Microscopic Theory of Random Lasing and Light Transport in Amplifying Disordered Media [Elektronische Ressource] / Regine Frank

rheinische_friedrich-wilhelms-universitat_bonn - Regine Frank

Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres

110 pages

English

Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres

A propos
Informations
Extrait

Sujets

Physik

Informations

Publié par	rheinische_friedrich-wilhelms-universitat_bonn
Publié le	01 janvier 2011
Nombre de lectures	14
Langue	English
Poids de l'ouvrage	1 Mo

Extrait

Microscopic Theory
of Random Lasing and Light Transport
in Amplifying Disordered Media
Dissertation
zur
Erlangung des Doktorgrades (Dr. rer. nat.)
der
Mathematisch-Naturwissenschaftlichen Fakult¨at
der
Rheinischen Friedrich-Wilhelms-Universit¨at Bonn
vorgelegt von
Regine Frank
aus Geislingen an der Steige
Bonn 2009Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen
Friedrich-Wilhelms-Universit¨at Bonn.
1.Referent: Prof. Dr. Johann Kroha
2.Referent: Prof. Dr. Kurt Busch
Tag der Promotion: 26. April 2010
Erscheinungsjahr: 2011Abstract
In the last decade Anderson Localization of Light and Random Lasing has attracted a variety of
interest in the community of condensed matter theory. The fact that a system so inartiﬁcial as a
thin layer consisting of dust particles, which have amplifying properties, can produce coherent laser
emission, is of a simple elegance, which promises new insights in fundamental physics. The Random
Laser consists of randomly distributed scatterers which have amplifying properties, embedded in an
either amplifying or a passive medium. There is no need of an external feedback mechanism, like in
the system of a conventional laser. Despite the striking chasteness of the eﬀect, there are still vivid
discussions about theory for the random laser, which has not been fully described yet. Whereas
there are several attempts to enter the subject numerically by considering cavity approximations,
this thesis is concerned with building a microscopically self-consistent theory of random lasing. In
order to design this method we had to study light localization eﬀects in random media including
absorptionandgain. WeincorporatedinterferenceeﬀectsbycalculatingtheCooperoncontributions
and we found, that we reach Anderson localized states for passive media. Mapping this theory
on a system which consists of laser active Mie-scatterers in a passive medium, we found that by
incorporation of the Cooperon, we loose the Anderson localization again, but the system is still
weakly localized, which is suﬃcient for the enhancement of population inversion and stimulated
emission. The description of a random lasing system is completed by coupling the analytically
derived microscopic transport theory to the laser rate equations of a four level laser and solving
the system numerically self-consistent. Finally we develop a generalized method for describing light
transport, multiple scattering and interference eﬀects in a translationally non-invariant system of
ﬁnite size analytically. The solution of this theory coupled to the rate equations gives us a closed
theory to calculate transport and lasing properties of a random lasing slab geometry.Contents
1 Introduction 7
2 Light Waves in Random Media 11
2.1 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Green’s Function Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Single Particle Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Two Particle Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Higher Moments of the Green’s Function . . . . . . . . . . . . . . . . . . . . 14
2.3 Scattering Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 The T Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Conﬁgurational Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.3 The Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Two Particle Quantities and Bethe Salpeter-Equation . . . . . . . . . . . . . . . . . 16
2.4.1 Incoherent and Coherent Contributions . . . . . . . . . . . . . . . . . . . . . 18
3 Light Transport in Inﬁnite Three Dimensional Disordered Media with Absorp-
tion or Ampliﬁcation 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Light Matter Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Setup and Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Light Propagation in Ladder Approximation . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Theory of Transport and Localization . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5.1 Expansion of Two-particle Green’s Function into Moments . . . . . . . . . . 29
3.5.2 Computing the coeﬃcients of the expansion into moments . . . . . . . . . . . 29
3.5.3 General Solution of the Bethe-Salpeter Equation . . . . . . . . . . . . . . . . 30
3.5.4 Vertex Function and Self-consistency . . . . . . . . . . . . . . . . . . . . . . . 32
3.6 ResultsandDiscussionoftheTransportTheoryininﬁnite3-DMediawithabsorption
and gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6.1 Discussion Transport Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6.2 Causality and Length Scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Phenomenological Random Laser 39
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Theory of a Diﬀusive Random Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5 Selfconsistent Microscopic Theory of Random Lasing 45
5.1 Single - Particle Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Light-Intensity Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3 Fourier transformed Bethe - Salpeter Equation . . . . . . . . . . . . . . . . . . . . . 51
5.4 Light Intensity Transport in Bounded Disordered Media with Absorption or Gain . 52
5.4.1 Solution by Moment Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 52
56 CONTENTS
5.4.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.4.3 Current Relaxation Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.5 Diﬀusion Pole Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.5.1 Selfconsistent Diﬀusion Coeﬃcient . . . . . . . . . . . . . . . . . . . . . . . . 58
5.6 Microscopic Theory of Random Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.6.1 Microscopic Determination of the Ampliﬁcation Rate of the Intensity . . . . 59
5.6.2 Numerics Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.7 Numerical Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.7.1 Film of ZnO at 50 % Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.7.2 Behavior of the Correlation Length for Various System Parameters . . . . . . 70
5.7.3 Behavior of the Diﬀusion Coeﬃcient for Various System Parameters . . . . . 75
5.7.4 Diﬀerent Filling Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6 Summary 83
Appendices 84
A Dyson Equations 85
B Disorder Averaged Full Single-Particle Green’s Function 87
′B.1 Analytic Calculation of G(~r,~r ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
C Two Particle Green’s Function -
The Intensity Correlator 93
D Calculating the Memory Kernel M(Ω) 97
E Technical Transformations 101
E.1 Transformation of the Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . 101
E.2 Transformation used for Single Particle Greens Function . . . . . . . . . . . . . . . . 101
Bibliography 103Chapter 1
Introduction
This thesis is concerned with light propagation, transport and ampliﬁcation in random media in
general, which is a wide area of research. Whereas we ﬁnd undamped propagation in the case of
vacuum, in the random medium we have to consider mainly two diﬀerent mechanisms of transport.
First the diﬀusive processes which lead to an exponential decay of the light intensity and second
the interference processes which may lead to Anderson localization. Finally the interplay of these
transport mechanisms may lead in amplifying media to an eﬀect which is called random lasing.
Figure 1.1: Schematic illustration of light transport in the three diﬀerent regimes. Shown is the
incident light pulse (intensity) in position space (upper row) at t = 0 and the same light pulse
after a ﬁnite time t > 0. In the ballistic regime, i.e. in the absence of disorder (left column),
light propagates with the bare speed of light, whereas in diﬀusive regime with weak disorder (center
column), the initial pulse decays slower. Finally in the localized regime, i.e. strong disorder, (right
column) the light pulse does not decay at all, light intensity is localized within a given a volume.
Random lasing is a phenomenon which has been discussed during the last decade for various dis-
ordered media. Optical gain in such a laser is either achieved by introducing a laser dye, or the
scattering structure is excited to deliver optical gai