is directly comparable.<br />
The precision of the experimental result is determined by the precision of the reference measurements.<br />
The order of the fitted polynomial must therefore be chosen such that the errors<br />
are not artificially reduced. Based on the observations on fig. 5.1 we use 3rd order polynomials<br />
to fit the data and obtain the calibration functions for the photodiodes.<br />
5.1.2 Calculating the albedo<br />
The albedo A of a multiple scattering sample is defined as the ratio of the (back-)scattered<br />
light power P out and the inserted power P in . The albedo of a sample without any losses would<br />
[b] ℵ = hebr. aleph<br />
45
5 Experiments<br />
E sample<br />
(t) / E half−space<br />
(t)<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
free space distribution<br />
half−space distribution<br />
−9.5 −9 −8.5 −8 −7.5 −7 −6.5 −6<br />
log 10<br />
(t [sec])<br />
Figure 5.2: Comparison of the albedos. The time-dependent albedos A(t) of an<br />
absorbing cut-out of the free space (solid line) and respectively a half-space (dashed<br />
line) at the position of the sample are similar enough to replace the latter by the former<br />
in the calculation of the total albedo. Calculations for teflon sample with radius R =<br />
20 mm, thickness L = 50 mm, diffusion coefficient D = 16500 m2 /s, absorption time<br />
τ = 3.3 ns, transport mean free path l ∗ = 220 µm, refractive index n = 1.35.<br />
be equal to one, meaning that the incoming light power P in is equal to the scattered power<br />
P out of a lossless sample. The albedo can therefore also be defined as<br />
A = P with losses<br />
P lossless<br />
=<br />
∫ ∫<br />
∫ surface ∫<br />
surface<br />
j with losses (⃗r ⊥ , t) d⃗r ⊥ dt<br />
j lossless (⃗r ⊥ , t) d⃗r ⊥ dt<br />
(5.3)<br />
In our multiple scattering samples, most of the loss of light energy is due to absorption. If it<br />
were the only reason for energy loss, the albedo could be easily calculated as<br />
A(τ) = 1 − 2(l∗ +z 0 )<br />
e− √<br />
Dτ<br />
(5.4)<br />
2(l ∗ +z<br />
√ 0 )<br />
Dτ<br />
with diffusion coefficient D, absorption time τ, and penetration depth z 0 .<br />
However, as the samples are not infinitely large, one can not a priori exclude that losses at the<br />
side and rear boundaries have significant influence on the photon distribution in the sample.<br />
Unfortunately there is no expression for the photon density distribution in a 3-dimensionally<br />
confined absorbing medium. Therefore we approximate the losses in a sample with finite<br />
radius R and thickness L by comparing the energy inside the non-absorbing infinite half-<br />
46
5.1 Conservation of energy in coherent backscattering<br />
Calculation with free space photon density distribution:<br />
0<br />
t = 10 −11 s<br />
0<br />
t = 10 −10 s<br />
0<br />
t = 10 −9 s<br />
1e8<br />
z [mm]<br />
10<br />
20<br />
30<br />
10<br />
20<br />
30<br />
10<br />
20<br />
30<br />
1e6<br />
1e4<br />
40<br />
40<br />
40<br />
1e2<br />
50<br />
−20 0 20<br />
r [mm]<br />
50<br />
−20 0 20<br />
r [mm]<br />
50<br />
−20 0 20<br />
0<br />
r [mm]<br />
Calculation with photon density distribution of the infinite half-space:<br />
0<br />
t = 10 −11 s<br />
0<br />
t = 10 −10 s<br />
0<br />
t = 10 −9 s<br />
10<br />
10<br />
10<br />
1e6<br />
z [mm]<br />
20<br />
30<br />
20<br />
30<br />
20<br />
30<br />
1e4<br />
40<br />
40<br />
40<br />
1e2<br />
50<br />
−20 0 20<br />
r [mm]<br />
50<br />
−20 0 20<br />
r [mm]<br />
50<br />
−20 0 20<br />
0<br />
r [mm]<br />
Figure 5.3: Calculated photon density distributions in the Teflon sample. The upper<br />
row shows the density distributions calculated as a cutout of free space, the lower row<br />
as a cutout of the infinite half-space. Note that the color axes are logarithmic. Assumed<br />
sample parameters: radius R = 20 mm, thickness L = 50 mm, diffusion coefficient<br />
D = 16500 m2 /s, absorption time τ = 3.3 ns, transport mean free path l ∗ = 220 µm,<br />
refractive index n = 1.35.<br />
47
5 Experiments<br />
space with the energy inside an absorbing cut-out of the half-space at the position of the<br />
sample:<br />
j with losses (⃗r ⊥ , t) = A(t) · j lossless (⃗r ⊥ , t) = j lossless (⃗r ⊥ , t) · ∫<br />
∫<br />
finite sample<br />
infinite half-space<br />
ρ τ (⃗r, t) d⃗r<br />
ρ τ→∞ (⃗r, t) d⃗r<br />
This ignores the fact that the photon density distribution is altered by the sample boundaries,<br />
but if the sample is large enough, only a few photons reach those boundaries anyway, so that<br />
the presence of the boundaries is of nearly no consequence for the density distribution.<br />
As solving eqns. 5.5 and 5.3 for the photon density distribution of the infinite half-space raises<br />
considerable algebraic and numeric difficulties, we fall back to the density distribution in free<br />
space, where the solution is easier to obtain. Fig. 5.2 shows that the albedos A(t) are similar<br />
enough to make this assumption, although the actual photon density distributions look quite<br />
different (fig. 5.3). The only difference in the albedos occurs when the photon cloud reaches<br />
the boundaries, which for the teflon sample in the figure happens at photon travel times<br />
around 10 to 100 ns. Even at this point the error is only a few percent, which presumably is<br />
not larger than the error that is already made by ignoring the altered density distribution.<br />
(5.5)<br />
Evaluating eqn. 5.5 gives the albedo<br />
∫ L<br />
1<br />
A(τ, L, R) =<br />
π(l ∗ +z 0 )<br />
0<br />
(√<br />
− K 0<br />
K 0<br />
(√<br />
4(l ∗ +z 0 ) 2 +z 2<br />
Dτ<br />
z 2<br />
Dτ<br />
)<br />
− 2K 0<br />
(√<br />
)<br />
+ 2K 0<br />
(√<br />
) (√<br />
R 2 +z 2<br />
Dτ<br />
+ K 0<br />
R 2 +4(l ∗ +z 0 ) 2 +z 2<br />
Dτ<br />
)<br />
− K 0<br />
(√<br />
)<br />
2R 2 +z 2<br />
Dτ<br />
−<br />
2R 2 +4(l ∗ +z 0 ) 2 +z 2<br />
Dτ<br />
)<br />
dz (5.6)<br />
where the K n (x) are modified Bessel functions of the second kind. The last step of integration<br />
is performed numerically.<br />
The transport mean free path l ∗ is the result of the evaluation of the backscattering data, for<br />
which the albedo is an input parameter. Therefore the correct value for l ∗ is not yet known<br />
when the albedo is calculated. Instead, we use l ∗ = 3D v<br />
≈ 3n effD<br />
c 0<br />
(see sec. 2.3), which should at<br />
least have the right order of magnitude.<br />
The resulting albedos for the samples examined in the backscattering experiments are given in<br />
tab. 5.1. One immediately observes the striking difference between the albedos of the titania<br />
samples, which are in excess of 99%, and the albedo of teflon, which is about 90%. The<br />
accuracy of the experiments therefore depends strongly on the accuracy of the teflon albedo.<br />
It is however nearly impossible to give an error for the calculated albedos, as the validity of the<br />
various assumptions is not known. Altogether, the error of the albedo mismatch is probably<br />
of the order of one or two percent.<br />
5.1.3 The correct theory of coherent backscattering<br />
The first steps towards a theoretical description of coherent backscattering have already been<br />
taken in sec. 2.1, where the intensity in a multiple scattering medium is presented as the<br />
48
5.1 Conservation of energy in coherent backscattering<br />
sample R [mm] L [mm] A (eqn. 5.4) A (eqn. 5.6)<br />
a-1 7.5 1.35 0.994 0.994<br />
a-2 7.5 1.03 0.995 0.995<br />
NIX-2 7.5 1.26 0.990 0.990<br />
NIX-3 7.5 1.75 0.992 0.992<br />
R700 (DuPont) 7.5 1.0 0.995 0.995<br />
R700 + gypsum 7.5 0.95 0.950 0.951<br />
R900 (DuPont) 7.5 0.5 0.994 0.994<br />
R902 (DuPont) 7.5 0.5 0.991 0.991<br />
S-25 7.5 0.5 0.993 0.993<br />
Teflon (D = 27500 m 2 /s) 20 50 0.901 0.890<br />
Teflon (D = 16500 m 2 /s) 20 50 0.922 0.918<br />
Teflon (D = 13300 m 2 /s) 20 50 0.930 0.927<br />
Ti-pure (Aldrich) 7.5 1.8 0.998 0.998<br />
Table 5.1: Albedos. The albedos for the samples with radius R and thickness L<br />
were calculated using eqns. 5.4 and 5.6. For other sample parameters see tab. 4.1 and<br />
sec. 4.2.2. The albedo of the teflon reference was calculated for the three different<br />
diffusion coefficients mentioned in sec. 4.2.2.<br />
solution of the wave equation by Green’s functions. This description would be exact, but<br />
it can not be solved for a system of millions and billions of randomly distributed particles,<br />
all of whose positions in addition would have to be known exactly. To turn it into a useful<br />
mathematical form, one therefore has to find a collective description of the multiple scattering<br />
medium by choosing exactly the right approximations.<br />
To describe multiple scattering, one must consider the products of Green’s functions corresponding<br />
to all possible scattering sequences. However, the coherent backscattering cone is<br />
observed in the averaged intensity, where most of the pairs of Green’s functions vanish, as<br />
their phase difference is large and random. In the so-called Drude-Boltzmann approximation,<br />
where the average over the product of two Green’s functions can be approximated by the<br />
product over the two averaged functions, the average intensity can therefore be written as<br />
∫∫ 〈Ψin<br />
I d (⃗r,⃗ r ′ ) ∝ (⃗r,⃗r 1 ) 〉 · 〈Ψ<br />
in(⃗r,⃗r ∗ 1 ) 〉 · Γ(⃗r 1 ,⃗r 2 ) · 〈G(⃗r<br />
2 ,⃗ r ′ ) 〉 · 〈G ∗ (⃗r 2 ,⃗ r ′ ) 〉 d⃗r 1 d⃗r 2<br />
where we introduce the structure factor or vertex function Γ(⃗r 1 ,⃗r 2 ), which takes into account<br />
all possible scattering sequences between the first and the last scattering site,⃗r 1 and⃗r 2 .<br />
One contribution that is not taken into account by the diffuson approximation are timeinverted<br />
photon paths, for which the phase difference also vanishes. It is given by<br />
∫∫ 〈Ψin<br />
I c (⃗r,⃗ r ′ ) ∝ (⃗r,⃗r 1 ) 〉 · 〈Ψ<br />
in(⃗r,⃗r ∗ 2 ) 〉 · Γ(⃗r 1 ,⃗r 2 ) · 〈G(⃗r<br />
2 ,⃗ r ′ ) 〉 · 〈G ∗ (⃗r 1 ,⃗ r ′ ) 〉 d⃗r 1 d⃗r 2<br />
In the limit of dilute systems with approximately spherical scatterers, where the Green’s<br />
functions are spherical waves and the transport mean free path is well described by the l ∗<br />
derived in sec. 2.3, this contribution is equivalent to the cooperon derived in sec. 2.7. The<br />
49
5 Experiments<br />
Figure 5.4: Contributions to the cooperon. The diagram on the left depicts the interference<br />
of two light waves running on time-inverted paths between ⃗r 1 and ⃗r 2 . The<br />
diagram on the right illustrates the light paths that include an additional scatterer in⃗r 0 .<br />
cooperon as defined above is often expressed in terms of the Hikami box H (A) as I c (⃗r) ∝<br />
∫∫ 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<br />
of the loop that is formed by direct and time-inverted path and is therefore also called<br />
a quantum crossing [10].<br />
The total power the cooperon distributes into the half-space above the sample, ∫ I c (θ) dΩ, is<br />
∝ 1/(kl ∗ ) 2 in leading order [9]. However, as Akkermans and Montambaux realized, there are<br />
two other contributions to the multiple scattered power which are of the same order in 1/(kl ∗ )<br />
and which therefore also must be taken into account. In contrast to the cooperon, where the<br />
interference occurs between time-inverted amplitudes, these additional contributions contain<br />
an additional scattering event in⃗r 0 (fig. 5.4).<br />
The additional contributions are defined analogous to the cooperon H (A) by the two dressed<br />
Hikami boxes H (B) and H (C) , which are complex conjugates:<br />
H (B) (⃗r,⃗ r ′ ,⃗r 1 ,⃗r 2 ) =<br />
∫ 〈Ψin<br />
= (⃗r,⃗r 1 ) 〉 · 〈Ψ<br />
in(⃗r,⃗r ∗ 0 ) 〉 · 〈G ∗ (⃗r 0 ,⃗r 2 ) 〉 · 〈G(⃗r<br />
2 ,⃗ r ′ ) 〉 · 〈G ∗ (⃗r 1 ,⃗r 0 ) 〉 · 〈G ∗ (⃗r 0 ,⃗ r ′ ) 〉 d⃗r 0<br />
Solving the integrals requires some mathematical ploys [9] and yields I (B+C)<br />
c ≈ −a/(kl ∗ ) 2 ·<br />
µ/(µ + 1), where µ = cos θ, and a is a fit parameter. Akkermans and Montambaux obtain<br />
a = 1.15 by fitting an intermediate result to experimental data <strong>from</strong> samples with wide<br />
backscattering cones [9, 19]. However, one can also use the fact that the cooperon together<br />
with the additional contributions now satisfies conservation of energy [8, 9, 10], so that<br />
∫<br />
I (A)<br />
c<br />
+ I (B+C)<br />
c d(⃗r − ⃗ r ′ ) = 0<br />
50
5.1 Conservation of energy in coherent backscattering<br />
1<br />
kl* = 2<br />
kl* = 4<br />
kl* = 10<br />
0.8<br />
cooperon<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
−90 −60 −30 0 30 60 90<br />
scattering angle [deg]<br />
Figure 5.5: Old and new theory. The graph shows the cooperon calculated with<br />
eqn. 2.15 (dashed lines) and including the additional contribution given in eqn. 5.7<br />
(solid lines). The dotted lines give the enhancement of the cooperon for the new theory<br />
(for the old theory the enhancement is always equal to 1). Parameters: wavelength<br />
λ = 590 nm, diffusion coefficient D = 15 m2 /s, reflectivity R = 0.5.<br />
The fit parameter a is therefore obtained by numerical integration of<br />
a =<br />
0<br />
∫ π/2<br />
0<br />
∫ π/2<br />
I (A)<br />
c<br />
1<br />
(kl ∗ ) 2 ·<br />
· sin θ dθ<br />
µ · sin θ dθ<br />
µ + 1<br />
After normalization analogous to eqns. 2.13, the additional contribution to the cooperon finally<br />
becomes<br />
α (B+C)<br />
c<br />
8π · a<br />
= −<br />
3 k 2 l ∗ (l ∗ + 2z 0 ) ·<br />
µ<br />
µ + 1<br />
(5.7)<br />
does not contribute at a specific angular value but it is rather spread out over the<br />
whole angular range, similar to the diffuson. In particular it reduces the enhancement of the<br />
coherent backscattering cone, so that the total cooperon α (A)<br />
c<br />
α (B+C)<br />
c<br />
+ α (B+C)<br />
c at the conetip at θ = 0<br />
51
5 Experiments<br />
is lower than 1 (fig. 5.5). At the wings of the backscattering cone an intensity cutback appears<br />
that balances the intensity enhancement of the cone.<br />
However, the effect of α (B+C)<br />
c is significant only for small kl ∗ . For kl ∗ 10 its influence is negligible,<br />
and the backscattering cone is well described by the cooperon α (A)<br />
c given in eqn. 2.15.<br />
In this regime the diffuson can be found <strong>from</strong> the large angle wings of the backscattering data<br />
[55]. For small kl ∗ the diffuson is hidden by the intensity cutback, and the proceeding devised<br />
above has to be used to extract the cooperon <strong>from</strong> the data.<br />
Small kl ∗ however means strongly scattering dense samples, where the scattering particles are<br />
in touching distance. In such systems, near-field effects become important for the scattering<br />
process. Strictly speaking, the random walk model developed for dilute systems and the<br />
deduction of the transport mean free path <strong>from</strong> the intensity distribution of Mie scattering<br />
and the anisotropy parameter 〈cos θ〉 are no longer valid. The kl ∗ obtained by fitting with the<br />
theory <strong>from</strong> Akkermans and Montambaux will therefore certainly have the correct order of<br />
magnitude, but might deviate slightly <strong>from</strong> the real value.<br />
5.1.4 The results of the test experiments<br />
To test the theoretical prediction made above, the coherent backscattering cones of various<br />
samples were measured with the wide angle setup. The diffusion coefficients lay in the range<br />
between 13 m2 /s and 250 m2 /s (see tab. 4.1), so that we expected to observe extremely wide<br />
backscattering cones with kl ∗ close to unity and a distinct intensity cutback as well as narrow<br />
cones with high kl ∗ where no cutback is visible.<br />
As the albedos of the titania samples are basically 1, the albedo mismatch of sample and<br />
reference is mainly given by the albedo of the teflon block. The latter was calculated with<br />
all three diffusion coefficients D = 27500 m2 /s, D = 16500 m2 /s, and D = 13300 m2 /s that can<br />
be derived <strong>from</strong> time of flight and small angle backscattering experiments, as discussed in<br />
sec. 4.2.2.<br />
The data quality of the wide angle backscattering experiments varies strongly, as the wide<br />
angle setup is extremely sensitive to parasitically scattered light. However, as stated above,<br />
the backscattering cone is evidentially caused by interference, so that the total amount of<br />
backscattered power must be the same both for the coherent and the incoherent addition of<br />
∫ π/2<br />
0<br />
a c (θ) sin θ dθ dφ of the cooperon is<br />
the backscattered intensity, and the integral E = ∫ 2π<br />
0<br />
necessarily zero. E can therefore be used as a criterion to judge the quality of the experimental<br />
data.<br />
Furthermore, it is also possible to substantiate the method to calculate the albedo that was<br />
developed earlier. In fig. 5.6 one observes that the data are approximately centered at E = 0<br />
if the albedo is calculated with D = 16500 m2 /s and D = 13300 m2 /s, but are shifted to E ≈<br />
−0.1 sterad for an albedo calculated with D = 27500 m2 /s. Therefore the data on average<br />
suggest conservation of energy for precisely those values of the diffusion coefficient which<br />
are obtained <strong>from</strong> the small angle measurements (see sec. 5.2.2). This strongly indicates that<br />
the values for the albedos are correct, although as proof further experiments on samples with<br />
other albedos than teflon or titania are necessary.<br />
52
5.1 Conservation of energy in coherent backscattering<br />
7<br />
6<br />
5<br />
#<br />
4<br />
3<br />
2<br />
1<br />
0<br />
−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<br />
integrated cooperon E [sterad]<br />
7<br />
6<br />
5<br />
#<br />
4<br />
3<br />
2<br />
1<br />
0<br />
−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<br />
integrated cooperon E [sterad]<br />
7<br />
6<br />
5<br />
#<br />
4<br />
3<br />
2<br />
1<br />
0<br />
−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<br />
integrated cooperon E [sterad]<br />
Figure 5.6: Distribution of backscattered energies. The distribution of backscattered<br />
light energies E centers around E ≈ 0 if D = 13300 m2 /s (top) and D = 16500 m2 /s<br />
(center) are used to calculate the albedo of teflon. For D = 27500 m2 /s (bottom) the<br />
distribution is clearly shifted away <strong>from</strong> zero, so that conservation of energy would not<br />
be obeyed in this case. Consequently, for all further evaluations the diffusion coefficient<br />
obtained <strong>from</strong> the small angle experiments, D = 13300 m2 /s, is used.<br />
53
5 Experiments<br />
1<br />
0.8<br />
α c<br />
(A)<br />
α c<br />
(A) + αc<br />
(B+C)<br />
R700 with ℵ=1<br />
R700 with ℵ=0.92<br />
cooperon<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
0 10 20 30 40 50 60 70 80 90<br />
scattering angle [deg]<br />
1<br />
0.8<br />
α c<br />
(A)<br />
α c<br />
(A) + αc<br />
(B+C)<br />
NIX−2 with ℵ=1<br />
NIX−2 with ℵ=0.93<br />
cooperon<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
0 10 20 30 40 50 60 70 80 90<br />
scattering angle [deg]<br />
Figure 5.7: Example wide angle backscattering data. Top: R700, bottom: NIX-2. If the<br />
different albedos of sample and reference are neglected (red), the data clearly violate<br />
conservation of energy and do not fit with any theory. The correctly evaluated data<br />
(orange) deviate significantly <strong>from</strong> α (A)<br />
c , but fit very well with α c = α (A)<br />
c + α (B+C)<br />
c within<br />
the region of confidence (which is only marginally larger than the data points and is<br />
therefore not drawn explicitly). For R700 one obtains kl ∗ = 2.7(3) and l ∗ = 250(30) nm,<br />
and kl ∗ = 6.4(3) and l ∗ = 600(30) nm for NIX-2. The integration over the cooperon<br />
yields E = −0.024(5) sterad for R700 and E = −0.013(7) sterad for NIX-2 if the error of<br />
the albedo is set to zero. E is zero within the margins of error if the error of the albedo<br />
is 1%.<br />
54
5.1 Conservation of energy in coherent backscattering<br />
0.8<br />
0.6<br />
0.4<br />
cooperon<br />
1<br />
0.5<br />
0<br />
theory<br />
R700 data<br />
0 30 60 90<br />
scattering angle [deg]<br />
a−1<br />
a−2<br />
NIX−2<br />
NIX−3<br />
R700<br />
R700+gypsum<br />
R900<br />
R902<br />
S−25<br />
Ti−pure<br />
approx. power ∫ α (A) dΩ<br />
c<br />
integrated cooperon E [sterad]<br />
0.2<br />
0<br />
−0.2<br />
−0.4<br />
−0.6<br />
cooperon<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
theory<br />
R700 data<br />
cooperon<br />
0.4<br />
0.2<br />
0<br />
theory<br />
S−25 data<br />
−0.2<br />
0 30 60 90<br />
scattering angle [deg]<br />
−0.1<br />
0 30 60 90<br />
scattering angle [deg]<br />
−0.8<br />
0 2 4 6 8 10 12 14 16 18 20<br />
kl*<br />
Figure 5.8: Distribution of backscattered energies. The quality of the data varies<br />
strongly. For the evaluation only data sets with E ≈ 0 can be used; data sets where<br />
the backscattered power differs strongly <strong>from</strong> zero do not obey conservation of energy<br />
and are distorted by experimental problems, as demonstrated by the insets. For the<br />
errorbars an error of ±1% of the albedo mismatch was assumed in addition to the<br />
measured variation of the data.<br />
55
5 Experiments<br />
As it seems to be the most well-founded value, all further evaluations were performed with<br />
the diffusion coefficient obtained <strong>from</strong> the small angle experiments, D = 13300 m2 /s. On good<br />
data sets with E ≈ 0 one can then observe the anticipated intensity cutback at the wings of the<br />
backscattering cone which balances the intensity enhancement of the cone (fig. 5.7). Especially<br />
for small kl ∗ the measured data therefore deviate considerably <strong>from</strong> α (A)<br />
c , which earlier was<br />
used to fit the backscattering data [22, 23, 47]. In contrast, the agreement with α (A)<br />
c + α (B+C)<br />
c is<br />
nearly perfect, except for a slight drop in the measured data at scattering angles close to 90 ◦ ,<br />
which however suspiciously looks as if it was caused by some technical problem in the setup.<br />
These findings confirm that the improved theory developed by E. Akkermans and G. Montambaux<br />
can indeed be applied to describe coherent backscattering, and conversely allow to<br />
judge the experimental data not only by conservation of energy, but also by their agreement<br />
with the theoretical curve. A combination of both criteria should hereby yield the best results,<br />
as good agreement with the theory can also be found for E that deviate somewhat <strong>from</strong> zero,<br />
and on the other hand even a few measurements with E ≈ 0 deviate <strong>from</strong> the theory (fig. 5.8).<br />
5.2 The coherent backscattering cone in high resolution<br />
The high resolution backscattering setup introduced in sec. 3.3 was developed as a possible<br />
means to investigate Anderson localization. The photons that are trapped on closed loops<br />
in the photon paths not only lead to a different path length distribution in time-resolved<br />
transmission experiments, but also change the shape of the coherent backscattering cone. This<br />
is because localization strongly reduces the number of photons emerging <strong>from</strong> long photon<br />
paths, which leads to a rounding of the tip of the backscattering cone similar to that caused<br />
by absorption (see sec. 2.7). At the same time the setup was built as a means to measure the<br />
transport mean free path of samples with high kl ∗ and consequently extremely narrow cones,<br />
which of course have other applications than the search for Anderson localization.<br />
5.2.1 Tip rounding of the coherent backscattering cone<br />
In time of flight experiments the titania powder R700 has shown an onset of localization<br />
[5, 6, 48], so that it was our first choice to test if the small angle setup is applicable in the<br />
search for Anderson localization. If the setup is indeed able to resolve the effect of localization,<br />
the measured rounding of the cone tip will deviate <strong>from</strong> the theoretical prediction for pure<br />
absorption given in eqn. 2.18. If on the other hand the rounding due to localization can not be<br />
observed, this is a strong indication that the concept of the small angle setup is not suitable to<br />
detect localization effects.<br />
To evaluate the small angle backscattering data of R700, the procedure proposed in sec. 3.3<br />
has to be altered a little: The backscattering cone of the titania powder is too wide to directly<br />
find the backscattering direction – the white region in the binary image is too frayed as that<br />
the center of mass could reliably identify the tip of the backscattering cone. Instead, the<br />
backscattering direction is derived <strong>from</strong> a teflon reference measurement, where the conetip<br />
can be found quite precisely (fig. 5.9). As the direction θ = 0 is determined only by the<br />
56
5.2 The coherent backscattering cone in high resolution<br />
2048<br />
3.7<br />
2048<br />
x 10 4<br />
4.4<br />
3.6<br />
4.2<br />
1024<br />
3.5<br />
3.4<br />
3.3<br />
3.2<br />
x 10 4 1 1024 2048<br />
1024<br />
4<br />
3.8<br />
3.6<br />
3.1<br />
3.4<br />
1<br />
1 1024 2048<br />
3<br />
1<br />
3.2<br />
Figure 5.9: CCD images of R700 and teflon. As the backscattering cone of R700 is very<br />
wide, it is not possible to find the backscattering direction θ = 0 <strong>from</strong> the CCD image<br />
(left). Instead, a teflon reference (right) has to be used, which gives the same position<br />
for the conetip if the optical path is not readjusted between the measurements.<br />
incoming laser beam and not by the sample, the backscattering directions of the titania sample<br />
and the teflon reference are identical.<br />
The resulting coherent backscattering cone of R700 is depicted in fig. 5.10. Although the<br />
measured data do not contradict the theory for coherent backscattering, it is obvious that it<br />
is impossible to derive more than the vague statement that the conetip is not triangular but<br />
somehow rounded. The data are much too noisy to draw any further conclusions.<br />
This disappointing result is however easy to explain. The small angle setup requires to have a<br />
lot of optical components between the sample and the camera. As none of these are ideal, they<br />
all scatter and give rise to the slightly inhomogenous and noisy background illumination that<br />
can be observed on the CCD images. This noise can not even be removed by the azimuthal<br />
average, as in the center the data are averaged over a comparatively low number of pixels.<br />
Therefore even low-intensity speckles or other disturbances are enough to hide the effect we<br />
are looking for.<br />
Altogether, it must be stated that the small angle setup in its present form can provide no<br />
information about Anderson localization. For investigations about this phenomenon other<br />
experiments like time of flight measurements have proven to be a lot more suitable [5, 6, 48].<br />
5.2.2 The transport mean free path of weakly scattering samples<br />
In terms of intensity resolution, measuring the transport mean free path of weakly scattering<br />
samples with large kl ∗ with the small angle setup is less challenging. The mean free path<br />
of teflon, our standard reference sample, can be estimated <strong>from</strong> eqn. 2.7 to be something<br />
like 220 µm, if the velocity of energy transport is given by the speed of light in a material<br />
57
5 Experiments<br />
3.65 x 104 scattering angle [deg]<br />
3.6<br />
theory (absorption)<br />
data average<br />
3.55<br />
intensity [a.u.]<br />
3.5<br />
3.45<br />
3.4<br />
3.35<br />
3.3<br />
3.25<br />
0 0.2 0.4 0.6 0.8 1<br />
3.62 x 104 scattering angle [deg]<br />
3.61<br />
theory (absorption)<br />
data average<br />
3.6<br />
intensity [a.u.]<br />
3.59<br />
3.58<br />
3.57<br />
3.56<br />
3.55<br />
3.54<br />
0 0.05 0.1 0.15 0.2<br />
Figure 5.10: Tip of the R700 cone. The azimuthal average of the backscattering data<br />
(bottom: detail) was fitted with the theory for an absorbing R700 sample (D = 15 m2 /s,<br />
τ = 2 ns, l ∗ = 250 nm, λ = 575 nm). The line width gives the approximate order of<br />
magnitude of the intensity difference between a localizing and a non-localizing sample.<br />
58
5.2 The coherent backscattering cone in high resolution<br />
intensity [a.u.]<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
1 mm<br />
2 mm<br />
3 mm<br />
4 mm<br />
5 mm<br />
6 mm<br />
7 mm<br />
9 mm<br />
10 mm<br />
11 mm<br />
12 mm<br />
13 mm<br />
14 mm<br />
15 mm<br />
16 mm<br />
0<br />
0 0.02 0.04 0.06 0.08 0.1<br />
scattering angle [deg]<br />
normalized intensity<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
1 mm<br />
2 mm<br />
3 mm<br />
4 mm<br />
5 mm<br />
6 mm<br />
7 mm<br />
9 mm<br />
10 mm<br />
11 mm<br />
12 mm<br />
13 mm<br />
14 mm<br />
15 mm<br />
16 mm<br />
0<br />
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04<br />
scattering angle [pixels]<br />
Figure 5.11: Focus accuracy. The position of the focussing lens was varied in a range<br />
of 16 mm around its nominal position 50 cm in front of the CCD camera. The lens<br />
is positioned correctly when the width of the teflon backscattering cone (top) and the<br />
mirror spot (bottom) is minimal, i.e. between 7 and 10 mm. For higher numbers the lens<br />
is too close to the camera, for lower numbers it too far away. (To make the measurements<br />
easier to compare the data were scaled to have intensities between 0 and 1.)<br />
59
5 Experiments<br />
220<br />
200<br />
180<br />
l* [µm]<br />
160<br />
140<br />
120<br />
100<br />
2 4 6 8 10 12 14 16<br />
lens position [mm]<br />
Figure 5.12: Transport mean free path in dependence of the lens position. Between<br />
7 and 10 mm the transport mean free path of teflon has a slight maximum of approximately<br />
180 µm, which can be clearly derived <strong>from</strong> the trend of the data. (The orange<br />
line was drawn as a guide to the eye.)<br />
with the refractive index of teflon, n = 1.35, and the diffusion coefficient is D = 16500 m2 /s,<br />
as suggested by the time of flight experiments (see sec. 4.2.2). With eqn. 2.16 one obtains<br />
a FWHM of about 0.01 ◦ , which corresponds to 12 pixels on the CCD chip. [47, 49] c report<br />
experimentally determined cone widths of about 0.03 ◦ , which would correspond to 35 pixels<br />
in our setup.<br />
This however is not that much wider than the intensity profile of the focus spot (see sec. 3.3.4),<br />
which therefore will have a considerable influence on the shape of the intensity enhancement.<br />
Consequently, the measured backscattering cone of teflon can be expected to be significantly<br />
widened, and the convolution of the fitted theory with the focus spot profile will become<br />
essential to yield correct results.<br />
The accuracy of the setup is for the most part determined by the divergence of the incoming<br />
laser beam (which should be as low as possible) and the correct position of the focussing lens<br />
in front of the camera. The former seems to be comparatively uncritical, as the beam can easily<br />
be adjusted to have a divergence lower than ±0.01 ◦ , which is the range where no significant<br />
change in the cone shape seems to occur.<br />
The appropriate distance between lens and camera is more difficult to determine. Fig. 5.12<br />
shows that the fitted values for the transport mean free path can fluctuate strongly, even<br />
if the lens–camera distance is set correctly. Hence it is reasonable to measure a series of<br />
backscattering cones where the setting is deliberately varied slightly for each take. The correct<br />
[c] In [49] there is obviously a typing error at this point.<br />
60
5.2 The coherent backscattering cone in high resolution<br />
1<br />
0.8<br />
l* = 175µm<br />
l* = 180µm<br />
l* = 185µm<br />
l* = 220µm<br />
teflon data<br />
intensity [a.u.]<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16<br />
scattering angle [deg]<br />
Figure 5.13: <strong>Coherent</strong> backscattering cone of teflon. The backscattering data can be<br />
fitted with a cone (i.e. eqn. 2.18) with l ∗ around 180 µm. From eqn. 2.7 one would expect<br />
l ∗ = 220 µm, which is only slightly narrower.<br />
value of l ∗ can then be read <strong>from</strong> the trend of the data, even if these contain some outliers that<br />
could otherwise be quite confusing.<br />
The reason for this tendency to produce unreliable data is that the intensity profile of the<br />
mirror reflex is rendered by extremely few data points. The convolution of mirror profile and<br />
theoretical cone shape, which is used to fit the data, is therefore prone to slight variations<br />
in the slope of the backscattering cone, which result in apparent variations of l ∗ . Although<br />
one instinctively tries to make the focus spot as small as possible to enhance the angular<br />
resolution, it is actually more preferable to go for larger focus spots, which are rendered more<br />
accurately. If the information about the intensity profile of the focus spot is incorporated in<br />
the data evaluation afterwards, this actually does not reduce the angular resolution of the<br />
experiment.<br />
For the teflon sample the data suggest l ∗ ≈ 180 µm, which is consistent with the results<br />
obtained with an older version of the setup [19, 47]. It is also not much lower than the<br />
220 µm that one calculates <strong>from</strong> eqn. 2.7 using the diffusion coefficient D = 16500 m2 /s. An<br />
estimate with D = 27500 m2 /s on the other hand would give a transport mean free path of<br />
about 370 µm, which agrees neither with the old nor the new experimental results. From<br />
the measured transport mean free path one can derive a diffusion coefficient D = 13300 m2 /s,<br />
which is probably the most correct of the three values, although it was not possible to confirm<br />
it in time of flight experiments for technical reasons.<br />
61
5 Experiments<br />
Altogether the teflon backscattering measurements confirm that the newly revised small angle<br />
setup yields reliable results for the transport mean free paths of weakly scattering samples.<br />
The difficulty is rather to properly determine the diffusion coefficient that enters in eqn. 2.7<br />
<strong>from</strong> which the expected value of l ∗ is calculated. The problem herein is a technical one, as<br />
only samples of a certain size fit into the time of flight setup. If in the future the diffusion<br />
coefficients of other weakly scattering samples are to be measured, an appropriate adaption<br />
of the setup should be considered.<br />
5.2.3 <strong>Coherent</strong> backscattering on fluidized beds<br />
Parallel to the experiments on weakly scattering solid samples like teflon, another field of application<br />
for the small angle setup arose <strong>from</strong> a cooperation with the group of Dr. M. Schröter<br />
of the Max Planck Institute (MPI) for Dynamics and Self-Organization in Göttingen. Working<br />
on statistical mechanics of granular media, their interest lies in the granular temperature of<br />
fluidized beds, which is proportional to the average speed of the particles.<br />
The group uses diffusing wave spectroscopy (DWS) to measure the typical timescale of the<br />
movements of the particles in the bed. To obtain a speed, they also needed a length scale for<br />
the transport. This scale is given by the transport mean free path l ∗ , which can for example<br />
be measured in coherent backscattering experiments.<br />
Setup modifications<br />
The small angle setup (fig. 3.5) was originally designed to take up the backscattering <strong>from</strong><br />
the horizontal top surface of the sample. This is not possible for fluidized beds, which have<br />
to be accessed <strong>from</strong> the side through the container walls (fig. 4.5), so that the setup has to be<br />
modified accordingly. Here it is important to position the circular polarizer directly in front of<br />
the container to avoid the polarization to be altered by additional optical components between<br />
beamsplitter and bed. As the light now transmits the container walls before and after being<br />
scattered in the fluidized bed, one has to take into account the refraction and reflection at the<br />
acrylic glass surfaces.<br />
In multiple scattering, the transition <strong>from</strong> the fluidized bed to the acrylic glass becomes noticeable<br />
in the average penetration depth z 0 = 2/3 · (1 + R)/(1 − R) · l ∗ , where R is the reflectivity<br />
of the sample–glass surface.<br />
In addition, the scattering angle θ CCD measured at the CCD camera has to be corrected for<br />
the refraction at the glass–air surface of the container (fig. 5.16) to be compared with the<br />
theoretical description of the backscattering cone:<br />
( )<br />
nair<br />
θ ms = arcsin · sin θ CCD ≈ n air<br />
· θ CCD (5.8)<br />
n glass n glass<br />
Although the light passes a circular polarizer before and after scattering, the ring structure of<br />
single scattering at Mie particles can also be found on the backscattering images (fig. 5.14).<br />
62
5.2 The coherent backscattering cone in high resolution<br />
1<br />
3.7<br />
3.6<br />
x 10 4 0 0.2 0.4 0.6 0.8 1<br />
15 x 107 scattering angle [deg]<br />
before polarizer<br />
after polarizer<br />
1024<br />
3.5<br />
3.4<br />
intensity [a.u.]<br />
10<br />
5<br />
3.3<br />
2048 3.2<br />
1 1024 2048<br />
0<br />
Figure 5.14: Single and multiple scattering. The CCD image (left) of the backscattering<br />
of a fluidized bed shows a pronounced ring structure caused by single scattering.<br />
Calculations with Mie theory (see fig. 5.15) show, that for circularly polarized light the<br />
central maximum is extinguished by a transition through a circular polarizer after the<br />
scattering event (right; the graph was calculated using the MATLAB code <strong>from</strong> the appendix).<br />
The backscattering image however still shows an intensity maximum at the<br />
center of the ring structure, which therefore must be the coherent backscattering cone.<br />
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛<br />
⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞<br />
S 11 + S 33<br />
1 −1 0 0 1 0 0 0 S 11 S 12 0 0 1 0 0 0 1 −1 0 0 1<br />
p = ⎜−S 11 − S 33<br />
⎟<br />
⎝ 0 ⎠ = 1 ⎜−1 1 0 0<br />
⎟<br />
2 ⎝ 0 0 0 0⎠ ·<br />
⎜0 0 0 1<br />
⎟<br />
⎝0 0 1 0⎠ ·<br />
⎜S 12 S 11 0 0<br />
⎟<br />
⎝ 0 0 S 33 S 34 ⎠ ·<br />
⎜0 0 0 1<br />
⎟<br />
⎝0 0 1 0⎠ · 1<br />
⎜−1 1 0 0<br />
⎟<br />
2 ⎝ 0 0 0 0⎠ ·<br />
⎜−1<br />
⎟<br />
⎝ 0 ⎠<br />
0<br />
0 0 0 0 0 −1 0 0 0 0 −S 34 S 33 0 −1 0 0 0 0 0 0 0<br />
p = LP · QWP · S · QWP · LP · p 0<br />
p S = S · QWP · LP · p 0<br />
⎛ ⎞ ⎛<br />
⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞<br />
S 11 S 11 S 12 0 0 1 0 0 0 1 −1 0 0 1<br />
p S = ⎜ S 12<br />
⎟<br />
⎝−S 34 ⎠ = ⎜S 12 S 11 0 0<br />
⎟<br />
⎝ 0 0 S 33 S 34 ⎠ ·<br />
⎜0 0 0 1<br />
⎟<br />
⎝0 0 1 0⎠ · 1<br />
⎜−1 1 0 0<br />
⎟<br />
2 ⎝ 0 0 0 0⎠ ·<br />
⎜−1<br />
⎟<br />
⎝ 0 ⎠<br />
−S 33 0 0 −S 34 S 33 0 −1 0 0 0 0 0 0 0<br />
Figure 5.15: Calculation of the Stokes vector for the experiment. Linearly polarized<br />
light with Stokes vector p 0 transmits a circular polarizer, which is represented by a<br />
combination of a linear polarizer LP and a quarter-wave-plate QWP, is scattered at the<br />
particle S, and again transmits the circular polarizer. The resulting Stokes vector p<br />
can be compared with the Stokes vector p S directly after the scatterer. The scattered<br />
intensities depicted in fig. 5.14 are given by the first elements of p and p S .<br />
63
5 Experiments<br />
Figure 5.16: Correction of the scattering angle. Left: Multiply scattered light <strong>from</strong> the<br />
medium with effective refractive index n eff enters the container wall (refractive index<br />
n glass ) with scattering angle θ ms . At the container–air interface this light is refracted. The<br />
scattering angle measured at the CCD camera is therefore θ CCD . Right: Singly scattered<br />
light <strong>from</strong> a particle with refractive index n particle in water is refracted at both surfaces<br />
of the container walls.<br />
The reason for this at first surprising observation is that the polarizer removes only the central<br />
maximum of the backscattered intensity distribution around θ = 0, which for the large<br />
particles in the fluidized bed is only about 0.2 ◦ wide.<br />
The singly scattered light is refracted at both surfaces of the acrylic glass wall of the container<br />
(fig. 5.16). The scattering angle at the CCD camera and the original scattering angle θ ss at the<br />
particle are therefore related like<br />
( )<br />
nair<br />
θ ss = arcsin · sin θ CCD ≈<br />
n water<br />
n air<br />
n water<br />
· θ CCD (5.9)<br />
This means that the signal at the CCD camera contains two superimposed contributions,<br />
which require different corrections to obtain the real scattering angles!<br />
The contribution of single scattering<br />
As explained before, the circular polarizer in the setup removes the central maximum of single<br />
scattering, but leaves the rest of its intensity pattern more or less unaltered. If the refractive<br />
indices of particles and water and the size distribution of the particles are known exactly, the<br />
intensity distribution of single scattering can be calculated using Mie theory (fig. 5.15), and<br />
can then be subtracted <strong>from</strong> the data.<br />
The background illumination of the backscattering image however is rather non-uniform and<br />
can therefore not be removed completely. This makes it an extremely delicate task to choose<br />
the correct parameters for the Mie distribution to subtract (fig. 5.17). We adapt the Mie distribution<br />
to the first order maximum and minimum of the azimuthally averaged data, where<br />
the variance of the data is still comparatively low.<br />
64
5.2 The coherent backscattering cone in high resolution<br />
intensity [a.u.]<br />
4000<br />
3000<br />
2000<br />
1000<br />
data profiles (f = 43%)<br />
data average (f = 43%)<br />
n water<br />
= 1.334 , n particle<br />
= 1.519 .<br />
n water<br />
= 1.334 , n particle<br />
= 1.520<br />
n water<br />
= 1.334 , n particle<br />
= 1.521<br />
n water<br />
= 1.334 , n particle<br />
= 1.522<br />
n water<br />
= 1.334 , n particle<br />
= 1.523<br />
n water<br />
= 1.333 , n particle<br />
= 1.519<br />
n water<br />
= 1.333 , n particle<br />
= 1.520<br />
n water<br />
= 1.333 , n particle<br />
= 1.521<br />
n water<br />
= 1.333 , n particle<br />
= 1.522<br />
n water<br />
= 1.333 , n particle<br />
= 1.523<br />
0<br />
−1000<br />
−2000<br />
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8<br />
scattering angle θ CCD<br />
[deg]<br />
Figure 5.17: Measured data and calculated Mie distributions. Due to non-uniform<br />
background lighting the data vary strongly for angles larger than 0.3 ◦ and can not be<br />
fitted properly. To get a reliable scaling, the Mie distributions were scaled to fit the first<br />
order minimum and maximum of the azimuthally averaged data, whose heights are<br />
marked in the graph. The first maximum of the calculated Mie distributions is shifted<br />
towards smaller scattering angles, which shows that the particle size distribution used<br />
for the calculations is slightly wrong.<br />
65
5 Experiments<br />
1<br />
0.8<br />
data − ( n water<br />
= 1.334 , n particle<br />
= 1.519 ) .<br />
data − ( n water<br />
= 1.334 , n particle<br />
= 1.521 )<br />
data − ( n water<br />
= 1.334 , n particle<br />
= 1.523 )<br />
intensity [a.u.]<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
0 0.05 0.1 0.15 0.2<br />
scattering angle θ ms<br />
[deg]<br />
Figure 5.18: <strong>Coherent</strong> backscattering cone of a fluidized bed. To obtain the coherent<br />
backscattering cone, three Mie distributions (stretched along the angular axis to match<br />
the slope of the first maximum of the data) with different zero levels were subtracted<br />
<strong>from</strong> the measured intensity distribution depicted in fig. 5.17. The angular width of the<br />
cone can be read <strong>from</strong> the graph to an accuracy of about 15%.<br />
The graph also shows that the actual distribution of particle sizes in the sample differs slightly<br />
<strong>from</strong> the one measured beforehand – either because the size distribution in the fluidized bed is<br />
not homogenous or because the size distribution changed during the experiments – as the first<br />
maxima of measured data and Mie theory do not fit. For the evaluation the Mie distribution<br />
was therefore stretched to match the measured intensity distribution. This is acceptable as the<br />
intensity characteristics are always the same at small angles; however, the zero level of the Mie<br />
distribution varies with the input parameters, so that the width of the backscattering cone is<br />
known only to an accuracy of about 15% (fig. 5.18).<br />
The transport mean free path of a fluidized bed<br />
The first backscattering experiments were performed on a fluidized bed of water and soda<br />
lime glass spheres with particle diameters around 150 µm. The transport mean free path was<br />
expected to be proportional to the distance between the particles in the fluidized bed. So the<br />
backscattering was measured for different flux rates in the bed, resulting in particle volume<br />
fractions between 39% and 58% (fig. 5.19). However, due to strong fluctuations in the data no<br />
real trend in the width of the backscattering cone can be observed – which does not necessarily<br />
mean that there is none. Here additional speckles <strong>from</strong> static particles stuck at the container<br />
surface do have a rather disadvantageous impact.<br />
66
5.2 The coherent backscattering cone in high resolution<br />
3.25 x 104 scattering angle θ CCD<br />
[deg]<br />
3.2<br />
3.15<br />
3.1<br />
f = 58%<br />
f = 49%<br />
f = 47%<br />
f = 43%<br />
f = 39%<br />
intensity [a.u.]<br />
3.05<br />
3<br />
2.95<br />
2.9<br />
2.85<br />
2.8<br />
0 0.05 0.1 0.15 0.2 0.25 0.3<br />
Figure 5.19: <strong>Backscattering</strong> for various volume fractions. The transport mean free path<br />
is expected to be proportional to the volume fraction of the fluidized bed. However, due<br />
to strong fluctuations in the data no real trend in the width of the backscattering cone<br />
can be observed.<br />
For the fluidized bed with f = 43% one reads <strong>from</strong> fig. 5.18 and with eqn. 2.16 a transport<br />
mean free path between 50 µm and 60 µm. This would mean that l ∗ is significantly shorter<br />
than the diameter of the particles in the fluidized bed, and it stands in strong contrast to<br />
reported mean free paths <strong>from</strong> similar experiments (e.g. [52, 17, 39, 41, 46, 51, 57]), which are<br />
all of the order of several particle diameters.<br />
As the present data are only first results and the experiments are still ‘work in progress’,<br />
there are all kinds of possible explanations for this surprising result, of both theoretical and<br />
experimental nature. These will have to be discussed and investigated if further experiments<br />
on fluidized beds or similar systems are planned for the future.<br />
67
68<br />
5 Experiments
6 Summary<br />
The focus of the work presented in this thesis lay on improvements of the experimental techniques<br />
for the investigation of multiple scattering phenomena, and in particular the coherent<br />
backscattering cone of visible light.<br />
This interference effect had been found to apparently violate the principle of conservation of<br />
energy: The total amount of backscattered energy can not exceed the energy which the light<br />
source inserts into the sample. The intensity enhancement of the cone should therefore be balanced<br />
by a corresponding intensity cutback, which however had never been observed neither<br />
in experiment nor in theory. Inaccuracies in the latter two being the only possible explanation,<br />
we reviewed our backscattering experiments and the evaluation of the backscattering<br />
data, and also tested an improved theory which had been worked out by E. Akkermans and<br />
G. Montambaux.<br />
The two theoreticians extended the theoretical description of the backscattering cone by two<br />
additional terms that contribute to the total backscattered energy in the same order as the<br />
cone itself. They lead to a cutback of the scattered intensity below the level of the incoherent<br />
addition of the backscattered intensity which balances the intensity enhancement of the cone.<br />
In the new theoretical description of the backscattering cone the energy is therefore conserved.<br />
The key to a precise measurement of the shape of the coherent backscattering cone was the<br />
correct calibration of the photodiodes of the wide angle setup. For this a teflon reference<br />
sample is used, the albedo of which differs <strong>from</strong> the titania albedo by about 10%. In earlier<br />
experiments this difference had been neglected, which led to a misrepresentation of the<br />
backscattering cone, so that conservation of energy seemed to be seriously violated. If however<br />
the albedos of sample and reference are considered in the evaluation, one can observe<br />
an intensity cutback at the wings of the cone, which balances the intensity enhancement in<br />
backscattering direction, and which also agrees with the predictions made by Akkermans and<br />
Montambaux.<br />
A second part of this work concerned the remodeling of the small angle setup for the measurement<br />
of the scattered intensity distribution in a small range around backscattering direction.<br />
One goal was to detect Anderson localization by the rounding of the tip of the coherent<br />
backscattering cone. For this an older CCD camera was replaced by a high resolving model,<br />
which improves the maximum intensity resolution by more than two orders of magnitude and<br />
the maximum angular resolution or respectively the maximum angular range by a factor of<br />
three. However, it turned out that the optical components between sample and camera cause<br />
too much extraneous light, so that the necessary intensity resolution can not be achieved.<br />
Still, the improved setup can be used to measure the transport mean free path of weakly<br />
scattering samples like for example teflon. For teflon one can measure a cone with FWHM ≈<br />
0.03 ◦ , corresponding to a transport mean free path l ∗ = 180 µm and a diffusion coefficient
6 Summary<br />
D = 13300 m2 /s. This agrees pretty well with results of measurements with the old setup and<br />
with the results of the wide angle experiments, which yield correct results if the diffusion<br />
coefficient obtained with the small angle setup is used for the evaluation. With these findings<br />
the reliability of the results of the small angle setup is confirmed.<br />
Another application for the small angle setup was to determine the transport mean free path<br />
of fluidized beds. As the scattering particles in these experiments are rather large and have a<br />
regularly spherical shape, the intensity distribution of the backscattered light is superimposed<br />
by the ring structure of the single scattering on the Mie particles. This structure can however<br />
be calculated theoretically and adapted to the measured curves, so that the backscattering<br />
cone can be extracted <strong>from</strong> the data.<br />
In first measurements the width of the backscattering cone led to a transport mean free path<br />
that was significantly shorter than the particle diameter of the scatterers. This is in strong<br />
contrast to the results of similar experiments, which are reported to be of the order of several<br />
particle diameters. The reason for this is still under discussion; however, the basic procedure<br />
for the evaluation of the backscattering data of fluidized beds was tested successfully.<br />
For future experiments, the combination of the two revised backscattering setups offers the<br />
unique possibility to measure kl ∗ over more than three decades, <strong>from</strong> samples at the transition<br />
to Anderson localization at kl ∗ ≈ 1 to weakly scattering samples with extremely narrow<br />
backscattering cones. Possible applications range <strong>from</strong> experiments on new, custom designed<br />
particles with small kl ∗ to measurements on foams or biological tissue. Some of these experiments<br />
are already planned for the near future, and will hopefully provide a step further<br />
towards fully understanding multiple scattering.<br />
70
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74
Figures and Tables<br />
Figures<br />
<strong>Backscattering</strong> of a fluidized bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .<br />
Title<br />
2.1 Coordinate system of single scattering . . . . . . . . . . . . . . . . . . . . . . . . 5<br />
2.2 Vibration ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7<br />
2.3 Polarization of a scattered wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />
2.4 <strong>Scattering</strong> cross section and scattering anisotropy . . . . . . . . . . . . . . . . . 9<br />
2.5 Random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />
2.6 <strong>Scattering</strong> in the presence of a single boundary . . . . . . . . . . . . . . . . . . 12<br />
2.7 <strong>Scattering</strong> in the presence of two parallel boundaries . . . . . . . . . . . . . . . 13<br />
2.8 Radiative transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />
2.9 Photon flux in backscattering geometry . . . . . . . . . . . . . . . . . . . . . . . 16<br />
2.10 Speckles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18<br />
2.11 Theorem of reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19<br />
2.12 Diffuson and cooperon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21<br />
2.13 <strong>Coherent</strong> backscattering with absorption . . . . . . . . . . . . . . . . . . . . . . 22<br />
2.14 <strong>Coherent</strong> backscattering with absorption – unnormalized cooperon . . . . . . . 23<br />
3.1 Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25<br />
3.2 Wide angle setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26<br />
3.3 Polarizer effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27<br />
3.4 Conetip rounding <strong>from</strong> absorption and localization . . . . . . . . . . . . . . . . 29<br />
3.5 Small angle setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30<br />
3.6 The optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31<br />
3.7 Sample shaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Figures and Tables<br />
3.8 Time of flight setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33<br />
4.1 Particle size and polydispersity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36<br />
4.2 Refractive index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37<br />
4.3 Reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38<br />
4.4 Titania and teflon samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40<br />
4.5 Fluidized bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41<br />
4.6 Particle size distribution in the fluidized bed . . . . . . . . . . . . . . . . . . . . 42<br />
5.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44<br />
5.2 Comparison of the albedos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46<br />
5.3 Calculated photon density distributions in the Teflon sample . . . . . . . . . . 47<br />
5.4 Contributions to the cooperon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50<br />
5.5 Old and new theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51<br />
5.6 Distribution of backscattered energies. . . . . . . . . . . . . . . . . . . . . . . . . 53<br />
5.7 Example wide angle backscattering data . . . . . . . . . . . . . . . . . . . . . . 54<br />
5.8 Distribution of backscattered energies . . . . . . . . . . . . . . . . . . . . . . . . 55<br />
5.9 CCD images of R700 and teflon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57<br />
5.10 Tip of the R700 cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58<br />
5.11 Focus accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59<br />
5.12 Transport mean free path in dependence of the lens position . . . . . . . . . . . 60<br />
5.13 <strong>Coherent</strong> backscattering cone of teflon . . . . . . . . . . . . . . . . . . . . . . . . 61<br />
5.14 Single and multiple scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63<br />
5.15 Calculation of the Stokes vector for the experiment . . . . . . . . . . . . . . . . 63<br />
5.16 Correction of the scattering angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 64<br />
5.17 Measured data and Mie distributions . . . . . . . . . . . . . . . . . . . . . . . . 65<br />
5.18 <strong>Coherent</strong> backscattering cone of a fluidized bed . . . . . . . . . . . . . . . . . . 66<br />
5.19 <strong>Backscattering</strong> for various volume fractions . . . . . . . . . . . . . . . . . . . . . 67<br />
76
Figures and Tables<br />
Tables<br />
4.1 Colloidal samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39<br />
5.1 Albedos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49<br />
77
78<br />
Figures and Tables
MATLAB codes<br />
Angular intensity distribution of single scattering<br />
The following MATLAB script calculates the angular distribution of the singly scattered intensity<br />
after the sequence ‘circular polarizer – spherical scatterer – circular polarizer’ in both<br />
backscattering setups, as derived in secs. 2.2 and 5.2.3.<br />
function mie_setup<br />
refractive_particle = 1.52 ;<br />
refractive_surround = 1.33 ;<br />
wavelength = 575e-9 ;<br />
diameter = 150e-6 ;<br />
% refractive index of the scatterer<br />
% refractive index of the surrounding medium<br />
% wavelength of the scattered light<br />
% particle diameter<br />
a = diameter / 2 ;<br />
m = refractive_particle / refractive_surround ;<br />
k = 2 * pi * refractive_surround / wavelength ;<br />
x = k * a ;<br />
angle = 180 : 0.01 : 181 ;<br />
theta = angle / 180 * pi ;<br />
mu = cos(theta) ;<br />
% or larger angular range for wide angle setup<br />
S1 = 0 ;<br />
S2 = 0 ;<br />
N = 50 + 1.03 * x ;<br />
N = fix (N) ;<br />
% avoids numerical overflow in Bessel functions<br />
for n = 1 : N<br />
A = J(n,m*x) ;<br />
B = J(n,x) ;<br />
C = dJ(n,x) ;<br />
D = dJ(n,m*x) ;<br />
E = H(n,x) ;<br />
F = dH(n,x) ;<br />
an = ( m * A * C - B * D ) / ( m * A * F - E * D ) ;<br />
bn = ( A * C - m * B * D ) / ( A * F - m * E * D ) ;
MATLAB codes<br />
end<br />
if n == 1<br />
p1 = 0 ;<br />
p = 1 ;<br />
t = mu ; % because t = n * mu * p - (n+1) * p1 = 1 * mu * 1 - 2 * 0<br />
else<br />
p0 = p1 ;<br />
p1 = p ;<br />
p = (2*n-1) / (n-1) * mu .* p1 - n / (n-1) * p0 ;<br />
t = n * mu .* p - (n+1) * p1 ;<br />
end<br />
S1 = S1 + (2*n+1) / (n*(n+1)) * (an*p + bn*t) ;<br />
S2 = S2 + (2*n+1) / (n*(n+1)) * (an*t + bn*p) ;<br />
S11 = 1/2 * ( S2 .* conj(S2) + S1 .* conj(S1) ) ;<br />
S33 = 1/2 * ( S1 .* conj(S2) + S2 .* conj(S1) ) ;<br />
I1 = S11 ;<br />
I2 = 1/2 * ( S11 + S33 ) ;<br />
% intensity after the scattering particle<br />
% after second transmission of circular polarizer<br />
plot (angle,I1,angle,I2)<br />
xlabel (’scattering angle [deg]’)<br />
ylabel (’intensity [a.u.]’)<br />
legend (’after scatterer’,’after circular polarizer’)<br />
end<br />
function J = J (n,x)<br />
J = sqrt(x*pi/2) * besselj(n+1/2,x) ;<br />
end<br />
function H = H (n,x)<br />
H = sqrt(x*pi/2) * ( besselj(n+1/2,x) + i * bessely(n+1/2,x) ) ;<br />
end<br />
function dJ = dJ (n,x)<br />
dJ = sqrt(pi/(2*x)) * ( (1+n) * besselj(n+1/2,x) - x * besselj(n+3/2,x) ) ;<br />
end<br />
function dH = dH (n,x)<br />
dH = sqrt(pi/(2*x)) * ( (1+n) * ( besselj(n+1/2,x) + i * bessely(n+1/2,x) ) ...<br />
- x * ( besselj(n+3/2,x) + i * bessely(n+3/2,x) ) ) ;<br />
end<br />
80
Evaluation of the wide angle data<br />
Evaluation of the wide angle data<br />
This MATLAB script calibrates and displays the data of the wide angle setup (secs. 3.2 and<br />
5.1), calculates the integrated cooperon E, and draws a theory curve which can be fitted to the<br />
data by varying the value for the transport mean free path l ∗ .<br />
Each data set consists of a series of reference measurements at different incoming laser powers<br />
and one measurement with the sample itself. The data are stored in ASCII files which contain<br />
the diode signals and their errors. The diode positions (in degrees) are read <strong>from</strong> the file<br />
‘diode angles.dat’. The filenames of the data files contain the average incoming laser powers<br />
and their errors (in arbitrary units), which are measured by the power meter in the setup.<br />
The albedo mismatch ℵ = A reference /A sample has to be calculated <strong>from</strong> eqn. 5.6, diffusion<br />
coefficient D and absorption time τ are measured in time of flight experiments (sec. 3.4), and<br />
the effective refractive index n eff is calculated using eqn. 4.1. With the variable ‘enhancement’<br />
the height of the theoretical cone can be adapted to that of the measured one.<br />
function wide_angle_evaluation<br />
%% Initialize variables and load files<br />
file_samp = ’C:\Examples\example_0.100+-0.001.dat’ ; % sample file<br />
path_ref = ’C:\Examples\’ ;<br />
% path reference files<br />
name_ref = ’teflon_01_0.200+-0.001.dat’ ;<br />
% arbitrary reference file<br />
wavelength = 590e-9 ;<br />
D = 15 ;<br />
tau = 2e-9 ;<br />
n_eff = 1.58 ;<br />
l = 500e-9 ;<br />
aleph = 0.890 ;<br />
enhancement = 0.9 ;<br />
% laser wavelength<br />
% diffusion coefficient of the sample<br />
% absorption time of the sample<br />
% effective refractive index of the sample<br />
% transport mean free path for theory fit<br />
% albedo mismatch<br />
% enhancement of the backscattering cone<br />
% angular positions of the photodiodes<br />
angles = load (’C:\Examples\diode_angles.dat’,’-ascii’) ;<br />
theta = angles * pi / 180 ;<br />
% sample file<br />
data_samp = load (file_samp,’-ascii’) ;<br />
parts = regexp (file_samp, ’_|+-|.dat’, ’split’) ;<br />
P_samp = parts ( size(parts,2) - 2 ) ;<br />
P_samp = str2double (P_samp:,1) ;<br />
Perror_samp = parts ( size(parts,2) - 1 ) ;<br />
Perror_samp = str2double (Perror_samp:,1) ;<br />
81
MATLAB codes<br />
% reference files<br />
index = regexp (name_ref, ’_’, ’start’) ;<br />
index = index (length(index) - 1) ;<br />
namebase = name_ref (1 : index) ;<br />
list = dir ([path_ref,namebase,’*.dat’]) ;<br />
data_ref = zeros (size(list,1),256,2) ;<br />
P_ref = zeros (size(list,1),1) ;<br />
for N = 1 : size(list,1)<br />
filename = [path,list(N).name] ;<br />
data_ref(N,:,:) = load (filename,’-ascii’) ;<br />
parts = regexp (list(N).name, ’_|+-|.dat’, ’split’) ;<br />
p_ref = parts ( size(parts,2) - 2 ) ;<br />
P_ref(N) = str2double (p_ref:,1) ;<br />
end<br />
% other data<br />
k = 2 * pi / wavelength ;<br />
c = 3e8 ; % speed of light<br />
l_approx = D * n_eff / c ;<br />
n_surr = 1 ; % refractive index of surrounding medium = air<br />
R = reflectivity (n_eff,n_surr) ;<br />
%% Calibrate data<br />
cone = zeros(256,1) ;<br />
cone_delta = zeros(256,1) ;<br />
for n = 1 : 256<br />
x = data_ref(:,n) ;<br />
y = P_ref * diffuson(theta(n),l_approx,R) ;<br />
temp1 = sortrows ([x,y],1) ; % sort by x<br />
x = temp1(:,1) ;<br />
y = temp1(:,2) ;<br />
warning off % many warnings because of defect photodiodes (causing NANs)!<br />
[p,s,mu] = polyfit (x,y,3) ;<br />
[cone(n),cone_delta(n)] = polyval (p,data_samp(n),s,mu) ;<br />
warning on<br />
end<br />
%% Fold data<br />
dummy = sortrows ([abs(theta), cone, cone_delta]) ;<br />
theta = dummy (:,1) ;<br />
cone = dummy (:,2) ;<br />
cone_delta = dummy (:,3) ;<br />
82
Evaluation of the wide angle data<br />
% remove NANs and data with huge errors <strong>from</strong> damaged photodiodes<br />
m = 1 ;<br />
for n = 1 : 256<br />
if isnan (cone(n))<br />
elseif cone_delta(n) > 0.1*P_samp<br />
else<br />
Cone(m) = cone(n) ;<br />
Delta(m) = cone_delta(n) ;<br />
Theta(m) = theta(n) ;<br />
m = m + 1 ;<br />
end<br />
end<br />
cone = Cone ;<br />
cone_delta = Delta ;<br />
theta = Theta ;<br />
clear Theta Cone Delta<br />
% fold data (i.e. -pi/2 < theta < pi/2 -> 0 < theta < pi/2)<br />
m = 1 ;<br />
n = 1 ;<br />
while n < size(theta,2)<br />
if theta(n+1) > theta(n)<br />
Theta(m) = theta(n) ;<br />
Theta(m+1) = theta(n+1) ;<br />
Cone(m) = cone(n) ;<br />
Cone(m+1) = cone(n+1) ;<br />
Delta(m) = cone_delta(n) ;<br />
Delta(m+1) = cone_delta(n+1) ;<br />
n = n + 1 ;<br />
else % theta(n) == theta(n+1)<br />
Theta(m) = theta(n) ;<br />
Cone(m) = (cone(n) + cone(n+1)) / 2 ;<br />
Delta(m) = 0.5 * sqrt ( (cone_delta(n)).^2 + (cone_delta(n+1)).^2 ) ;<br />
n = n + 2 ;<br />
end<br />
m = m + 1 ;<br />
end<br />
cone = Cone ;<br />
cone_delta = Delta ;<br />
theta = Theta ;<br />
%% Derive experimental and theoretical cones<br />
% Correct cone with albedo mismatch<br />
corr_cone = aleph * cone ;<br />
corr_cone_delta = aleph * cone_delta ;<br />
83
MATLAB codes<br />
% Derive experimental diffuson and cooperon<br />
exp_diffuson = diffuson (theta,l,R) * P_samp ;<br />
exp_diffuson_delta = diffuson (theta,l,R) * Perror_samp ;<br />
exp_cooperon = (corr_cone - exp_diffuson) / P_samp ;<br />
exp_cooperon_delta = ...<br />
sqrt ( (corr_cone_delta / P_samp).^2 ...<br />
+ (exp_diffuson_delta / P_samp).^2 ...<br />
+ ((corr_cone - exp_diffuson) / P_samp^2 * Perror_samp).^2 ) ;<br />
% Calculate theory curve<br />
theory_theta = 0 : 0.001*pi : pi/2 ;<br />
theory_cooperon = enhancement * cooperon (theory_theta,k,l,R,D,tau) ;<br />
%% Calculate energy<br />
e_theta = [ 0 , theta , pi/2 ] ;<br />
e_cooperon = [ 1 , exp_cooperon , 0 ] ;<br />
e_cooperon_delta = [ 0 , exp_cooperon_delta , 0 ] ;<br />
E = 2*pi ...<br />
* sum ( ( e_theta(:,2:end) - e_theta(:,1:end-1) ) ...<br />
* 1/2 .* ( e_cooperon(:,1:end-1) .* sin(e_theta(:,1:end-1)) ...<br />
+ e_cooperon(:,2:end) .* sin(e_theta(:,2:end)) ) ) ;<br />
E_delta = 2*pi ...<br />
* sqrt ( sum ( ( 1/2 * ( e_theta(1,2:end) - e_theta(1,1:end-1) ) ...<br />
.* e_cooperon_delta(1:end-1) ).^2 ...<br />
.* sin (1/2 * ( e_theta(1,2:end) + e_theta(1,1:end-1) )) ...<br />
+ ( 1/2 * ( e_theta(1,2:end) - e_theta(1,1:end-1) ) ...<br />
.* e_cooperon_delta(2:end) ).^2 ...<br />
.* sin (1/2 * ( e_theta(1,2:end) + e_theta(1,1:end-1) )) ) ) ;<br />
disp ([’E = ’,num2str(E),’+-’,num2str(E_delta)])<br />
%% Plot data<br />
angles = theta / pi * 180 ;<br />
theory_angles = theory_theta / pi * 180 ;<br />
plot ( angles, exp_cooperon, ’b’, ...<br />
theory_angles, theory_cooperon, ’r’, ...<br />
angles, exp_cooperon+exp_cooperon_delta, ’:b’, ...<br />
angles, exp_cooperon-exp_cooperon_delta, ’:b’ )<br />
xlabel (’scattering angle [deg]’)<br />
ylabel (’cooperon’)<br />
legend (’backscattering data’,’theory’)<br />
end<br />
84
Evaluation of the wide angle data<br />
function d = diffuson (theta,l,R)<br />
z0 = 2/3 * l * (1 + R) / (1 - R) ;<br />
mu = cos(theta) ;<br />
d = ( mu .* (z0/l + mu./(mu+1)) ) ./ (z0/l + 1/2) ;<br />
end<br />
function c = cooperon (theta,k,l,R,D,tau)<br />
z0 = 2/3 * l * (1 + R) / (1 - R) ;<br />
mu = cos(theta) ;<br />
q = k * sin(theta) ;<br />
L_a = sqrt (D * tau) ;<br />
q_a = sqrt (q.^2 + 1/(D*tau)) ;<br />
c = ( l * (l + L_a)^2 * ( (1 - exp(-2*q_a*z0)) ./ ...<br />
(q_a*l) + 2*mu ./ (mu+1) ) ) ...<br />
./ ( L_a^2 * ( l + (1 - exp(-2*z0/L_a)) * L_a ) * ...<br />
( l*q_a + (mu+1) ./ (2*mu) ).^2 ) ;<br />
c_add = (8 * pi * a(k,l,R)) / (3 * k^2 * l * (l + 2 * z0)) * mu ./ (mu + 1) ;<br />
c = c - c_add ;<br />
end<br />
function a = a (k,l,R)<br />
theta = 0.00001*pi : 0.00001*pi : pi/2 ;<br />
z0 = 2/3 * l * (1 + R) / (1 - R) ;<br />
mu = cos(theta) ;<br />
q = k * abs(sin(theta)) ;<br />
E_A = trapz ( theta, ...<br />
3/(8*pi) * 1./(q*l + (mu+1)./(2*mu)).^2 ...<br />
.* ( (1 - exp(-2*q*z0))./(q*l) + (2*mu)./(mu+1) ) ...<br />
.* sin(theta) ) ;<br />
E_BC = trapz ( theta, ...<br />
1/(k*l)^2 * mu./(mu+1) ...<br />
.* sin(theta) ) ;<br />
a = E_A / E_BC ;<br />
end<br />
85
MATLAB codes<br />
function R = reflectivity (n1,n2)<br />
if n1 > n2<br />
theta_totalreflection = asin (n2/n1) ;<br />
else<br />
theta_totalreflection = pi/2 ;<br />
end<br />
theta1 = 0 : pi/100 : theta_totalreflection ;<br />
theta2 = theta_totalreflection+pi/100 : pi/100 : pi/2 ;<br />
theta = [ theta1, theta2 ] ;<br />
R_parallel = ( n1 * cos(theta1) ...<br />
- n2 * sqrt( 1 - (n1 / n2 * sin(theta1)).^2 ) ).^2 ./ ...<br />
( n1 * cos(theta1) ...<br />
+ n2 * sqrt( 1 - (n1 / n2 * sin(theta1)).^2 ) ).^2 ;<br />
R_parallel = [ R_parallel , ones(size(theta2)) ] ;<br />
R_perp = ( n1 * sqrt( 1 - (n1 / n2 * sin(theta1)).^2 ) ...<br />
- n2 * cos(theta1) ).^2 ./ ...<br />
( n1 * sqrt( 1 - (n1 / n2 * sin(theta1)).^2 ) ...<br />
+ n2 * cos(theta1) ).^2 ;<br />
R_perp = [ R_perp , ones(size(theta2)) ] ;<br />
C1 = trapz ( theta , ...<br />
(R_parallel + R_perp) / 2 .* sin(theta) .* cos(theta) ) ;<br />
C2 = trapz ( -theta , ...<br />
(R_parallel + R_perp) / 2 .* sin(-theta) .* (cos(-theta)).^2 ) ;<br />
R = ( 3 * C2 + 2 * C1 ) / ( 3 * C2 - 2 * C1 + 2 ) ;<br />
end<br />
86
Evaluation of the small angle data<br />
Evaluation of the small angle data<br />
The following MATLAB code is a very simplified version of the evaluation program for the<br />
small angle setup; it will not work with all data. The script finds the backscattering direction<br />
θ = 0 in the CCD image and calculates either an azimuthal average or a radial profile section<br />
of the backscattering cone.<br />
The signals of the CCD chip and their errors (se = standard error) are stored in two uint16<br />
binary files. Depending on the preprocessing the numbers can have big (ieee-be) or little<br />
(ieee-le) endian.<br />
function small_angle_evaluation<br />
%% Open file<br />
endian = ’ieee-be’ ;<br />
% alternative: ’ieee-le’<br />
fid = fopen (’C:\Examples\example.i16’, ’r’) ;<br />
matrix = fread (fid, [2048,2048], ’uint16’, 0, endian) ;<br />
fclose (fid) ;<br />
fid = fopen (’C:\Examples\example_se.i16’, ’r’) ;<br />
errormatrix = fread (fid, [2048,2048], ’uint16’, 0, endian) ;<br />
fclose (fid) ;<br />
%% Find peak<br />
% the image has to be scaled to have intensities in [0,1] by dividing by the<br />
% largest possible intensity (i.e. 65535)<br />
dummy = matrix / 65535 ;<br />
% turn matrix into black-and-white picture (conetip forms white blob,<br />
% everything else is black)<br />
n = 5 ;<br />
dummy = medfilt2 (dummy, [n,n]) ;<br />
maximum = max ( max (dummy) ) ;<br />
level = 0.95 * maximum ;<br />
bwimage = im2bw (dummy, level) ;<br />
% smooth picture so that fluctuations do<br />
% not disturb the search for the cone<br />
% find center of the blob => scattering angle theta = 0<br />
dummy = bwlabel (bwimage) ; % label connected components in binary image<br />
regions = regionprops (dummy, ’Area’, ’Centroid’) ; % measure properties<br />
% of image regions<br />
[area,index] = max ( [ regions.Area ] ) ;<br />
87
MATLAB codes<br />
position = regions(index).Centroid ;<br />
position = round (position) ;<br />
% position = center of mass of blob area;<br />
% position(1) = x, position(2) = y !!!<br />
%% Calculate profile or average<br />
mode_selector = ’average’ ;<br />
% alternative: ’profile’<br />
if strcmp(mode_selector, ’average’)<br />
range = 1 ; % change value to average over a larger radial range<br />
[intensity,error,pixels] = ...<br />
calculate_average (matrix, errormatrix, position, range) ;<br />
else<br />
direction = 0 ; % change value to select other direction<br />
range = 1 ; % change value to average over a larger radial range<br />
[intensity,error,pixels] = ...<br />
calculate_profile (matrix, errormatrix, position, direction, range) ;<br />
end<br />
%% Display result<br />
plot ( pixels, intensity )<br />
hold on<br />
plot ( pixels, intensity-error, ’:’ )<br />
plot ( pixels, intensity+error, ’:’ )<br />
hold off<br />
xlabel (’scattering angle [pixels]’)<br />
ylabel (’intensity [a.u.]’)<br />
end<br />
function [intensity,error,pixels] = ...<br />
calculate_average (matrix, errormatrix, position, range)<br />
% create carthesian coordinate system<br />
X = repmat(1:2048,2048,1) ;<br />
Y = X’ ;<br />
% shift origin of the coordinate system to ’position’<br />
X = X - position(1) ;<br />
Y = Y - position(2) ;<br />
% turn carthesian coordinate system into polar coordinates<br />
[theta,rho] = cart2pol (X,Y) ; % range of theta: [-pi,pi]<br />
88
Evaluation of the small angle data<br />
% reshape matrices into vectors<br />
matrix = reshape (matrix,[],1) ;<br />
errormatrix = reshape (errormatrix,[],1) ;<br />
rho = reshape (rho,[],1) ;<br />
% sort vectors for rho<br />
sorted = sortrows ([rho,matrix,errormatrix], 1) ;<br />
rho = sorted(:,1) ;<br />
matrix = sorted(:,2) ;<br />
errormatrix = sorted(:,3) ;<br />
% define ranges for the radial average<br />
rho = rho / range ;<br />
rho = round (rho) ;<br />
% radial = values of rho without repetitions, sorted in ascending order<br />
[radial,first] = unique (rho, ’first’) ; % first = first occurrence of each<br />
% unique value in rho<br />
[radial,last] = unique (rho, ’last’) ; % last = last occurrence of each<br />
% unique value in rho<br />
% average of the values of matrix and errormatrix<br />
weight = 1 ./ errormatrix.^2 ;<br />
weighted_matrix = matrix .* weight ;<br />
intensity = zeros (size(radial,1), 1) ;<br />
error = zeros (size(radial,1), 1) ;<br />
for k = 1 : size(radial,1)<br />
intensity(k) = sum ( weighted_matrix(first(k):last(k)) ) ...<br />
/ sum ( weight(first(k):last(k)) ) ;<br />
error(k) = 1 / sqrt ( sum ( weight(first(k):last(k)) ) ) ...<br />
* 1 / sqrt ( last(k)-first(k)+1 ) ;<br />
end<br />
% define scale<br />
pixels = radial * range ;<br />
end<br />
function [intensity,error,pixels] = ...<br />
calculate_profile (matrix,errormatrix,position,direction,range)<br />
% calculate direction angle<br />
angle = direction / 180 * pi ;<br />
if angle < 0<br />
angle = angle + 2 * pi ; % angles between 0 and 2*pi<br />
89
MATLAB codes<br />
end<br />
% calculate profile endpoints at the image borders<br />
if 0
Evaluation of the small angle data<br />
% calculate intensity profile<br />
part1 = improfile ( matrix, [endpoints(1),position(1)] , ...<br />
[endpoints(3),position(2)] ) ;<br />
part2 = improfile ( matrix, [position(1),endpoints(2)] , ...<br />
[position(2),endpoints(4)] ) ;<br />
part2 = part2(2:end) ; % first pixel is already in part1 !<br />
intensity = [part1;part2] ;<br />
% calculate error profile<br />
part1 = improfile ( errormatrix, [endpoints(1),position(1)] , ...<br />
[endpoints(3),position(2)] ) ;<br />
part2 = improfile ( errormatrix, [position(1),endpoints(2)] , ...<br />
[position(2),endpoints(4)] ) ;<br />
part2 = part2(2:end) ; % first pixel is already in part1 !<br />
error = [part1;part2] ;<br />
% calculate scale<br />
if range > 1<br />
modulo1 = mod ( size(part1,1) - ceil(range/2) , range ) ;<br />
modulo2 = mod ( size(part2,1) - floor(range/2) , range ) ;<br />
intensity = intensity ( modulo1+1 : size(part1,1)+size(part2,1)-modulo2 ) ;<br />
intensity = reshape (intensity, range, []) ;<br />
error = error ( modulo1+1 : size(part1,1)+size(part2,1)-modulo2 ) ;<br />
error = reshape (error, range, []) ;<br />
weight = 1 ./ error.^2 ;<br />
weighted_intensity = intensity .* weight ;<br />
intensity = sum (weighted_intensity, 1) ./ sum (weight, 1) ;<br />
intensity = intensity’ ;<br />
error = 1 / sqrt(range) .* 1 ./ sqrt ( sum (weight, 1) ) ;<br />
error = error’ ;<br />
else<br />
modulo1 = 0 ;<br />
end<br />
length = sqrt ( ( x_end - x_start )^2 + ( y_end - y_start )^2 ) ;<br />
number = size ( intensity , 1 ) ;<br />
pixels = 1 : length/number : length ;<br />
pixels = pixels - pixels( ceil ( ( size(part1,1) - modulo1 ) / range ) ) ;<br />
pixels = pixels’ ;<br />
end<br />
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