Governing the Speed of a Light Signal in Optical Fibers: Brillouin

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Governing the Speed of a Light Signal in Optical Fibers:
Brillouin Slow and Fast Light
THÈSE NO 4459 (2009)
PRÉSENTÉE le 9 juillet 2009
À LA FACULTé SCIENCES ET TECHNIQUES DE L'INGÉNIEUR
INSTITUT DE GENIE ELECTRIQUE ET ELECTRONIQUE
PROGRAMME DOCTORAL EN PHOTONIQUE
ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE
POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES
PAR
Sang Hoon Chin
acceptée sur proposition du jury:
Prof. R. Salathé, président du jury
Prof. L. Thévenaz, directeur de thèse
Prof. M. Gonzalez Herraez, rapporteur
Dr R. Houdré, rapporteur
Prof. J. Khurgin, rapporteur
Suisse
2009
To my parents
To my wife Uhhee Song
To my son See-Won and my daughter Rose
Abstract
Dynamic control of the speed of a light signal, based on stimulated Brillouin scattering in
optical fibers, was theoretically studied and also experimentally demonstrated as the core
object of this thesis. To date, slow light based on stimulated Brillouin scattering has shown
the unmatched flexibility to offer an efficient timing tool for the development of all-optical
future router. Nevertheless, the seeming perfect Brillouin slow light suffered from three
major obstacles: naturally narrow signal bandwidth, strong change of signal amplitude, and
significant signal distortion. The essential contribution of this work has been mostly
dedicated to resolve all those impairments so as to make Brillouin slow light a completely
operating all-optical delay line for practical applications. Actually, high capability of tailoring
the spectral distribution of the effective Brillouin resonance makes possible to resolve
partially or completely all those problems.
First of all, a broadband spectral window was passively obtained in between two
Brillouin gain/loss resonances by simply appending two segments of fibers showing different
Brillouin frequency shifts. The global Brillouin gain of the concatenated fibers manifests a
gain/loss doublet resonance showing a broad window in between gain/loss peaks. In practice,
this configuration has a crucial advantage that it removes the need of the pump modulation,
generally used to create a polychromatic pump source. Therefore, a broadband Brillouin slow
and fast light was simple realized with a reduced distortion.
Secondly, the signal amplification or attenuation associated to the signal delay was
completely compensated by superposing Brillouin gain and loss resonances with identical
depth but different width. As a result, the Brillouin gain led to effectively a spectral hole in
the center of the broadband absorption and opened a transparent window while the sharp
change in the refractive index was preserve. This way it makes possible to realize zero-gain
Brillouin slow light. This configuration was also exploited to produce Brillouin fast light with
a total absence of signal loss, simply by swapping the spectral position of the two pumps.
At last, a signal was continuously delayed through a Brillouin fiber delay line
without any distortion. Due to the strong induced dispersion, pulse broadening is a major
ABSTRACT
iv
difficulty in all slow light systems and it is impossible to compensate such broadening using a
linear system. Therefore, a conventional Brillouin slow light system was combined with a
nonlinear optical fiber loop mirror that gives a nonlinear quadratic transmission. Using this
configuration, the inevitable pulse broadening was completely compensated at the output of
the loop and a signal was delayed up to one symbol without any distortion
Brillouin slow light systems were further studied in the spectral domain. For a given
Brillouin resonance the spectrum of a light pulse was optimized to better match the Brillouin
bandwidth. When the time envelope of a pulse was properly shaped, it was clearly observed
that the spectral width of the pulse became minimized while preserving the pulse duration.
This way the maximum time delay through Brillouin slow light could be enhanced for a fixed
pulse width.
Brillouin fast light was even realized in total absence of any pump source, which is a
plain requirement for the generation of Brillouin slow or fast light. This self-generated delay
line, key contribution of this thesis, relies on both spontaneous and stimulated Brillouin
scattering in optical fibers. In this implementation, a light signal was strongly boosted above
the so-called Brillouin threshold, so that most power of the signal was transferred to a
backward propagating wave, namely the generation of amplified spontaneous Stokes wave.
Since the center frequency of the intense Stokes wave is below the signal frequency by
exactly Brillouin shift of the fiber used, this wave led to a Brillouin loss band centered at the
signal frequency. Consequently, the signal experienced fast light propagation and the
propagation velocity of the signal was self-controlled, simply by varying the signal input
power. This technique has many practical advantages such as its high simplicity of the
configuration and an invariant signal power in the output of this delay line. Additionally, this
system self-adapts the signal bandwidth as the spectrum of the amplified Stokes wave
matches the spectral distribution of the signal.
An alternative method to generate all-optical delay line was proposed instead of slow
light. This scheme makes use of the combination of wavelength conversion and group
velocity dispersion. This type of delay line was mainly aimed at improving the storage
capability of delaying element. The wavelength of a signal was simply and efficiently
converted at a desired wavelength using cross gain modulation in semiconductor optical
amplifiers. Then the converted signal was delivered to a high dispersive medium and arrived
at the end of the medium with relative time delay due to the group velocity dispersion. A
fractional delay of 140 was continuously produced through this delay line for a signal with a
duration of 100 ps, preserving signal bandwidth and wavelength.
The effect of slow light on linear interactions between light and matter was
experimentally investigated to clarify the current scientific argument regarding slow light-
ABSTRACT
v
enhanced Beer-Lambert absorption. It was predicted that real slowing of the light group
velocity could enhance the molecular linear absorptions so as to improve the sensitivity of
this type of sensing. However, the experimental results unambiguously show that material
slow light (slow light in traveling wave media) does not enhance the Beer-Lambert
absorption.
Keywords
Fiber optics, Optical fibers, Nonlinear fiber optics, Brillouin scattering, Pulse propagation,
Amplifiers and oscillators
Version abrégée
L’étude théorique et la démonstration expérimentale du contrôle dynamique de la vitesse
d’un signal lumineux, basé sur la diffusion Brillouin stimulée dans les fibres optiques,
représente l’essentiel de ce travail de thèse. À ce jour, la lumière lente obtenue par la
diffusion Brillouin stimulée a démontré une souplesse inégalée pour réaliser la fonction de
temporisation destinée au développement des futurs routeurs tout-optiques. Néanmoins,
malgré ses atouts évidents, la lumière lente Brillouin présente trois défauts majeurs: une
bande passante du signal naturellement étroite, une forte altération de l'amplitude du signal et
une distorsion significative de l’enveloppe du signal. La contribution essentielle de ce travail
réside dans la résolution de ces problèmes, afin de réaliser avec la lumière lente Brillouin une
ligne de retard tout-optique pour des applications pratiques. En fait, la possibilité de modifier
la distribution spectrale effective de la résonance Brillouin permet de résoudre partiellement
ou complètement tous ces problèmes.
En premier lieu, une fenêtre spectrale large-bande a pu être obtenue, de manière
passive, entre deux résonances Brillouin en gain ou perte, en juxtaposant simplement deux
segments de fibres avec des décalages fréquentiels Brillouin différents. Le gain Brillouin
global des fibres concaténées se présente sous la forme d’une double résonance gain/perte
présentant un large intervalle spectral entre les pics gains/pertes. Dans la pratique, cette
configuration a l’énorme avantage de supprimer le besoin de moduler la pompe - modulation
généralement utilisée pour créer une pompe polychromatique. Une lumière lente et rapide
Brillouin large bande a été réalisée de façon simple avec une distorsion réduite en utilisant
cette configuration.
Deuxièmement, l'atténuation ou l’amplification du signal, associées à la création de
l’effet retard sur le signal, ont été entièrement compensée en superposant des résonances en
gain et perte Brillouin d’égale amplitude, mais de largeur différente. Par conséquent, le gain
Brillouin superposé à la perte conduit pratiquement à la formation d’un trou spectral au
centre de la large bande d'absorption, créant ainsi une fenêtre de transparence tout en
VERSION ABRÉGÉE
viii
préservant la variation accentuée de l'indice de réfraction. De cette façon, il est possible de
réaliser une lumière lente Brillouin à gain nul. Cette configuration est également exploitée
pour produire de la lumière Brillouin rapide avec une totale absence d’atténuation du signal,
en échangeant simplement la position spectrale des deux pompes.
Enfin, un signal a été continûment retardé sans aucune distorsion par le biais d'une
ligne de retard Brillouin fibrée. En raison de la forte dispersion induite, l'élargissement de
l'impulsion est une pénalité majeure dans tous les systèmes de lumière lente et il est
impossible de compenser un tel élargissement en utilisant un système linéaire. Pour y
remédier, un miroir non linéaire, réalisé avec une boucle de fibre optique, a été associé à un
système conventionnel à base de lumière lente Brillouin, pour donner une transmission
quadratique non-linéaire. En utilisant cette configuration, l'inévitable l'élargissement de
l'impulsion a été entièrement compensé à la sortie de la boucle et un signal a pu être retardé
sans aucune distorsion sur une durée équivalente à un symbole.
Les systèmes à lumière lente Brillouin ont également été étudiés dans le domaine
spectral. Pour une résonance Brillouin donnée, le spectre d'une impulsion lumineuse a été
optimisé pour mieux correspondre à la largeur utile de la résonance Brillouin. Lorsque
l’enveloppe temporelle d'une impulsion est correctement mise en forme, il a été clairement
observé que la largeur spectrale de l'impulsion est minimisée tout en préservant la durée
temporelle de l'impulsion. De cette manière, le retard maximal par la lumière lente Brillouin
peut être potentiellement augmenté pour une largeur d'impulsion fixée.
Une lumière rapide Brillouin a pu même être obtenue en l'absence totale de toute
pompe, le pompage étant jusqu’ici considéré comme une condition sine qua non pour la
production de lumière lente ou rapide Brillouin. Cette ligne à retard auto-générée repose sur
les diffusions Brillouin spontanée et stimulée dans les fibres optiques. Pour cette mise en
œuvre, un signal lumineux a été fortement amplifié au-dessus du seuil Brillouin, de sorte que
la majeure partie de la puissance du signal a été transférée à une onde contrapropagative, à
savoir la génération d'ondes de Stokes spontanée amplifiée. Puisque la fréquence centrale de
l'onde Stokes intense est décalée en-dessous de la fréquence du signal d'une valeur
correspondant exactement au décalage Brillouin de la fibre utilisée, cette onde conduit à la
formation d’une résonance de perte Brillouin centrée sur la fréquence du signal. Par
conséquent, le signal se trouve en condition pour une propagation en lumière rapide et la
vitesse de propagation du signal a pu être autocontrôlée, en faisant simplement varier le
niveau du signal d'entrée. Cette technique a de nombreux avantages pratiques, tels une grande
simplicité de la configuration et une puissance de signal constante en sortie de la ligne à
retard. En outre, ce système adapte automatiquement la bande passante de la résonance de
VERSION ABRÉGÉE
ix
perte au signal, puisque le spectre de l'onde Stokes amplifiée correspond à la distribution
spectrale du signal.
Une autre méthode pour générer des lignes à retard tout-optiques a été proposée en
lieu et place de la lumière lente. Ce procédé utilise une combinaison de conversion de
longueur d'onde et de dispersion de vitesse de groupe. Ce type de ligne à retard vise
principalement à améliorer la capacité de stockage d'un élément à retard. La longueur d'onde
d'un signal a pu être simplement et efficacement convertie vers la longueur d'onde désirée en
utilisant l’intermodulation de gain des amplificateurs optiques à semi-conducteurs. Puis, le
signal converti est injecté dans un milieu hautement dispersif pour obtenir à la sortie du
milieu un retard temporel relatif dû à la dispersion de la vitesse de groupe. Un retard
normalisé de 140, continûment ajustable, a été produit par le biais de cette ligne à retard pour
un signal d’une durée de 100 ps, sans modification de la largeur de bande du signal et de sa
longueur d'onde.
L'effet de la lumière lente sur les interactions linéaires entre la lumière et la matière a
été expérimentalement étudié afin de clarifier l’actuelle conjecture scientifique pour savoir si
la lumière lente augmente l’absorption moléculaire de type Beer-Lambert ou pas. Il a été
prédit que le ralentissement de la vitesse de groupe de la lumière pourrait renforcer
l'absorption linéaire moléculaire de façon à améliorer la sensibilité de ce type de détection.
Toutefois, les résultats expérimentaux montrent clairement que la lumière lente de type
matériel (lumière lente produite lorsqu’une onde traverse un matériau) ne renforce pas
l'absorption de type Beer-Lambert.
Mots clés
optique nonlinéaire, optique fibre, optique fibre nonlinéaire, diffusion Brillouin, propagation
d’impulse, amplificateurs et oscillateurs
Acknowledgement
It is my great pleasure to take an opportunity to acknowledge those people who gave me
precious help during my Ph.D study. They are quite broadly ranged. They could be around
my academic life and also around my private life. Actually, I don’t know who I have to start
from, but let me start from academic area. Prof. Luc Thévenaz, my wonderful supervisor,
must be the first who I have to appreciate. First of all, he gave me this great chance to make
Ph.D work at EPFL in Switzerland and guided me always to the right direction. Technical
discussion with him was always fruitful, and showed me some challenges and motivations to
work. The most important thing is that any second during the discussion was precious and
worth doing it. To tell the truth, during discussion I always felt like I am growing up. I
believe that he is an extraordinary scientist and it is him who made me a scientist. In addition,
he was a very good counselor about family life. Since I have a family, actually I had two jobs,
working as a student at school during day and working as a husband and a father of a boy at
home during evening. He always gave me good advice about what a family is supposed to be.
Officially, I have only one supervisor, but personally, I have a second supervisor for
my Ph.D work. He is Prof. Miguel González-Herráez at Alcala in Spain, one of the juries in
my Ph.D exam. During my Ph.D work, he was always next to me, not physically but through
email we spent a lot of time to deeply discuss about my experimental results and to devise
new ideas. When I didn’t follow the discussion well, he didn’t save his time before he made
me understood the physics behind experiments and mathematical analysis like Luc did. I like
to give special thanks to him. And, also, I like to thank all juries in my Ph.D exam, Prof.
René Salathé at EFFL, Prof. Jacob Khurgin at Johns Hopkins University in USA and Dr.
Romuald Houdré at EFPL. I learned a lot from their questions and comments. They made me
look at my work at different angles. It was very nice. Moreover, I like to thank Prof. Moche
Tur at Tel Avi University in Israel. I had a discussion with him, just 4 days before my exam,
about my presentation for Ph.D exam. I got very precious wisdoms and comments, which
improved the quality of my presentation by order of magnitude.
I remember the moment when I just arrived at Luc’s laboratory. Frankly speaking, I
was so afraid of all electrical equipments and optical components because I was not at all
familiar with such stuff and even I didn’t know how to operate and which can be helpful for
my experiments. Whenever I got in such trouble my brothers (Dario Alasia, Jean-Philippe
Besson and Mario Mattiello) came to me to solve all problems, showing me clear explanation.
The valuable friendship with them gave me tremendous pleasure while living in Switzerland.
One of my best moments at school was a coffee break with them, speaking about our lives,
our babies, our bright futures and our dreams. I never forget them. There are still many
people who I have to thank. Paulou Pierrette, secretary of photonics doctoral school, helped
me a lot when I settled down my new life in Switzerland. She spent lots of time to find an
apartment for me. I really appreciate it. Of course, all members of our group (Jean-Charles,
Stella, Isabelle and Nikolay) deserve appreciated. I enjoyed a lot having lunch and coffee
with them. Two technicians, Frédo and Pascal, were very kind to me, supporting all
electronic elements I needed. Also, I like to thank Holger Ficher, Sébastien Equis and José
Dintinger for hanging out and also technical discussion, and all people in the laboratory
NAM as well.
Today I am so sad since I can not share this pleasure with my parents. Both of them
passed away, especially my father passed away just two weeks before the deadline for the
submission of my dissertation. For this moment I had sunk into mental chaos, but now
everything is ok. In fact, this event makes me much stronger and work harder. I really
appreciate their way to raise me. They taught me how to interact with people and how to
behave among people from school, society, etc. They always said, “Save money, energy,
whatever. Be honest and don’t tell a lie. The honest is your best friend and it will bring
people to you.” Yes, of course, it is very easy to say, but not easy to act so. But, I saw them
living this way. Neither of them was working in academic fields. Actually, in my family, I
am the only one who did Ph.D study. They don’t know the nature of the academic life, but
always encouraged me a lot for my study and appreciated my decision whatever I made by
myself. They were really good teachers in my life, ever. Also, I like to appreciate my brother
and sisters’ help and support. They always encouraged me in my whole life and gave me their
confidence.
My sweet wife should deserve wonderful thanks, during the last three years and half,
for rolling up my sleeves and for letting me concentrate on my work. It is for sure difficult to
live abroad without any family, but only with her husband who is not taking care of home
very well but only working for Ph.D. Even, having a baby (See-Won) and raising him was
mainly her task in Switzerland. I tried to help her as much as possible, but I don’t think that it
was enough help for her. So, through this time, I like to thank her again and I like to say to
her, “I LOVE YOU.” Also, I like to give many thanks to my first son, See-Won for not
making huge accidents (^^;). Also, my parents-in-law spent bunch of time to pray for my
success. I really appreciate it.
I must thank Heui-sook Weman, the mother of Léo and the wife of Helges Weman.
Their help was endlessly flowing into my family. Actually, thanks to them we didn’t have
any time, even a second to feel lonely and to be bored. First of all, their regular invitation for
dinner was fantastic and always delicious. My wife enjoyed a lot the foods when she was
pregnant. That’s why my son is very healthy today, I guess. ^^. Actually, the meal is only just
tiny part during dinner. Talking about the life, Korea’s politics, international cultures and so
on was really great. We learned a lot while talking together. It is true that Heui-sook is not
only a good cook, but also a very good counselor about how to build up my family. I really
appreciate whatever she did for my family. Also, I like to thank many Korean people from
EPFL and the church. Especially, the bible study with Korean people, living in Lausanne,
was memorial event.
Finally, I like to acknowledge the support from the Swiss National Science
Foundation through project 200021-109773 and 200020-121860, and the European
Community's Seventh Framework Program [FP7/2007-2013] under grant agreement
n° 219299 (GOSPEL project). The study about ‘Slow-light effect on Beer-Lambert
absorption’ was realized in the framework of the European COST Action 299 “FIDES”.
Contents
1 Introduction ·············································································································
1
Bibliography ·············································································································
5
2 Optical signal propagation in a dispersive medium ············································
7
2.1
Signal propagation ···························································································
7
2.1.1 Phase and group velocity ······································································
8
2.1.2 Signal velocity ······················································································
12
2.1.3 Signal propagation in a dispersive medium ··········································
12
2.2
Dispersion management in an optical medium ················································
14
2.3
Optical resonances in optical fibers ·································································
15
2.3.1 Nonlinearities in optical fibers ·····························································
17
2.3.2 Four-wave mixing ·················································································
16
2.3.3 Narrow band optical parametric amplification ·····································
18
2.3.4 Stimulated Raman scattering ································································
19
2.3.5 Coherent population oscillation ····························································
19
Bibliography ·············································································································
21
3 Brillouin scattering ·································································································
25
3.1
3.2
Linear light scattering ······················································································
26
3.1.1 Generalities ···························································································
26
3.1.1.1 Perturbed wave equation ························································
28
3.1.2 Rayleigh scattering ···············································································
28
3.1.3 Spontaneous Brillouin scattering ··························································
29
Stimulated Brillouin scattering ········································································
33
3.2.1 Electrostriction ······················································································
33
CONTENTS
xvi
3.2.2 Stimulated scattering process ·······························································
34
3.2.2.1 SBS-induced gain resonance ·················································
35
3.2.2.2 SBS-induced loss resonance ··················································
38
3.2.2.3 Usual simplifications in the description of the SBS process ··
38
3.2.3 Graphical illustration of stimulated Brillouin scattering ······················
40
Bibliography ·············································································································
41
4 Brillouin slow & fast Light in optical fibers ·························································
43
4.1
4.2
4.3
4.4
4.5
4.6
Basic Brillouin slow light system ····································································
44
4.1.1 Analytical model of Brillouin slow light ···············································
44
4.1.1.1 Signal delay via stimulated Brillouin scattering ····················
44
4.1.1.2 Signal distortion in Brillouin slow light ································
47
4.1.2 Principle of Brillouin slow light ···························································
49
Broadband Brillouin slow light ·······································································
52
4.2.1 Arbitrary bandwidth by pump modulation ···········································
52
4.2.2 Gain-doublet by a bichromatic pump ···················································
54
4.2.3 Gain-doublet by a concatenated fiber ···················································
56
4.2.3.1 Direct consequence of the linearity of Brillouin ···················
amplification
57
4.2.3.2 Broadband window in the center of gain doublet ··················
59
Transparent Brillouin slow light ······································································
64
4.3.1 Principle ································································································
64
4.3.2 Transparent Brillouin gain resonance ···················································
66
4.3.3 Signal delay with small amplitude change ··········································
68
Optimized shape of signal for Brillouin slow light ·········································
72
4.4.1 Optimization of the signal spectral width ·············································
72
4.4.2 Enhanced signal delay ··········································································
74
Reduced broadening signal delay in Brillouin slow light ································
77
4.5.1 Compensation of signal distortion ························································
78
4.5.2 Nearly non-broadened signal delay ······················································
80
Self-advanced Brillouin fast light ····································································
83
4.6.1 Noise-seeded stimulated Brillouin scattering ·······································
83
4.6.2 Characteristics of spontaneous Stokes wave ········································
85
4.6.3 Self-advanced signal propagation ·························································
87
CONTENTS
xvii
4.6.4 Self-adapted signal bandwidth ······························································
90
Dispersive delay line based on wavelength conversion ··································
92
4.7.1 Wavelength conversion using cross gain modulation····························
92
4.7.2 Continuous control of large fractional delay ·······································
93
Bibliography ·············································································································
97
5 Slow light and linear light-matter interactions ····················································
101
4.7
5.1
Principle ···········································································································
102
5.2
Group velocity change through the fiber gas cell ············································
103
5.3
Effect of slow light on Beer-Lambert absorption ············································
105
Bibliography ·············································································································
109
6 Conclusions and Perspectives ················································································
111
Bibliography ·············································································································
114
List of Figures
2.1
Waveforms of plane wave propagating in space at times t and t+Δt
8
2.2a
On left temporal evolution of the electric field of a monochromatic plane wave
9
that is an infinite sine wave and on right its spectrum obtained by a Fourier
transform, showing a Dirac distribution δ(ωo) with zero spectral width.
2.2b
On left temporal evolution of the electric field of a Gaussian shaped pulse
9
representing an optical signal, and on right its spectrum is presented, showing
a Gaussian distribution with a spectral width proportional to Г-1.
2.3
Schematic representation of the generation of an optical pulse. The peak of the
10
pulse appears at the position, where a large number of frequency components
are all in phase.
2.4
Dispersion curve of an optical medium, showing the representation of the
11
phase velocity vp and the group velocity vg in the dispersion diagram.
2.5
Relationship between either an absorption or a gain narrowband resonance and
15
the refractive index, governed by the Kramers-Kronig relations. The strong
dispersion in the vicinity of the absorption and gain resonances induces fast
light or slow light propagation, respectively.
2.6
Four-wave mixing interaction in optical fibers. When the input electric field is
17
composed of two distinct optical waves at frequencies of ω1 and ω2 the
interaction of the two waves generates two new frequency components at
frequencies ωs and ωas.
2.7
(a) CPO realized in a simple two-level system. (b) Relevant energy levels in
erbium-doped optical fibers used for CPO slow and fast light. T2 is a simple
dipole moment dephasing time, which determines the spectral width of the
absorption band.
20
LIST OF FIGURES
3.1
Typical spectral components of spontaneous scattering in an inhomogeneous
xx
27
medium.
3.2
Illustration of Stokes scattering in terms of dispersion relations.
31
3.3
Illustration of anti-Stokes scattering in terms of dispersion relations.
31
3.4
Measured spectral profile of spontaneously scattered Brillouin Stokes and
32
anti-Stokes waves in optical fibers. A fraction of the pump wave was also
simultaneously detected for a clear demonstration of frequency shifts of the
scattered waves.
3.5
The Brillouin gain resonance with a Lorentzian shape and the associated phase
37
shift.
3.6
The Brillouin loss resonance with a Lorentzian shape and the associated phase
38
shift.
3.7
Schematic representation of the four different effects involved in SBS as a
40
parametric process. The successive realization of this feedback loop reinforces
the energy transfer (counterclockwise succession).
4.1.1
Schematic illustrations of the principle of group velocity control using
46
stimulated Brillouin scattering in optical fibers. On left, SBS gain resonance at
frequency -νB below the pump frequency induces a large normal dispersion
across the gain band, which is responsible for signal delay. On the contrary, on
right, the generation of signal advancements in the vicinity of a SBS loss
resonance is depicted.
4.1.2
Transformation of signal after propagating through a Brillouin slow light
48
medium, where T(w) represents the transfer function of Brillouin slow light.
IFT; inverse Fourier transform.
4.1.3
Schematic diagram to generate Brillouin slow and fast light in optical fibers.
49
VOA; variable optical attenuator, OI; optical isolator.
4.1.4
Measured time delays for a 1 MHz sine modulated signal as a function of the
50
signal gain. The experiment was repeated for different signal input powers at
3 μW and 23 μW with star and square symbols, respectively.
4.2.1
The effective Brillouin gain spectrum geff (ν), resulting from the convolution
of the pump power spectrum and the intrinsic Brillouin gain spectrum.
52
LIST OF FIGURES
4.2.2
Effective SBS gain spectra induced by a randomly modulated pump source.
xxi
53
(a) The first experimental demonstration of the broadband Brillouin gain
resonance, and (b) the maximum achievable SBS bandwidth limited by the
overlap of the SBS loss resonance. The right wing of the gain is canceled out
by the identical left wing of the loss, resulting in the maximum bandwidth
ΔνB ≈ νB. (c) However, when introducing another pump (pump2) the SBS
gain2 generated by the pump2 compensates the SBS loss1, which allows the
further extension of the effective bandwidth of the SBS gain1.
4.2.3
Schematic diagram to produce a Brillouin gain/loss doublet using a two-tone
54
pump generated by external modulation. EOM; electro-optic modulator, OI;
optical isolator.
4.2.4
(a) The SBS doublet generated by the spectrally broadened bichromatic pump
56
source. (b) The time waveforms of the signal pulses after propagating through
the fiber for different pump powers, showing clear signal advancement.
4.2.5
Principle of the single-pumped passive configuration to generate a SBS gain
57
or loss doublet. A partial overlap of the gain spectra can be created by using a
spectrally broadened pump, as shown on the bottom situation.
4.2.6
Experimental setup to realize fast light propagation with low distortion, by
59
appending two optical fibers showing different Brillouin shift and a spectrally
broadened pump laser. EDFA: erbium doped fiber amplifier, VOA: variable
attenuator; EOM: electro-optic modulator, PC: polarization controller.
4.2.7
The spectral profiles of gain-doublets as a function of frequency for different
60
spectral widths of the pump while the pump power is kept constant.
4.2.8
Normalized temporal traces of the signal pulses after propagating through the
61
concatenated fibers for different pump powers, showing clear signal
advancements.
4.2.9
Signal advancements for a 1 MHz sine modulated signal as a function of the
61
pump power in the optimum delay-bandwidth conditions (resonance
separation: 120 MHz, effective resonance width 40 MHz).
4.2.10 Signal delays for 1 MHz sine modulated signal with respect to the pump
power after propagating through two different optical fibers. Insert shows the
Brillouin loss doublet created in the Brillouin loss regime.
62
LIST OF FIGURES
4.3.1
Principle of the experimental configuration to generate transparent spectral
xxii
65
resonances where two distinct optical pumps were used to produce Brillouin
gain and loss, respectively.
4.3.2
Experimental setup to realize transparent slow light via optical fibers, by
66
spectrally superposing a gain spectrum over a loss spectrum, generated from
distinct sources showing different linewidths. VOA: variable attenuator; BPF:
band-pass filter; FBG: fiber Bragg grating; EDFA: Erbium-doped fiber
amplifier.
4.3.3
Variation of the amplitude of the probe signal as a function of frequency after
67
propagation through a 2-km fiber, showing the achievement of a wellcompensated SBS gain/loss profile.
4.3.4
Time traces of signals after propagation in a fiber with a transparent profile for
68
different pump1 powers, showing a clear delay and a minor amplitude change.
Traces are non-normalized and measured in unmodified experimental
conditions. Arrows indicate the pulses peak position.
4.3.5
Delays and amplitudes for a 1 MHz sine modulated signal as a function of the
69
pump1 power in a transparent slow light configuration. The pump2 power is
12 times larger than the pump1 power.
4.3.6
Advancements and amplitudes for a 1 MHz sine modulated signal as a
71
function of pump1 power in a transparent fast light configuration. Power of
Pump2 is 8 times larger than power of Pump 1. The insert shows the measured
SBS gain/loss profile in this configuration.
4.4.1
Signal pulses under study (exponential, Gaussian and rectangular time
73
distribution from left to right with corresponding spectra on bottom chart).
Numerical and measured results are shown in solid and dashed lines,
respectively.
4.4.2
Experimental setup to demonstrate the effect of the pulse shape on the time
74
delay. FBG: fiber Bragg grating, EDFA: erbium doped fiber amplifier, VOA:
variable optical attenuator, PC: polarization controller.
4.4.3
Normalized time traces of the signal pulses with pump power at 0 mW,
75
20 mW, 35 mW and 50 mW, showing a clear time-delay dependence on the
signal shape.
4.4.4
Comparison of the temporal delays as a function of the pump power for the
signal pulses with three different shapes.
76
LIST OF FIGURES
4.5.1
Schematic diagram of an attenuation imbalanced nonlinear optical fiber loop
xxiii
79
to compress the shape of signal. E1 and E2 present clock and counter-clock
wise electric fields of pulses, respectively. DSF; dispersion shifted fiber, PC;
polarization controller and α; an attenuation factor.
4.5.2
Experimental setup to produce non-distorted signal delays, by combining a
80
nonlinear generation element with a typical Brillouin slow light system.
EDFA; erbium doped fiber amplifier, EOM; electro-optic modulator, FBG;
fiber Bragg grating, VOA; variable optical attenuator, DSF; dispersion shifted
fiber, PC; polarization controller and α; an attenuation factor.
4.5.3
(a) Normalized waveforms of signals that experienced time delays through
81
SBS slow light and (b) normalized waveforms of transmitted signals through a
saturable absorber, showing noticeable pulse compression.
4.5.4
Factional delays and broadening factors of signal pulses, respectively, with
82
square and star symbols as a function of signal gain when the nonlinear loop
mirror is present (filled symbols) or absent (opened symbols).
4.6.1
Principle of the configuration to generate self-advanced fast light. The signal
84
power is high enough to generate a strong amplified spontaneous Stokes wave,
which in turn depletes the signal wave. The depletion is assimilated to a
narrowband loss spectrum.
4.6.2
Experimental configuration to realize the self-pumped signal advancement
86
based on both amplified spontaneous and stimulated Brillouin scattering.
EOM; electro-optic modulator, EDFA; Erbium-doped fiber amplifier, VOA;
variable optical attenuator, DSF; dispersion shifted fiber.
4.6.3
(a) Measured optical powers of the Stokes waves and transmitted signals.
87
(b) Linewidths of the generated Brillouin Stokes waves recorded in the ESA,
by use of the delayed homo-heterodyne system.
4.6.4
Temporal traces of the signal pulse after propagating through the dispersion
88
shifted fiber for different input signal powers, showing clear advancements.
4.6.5
Temporal advancements of the signal pulses as a function of the signal
average power, showing logarithmic dependence of delay on the signal power.
88
LIST OF FIGURES
4.6.6
xxiv
(a) Temporal traces of data streams for a signal power below the critical power
89
(solid line) and at maximum signal power realized in our setup (dashed line).
(b) Signal advancement as a function of the average signal power, showing the
logarithmic dependence over the Brillouin critical power at 10 dBm.
4.6.7
(a) Measured spectra of the Stokes emission by the delayed self-homodyne
90
technique, for different signal widths at a constant normalized repetition rate.
(b) Measured linewidth of back-sacttered Stokes wave as a function of the
measured signal bandwidth.
4.7.1
Schematic diagram of the principle to generate the XGM wavelength
93
conversion, in which the pump and signal waves counter-propagate through
semiconductor optical amplifier.
4.7.2
Experimental setup to generate a wide range of signal delays, using
94
wavelength conversion through semiconductor optical amplifier and group
velocity dispersion in optical fibers.
4.7.3
Time waveforms of the delayed signal pulse trains while the TLS wavelength
95
was swept from 1550 nm to 1556 nm by 2 nm steps, showing clear delays of
the signal pulse.
4.7.4
The relative signal delays as a function of the TLS wavelength and the red
95
th
curve represents the result of the 4 order fitting.
4.7.5
The associated signal broadening to achieve 1000-bits delay for a transform
96
limited Gaussian pulse with a width of 100 ps FWHM and the required
wavelength change as a function of group velocity dispersion.
5.1
(a) SEM image of the solid-core microstructured photonic crystal fiber. 103
(b) Calculated mode field distribution of the fundamental mode and (c) the
small evanescent fraction of the guided field present in air holes.
5.2
The experimental setup to verify the effect of slow light on BLB absorption. 104
EDFA; erbium doped fiber amplifier, VOA; variable optical attenuator, PC;
polarization controller, EOM; electro-optic modulator, FBG; fiber Bragg
grating.
5.3
(a) Time waveforms of the signal pulse after propagating through the PCF for 105
the different pump powers and (b) the time delays achieved in this Brillouin
delay line as a function of pump power and the associated slow-down factor.
5.4
Variation of the signal amplitude in logarithmic scale after propagating 106
through the PCF gas cell for different pump powers.
LIST OF FIGURES
5.5
xxv
Measured optical power loss at the peak attenuation due to the Beer-Lambert 106
absorption as a function of the slow-down factor. The error bars show the
measured standard deviation on the attenuation measurement and the red line
represents the hypothetical response expected for an absorption coefficient
inversely proportional to the group velocity.
5.6
Amplitude variation of the pulsed signal in logarithmic scale after propagating 107
through the PCF gas cell for different pump powers.
5.7
Measurement of pulsed signal power loss at the peak attenuation due to the 107
Beer-Lambert absorption as a function of the slow-down factor.
List of Tables
3.1
Typical characteristics describing several light scattering processes in liquids.
27
The gain concerns the stimulated version of the processes in silica.
3.2
Characteristics of the Brillouin spectrum at different pump wavelengths. In all
33
situations, the acoustic velocity in silica was considered as a constant,
va=5775 m/s.
4.1
Values of the FWHM time-bandwidth product K for various pulse shapes.
u(t) is the unit step function at the origin..
73
Chapter 1
Introduction
The physics of wave propagation in a dispersive medium has a long history that goes back to
more than a century. Lord Rayleigh introduced the concept of group velocity when he
discovered that dispersive materials could modify the propagation velocities of wave packets,
and predicted the possibilities to make the group velocity faster than the phase velocity in
sound waves. The transformation of this theory from sound to light wave led to the apparent
contradiction with the fundamental principles of special relativity and causality. Such a
problem has been already mentioned by A. Sommerfeld in 1910’s and extensively debated in
the scientific community. To resolve this problem, A. Sommerfeld and L. Brillouin
introduced the concept of signal velocity that must be distinguished from group velocity.
They noted that a signal undergoes severe distortion while propagating through a material
with large anomalous dispersion, so that group velocity turns meaningless in terms of
information propagation. Thus they concluded that a signal can not propagate at a velocity,
faster than c (c being the light velocity in vacuum).
Slow light has been coined in modern science for a phenomenon that allows a light
signal to propagate with a reduced group velocity. Although slow light was initially
motivated by scientific curiosity, this technique has been rapidly developed for many
potential applications for which slow light has been identified as a solution. As a matter of
fact, the possibility to exert an optical control on the group velocity of an optical signal could
have important implications in modern applications such as high capacity data networks and
their associated modern technologies: all-optical signal processing, optical buffering and
quantum computing, just to mention a few. Over the last decade, successful experiments on
slow light propagation has been demonstrated with an astonishing control of the group
velocity, namely the velocity at which a signal travels in a material, from nearly stopping it to
exceeding the vacuum velocity c or even reaching negative velocities.
CHAPTER 1. INTRODUCTION
2
So far, slow light has been realized in a wide variety of materials with different
physical phenomena, i.e. ultra-cold or -hot atomic gases [1-5], crystalline solids [6,7], microcavities [8,9], semiconductors [10,11], optical fibers [12-15] and photonic crystal structures
[16,17]. Apparently a wide diversity of slow light schemes has been demonstrated, but it
must be pointed out that all schemes are characterized by one common feature: the presence
of one or multiple strong resonances to obtain sufficient dispersion in the media. Besides,
some recent works have already explored the fundamental limits of the maximum time delay
achievable in slow light devices. For instance, fractional delay (the ratio of time delay to
signal duration) or delay-bandwidth product and significant distortion imposed onto the
temporally delayed signals after passing through a slow light device have been studied to
evaluate the potential applications in telecommunication systems.
L. Hau et al. demonstrated, in 1999, the first pioneering slow light propagation in
ultra-cold atomic gases as cold as 450 nK to obtain a Bose-Einstein condensate [1]. The light
velocity was spectacularly reduced down to 17 m/s, equivalent to the speed of a bike, in the
vicinity of a narrowband electromagnetically induced transparency (EIT). The extreme
environmental conditions were, however, soon identified as a scientific challenge to be
solved by the photonic community. M. Bigelow et al. proposed, in 2003, a novel solution to
overcome this issue, using a solid crystal [7]. They created a spectral hole as narrow as
612 Hz FWHM within the absorption band of alexandrite crystal, using the quantum effect of
coherent population oscillations (CPO). Then they spectrally placed a signal in the center of
the spectral hole and observed successfully, at room temperature, both ultra-slow and
superluminal propagation of light in a solid crystalline. Although this technique resulted in
remarkable reduction of the group velocity, it seems still far from acceptable for real
application systems since it operates at a very well-defined wavelength and the achievable
time delay in this system is very limited to around 10 % of the signal duration. Besides, the
bandwidth of the spectral hole restricted inherently the signal bandwidth to the kHz range.
Therefore, the scientific challenge to generate practical control of signal delay remains still
fascinating.
A significant step towards real applications was achieved by Song et al. in 2005
when slow light was experimentally and efficiently realized in an optical fiber using
stimulated Brillouin scattering (SBS) [13]. It has been then experimentally proved that
Brillouin slow light can readily reproduce most of the former results in slow and fast light,
showing group velocities from as slow as 71000 km/s to superluminal or even negative group
velocities. One can argue that the scale of reduction in group velocity through Brillouin slow
light is much far from being as large as the reduced group velocities observed in the slow
light systems based on electric transitions such as EIT or CPO. However, after introducing
CHAPTER 1. INTRODUCTION
3
the concept of fractional delay, this argument was no more issued since fractional delay has
been the parameter of interest for real applications.
To date, Brillouin slow light systems have proved to be an unmatched and
unprecedented flexible timing tool as a result of their unique spectral tailoring capability.
What makes Brillouin slow light a particularly attractive delay line lies in the high flexibility
of SBS, essentially the possibility to be achieved in any type of optical fiber and at any
wavelength. Moreover, Brillouin slow light presents some inherent advantages: room
temperature operation, high potential for large signal bandwidths, simple tabletop
configuration, and seamless integration in fiber optic communication systems. Despite the
apparent perfection of Brillouin slow light, this scheme requires novel solutions to resolve
intrinsic problems. First, since the linewidth of natural Brillouin resonance is as narrow as 30
MHz, the signal bandwidth in optical communication systems is strictly restricted up to 50
Mbit/s. Second, the amplitude of the delayed signal is strongly amplified or attenuated by the
associated Brillouin gain or loss, respectively. Third, the signal suffers from significant
distortion, expressed by signal broadening, since the signal experiences frequency-dependent
gain and phase distortion through slow-light. Signal distortion is an intrinsic deadlock in any
kind of slow light system.
The objective of this thesis work has been to propose possible solutions to the three
major bottlenecks in SBS delay-lines: signal bandwidth, signal amplification and signal
distortion, so as to enable the full exploitation of Brillouin slow light for all-optical signal
delaying and processing. In this thesis, several novel configurations were introduced to
realize broadband Brillouin slow light, nearly zero-gain Brillouin slow light and nondistorted Brillouin slow light. Moreover, it is found that the achievable signal delay via
Brillouin slow light was clearly enhanced when a signal shape is optimized, so that the
spectral width of the signal is minimized to precisely match the bandwidth of a given
Brillouin gain resonance. Departing from typical Brillouin slow light systems, it could be
clearly observed that the propagation velocity of signal could be speeded up using an
extremely simple configuration and in absence of any distinct pump source. This scheme is
referred to as self-advanced Brillouin fast light. In addition, an alternative delaying technique
was devised in order to match the real data rate in telecommunication systems (up to tens of
Gbit/s), which makes use of the combination of wavelength conversion and group velocity
dispersion in a material. Finally, the effect of slow light on linear light-matter interaction was
experimentally investigated to verify the recent scientific question if slow light can enhance
the Beer-Lambert absorption while light propagates through a gas cell.
CHAPTER 1. INTRODUCTION
4
Organization of this thesis
The content of this dissertation is addressed in four distinct chapters and a conclusion.
Chapter.2 defines and addresses the different characteristic velocities of a light signal for a
better understanding of the underlying concepts behind slow light. In the second part, an
analytical description of the propagation of a light signal in a dispersive medium is discussed.
The evolution of the signal envelope during propagation is obtained by a representation in the
Fourier domain. In the last part, the types of spectral resonances, which can be obtained in
optical fibers for the generation of slow light, will be briefly reviewed.
Chapter 3 is devoted to the physics of Brillouin scattering in a material. The generation of
spontaneous scattering will be described in terms of the thermodynamic nature and then
stimulated Brillouin scattering will be presented as a parametric process of two counterpropagating optical waves and acoustic wave.
Chapter 4 is the core of this thesis. The essential contributions of this thesis work to Brillouin
slow light systems are experimentally demonstrated to propose adequate solutions for
resolving the critical issues related to signal bandwidth, signal amplification and signal
distortion. The actual limitations of Brillouin slow light are also briefly discussed. Another
approach to realize an all-optical tunable delay-line is followed, resulting in very large signal
delays suitable for high speed network systems.
Chapter 5 is dedicated to clarify the possibilities of slow light to enhance the linear
interactions between light and matter.
Chapter 6 will present the conclusions and perspectives of this fascinating technique.
CHAPTER 1. BIBLIOGRAPHY
5
Bibliography
[1]
L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17
meters per second in an ultracold atomic gas,” Nature, 397, 594-598 (1999).
[2]
M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R.Welch, M. D. Lukin, Y.
Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced
nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82,
5229-5232 (1999).
[3]
C. Liu, Z. Dutton, C. H. Behroozi and L. V. Hau, “Observation of coherent optical
information storage in an atomic medium using halted light pulses,” Nature, 409, 490493 (2001).
[4]
M. D. Stenner, D. J. Gauthier and M. A. Neifeld, “The speed of information in a fastlight optical medium,” Nature, 425, 695-698 (2003).
[5]
L. J. Wang, A. Kuzmich and A. Dogariu, “Gain-assisted superluminal light
propagation,” Nature, 406, 277-279 (2000).
[6]
A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R.
Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett.
88, 023602 (2002).
[7]
M. S. Bigelow, N. N. Lepeshkin and R. W. Boyd, “Superluminal and slow light
propagation in a room-temperature solid,” Science, 301, 200-202 (2003).
[8]
A. Yariv, Y. Xu, R. K. Lee and A. Scherer, “A coupled resonator optical waveguide: a
proposal and analysis,” Opt. Lett. 24, 711-713 (1999).
[9]
J. E. Heebner and R. W. Boyd, “Slow and fast light in resonator-coupled waveguide,” J.
Mod. Opt. 49, 2629-2636 (2002).
[10] P. C. Ku, F. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. Wang, S. W.
Chang and S. L. Chuang, “Slow light in semiconductor quantum wells,” Opt. Lett. 29,
2291-2293 (2004).
[11] J. Mork, R. Kjaer, M. van der Poel and K. Yvind, “Slow light in a semiconductor
waveguide at gigahertz frequencies,” Opt. Express, 13, 8136-8145 (2005).
[12] N. Brunner, V. Scarani, M. Wegmuller, M. Legré, and N. Gisin, “Direct measurement of
superluminal group velocity and signal velocity in an optical fiber, Phys. Rev. Lett. 93,
203902 (2004).
[13] K. Y. Song, M. G. Herráez, and L. Thévenaz, “Observation of pulse delaying and
advancement in optical fibers using stimulated Brillouin scattering,” Opt. Express, 13,
82-88 (2005).
CHAPTER 1. BIBLIOGRAPHY
6
[14] J. E. Sharping, Y. Okawachi, and A. L. Gaeta, “Wide bandwidth slow light using a
Raman fiber amplifier,” Opt. Express, 13, 6092-6098 (2005).
[15] D. Dahan and G. Eisenstein, “Tunable all optical delay via slow and fast light
propagation in a Raman assisted fiber optical parametric amplifier: a route to all
optical buffering,” Opt. Express, 13, 6234-6249 (2005).
[16] A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic
crystal waveguides,” Appl. Phys. Lett. 85, 4866-4868 (2004).
[17] Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow
light on a chip with photonic crystal waveguides,” Nature, 438, 65-69 (2005).
Chapter 2
Optical signal propagation in a
dispersive medium
Prior to the discussion about the recent progress that leads to a control of the speed of a light
signal, it is important to consider how to define the characteristics of the velocity of a light
wave propagating in a dispersive medium. In early works [1,2], the light speed has been
characterized in at least eight different ways: phase velocity, group velocity, information (or
signal) velocity, centro-velocity, energy velocity, relativistic velocity, ratio-of-units velocity,
correlation velocity and even more. This section, however, will introduce the first three
velocities which are the most useful quantities for real photonic communication systems, and
in particular group velocity will be discussed in detail.
2.1 Signal propagation
Let us consider the light propagation in an optical waveguide. Once either the electric or
magnetic field experiences a perturbation at some point in space, an electromagnetic wave is
generated and travels from that point. Like all electromagnetic phenomena, the propagation
of light is completely governed by the classical Maxwell’s equations. For a nonconducting
medium without free charges these equations are given as [3,4]:
∇ × E = −∂B / ∂t , B = μo H + M
∇ × H = −∂D / ∂t , D = ε o E + P
∇⋅D = 0
∇ ⋅ B = 0,
( 2.1)
( 2.2)
( 2.3)
( 2.4)
where E and H represent the electric and magnetic field vectors, respectively, and D and B
correspond to electric and magnetic flux densities. By applying a curl operator to Eq.(2.1)
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
8
and Eq.(2.2) the electric and magnetic fields can be decoupled in these equations and thus a
light wave can be derived for a single field E or B. As a result, the propagation equation for
optical fields can be written as:
∇2 E = −
1 ∂2E
,
c 2 ∂t 2
1
= μoε o ,
c2
(2.5)
where the light velocity c in vacuum is determined by the electric and magnetic permittivity
of the waveguide material, εo and μo, respectively. In isotropic and dielectric media, the
vacuum constants εo and μo must be replaced by the corresponding constants for the medium
ε and μ. Thereby, the propagation speed of the optical fields in a material is given by
v = (με)-1/2. This allows the introduction of the index of refraction n, which represents the
ratio between the light speeds in vacuum and in a medium, n=c/v. A particular well known
solution of the second order differential equation Eq.(2.5) in an homogeneous medium is a
monochromatic plane wave:
E ( z , t ) = Re{Eo exp[i (ω t − k z )]},
(2.6)
where ω and k are the angular frequency and wavevector, respectively. The wavelength λ is
usually described as the measured distance over which a wave propagates in one complete
cycle, λ =2π/k. The wave equation (2.5) describes the propagation of an electric field E along
the longitudinal axis z. At a given time the amplitude of this field varies sinusoidally with the
distance z, as shown in Figure.2.1, and at a given point z in space the electric wave varies
harmonically with time t.
E
E(z,t)
E(z,t+Δ t)
Δz
Z
Figure 2.1: Waveforms of plane wave propagating in space at times t and t+Δt.
2.1.1 Phase and group velocity
Phase velocity
It must be pointed out that a monochromatic plane wave represented by a simple cosine
function has essentially an infinite duration and its spectral content is given by a Dirac
distribution δ (ωo) since it contains only one angular frequency ωo. Let us assume that a
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
9
monochromatic plane wave propagates through a medium with refractive index n. The phase
of this wave is straightforwardly given by:
φ = ωo t − kz ,
(2.7)
and the phase velocity is in general defined as the velocity at which particular points of
constant phase travel through the medium for the plane wave. Thus it is given as:
vp =
ωo
c
= .
k n
(2.8)
If the phase is known at defined positions at a given time, the phase of a light wave
propagating over a certain depth in the medium can be recovered at a later time using the
phase velocity. In order words, the phase velocity indicates only how the phase of the
monochromatic plane wave is delayed in the medium.
Group velocity
A light signal such as an optical pulse has totally different properties when compared to a
sinusoidal wave in terms of duration and spectral distribution. A pulse can be simply
constructed by multiplying Eq.(2.6) by a bell-shaped function. This way a Gaussian-shaped
pulse can be obtained by:
⎧⎪
⎡ 1 ⎛ t ⎞2
⎤ ⎫⎪
E (0, t ) = Re ⎨ Eo exp ⎢ − ⎜ ⎟ + iωot ⎥ ⎬ ,
⎢⎣ 2 ⎝ to ⎠
⎥⎦ ⎭⎪
⎪⎩
(2.9)
where to is a factor related to the duration of the Gaussian envelope. Figure.2.2 depicts its
temporal envelope and spectrum compared to those of a monochromatic wave, where the
FT
Delta function
δ(ωo)
ωo
t
ω
Figure.2.2a: On left temporal evolution of the electric field of a monochromatic plane wave that is an infinite
sine wave and on right its spectrum obtained by a Fourier transform (FT), showing a Dirac distribution δ(ωo)
with zero spectral width.
FT
t
ωo
ω
Figure 2.2b: On left temporal evolution of the electric field of a Gaussian shaped pulse representing an optical
signal, and on right its spectrum is presented, showing a Gaussian distribution with a spectral width
proportional to Г-1
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
10
spectra of the waves are simply obtained by a Fourier transform of the envelope function of
the signal. It is important to discuss the spectral components of a light signal. Once again a
monochromatic plane wave contains only one angular frequency ωo, whereas a light signal
naturally requires a full set of frequency components. Therefore, the electric field of the
signal is given by the superposition of the electric fields of a set of monochromatic waves:
E ( z , t ) = ∑ Em exp [i (ωm t − km z ) ] .
(2.10)
m
When the waves are all added up in phase the peak of the signal appears and moves forward
at group velocity vg, as shown in Figure.2.3.
E1 = sin (ω1t − k1 z )
E2 = sin (ω2t − k2 z )
E3 = sin (ω3t − k3 z )
E = ∑ sin (ωmt − km z )
m
E ( z, t )
2
Figure 2.3: Schematic representation of the generation of an optical signal. The peak of the signal appears at
the position, where a large number of frequency components are all in phase.
When electromagnetic fields interact with atoms and molecules in a material the
optical response of the medium depends in general on the angular frequency ω. Thereby, the
index of refraction n(ω) becomes a function of the angular frequency. Such a phenomenon is
designated as material dispersion. In addition, the confinement of light in a waveguide
structure gives rise to another type of dispersion, referred to as waveguide dispersion, since
the propagation constant β is dependent on the angular frequency. Now let us launch an
optical signal into a dispersive medium. What will be the effect of dispersion on the signal? If
the medium has a negligibly small dispersion or is vacuum the signal will travel without any
distortion through the medium at the phase velocity as a sinusoidal wave does. Otherwise, in
a dispersive medium phase velocity does not describe properly the process of signal
propagation. This distinction between phase velocity and group velocity was first discovered
by L. Rayleigh [5]. More details on signal propagation in the dispersive medium will be
discussed in section 2.2.
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
11
To mathematically understand how a signal propagates through a medium, one may
start from the basic expressions for the wavevector k=2π/λ and the wavelength λ=2πc/ω in
the medium. Due to the dispersion in the medium each frequency component ω contained in
the signal propagates at its own phase velocity and the phase of the signal is given by:
φ=
nω z
−ω t.
c
(2.11)
After the signal propagates a certain distance through the medium, the phase must be constant
to the first order approximation in the angular frequency ω, so dφ/dω=0:
d n(ω ) ω z n z
+
− t = 0.
dω c
c
(2.12)
The traveling distance z of the signal is determined as z=vg t, where vg is defined as
vg =
⎛
dω
c
ω dn(ω ) ⎞
=
≈ v p ⎜1 −
⎟,
dk n(ω ) + ω dn(ω ) / dω
⎝ n(ω ) dω ⎠
(2.13)
showing the relationship between phase velocity and group velocity. The second term
depends on the dispersion characteristics of the medium and modifies the propagation
velocity of the signal. It leads to the possibility to generate slow light propagation in presence
of normal dispersion (dn/dω > 0) and fast light propagation in anomalous dispersion
(dn/dω < 0). Figure.2.4 depicts the visual distinction between the two velocities in a
dispersive medium, using a dispersion diagram. The phase velocity vp is simply calculated as
the slope of a straight line from the origin to the point P at an angular frequency ωo while the
group velocity vg is determined by the slope of the line tangential to the curve at the same
frequency ωo.
ω
⎛ω⎞
⎟
⎝ k ⎠ ωo
ν p (ω o ) = ⎜
ωo
P
⎛dω ⎞
⎟⎟
⎝ d k ⎠ω o
ν g (ω o ) = ⎜⎜
0
k
Figure 2.4: Dispersion curve of an optical medium, showing the representation of the phase velocity vp and the
group velocity vg in the dispersion diagram.
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
12
2.1.2 Signal velocity
Einstein claimed in his theory of special relativity that no light can travel with velocity
higher-than-c. However, looking more carefully at Eq.(2.12), one may see an apparent
contradiction to the special relativity. In a medium with anomalous dispersion where vg > vp,
a light signal seems to show superluminal propagation through the material or even negative
velocity. Sommerfeld first pointed out this contradiction and it has then been of interest to
discuss it extensively. Addressing to solve this issue, Sommerfeld and Brillouin introduced
the concept of signal velocity (or information velocity) and claimed that there must be a
distinction between signal velocity and group velocity. They found that in a region of small
absorption or dispersion, the signal velocity doesn’t differ much from the group velocity
whereas in the presence of strong absorption large differences between the two velocities can
appear. Unlike other velocities, it is not easy to analytically formulate the signal velocity
because defining a signal is very ambiguous. Nevertheless, a signal has been defined in
various ways, i.e. Brillouin stated that a signal is a short isolated succession of wavelets, with
the system at rest before the signal arrived and also after it has passed. Eventually, one
intends to impose information onto the non-zero part of a signal with an arbitrary shape.
Now let a signal proceed through a material and let us detect two instants, ta and tb
when the non-zero part of the signal passes point A and point B (B being situated further
along the path of propagation), respectively. Just like other velocities are defined the signal
velocity is given by the ratio of the distance and the time. The definition of the signal velocity
in this manner, however, raised difficulties with the principle of causality, because when
considering superluminal propagation or a negative velocity the effect would happen before
its cause. Since the work of Sommerfeld and Brillouin, many researches [5-16] have been
done to verify whether or not the information velocity could exceed the vacuum velocity c
and also if the group velocity could represent the signal velocity near a strong absorption. It is
now widely accepted that for signals in slow light conditions signal velocity is considered as
group velocity, but in ultra fast light conditions when the signal velocity is faster than c the
information encoded in the signal does not exceed the vacuum velocity c and thus propagates
at its own velocity that turns distinct from the group velocity [12,15]. However, this argument
is still discussed openly among scientists.
2.1.3 Signal propagation in a dispersive medium
To further study the signal propagation through a dispersive medium, let us consider a
Fourier transform limited Gaussian pulse for simplicity. If one defines the initial electric field
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
13
of the signal E(0,t), its initial spectral distribution Ẽ(0,ω) is essentially related to the envelope
function after performing a Fourier transform:
⎡ 1 ⎛ t ⎞2
⎤
E (0, t ) = Eo exp ⎢ − ⎜ ⎟ + iωo t ⎥ ,
⎢⎣ 2 ⎝ to ⎠
⎥⎦
(2.14)
⎡ 1
⎤
E (0, ω ) = E o exp ⎢ − (ω − ωo ) 2 to 2 ⎥ .
⎣ 2
⎦
(2.15)
When the signal travels in a transparent medium, its spectrum is modified due to the
dispersion characteristics of the material. The signal propagation can be readily predicted in
the Fourier domain once the transfer function of the material is well defined, which is
normally given as T(ω) = exp[-ik(ω)]. After the signal propagates a distance z inside the
optical medium, the signal spectrum in the output evolves according to the relation
E ( z , ω ) = E (0, ω ) exp [ −i k (ω ) z ] ,
k (ω ) = nω / c ,
(2.16)
where k is the wavevector. It is convenient to expand the wavevector in a Taylor series with
respect to angular frequency, for which the signal should fulfill the condition, Δω << ωo,
(Δω being the spectral width of the signal and ωo being the center frequency). This condition
results in an approximate analytical solution for the signal propagation. Substituting the
Taylor expansion of the wavevector
k (ω ) = ko (ωo ) + k1 (ω − ωo ) + 12 k2 (ω − ωo ) 2 + 16 k3 (ω − ωo )3 +"
(2.17)
⎛ d k (ω ) ⎞
⎛ d 2 k (ω ) ⎞
k1 = ⎜
⎟ , k2 = ⎜
⎟
2
⎝ d ω ⎠ωo
⎝ d ω ⎠ωo
into Eq.(2.15), the signal spectrum becomes
1
i
⎡
⎤
E ( z , ω ) = exp ⎢ −i k (ωo ) z − i k1 (ω − ωo ) z − ( to 2 + i k2 ) (ω − ωo ) 2 z + k3 (ω − ωo )3 z ⎥ . (2.18)
2
6
⎣
⎦
The parameters k1 and k2 represent the group velocity and the group velocity dispersion,
respectively, and k3 is related to the dispersion slope. If the carrier frequency is far away from
the zero dispersion frequency of the medium the contribution of higher order terms for the
dispersion turns negligible [17].
Neglecting the third order dispersion term k3, the time waveform of the signal after
propagation through the medium is then obtained using the inverse Fourier transform of
Eq.(2.18). As a result, the time evolution of the signal is analytically derived as:
2
⎡
⎡
⎛
⎛
⎞ ⎤
z ⎞⎤
z
2
⎥.
exp ⎢ −2 ( to + i k2 z ) ωo ⎜ t −
E ( z , t ) = A exp ⎢ −iωo ⎜ t −
⎜ v (ω ) ⎟⎟ ⎥⎥
⎜ v (ω ) ⎟⎟ ⎥
⎢
p
o ⎠⎦
g
o ⎠
⎝
⎝
⎣⎢
⎣
⎦
(2.19)
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
14
The first exponential term takes into account the phase delay of the signal with respect to the
angular frequency ωo after traveling a distance z, but it doesn’t have any visible effect on the
temporal distribution of the signal and doesn’t carry any information since it modifies only
the phase of the sinusoidal waves. The second exponential term, however, is more interesting
to discuss as far as the signal propagation is concerned. The shape of the signal remains
Gaussian, but it is clearly observed that the peak of the signal is delayed by an amount z/vg
depending on the group velocity vg or group index ng (ng being c/vg) after the signal has
passed through the medium with a length z. Now let us consider the second order dispersion
term k2, referred to as group velocity dispersion (GVD). In general, the quadratic dependence
of the refractive index on frequency gives rise to GVD, which causes normally a signal
broadening if k2≠0 unless the signal has an initial frequency chirp with opposite sign to the
value of GVD. The higher order dispersion terms in the material may also induce
supplementary phase distortion if they turn non negligible when compared to k2. Therefore, in
real slow light systems, the signal experiences both time delay and distortion. It will be
discussed later in detail.
2.2 Dispersion management in an optical medium
As previously mentioned, the dispersive characteristics of an optical medium are the crucial
parameters to produce dynamic time delays for a light signal. All that is required to
manipulate the group velocity is only the presence of spectral resonances in the material,
which gives rise to a rapid change of the group index as a function of frequency. Actually,
this is readily obtained within absorption or gain bands that show a complex transfer function.
According to Kramers-Kronig relations the real part of the frequency-dependent refractive
index nr is related to the absorption α within the material by [18]:
nr (ω ) =
α (ω ) = −
2
π
2ω
π
∞
p.v. ∫ d ω ′
0
∞
ω ′ α (ω ′)
ω ′2 − ω 2
p.v. ∫ d ω ′
0
nr (ω ′)
,
ω ′2 − ω 2
(2.20)
(2.21)
where p.v. represents the Cauchy’s principal value. An analysis of these relations shows that
a spectrally narrow absorption tends to induce a sharp transition of the refractive index in the
material, which in turn leads to a strong anomalous dispersion (dn/dω<<0) associated with
signal advancement or fast light. On the contrary, a peak or gain band will create a strong
normal dispersion (dn/dω>>0) in the material, resulting in signal delay or slow light. This
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
15
situation for a Lorentzian shaped absorption is graphically illustrated in Figure.2.5. The steep
linear variation of the refractive index, making dn/dω large in absolute value, induces in turn
a strong change of the group index. Thereby, a large change of the relative time delay for a
pulse can be observed after the pulse propagates through the material.
As a result, the key point to obtain pulse delays or advancements in slow light
systems is to find some physical processes that can provide optical resonances showing the
necessary spectral features, namely a narrow bandwidth and a strong amplitude.
α
Kramers-Kronig
relations
nr
ng = n + ω
dn
dω
ng
Advancement
Delay
Figure 2.5: Relationship between either an absorption or a gain narrowband resonance and the refractive
index, governed by the Kramers-Kronig relations. The strong dispersion in the vicinity of the absorption and
gain resonances induces fast light or slow light propagation, respectively.
2.3 Optical resonances in optical fibers
A flurry of schemes for tailoring the dispersive properties of an optical medium has been
actively developed in diverse materials over the last decade, in order to reduce and control
the group velocity of a signal [19]. In early works, the most widely used methods to engineer
the dispersion have been conducted using atomic resonances in gas atoms. However, optical
fibers have recently started to stimulate this research in modern photonic systems since they
show unique properties, such as low-loss transmission, flexible integration with most optical
transmission systems, and operational wavelengths at telecommunication windows.
In this section, a brief overview of physical mechanisms that are responsible for
creating either absorption or gain spectral resonances in optical fibers will be presented. So
far, the most efficient approach is to take advantage of optical interactions accompanied with
nonlinear optical phenomena. Once a strict and necessary phase matching condition in such
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
16
processes is satisfied to generate an idler wave, energy transfer occurs through the idler wave
form one optical wave to another optical wave, usually referred to as pump and signal waves,
respectively. If the signal wave benefits from the process of the energy transfer to experience
a linear gain, the propagation speed of the signal through the medium will be reduced and
will be subject to slow light. On the contrary, an optical wave which undergoes a linear loss
from this process will propagate at an increased group velocity, so that fast light will be
observed in transmission. Since the title of this thesis refers to Brillouin slow and fast light,
both Brillouin gain and loss resonances will be discussed in detail in Chapter.3.
2.3.1 Nonlinearities in optical fibers
The predominant schemes to manage dispersion characteristics in optical fibers rely on
nonlinear effects. For an intense electromagnetic field, the response of any dielectric to light
becomes nonlinear. More specifically, the polarization P induced in a medium becomes
nonlinear with the applied field, and thus nonlinear terms in the electric field E begin to
appear in the material response. The magnitudes of the high-order dependence on the electric
field are determined by the nonlinear susceptibilities χj [18],
P = ε o ( χ1 E + χ 2 EE + χ 3 EEE +")
= PL1 + PNL2 + PNL3 +" ,
where εo is the vacuum permittivity. The polarization P consists of the linear
susceptibility χ1 and nonlinear susceptibilities χ2 and χ3. The linear susceptibility provides
the dominant contribution to the polarization and its effect is included in the index of
refraction n and the absorption coefficient α. On the other hand, the nonlinear parametric
processes are categorized as second- or third-order processes depending on whether χ2 and χ3
is involved, respectively. In isotropic media at the molecule level, the second-order
susceptibility χ2 vanishes completely in the dipole approximation [18,20]. As optical fibers
are constituted of amorphous silica, the second-order nonlinear interactions such as secondharmonic generation and sum-frequency generation can not be observed in fibers in a passive
way [21]. However, the third-order parametric processes can occur in both isotropic and
anisotropic media and is scaled by the third-order susceptibility χ3. As a consequence, one
can produce in fibers the nonlinear effects such as third-harmonic generation, four-wave
mixing and parametric amplification [21].
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
17
2.3.2 Four-wave mixing
Wave mixing turns out to be one of the most important phenomena in modern nonlinear
optics. When two or more waves simultaneously transmit in a nonlinear medium the
interactions between the waves tend to produce new frequency components that were not
present in the incident field. Due to this particular feature, four-wave mixing attracted
considerable attention in wavelength division multiplexed communication systems for
potential applications such as the development of tunable light source and the efficient
implementation of wavelength conversion [22-25]. Actually, four-wave mixing comes from
the nonlinear response of bound electrons to the applied optical field and refers to the
interactions of four different waves through the third-order nonlinear polarization. In
quantum mechanical terms, when the phase matching condition is satisfied, photons from one
or more waves are annihilated by modulated refractive index and are then transferred at
different frequencies. As a result, it leads to the generation of idler waves while energy and
momentum are conserved during this process. These conditions are usually satisfied with
high efficiency in the vicinity of the zero dispersion wavelength of the material as it allows
the waves to propagate at similar group velocities.
For the four-wave mixing process, three distinct waves at frequencies ω1, ω2 and ω3
are present in the incident field and can be coupled to give a fourth wave (signal wave). If the
signal wave is created at frequency ω4=ω1+ω2+ω3, this nonlinear process will be referred to
as third harmonic generation in the particular case: ω1=ω2=ω3, but it is rarely observed in
optical fibers with high efficiencies. In practice, a signal wave generated at frequency
ω4=ω1+ω2-ω3 is the most widely used situation to generate wavelength conversion. Once the
frequency matching condition (ω3+ω4=ω1+ω2) and the associated phase matching condition
(k3+k4=k1+k2) occur, photons from the incident waves are annihilated to simultaneously create
photons at frequencies ω3 and ω4. In the particular case when only two distinct waves at
Δω
Δω
ωs
Δω
ω1
ω2
ωas
ω
Figure 2.6: Four-wave mixing interaction in optical fibers. When the input electric field is composed of two
distinct optical waves at frequencies of ω1 and ω2 the interaction of the two waves generates two new
frequency components at frequencies ωs and ωas.
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
18
frequencies ω1 and ω2 (ω1 < ω2) are present in the incident field, it is relatively
straightforward to satisfy the strict phase matching conditions. This partially degenerate case
is the most relevant in optical fibers. The refractive index of the fiber is modulated by the
beat frequency between the two waves, Δω = ω2 - ω1. As a result, new frequency components,
which were not present in the incident electric field, are created at the frequency of
ωs = ω1 - Δω = 2ω1 - ω2 (Stokes wave) and ωas = ω2 + Δω = 2ω2 - ω1 (anti-Stokes wave), as
shown in Figure 2.6. The whole nonlinear interaction is described by a model coupling the
three waves [22,26]. When a weak signal at frequency ωs is simultaneously launched into the
fiber the signal will be amplified, and such amplification is called parametric gain. However,
it is still difficult to maintain the phase matching conditions along the fiber since microscopic
fluctuations in material density such as birefringence and chromatic dispersion increase the
phase mismatch. Therefore, in practice, the parametric gain using 4-wave mixing can hardly
induce a sharp change of refractive index to realize slow and fast light in optical fibers.
2.3.3 Narrow band optical parametric amplification
Brillouin slow light experimentally demonstrated some impressive results on signal delays
for the first time in fibers. Nevertheless, photonic community identified the narrowband
feature of the Brillouin resonance as an issue and explored new schemes to increase the
capacity of digital data streams. D. Dahan et al. [27,28] proposed the use of narrow band
optical parametric amplification as optical resonances. The experimental setup made use of
the combination of parametric amplification and Raman amplification in optical fibers; the
coupling of two nonlinear effects offers the possibility to modify the spectral profile of
resonances. Therefore, both signal delay and advancement were observed in this system with
a large bandwidth at tens of Gbit/s rates for real communication applications.
The parametric gain in optical fibers has been used to generate parametric optical
resonances or amplifiers. According to the theoretical prediction of the three-wave mixing
model, partially degenerated four-wave mixing allows to transfer energy from a strong pump
wave to a weak signal and an idler wave. The parametric gain via this process differs much if
the signal and idler waves are launched into the fiber simultaneously with the pump, so that
one can control the direction of the energy flow by adjusting the phase mismatching between
the three waves. In particular, the signal can be either amplified or attenuated depending on
the relative input phase between the incident waves. On the contrary, in the case where a
signal power is relatively negligible compared to a pump power, the signal amplitude will
start to be amplified. Such parametric gain is generally a linear function of the pump power
and fiber length under phase matching conditions, thus this system could offer a linear all-
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
19
optical delay line. Moreover, the spectral bandwidth of parametric amplification is related to
the fiber length and group velocity dispersion [20], so that this delay line can be properly
designed for a required data capacity.
2.3.4 Stimulated Raman scattering
As a solution to overcome the narrow signal bandwidth of Brillouin slow light, stimulated
Raman scattering in optical fibers was utilized by J. E. Sharping et al. to produce a large
bandwidth slow light [29]. In this technique, an all-optical delay line was realized for ultrashort optical signals spectrally placed in the center of a Raman gain resonance. The group
velocity of the signal was then continuously controlled by the wavelength and power of the
pump wave. As a result, a signal pulse with transform-limited duration of 430 fs was
temporally delayed up to 85 % of its initial pulsewidth in this experiment.
Stimulated light-scattering processes have attracted a large interest as efficient
nonlinear phenomena that can provide broadband amplifiers and tunable lasers. In such
processes, optical fibers play an active role as nonlinear materials since they involve
molecular vibrations or acoustic phonons that are responsible for Raman and Brillouin
scattering, respectively. Stimulated Raman scattering is similar to stimulated Brillouin
scattering in the sense that it is an inelastic process. A fraction of pump power is rapidly
transferred to vibrational modes of media and to a so-called Stokes wave whose frequency is
lower than the pump frequency. Since this process accompanies the interactions between
photons and optical phonons of the material structure, the energy difference between the
pump and the Stokes waves is significantly large. This yields a frequency shift for the Stokes
wave of about 10 THz, namely three orders of magnitude greater than Brillouin shift. Further
details on Raman scattering can be found in [18,20].
2.3.5 Coherent population oscillation
Coherent population oscillation (CPO) is a quantum effect that can generate a spectrally
narrow hole in the center of an absorption profile due to a wave-mixing interaction. Such a
dip due to CPO was first predicted in 1967 from the interaction between two optical beams in
a saturable absorber [30]. The type of spectral hole burning was for the first time
experimentally observed in a ruby crystal and the spectral width of 37 Hz was measured at
half width at half maximum [31]. Recently, this narrow spectral feature has been exploited
for slow and fast light propagation at room temperature in semiconductor structures [32-37]
CHAPTER 2. OPTICAL SIGNAL PROPAGATION IN A DISPERSIVE MEDIUM
20
~ 1/T2
b absorption band
b
metastable c
τba = T1
a ground state
(a)
τbc
τca = T1
a
(b)
Figure 2.7: (a) CPO realized in a simple two-level system. (b) Relevant energy levels in erbium-doped optical
fibers used for CPO slow and fast light. T2 is a simple dipole moment dephasing time, which determines the
spectral width of the absorption band.
such as quantum wells, quantum dots and quantum structure optical amplifiers, in solid
crystals [38,39] and even in optical fibers [40-42].
The principle of CPO is in general described in a two-level system as depicted in
Figure.2.7. When a strong pump wave illuminates a saturable medium, the population of the
ground state is excited to the first absorption band. Then the population returns to the ground
state in a few milliseconds (T1). A weak probe wave is then launched to the medium and
interferes with the pump. The induced beating signal makes the population of electron
oscillate between the ground state and the excited state. Due to the long decay time T1, the
population oscillations are only appreciable under the condition that the product of δ and T1
is ~1, where δ is a beating frequency between the pump and probe waves. When this
condition is satisfied, the pump wave can efficiently scatter off the ground state population
into the probe wave. As a result, a spectral hole appears at the center of the probe frequency
and the spectral width of the hole is inversely proportional to the population relaxation
lifetime T2.
Recently, CPO was also observed in single-mode erbium-doped fibers (EDF) [40-42].
In practical point of view, EDF gives rise to longer interaction lengths and stronger effects
for coherent population oscillations. In addition, since EDF amplifier plays the role of both
an absorber and an amplifier depending on the pump power, this system opened the
possibilities to realize slow and fast light and experimentally demonstrated the superluminal
propagation with negative group velocity [41]. In fact, the erbium atoms in fibers are
effectively a three-level system with a strong absorption band as shown in Figure.2.6, but the
population of the absorption band decays very rapidly to a metastable level in a few
picoseconds [20]. Thereby, such system can be regarded as a two-level system.
CHAPTER 2. BIBLIOGRAPHY
21
Bibliography
[1]
R. L. Smith, "The velocities of light," Am. J. Phys., 38, 978 (1970).
[2]
S. C. Bloch, "Eight velocity of light," Am. J. Phys., 45, 538 (1977).
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[12] J. C. Garrison, M. W. Mitchell, R. Y. Chiao, and E.L. Bolda, "Superluminal signals:
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[16] M. D. Stenner, D. J. Gauthier, and M.A. Neifeld, "The speed of information in a fastlight optical medium," Nature, 425, 695-698 (2003).
[17] G. P. Agrawal, Fiber-optic Communication System, 3rd ed, (John Wiley & Sons, Inc.,
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[18] R. W. Boyd, Nonlinear Optics, 2nd ed., (Academic Press, New York, 2003).
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CHAPTER 2. BIBLIOGRAPHY
22
[20] G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed., (Academic Press, San Diego, 1995).
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CHAPTER 2. BIBLIOGRAPHY
23
[35] F. Ohman, K. Yvind, and J. Mork, "Voltage-controlled slow light in an integrated
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Chapter 3
Brillouin Scattering
Since the invention of the laser in the early 1960’s, the optical response of a transparent
medium to light has started to be considered as nonlinear. When the optical intensity of the
incident light increases to reach a critical level the light-matter interaction in the medium
starts to build up nonlinear effects during the light propagation [1,2]. In optical fibers,
nonlinear effects manifest through some parametric processes such as four-wave mixing,
self-phase modulation and cross-phase modulation. In these nonlinear phenomena, the
refractive index of the fiber is modulated with respect to the light intensity and the induced
index grating scatters photons from the incident lights. Light scattering such as stimulated
Raman and Brillouin scattering can also occur in fibers as a result of parametric interaction of
light waves through optical or acoustic phonons (molecular vibrations). Due to the active
participation of the nonlinear medium in these processes, a strict phase matching condition
can be straightforwardly satisfied. Once the linear dispersion relations between light and
phonons are matched, the incident light is strongly scattered and the scattered light is
frequency-shifted. In this chapter, Brillouin scattering in silica will be discussed in detail, in
terms of both spontaneous and stimulated processes.
CHAPTER 3. BRILLOUIN SCATTERING
26
3.1 Linear light scattering
3.1.1 Generalities
Light scattering is known as a general physical process in an optical medium. When one or
more localized non-uniformities are present in the medium, light intends to be forced to
deviate from a straight trajectory while being transmitted through the medium. In other words,
light scattering occurs as a consequence of fluctuations in the optical properties of the
medium. Nevertheless, light can be scattered under conditions such that the optical properties
of the medium are not modified by the incident light. This scattering is referred to as
spontaneous or linear scattering. In optical fibers, the type of spontaneous scattering is more
concretely identified in terms of spectral components of the scattered light. It is separated in
two main categories: elastic scattering or inelastic scattering. However, the universal
principles of energy and momentum conservations are preserved in all cases.
• In elastic scattering, the induced light from a scattering process has an identical spectral
profile to the incident light.
• In inelastic scattering, energy exchange occurs between light and the dielectric medium,
which leads to a frequency shift between incident and scattered lights.
Under most general circumstances, the scattered light shows spectral distributions as depicted
in Figure.3.1, where Brillouin, Rayleigh, Rayleigh-wing and Raman features are depicted [1].
Note that the peak amplitude and the spectral width of the scattered light are not correctly
scaled, but Table 3.1 provides some physical parameters for those processes. By definition,
the down-shifted frequency components are referred to as Stokes components while the upshifted components are called anti-Stokes waves. Different spectral characteristics of the
scattered light are observed for different scattering processes since distinct types of
interaction between the radiation and the media are respectively involved. The four different
scattering phenomena can be briefly described as [1]:
• Brillouin scattering is an inelastic process. Light is scattered by a traveling density
perturbation associated with propagating pressure waves in the medium. Formally, it is
considered as light scattering by the interaction of photons with acoustic phonons in the
medium. The frequency of the acoustic mode is relatively low, so the induced frequency
shift is ranged in the order of 1010 Hz below the incident frequency.
• Rayleigh scattering is a quasielastic process, which originates from non-propagating
density fluctuations. Formally, it can be described as light scattering resulting from
CHAPTER 3. BRILLOUIN SCATTERING
27
fluctuation of material entropy, i.e. the degree of molecular organization states. This effect
doesn’t occur in mono-crystal structure, but in amorphous media such as optical fibers it
turns to be an important phenomenon.
• Rayleigh-wing scattering is scattering in the wings of the Rayleigh line. In general, it
results from fluctuations in the orientation of asymmetric molecules. Since the molecular
reorientation process is very rapid, the wings spread over a very large spectral width.
• Raman scattering is a highly inelastic process. The interaction between light and the
vibrational modes of molecules in the media scatters the light. In particular, optical
phonons of the material structure diffract photons from the light. The scattered photons are
then transferred at frequency, which is largely shifted from the incident light frequency by
some 1013 Hz.
anti-Stokes components
Stokes components
Rayleigh
Brillouin
Brillouin
Raman
Raman
Rayleigh-wing
vo
v
Figure 3.1: Typical spectral components of spontaneous scattering in an inhomogeneous medium.
Shift [cm-1]
Linewidth [cm-1]
Relaxation time [s]
Gain [m/W]
Brillouin
1
5∙10-3
10-9
2∙10-13
Rayleigh
0
5∙10-4
10-8
Rayleigh-wing
0
5
10-12
Raman
1000
5
10-12
Process
5∙10-11
Table.3.1: Typical characteristics describing several light scattering processes in liquids. The gain concerns
the stimulated version of the processes in silica.
Since the last two processes are not directly relevant to the content of this thesis, a full
description of these scattering processes will be devoted to References [1-4], but only the
scattering processes related to static and propagating density fluctuations are discussed in the
next sections.
CHAPTER 3. BRILLOUIN SCATTERING
28
3.1.1.1 Perturbed wave equation
When light propagates in optical media, the wave equation (2.5) has to take into account the
polarization field P in order to describe the material response to the light and is modified as:
∇2 E −
1 ∂2 E
∂2 P
= μo 2 ,
2
2
c ∂t
∂t
(3.1)
For homogeneous and isotropic media, the polarization field is proportional to the applied
electric field E and to the associated dielectric susceptibility of the media χ: P=εoχE.
Substituting this linear dependence into Eq.(3.1), the wave equation is expressed as:
∇2 E −
n2 ∂ 2 E
= 0,
c 2 ∂t 2
(3.2)
where n = 1 + χ = ε / ε o presents the refractive index, ε being the dielectric constant. The
wave equation (3.2) in an optical medium is identical to Eq.(2.5), excepted the propagation
velocity of light: in the medium, the vacuum velocity c is replaced by v=c/n. Nevertheless, at
the scale of 10-10 m (~atomic radius), a real optical medium can no longer be considered
homogeneous since microscopic variations of density along the media will give rise to
inhomogeneities along the direction of light propagation. Therefore, the constant scalar
dielectric susceptibility χ can be conveniently replaced by a tensor χ+Δχ(r,t) [5], so that after
substituting P=εoχE +εoΔχE = εoχE + ΔεE, Eq.(3.2) becomes
∇2 E −
n2 ∂ 2 E
∂2
=
μ
(Δε E ).
o
c 2 ∂t 2
∂t 2
(3.3)
The right hand side of this perturbed wave equation is responsible for the mechanism of light
scattering. The perturbation term Δε can be decomposed into scalar and tensorial distributions
as follows: Δε(r,t) = Δεs(r,t) + Δεt(r,t). The diagonal scalar term Δεs describes the fluctuations
in thermodynamic quantities such as density, temperature, entropy or pressure. Formally,
such fluctuations give rise to light scattering such as Brillouin and Rayleigh scattering. On
the contrary, the purely off-diagonal term Δεt can be further decoupled in symmetric and
asymmetric contributions, both statistically independent and intrinsically related to the optical
polarisability of the medium [4]. The symmetric part leads to Rayleigh-wing scattering while
the asymmetric one gives Raman scattering.
3.1.2 Rayleigh scattering
In the perturbed wave equation (3.3), the dielectric constant Δε taken as a scalar quantity is
considered mainly as a function of material density. In linear regime, the presence of light in
CHAPTER 3. BRILLOUIN SCATTERING
29
the medium causes an extremely weak heating while interacting with the material [3], so that
the temperature dependence can be ignored. Thus one can write:
Δε =
∂ε
Δρ .
∂ρ
(3.4)
The fluctuations of the material density Δρ can also be expanded as a function of two
independent quantities, pressure p and entropy s, so that:
Δρ =
∂ρ
∂ρ
Δp +
∂p s
∂s
Δs .
(3.5)
p
The first term describes adiabatic density fluctuations (i.e. acoustic wave) and gives rise to
Brillouin scattering. The second term corresponds to isobaric density fluctuations (i.e.
temperature or entropy variations at a constant pressure) and is responsible for Rayleigh
scattering. Entropy variations are usually described by a diffusion equation, given as:
ρ cp
∂Δs
− κ∇ 2 Δs = 0 ,
∂t
(3.6)
where cp and κ present the specific heat at a constant pressure and the thermal conductivity,
respectively. The general solution of Eq.(3.6) is:
Δs ( r , t ) = Δso exp[ −δ t ]exp[ −i ( q ⋅ r )],
(3.7)
where q is the wavevector of the scattered light. This fluctuations are damped at a rate of
δ=κq2/ρcp. Here, one can see that the entropy wave does not propagate along the media. Now,
let us replace Δs in Eq.(3.5) by the general expression from Eq.(3.7) while discarding the first
term of Eq.(3.5). This shows that the term of the induced polarization has only the same
frequency component ω as the incident light frequency; thereby the scattered light remains
spectrally unchanged.
3.1.3 Spontaneous Brillouin scattering
The equation of motion for pressure fluctuations Δp, which is the first term in Eq.(3.5), is
usually described as an acoustic wave. In general, the induced acoustic wave is governed by
the following equation [1,3]:
∂ 2 Δp
∂Δp
− Γ∇2
− va 2∇2 Δp = 0,
2
∂t
∂t
(3.8)
where Γ is the acoustic damping coefficient, related to the shear and bulk viscosity
coefficients in the medium. Moreover, the sound velocity in the medium va can be expressed
as a function of thermodynamic properties as:
CHAPTER 3. BRILLOUIN SCATTERING
va =
30
K
ρ
=
1
,
Cs ρ
(3.9)
where K and Cs denote the bulk modulus and the adiabatic compressibility, respectively. The
general solution of Eq.(3.8) can be found as a propagation wave:
Δp(r , t ) = Δpo exp[i (q ⋅ r − Ωt )],
(3.10)
and here the phonon dispersion relation, Ω = |q|va must be satisfied to give rise to an acoustic
wave. Inserting Eq.(3.10) into the perturbed term of Eq.(3.3), the wave equation for the
scattered light is obtained as:
∇2 E −
n2 ∂ 2 E γ eCs
= 2 Δpo Eo [(ω − Ω)2 exp[i(k − q) ⋅ r − i(ω − Ω)t ]
c 2 ∂t 2
c
(3.11)
+ (ω − Ω)2 exp[i(k + q) ⋅ r − i(ω + Ω)t ],
where γe is an electrostrictive constant, defined as γ e = ρo (∂ε / ∂ρ ) .
Spontaneous Brillouin Stokes scattering
The first term in the right part of Eq.(3.11) represents a Stokes wave in Brillouin scattering as
an oscillating component. To satisfy energy and momentum conservations, the Stokes wave
must have a wavevector of k'=k-q and an angular frequency of ω'=ω-Ω. Indeed, three waves
are involved in this scattering interaction: the incident wave at frequency ω, the scattered
wave (Stokes wave) at frequency ω' and the acoustic wave at frequency Ω. In principle, the
three waves are entirely coupled together by means of dispersion relations, ω=|k|c/n,
ω'=|k'|c/n and Ω=|q|va. Since the frequency of the sound wave Ω is much smaller than the
optical frequencies ω and ω', one can assume that |q|=2|k|sin(θ/2) with the approximation that
|k|≈|k'|, as illustrated in Figure.3.2. Using the dispersion relations, the frequency of the
acoustic wave can be expressed by:
Ω = 2 k va sin(θ / 2) = 2 nω
va
sin(θ / 2) .
c
(3.12)
In this equation, it is clearly observed that the Stokes frequency shift engendered by Brillouin
scattering is highly dependent on the scattering angle θ. In practice, the frequency shift is
maximum in the backward direction (when θ=π) and is equal to zero in the forward direction
(when θ=0). For this reason, the Stokes wave is usually generated by a receding acoustic
wave (co-propagating with the incident wave). Therefore, the Brillouin Stokes wave
propagates in the opposite direction to the incident light and the Brillouin frequency shift is
given by:
νB =
2 nva
Ω
.
=
λ
2π
(3.13)
CHAPTER 3. BRILLOUIN SCATTERING
31
In a quantum point of view, this process can be represented as photons from the incident light
that are annihilated by the acoustic wave and scattered photons that are created at frequency
-νB below the incident frequency.
k' = k - q
ω' = ω - Ω
q
q, Ω
k, ω
k'
θ
k
Figure.3.2: Illustration of Stokes scattering in terms of dispersion relations.
Spontaneous Brillouin anti-Stokes scattering
An analogous analysis to the Stokes wave can be developed for the second term in the right
part of Eq.(3.11). This process is responsible for the generation of an anti-Stokes wave. In the
same manner, the anti-Stokes wave has a wavevector of k'=k+q and an angular frequency of
ω'=ω+Ω. As illustrated in Figure.3.3, this wave is visualized as light scattered from an
oncoming acoustic wave (counter-propagating with the incident wave). Quantummechanically, Brillouin anti-Stokes is described as simultaneous absorption of both incident
photons and acoustic phonons, followed by the emission of photons at a frequency +νB above
the incident frequency.
k' = k + q
ω' = ω + Ω
q
q, Ω
k, ω
k'
θ
k
Figure.3.3: Illustration of anti-Stokes scattering in terms of dispersion relations.
Figure.3.4 depicts the measured spectrum of spontaneous Brillouin scattering in optical fibers,
where the center peak represents the pump wave. It is clearly shown that the spontaneous
Stokes and anti-Stokes waves are equally shifted from the pump frequency with identical
amplitude.
CHAPTER 3. BRILLOUIN SCATTERING
32
Amplitude, a.u.
-60
-70
Brillouin
Stokes
-80
Brillouin
anti-Stokes
-90
-100
-30
-20
-10
0
10
20
30
Frequency, GHz
Figure.3.4: Measured spectral profile of spontaneously scattered Brillouin Stokes and anti-Stokes waves in
optical fibers. A fraction of the pump wave was also simultaneously detected for a clear demonstration of
frequency shifts of the scattered waves.
The damping parameter Γ in eq. (3.8) for the acoustic wave has been ignored so far. If this
term is inserted in the complete analysis of Brillouin scattering, a spectral distribution
obviously appears for the light scattered into direction θ. The spectral width of Brillouin
scattering at full width at half maximum (FWHM) is given by:
2
δω = Γ q = 4 Γ n 2
ω2
c2
sin(θ / 2).
(3.14)
Due to the damping characteristics, the acoustic wave only propagates over several cycles of
the incident light and its intensity undergoes an exponential decrease as:
2
2
Δ p (t ) = Δ p (0) exp[ −t / τ p ],
(3.15)
where τp=(Γ/|q|2)-1 is the average life time of acoustic phonons in media, inversely depending
on the acoustic damping coefficient. In the Fourier domain, the backscattered light has a
spectral profile with a Lorentzian line shape with a FWHM of:
δω
1
.
=
2π πτ p
(3.16)
The phonon life time varies substantially with the frequency of the acoustic wave, and thus
with the frequency of the incident light [2]. Table 3.1 represents some spectral characteristics
of Brillouin scattering in silica with respect to the incident wavelength. The velocity of
acoustic wave in standard single-mode fibers is estimated to be va=5775 m/s in this table.
CHAPTER 3. BRILLOUIN SCATTERING
Wavelength [nm]
33
Brillouin shift [GHz]
Brillouin linewidth [MHz]
514
34.0
100
832
21.6
70
1320
12.8
35
1550
10.8
29
Table.3.2: Characteristics of the Brillouin spectrum at different pump wavelengths. In all situations, the
acoustic velocity in silica was considered as a constant, va=5775 m/s.
3.2 Stimulated Brillouin scattering
Spontaneous Brillouin scattering was discovered in 1922 by Leon Brillouin [6] as light is
scattered by the thermally fluctuating density in a medium. However, the advent of the laser
in the early 60’s changed the physical view on Brillouin scattering since the intense coherent
optical wave significantly enhanced the scattering process. Consequently, more photons from
the laser beam are annihilated whereas the Stokes wave experiences an exponential growth
due to the amplitude growth of the acoustic wave. This regime corresponds to the
phenomenon referred to as stimulated Brillouin scattering (SBS). SBS as a nonlinear
scattering was first observed in 1964 by Chiao et al. in quartz and sapphire [7].
3.2.1 Electrostriction
Electrostriction is basically described as the tendency of a material to become compressed in
the presence of an electric field. As a consequence of the maximization of the potential
energy, molecules in a medium tend to be attracted into the regions where an intense electric
field is localized. The potential energy density of media u is proportional to the dielectric
constant as:
u=
ε
(3.17)
E 2.
8π
When the material density is perturbed by an amount Δρ, the associated dielectric constant
also changes from the origin value εo to the value εo+Δε, where
Δε =
∂ε
Δρ .
∂ρ
(3.18)
Therefore, the energy density Δu can be expressed according to the variation of the dielectric
constant:
Δu =
E2
E 2 ⎛ ∂ε ⎞
Δε =
⎜ ⎟ Δρ .
8π
8π ⎝ ∂ρ ⎠
(3.19)
CHAPTER 3. BRILLOUIN SCATTERING
34
According to the first law of thermodynamics, the change in energy Δu must be equal to the
work Δw, which is performed to compress the media:
Δw = pst
ΔV
Δρ
.
= − pst
ρ
V
(3.20)
The electrostrictive pressure pst represents the contribution of the electric field to the pressure
in media. Since Δu=Δw, by equating Eq.(3.19) and Eq(3.20), the electrostrictive pressure can
be derived as:
⎛ ∂ε ⎞ E 2
E2
,
pst = − ρ ⎜ ⎟
= −γ e
8π
⎝ ∂ρ ⎠ 8π
(3.21)
where γ e = ρo (∂ε / ∂ρ ) was already defined as the electrostrictive constant in section 3.1.1.
Since pst has a negative value, the net pressure turns to be reduced in the regions where the
electric field is intense. As a result, the molecules will be naturally pulled into those regions
within the medium, so that the material density will be locally compressed in the medium. In
general, the molecule displacement is very slowly processed compared to the frequency of
the optical waves, so that the effect on the material must be characterized by the time
averaged value <E2> instead of the instantaneous field E2 in Eq.(3.21).
3.2.2 Stimulated scattering process
The process of SBS is classically described as a nonlinear interaction between a pump wave
at frequency νp and a Stokes wave at frequency νS. The two waves are coupled by nonlinear
polarizations. It must be pointed out that the higher order polarization associated to SBS is
not caused by the nonlinear susceptibilities induced by a parametric process, but by a sound
wave induced by mean of electrostriction. Therefore, the description of SBS starts from the
propagation of the three interacting waves using the following equations:
∇2 E p −
2
∂ 2 Pp NL
n2 ∂ E p
μ
=
o
c 2 ∂t 2
∂t 2
(3.22a)
∇2 ES −
∂ 2 PS NL
n2 ∂ 2 ES
μ
=
o
c 2 ∂t 2
∂t 2
(3.22b)
∂ 2 Δρ
∂Δρ
+ Γ∇ 2
− va ∇ 2 ρ = −∇ ⋅ f ,
∂t 2
∂t
(3.22c)
where Ep and ES are the amplitudes of the electric fields (pump and Stokes waves),
respectively. Notice that, here, Δρ denotes the density variation rather than the pressure
variation to describe the acoustic wave. The equations (3.22) can in turn be all coupled
through the electrostrictive force f using the material constitutive relations given by:
CHAPTER 3. BRILLOUIN SCATTERING
P NL (r , t ) = Δε E (r , t ) =
35
γe
Δρ E (r , t )
ρo
2
1
Δ⋅ f (r, t ) = γ e∇2 Ep (r, t ) + ES (r, t ) .
2
(3.23a )
(3.23b)
To simplify the coupled equations involved in SBS, one can assume that:
• The state of polarization (SOP) of the pump wave is parallel to the SOP of the Stokes wave,
designated by the unit polarization vector e.
• The fiber attenuation α is considered negligible while the acoustic damping coefficient Γ is
maintained in the expressions.
• The system is in steady-state conditions, where the life time of acoustic wave is so short
that it can be neglected when compared to that of the pump wave.
• The pump wave is not depleted during the SBS process.
These simplifications will be discussed later in detail.
3.2.2.1 SBS-induced gain resonance
Stimulated Stokes scattering can be observed under the conditions that the dispersion
relations fulfill kp=kS+Ω. The conditions indicate that the acoustic wave co-propagates with
the pump in the direction of +z. In unperturbed situations (when f = 0), the three waves
propagate in the optical medium as independent plane waves represented by the exponential
terms in Eq.(3.24). On the contrary, the presence of the force term f (the so-called
electrostriction) will modify the amplitudes of the waves (Ep, ES and A) through coupling
effects. In such circumstance, the slowly-varying envelope approximation can be applied to
the propagation equations for the waves, resulting in:
E p = e E p ( z, t ) exp[i (ω p t − k p z ) ]
(3.24a)
ES = e ES ( z , t ) exp[i (ωS t + kS z ) ]
(3.24c)
Δρ = A( z, t ) exp[i (Ω t − q z )].
(3.24b)
Let us substitute Equations (3.24) into Equations (3.23). By retaining only the resonant terms
for each wave, one can obtain:
Pp NL = e
PS NL = e
∇⋅ f =
γe
A( z, t ) ES ( z, t ) exp[i (ω pt − k p z ) ]
2 ρo
γe
A∗ ( z, t ) E p ( z, t ) exp[i(ωS t + kS z ) ]
(3.25b)
[ E p ( z, t ) ES∗ ( z, t )]exp[i(Ω t − q z ) ].
(3.25c)
2 ρo
γ e q2
2
(3.25a )
CHAPTER 3. BRILLOUIN SCATTERING
36
In turn, substituting Equations (3.24) and (3.25) into Equations (3.22) results in three coupled
equations responsible for the description of stimulated Brillouin scattering. Due to the
slowly-varying envelope approximation, a simplified form to the coupled equations can be
found:
k pγ e
⎡∂ n ∂⎤
⎢⎣ ∂z + c ∂t ⎥⎦ E p = i 4 ερ A ES
o
(3.26a)
kS γ e ∗
⎡ ∂ n ∂⎤
⎢⎣ − ∂z + c ∂t ⎥⎦ ES = i 4 ερ A E p
o
(3.26b)
⎡ ∂ 2Ω − i Γq2 ∂ Γq2 ⎤
qγ e
∗
+
⎢ +
⎥ A = i 2 E p ES .
2Ω va ∂t 2va ⎦
4 va
⎣ ∂z
(3.26c)
Since the acoustic wave is rapidly damped, the mean free path of acoustic phonons is
typically very small (~10-4 m) compared to the distance over which optical fields vary
significantly. Therefore, the spatial derivative term in Eq.(3.26c) can be conventionally
dropped. The time derivative term can also vanish as assuming the steady-state conditions.
As a result, Eq.(3.26c) can be substantially simplified and the amplitude of the acoustic wave
is in turn given by:
A=i
qγ e
E p ES∗ ,
2Γ B va
(3.27)
where ΓB=Γq2 denotes the bandwidth of Brillouin scattering. Inserting this value into the
equations (3.26) leads to two coupled equations relating the pump and Stokes waves:
∂E p
∂z
=−
q k p γ e2
2
ES E p
8 ερo Γ B va 1 − i(2Δν / Δν B )
(3.28a )
2
E p ES
q kS γ e2
∂ES
=−
,
∂z
8 ερ o Γ B va 1 − i (2Δν / Δν B )
(3.28b)
Analyzing the form of these equations, one can see two main phenomena occurring to the
optical waves during the SBS process: energy transfer and nonlinear phase between the pump
and Stokes waves can be predicted according to the real and imaginary part of the equations,
respectively. This system can be further simplified by introducing the intensities of the
optical waves in the coupled equations while defining I = ½nεoc|E|2:
∂I p
∂z
= − g B (ν ) I p I S
∂I S
= − g B (ν ) I S I P ,
∂z
where gB(ν) represents the Brillouin gain spectrum. It is defined as:
(3.29a)
(3.29b)
CHAPTER 3. BRILLOUIN SCATTERING
37
( Δν B / 2)
,
2
2
(ν − Δν ) + ( Δν B / 2 )
2
g B (ν ) = g B
(3.30)
where gB is the Brillouin gain coefficient. Equivalent expressions of gB in terms of the
constitutive parameters are shown in literatures
gB =
q k p2γ e2
1
8ερo Γ B va nε o c
=
2π n7 p122
,
cλ p2 ρ Δν B va
(3.31)
where λp denotes the pump wavelength and p12 is the longitudinal elasto-optic coefficient. In
bulk silica, the measurable value of gB is approximately 5⋅10-11 m/W. The SBS process
scatters photons off the pump wave so as to create the amplification at the Stokes wave. The
gain spectrum has inherently a Lorentzian distribution centered at the Stokes shift and a
spectral width ΔvB at full width at half maximum, as depicted in Figure.3.5.
vB
Δv
Brillouin gain
ΔvB
Pump
Phase shift
vp
v
Figure 3.5: The Brillouin gain resonance with a Lorentzian shape and the associated phase shift.
The equations (3.29) describe the spatial evolution of the two waves, resulting in an
analytical solution for Stokes wave [8,9] as:
I S ( z ) = I S ( L) exp[ g B I p (0)( L − z )],
(3.32)
where the Stokes wave is launched into the material at z=L. It is clearly seen that the Stokes
wave experiences an exponential growth as a function of the pump intensity while
propagating in the opposite direction to the pump wave.
It is also interesting to discuss the transition of the phase shift associated to the amplification
through SBS. By taking into account the imaginary parts of equations (3.28), the SBSinduced phase shifts for the two waves can be expressed as:
∂Φ SBS
p
∂z
1
(2Δν / Δν B )
= − gB IS
2
1 + (2Δν / Δν B )2
∂Φ SBS
1
(2Δν / Δν B )
S
.
= − gB I p
2
1 + (2Δν / Δν B ) 2
∂z
(3.33a)
(3.33b)
CHAPTER 3. BRILLOUIN SCATTERING
38
The phase shift is usually defined in the manner that a positive ΦSBS causes an additional
positive delay. As a result, positive Δv makes optical waves retarded while the waves are
advanced with negative Δv.
3.2.2.2 SBS-induced loss resonance
SBS-induced loss resonance is mathematically described in the same manner as the Stokes
scattering. The only difference is that the sign of Brillouin gain is negative and the
frequencies of the two waves are swapped (νAS > νp). The interaction between two counterpropagating waves, pump and anti-Stokes, fulfills the following equations:
∂I p
∂z
= − g B (ν ) I p I AS
(3.34a)
∂I AS
= + g B (ν ) I AS I P ,
∂z
(3.34b)
which are formally similar to the equations (3.29). Nevertheless, the change of sign in
Eq.(3.34b) implies that the anti-Stokes signal undergoes an attenuation while propagating in
the fibers. The spectrum of Brillouin absorption is identical to that of Brillouin gain as a
Lorentzian distribution and its characteristics are depicted in Figure.3.6.
vB
Pump
Δv
Brillouin gain
Phase shift
ΔvB
vp
v
Figure 3.6: The Brillouin loss resonance with a Lorentzian shape and the associated phase shift.
3.2.2.3 Usual simplifications in the description of the SBS process
Parallel polarizations
If the states of polarization (SOP) of the pump and Stokes waves are not parallel, the
efficiency of SBS tends to significantly decrease. The scalar product of the polarization
vectors |ep⋅eS| is no longer equal to unity, so that Eq.(3.27) has to be replaced by:
A = −i e p ⋅ eS
E p ES∗
qγ e
.
2Γ B va 1 − i(2Δν / Δν B )
(3.35)
CHAPTER 3. BRILLOUIN SCATTERING
39
Therefore, the polarization mixing efficiency ηp=|ep⋅eS|2 manifests the efficiency of Brillouin
gain in Eq.(3.30) as it is added in the equation:
( Δν B / 2 )
.
2
2
(ν − Δν B ) + ( Δν B / 2)
2
g B (ν ) = η p g B
(3.36)
The effect of ηp is so significant that it can completely suppress the SBS phenomenon when
the SOPs of the two waves are relatively orthogonal.
No fiber attenuation
The most distinguished feature of the optical fibers is the close-to-transparent characteristics.
However, a light wave experiences some linear attenuation during propagation in fibers. The
fiber attenuation is mainly due to Rayleigh scattering and in standard single-mode fibers the
attenuation α is approximately 0.18 dB/km at 1550 nm. When considering this parameter, the
equations (3.29) has to be modified as:
∂I p
∂z
= − g B (ν ) I p I S − α I p
∂I S
= − g B (ν ) I S I P + α I S .
∂z
(3.37 a)
(3.37b)
Unfortunately, exact analytical solutions for these coupled equations are not possible [10],
but numerical implementations for the equations (3.26) while neglecting the fiber attenuation
show good agreement with the experimental results.
Steady-state conditions
The amplitude of the acoustic wave A in Eq.(3.27) was calculated under steady-state
conditions. In such conditions, the optical waves must consist of essentially DC components,
thereby the life time of phonons tphonon can be neglected. However, when the waves are
intensity-modulated, i.e. pulse operations, the Brillouin gain turns to be time dependent. If
duration of the pulse tpulse is significantly longer than tphonon, which represents a quasi-DC
regime, these conditions are still valid. On the contrary, if tpulse < tphonon, one must take into
account the dynamics of the acoustic wave.
Non-depleted pump
In the assumption in which the intensity of the pump in Eq.(3.29) wave remains constant
while transmitting in optical fibers, namely non-depleted pump, the Stokes wave takes
benefits from energy transfer in the SBS process and undergoes an exponential growth in
Eq.(3.32). The SBS gain shows then a logarithmic dependence on the pump intensity, so that
as far as the signal gain is concerned this system can be regarded as a linear system. However,
if the Stokes wave grows to a critical intensity comparable to that of the pump wave, the
pump must start to be significantly depleted and this system turns to be a nonlinear system.
CHAPTER 3. BRILLOUIN SCATTERING
40
3.2.3 Graphical illustration of stimulated Brillouin scattering
Figure.3.7 depicts the schematic illustration of stimulated Brillouin scattering (SBS) as a
parametric process of two counter-propagating optical waves and acoustic wave in optical
fibers. Here, the signal wave at frequency νs propagates in the opposite direction to the pump
wave at frequency νp. Once the frequency separation between two waves is equal to the
Brillouin frequency of the fiber or the frequency of acoustic wave, namely phase matching
conditions (νp=νs+νB), the interference between the pump and signal waves generates an
optical beating resonant at frequency νB so as to reinforce the acoustic wave. In addition, due
to the phenomenon of electrostriction, the material density is strongly fluctuated along the
fiber and the refractive index turns to be modulated along the fiber core. Then this nonlinear
interaction induces a dynamic Bragg-type grating in the fiber core, which co-propagates with
the pump wave and this grating in turn diffracts photons from the pump to the signal, thereby
resulting in the signal amplification. The whole process is self-sustained in a complete
feedback loop and enhances the Brillouin scattering process: it is referred as stimulated
Brillouin scattering.
Pump wave
Optical
energy transfer
Signal wave
Electrical
Self-sustained
Loop
energy density
Refractive
index change
Energy density
Figure 3.7: Schematic representation of the four different effects involved in SBS as a parametric process. The
successive realization of this feedback loop reinforces the energy transfer (counterclockwise succession).
CHAPTER 3. BIBLIOGRAPHY
41
Bibliography
[1]
R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, New York, 2003).
[2]
G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, 1995).
[3]
I. L. Fabelinskii, Molecular Scattering of Light (Plenum Press, New York, 1968).
[4]
Y. R. Shen, The principle of nonlinear optics (Wiley, New York, 1984).
[5]
L. D. landau and E. M. Lifshitz, Statistical physics, 3rd ed. (Butterworth-Heinemann,
1984).
[6]
L. Brillouin, “Diffusion of light and X-rays by a transparent homogeneous body. The
influence of thermal agitation,” Ann. phys. 17, 88-122 (1922).
[7]
R. Y. Chiao, C. H. Townes and B. P. Stoicheff, “Stimulated Brillouin scattering and
coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592-596 (1964).
[8]
R. H. Enns and I. P. Batra, “Saturation and depletion in stimulated light scattering,”
Phys. Lett. A, 28, 591-592 (1969).
[9]
C. L. Tang, “Saturation and spectral characteristics of the Stokes emission in the
stimulated Brillouin process,” J. Appl. Phys. 37, 2945-2955 (1966).
[10] L. Chen and X. Bao, “Analytical and numerical solutions for steady state stimulated
Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65-70 (1998).
Chapter 4
Brillouin slow & fast light in optical
fibers
Slow & fast light based on stimulated Brillouin scattering (SBS) in optical fibers has been
experimentally and efficiently demonstrated at EPFL, Switzerland [1,2] in 2004, and
independently a few months later at Cornell University. Unlike the experimental conditions
and the operational characteristics of the former slow and fast light systems [3-7], Brillouin
slow light was readily realized in standard optical fibers with a simple benchtop configuration.
Moreover, its room-temperature operation at any wavelength has received tremendous
interest from the optical communication society for fascinating potential applications such as
all-optical delay lines, optical buffers and signal synchronizations. To date, Brillouin slow
light has already shown the possibilities to manipulate the group velocity of a light signal
from as slow as vg=71000 km/s to superluminal propagation, even reaching negative
velocities [2]. However, Brillouin slow light is not widely exploited in a continuous data
stream for communication applications due to some inherent features of this system: narrow
signal bandwidth, strong change of signal amplitude and significant signal distortion. So far,
these limitations have been extensively investigated to improve the Brillouin slow light
system, as optimizing the gain spectral profile and the induced dispersion in the material.
This thesis work was also aimed at proposing an adequate solution for each issue to make
Brillouin slow light a completely operating all-optical delaying system.
CHAPTER 4.1. BASIC BRILLOUIN SLOW LIGHT SYSTEM
44
4.1 Basic Brillouin slow light system
4.1.1 Analytical model of Brillouin slow light
4.1.1.1 Signal delay via stimulated Brillouin scattering
As mentioned in the previous chapter, stimulated Brillouin scattering in optical fibers is
usually described as a nonlinear interaction between two counter-propagating waves, a strong
pump at frequency νp and a weak probe at frequency νs. Under particular phase matching
conditions, the two waves are coupled by means of an acoustic wave induced by the
phenomenon of electrostriction. Using the slowly varying envelope approximation, the SBS
interaction is fully described by three coupled amplitude equations [8,9]:
∂E p
∂z
−
+
n ∂E p
= i g1 ES A
c ∂t
∂ES n ∂ES
+
= i g1 E p A∗
∂z c ∂t
∂A ⎛ Γ B
⎞
+⎜
− i δω ⎟ A = i g 2 E p ES∗ ,
∂t ⎝ 2
⎠
(4.1a)
(4.1b)
(4.1c)
where the fiber loss α is neglected. Ep, Es and A are the field amplitudes of the pump, the
probe and the acoustic waves, respectively; n is the refractive index of the fiber; ΓB/2π
represents the bandwidth of the Brillouin resonance at full width at half maximum; δω is the
detuning of the probe frequency from the center frequency of the Brillouin gain resonance,
δω=ωs-ωp+ΩB; ΩB is the Brillouin frequency shift of the optical fiber; g1= ωpγe/(4cnρo) and
g2=γeΩB/(2cnva2). To derive analytical solutions of Equations (4.1), let us consider a weak
interaction regime, so that the continuous-wave pump remains undepleted during the SBS
process. Moreover, assuming steady state conditions, the equations can be simplified as:
−
∂Es n ∂Es
+
= i g1 E p A∗
∂z c ∂t
A∗ =
−i g 2 E ∗p Es
Γ B / 2 + i δω
.
(4.2a )
(4.2b)
After substituting Eq.(4.2b) into Eq.(4.2a), the spatial amplitude evolution of the probe wave
is obtained as:
g1 g 2 I p ⎞
∂Es ⎛ nω
= ⎜i
−
⎟ Es .
∂z ⎝ c
Γ B / 2 + iδω ⎠
(4.3)
CHAPTER 4.1. BASIC BRILLOUIN SLOW LIGHT SYSTEM
45
By simply integrating Eq.(4.3) with respect to space, the dispersion ks(ω) induced by the
stimulated Brillouin scattering process can be expressed as:
ks (ω ) = n
ω
c
go I p
⎞ ω
i⎛
− ⎜
⎟ ≡ n′s .
2 ⎝ 1 + i 2δω / Γ B ⎠ c
(4.4)
From the form of Eq.(4.4) as a complex response function, it is clearly seen that the SBS
process generates a pure gain resonance with a Lorentzian shape, centered at the probe
frequency. The effective complex index n's is given by:
n′s = n − i
go I p
c ⎛
⎜
2ω ⎝ 1 + i 2δω / Γ B
⎞
⎟,
⎠
(4.5)
and the gain coefficient gs(ω), the refractive index ns(ω) and group index ng(ω) can be
expressed as a function of frequency deviation, respectively:
go I p
⎛ 2ω ⎞
g s (ω ) = − ⎜
⎟ Im ( ns′ ) =
1 + 4δω 2 / Γ 2B
⎝ c ⎠
ns (ω ) = Re {ns′ } = n +
ng (ω ) = ns + ω
c go I p
ω
δω / Γ B
1 + 4δω 2 / Γ 2B
c g o I p 1 − 4δω 2 / Γ 2B
dns
= n+
Γ B (1 + 4δω 2 / Γ 2 )2
dω
B
(4.6)
(4.7)
(4.8)
Figure 4.1.1 represents the schematic illustrations, showing the modification of the group
velocity in a material, using stimulated Brillouin scattering in optical fibers. When a
monochromatic pump wave at frequency νp propagates through the fiber, an efficient
narrowband amplification or attenuation can be created by the SBS process, as shown in
Figure 4.1.1a. In a standard optical fiber the Brillouin gain resonance gs with a Lorentzian
shape is spectrally centered at frequency -νB (~11 GHz) below the pump frequency and has a
spectral FWHM width 30 MHz [10]. According to Kramers-Kronig relations, a sharp
transition in the effective refractive index ns is accompanied in the vicinity of the gain
resonance, as shown in Figure.4.1.1b. Then the large normal dispersion (dns/dω >0) induced
by SBS gives rise to a strong change in the group index ng, resulting in an increase in group
index, as shown in Figure.4.1.1c. As a result, the group velocity vg=c/ng at which a signal
travels through the fiber decreases, leading to the generation of slow light and signal delay.
Fast light and signal advancement can also be produced using SBS, simply by swapping the
frequencies of the pump and signal waves. As depicted in the right side of Figure.4.1.1, SBS
can generate a narrowband loss resonance at frequency +νB above the pump frequency and a
large anomalous dispersion across the loss band. Therefore, the group index in the center of
CHAPTER 4.1. BASIC BRILLOUIN SLOW LIGHT SYSTEM
vp - vB
vp
46
vp + vB
(a)
gs
v
Kramers-Kronig
relations
(b)
ns
v
ng = n + ω
(c)
ng
dn
dω
v
delays
advancements
Figure.4.1.1: Schematic illustrations of the principle of group velocity control using stimulated Brillouin
scattering in optical fibers. On left, SBS gain resonance at frequency -νB below the pump frequency induces a
large normal dispersion across the gain band, which is responsible for signal delay. On the contrary, on right,
the generation of signal advancements in the vicinity of a SBS loss resonance is depicted.
the absorption is significantly reduced, and thus it allows a signal to propagate in optical
fibers with an increased group velocity.
To estimate the amount of delay that a signal will experience in a Brillouin delay line,
let us consider a standard single-mode fiber with a length L as a Brillouin gain medium, over
which a signal propagates. After passing through the fiber, the signal arrives at the end of the
fiber with a transit time T=L/vg. Due to the dependence of the group index on the induced
dispersion in Eq.(4.7), the relative transit time of the signal through the fiber can be varied
through the group index change, and thus the group velocity change. In consequence, the
relative signal delay ΔTd between situations with and without the pumping is given by:
ΔTd =
L
G 1 − 4δω 2 / Γ 2B
Δng =
c
Γ B (1 + 4δω 2 / Γ 2 )2
B
G
(1 − 12δω 2 / Γ2B )
ΓB
(4.9)
when 4δω 2 / Γ 2B 1,
where G=goIpL is relevant to the Brillouin gain, Ip is the pump intensity and L is the fiber
length. In the exact center of the resonance (δω=0), the relative signal delay achievable in
Brillouin slow light is then simplified as:
ΔTd G
.
ΓB
(4.10)
CHAPTER 4.1. BASIC BRILLOUIN SLOW LIGHT SYSTEM
47
It is clearly shown that the relative signal delay ΔTd via a Brillouin fiber delay line has a
logarithmic dependence on the net signal gain in most implementations, since the G
exponentiated represents the net signal gain. On the other hand, it is inversely proportional to
the bandwidth of a given Brillouin resonance, so that a narrower gain resonance will result in
a larger signal delay. Nevertheless, using the SBS process a dynamic control of the speed of a
light signal can be all-optically produced in optical fibers, since a signal can actually be
linearly delayed or accelerated using the Brillouin interaction by simply and proportionally
varying the pump intensity Ip.
4.1.1.2 Signal distortion in Brillouin slow light
In absence of gain saturation or pump depletion a Brillouin fiber delay line is usually
described as a stationary linear system as far as the signal transformation is concerned. The
transfer function of the Brillouin slow light system T(ω) can be expressed as:
⎡G ⎛
⎞
1
2ω / Γ B ⎤
⎥
T (ω ) = exp [i k s (ω ) L ] = exp ⎢ ⎜
⎟−i
2
2
⎢⎣ 2 ⎜⎝ 1 + ( 2ω / Γ B ) ⎟⎠ 1 + ( 2ω / Γ B ) ⎥⎦
(4.11)
= exp ⎡⎣G (ω ) + i Φ (ω ) ⎤⎦ ,
where L is the effective length of the fiber used as a Brillouin gain medium. Since the
Brillouin resonance has a complex response, the amplitude and the phase of the signal is
transformed by the real and the imaginary parts of T(ω), respectively. When (2ω/ΓΒ)2 « 1, the
real and imaginary parts of the transfer function of the delaying system can be further
explicitly expressed as:
G (ω ) =
(
G
2
4
6
1 − ( 2 / ΓB ) ω 2 + ( 2 / ΓB ) ω 4 + ( 2 / ΓB ) ω 6 + "
2
(
)
Φ (ω ) = − ( 2 / Γ B ) ω − ( 2 / Γ B ) ω 3 + ( 2 / Γ B ) ω 5 +" .
3
5
)
(4.12a)
(4.12b)
The frequency dependence of the signal gain is responsible for a spectral filtering effect (lowpass filter) and distorts the signal shape. Therefore, a temporal broadening of signal is
essentially observed when a signal emerges from the Brillouin delay line. This is referred to
as gain broadening, BG. On the other hand, due to the large induced dispersion associated to
the imaginary part of T(ω), the higher order dispersion terms (except the first term, which is
directly responsible for the signal delaying) inherently produces an additional phase
distortion, referred to as dispersion broadening, BD. Therefore, the total pulse broadening,
Btotal must be calculated by the geometrical sum of the two broadening factors:
B2total = B2G + B2D. It is clearly shown that the amplitude response G(ω) contains only even
CHAPTER 4.1. BASIC BRILLOUIN SLOW LIGHT SYSTEM
48
terms in its polynomial expansion with respect to angular frequency while the phase response
Φ(ω) is anti-symmetric and contains only odd terms. In other words, these two broadening
effects can not mutually cancel to compensate the signal distortion. Moreover, since there is
no quadratic dependence in the phase response, the introduction of ordinary chromatic
dispersion into the Brillouin slow light system can only bring further distortion. Signal
distortion essentially accompanies signal delay in any kind of linear slow light systems.
Let us launch a transform-limited Gaussian pulse as a light signal into a Brillouin
slow light system. The amplitude of the initial electric field is defined as:
⎡ 1 ⎛ t ⎞2 ⎤
E ( t ) = exp ⎢ − ⎜ ⎟ ⎥ ,
⎢⎣ 2 ⎝ to ⎠ ⎥⎦
(4.13)
where to is related to the pulse duration. The pulse duration at full width at half maximum
(FWHM) is given by tin = 2to ln 2 . The spectrum of the pulse can be simply obtained by the
Fourier transform of the envelope function:
⎡ ω 2to2 ⎤
E (ω ) = 2π to2 exp ⎢ −
⎥.
⎣ 2 ⎦
(4.14)
The spectral width (FWHM) of the initial pulse is given by Δωin = 2 ln 2 / to. The propagation
of the different frequency components forming the signal pulse depends on the dispersive
properties in the fiber modified by SBS amplification, so that the spectrum of the pulse
exiting from the slow light medium is given as:
E out (ω ) = T (ω ) ⋅ E in (ω ) = exp [G (ω ) ] ⋅ exp [i Φ (ω ) ] ⋅ E in (ω )
(4.15)
⎡ ⎛1
⎛ 2
⎞⎤
2G ⎞
8
= exp [G ] exp ⎢ − ⎜ to2 + 2 ⎟ ω 2 − i ⎜ ω − 3 ω 3 ⎟ ⎥ .
ΓB ⎠
ΓB
⎝ ΓB
⎠⎦
⎣ ⎝2
Figure 4.1.2 depicts a simple sketch of the pulse propagation in the Brillouin slow light
system. Unfortunately, the electric field of the output pulse can not be analytically expressed
by performing the inverse Fourier transform of Eq.(4.15). However, if it is assumed that the
output pulse maintains its Gaussian shape, the FWHM width of the output pulse can be
Input signal
Output signal
SBS slow light medium
tin
⎡ 1 ⎛ t ⎞2 ⎤
Ein = exp ⎢ − ⎜ ⎟ ⎥
⎢⎣ 2 ⎝ to ⎠ ⎥⎦
tout
Transfer function
T(ω)=eG(ω) eiΦ(ω)
{
}
Eout = IFT T (ω ) ⋅ E (ω )
Figure.4.1.2: Transformation of signal after propagating through a Brillouin slow light medium, where T(ω)
represents the transfer function of Brillouin slow light. IFT; inverse Fourier transform.
CHAPTER 4.1. BASIC BRILLOUIN SLOW LIGHT SYSTEM
49
approximated by a general Heisenberg’s uncertainty relation. According to this relation, the
product of the spectral and temporal widths for chirped-free Gaussian pulse is 0.441 [11].
The power spectrum of the output pulse is represented by the modulus of Eq.(4.15) and
results in the spectral width (FWHM):
Δωout = 2
ln 2
.
t + 4G / Γ 2B
(4.16)
2
o
As a consequence, the pulse-broadening factor B is found as:
B=
tin Δωout
4
16 ln 2
16 ln 2
=
= 1+ 2 2 G = 1+ 2 2 G = 1+ 2
( ΔTd ) .
tout Δωin
to Γ B
tin Γ B
tin Γ B
(4.17)
It must be noticed that this broadening factor takes into account only gain broadening caused
by the non-flat spectral amplification. For this reason, it can be predicted that a short pulse
whose spectral width is considerably broader than the bandwidth of a given SBS resonance
can not be delayed or advanced without a significant distortion.
4.1.2 Principle of Brillouin slow light
Figure.4.1.3 depicts the schematic principle to generate slow and fast light using stimulated
Brillouin scattering in optical fibers. A basic Brillouin slow light system consists of a very
simple configuration, in which a continuous-wave (CW) pump is launched into one end of an
optical fiber with Brillouin shift νB via a fiber circulator and a signal is simultaneously
introduced into the other end of the fiber. When the pump propagates through the fiber,
acoustic phonons are thermally excited and scatter off photons from the pump light to
produce a SBS gain resonance at frequency -νB below the pump frequency. The spectral
FWHM width of Brillouin scattering is typically ~30 MHz for a standard single-mode fiber.
Pump
Signal in
Brillouin shift vB
VOA
OI
Slow light
vas
vp
vs
vp
Fast light
Delayed signal out
Figure.4.1.3: Schematic diagram to generate Brillouin slow and fast light in optical fibers. VOA; variable
optical attenuator, OI; optical isolator.
CHAPTER 4.1. BASIC BRILLOUIN SLOW LIGHT SYSTEM
50
Besides the gain band, the pump creates a SBS loss resonance with an identical bandwidth at
frequency +νB above the pump frequency. When the signal pulse is spectrally placed at center
of the gain or absorption (particular phase matching conditions) the group velocity of the
signal is efficiently reduced or increased, respectively. Since the amplitudes of the resonances
can be controlled with the pump power, the amount of signal delay can be precisely
determined, simply by varying the pump power using an optical variable attenuator before it
enters into the fiber.
Maximum time delay due to pump depletion
In a small signal regime, Brillouin slow light is usually regarded as a linear delay line,
showing a linear dependence of time delay on logarithmic signal gain, as expressed in
Eq.(4.10). However, the maximum time delay achievable through the delay line is in
principle limited by pump depletion. When the Brillouin system undergoes the pump
depletion the signal gain starts to be saturated, and it may occur that the significantly
amplified signal generates another Stokes wave by self-depletion [8,9,12]. Consequently, the
signal delay turns to be decoupled from the signal gain. Figure.4.1.4 demonstrates
experimentally the effect of pump depletion on Brillouin slow light. To accurately determine
the amount of the achieved time delay, a sinusoidally modulated wave at 1 MHz was used as
a signal. The phase of the sine wave after propagating through the Brillouin gain medium was
measured as a function of the pump power. According to the phase difference obtained with
and without pumping, the signal delays that the signal experienced through the Brillouin
30
Time delay, ns
25
Signal input power
3 μW
23 μW
20
15
10
5
0
0
5
10
15
20
25
30
35
40
45
Signal gain, dB
Figure.4.1.4: Measured time delays for a 1 MHz sine modulated signal as a function of the signal gain. The
experiment was repeated for different signal input powers at 3 μW and 23 μW with star and square symbols,
respectively.
CHAPTER 4.1. BASIC BRILLOUIN SLOW LIGHT SYSTEM
51
delay line were obtained and plotted as a function of signal gain in Figure.4.1.4. This
experiment was performed with two different levels of signal input power at 3 μW and
23 μW while preserving the same environmental conditions. In both cases, the signals are
clearly delayed in a proportional manner with the signal gain in a small gain regime, showing
good agreements with the theoretical results, and proving the linearity of the system.
However, in a large gain regime, the signal delay tends to saturate with the signal gain and to
even decrease for larger gains. The time delay is totally decoupled from the signal gain when
the signal is so amplified that it can generate another Brillouin Stokes wave at frequency -vB
below the signal frequency. For this reason, the maximum time delays achieved in this
experiment were different for the two signals with different input powers. In consequence, for
a given Brillouin slow light system, the maximum time delay can be simply extended by
decreasing the signal input power. However, the maximum Brillouin gain for a signal is
strictly limited, on account of amplified spontaneous Brillouin scattering. If the pump power
increases so much to reach a critical level, thermally excited acoustic phonons scatter photons
in the pump at the entrance of the fiber. In turn, this spontaneous Stokes wave is significantly
amplified and depletes the pump, saturating energy transfer from the pump to the signal.
CHAPTER 4.2. BROADBAND BRILLOUIN SLOW LIGHT
52
4.2 Broadband Brillouin slow light
To date, Brillouin slow light has been considered as a promising technique to provide alloptical buffers for future optical routers. However, the inherently narrow bandwidth of
stimulated Brillouin scattering in optical fibers limits the full implementation of this system
in practical applications. Actually, the bandwidth of SBS in fibers can be passively broadened
by changing the intrinsic properties of the fiber such as doping concentration on the fiber core,
but to an inefficient extent. The natural linewidth of SBS in a standard optical fiber is as
narrow as 30 MHz due to the slow decaying feature of acoustic phonons and it limits the data
rates in delay lines up to ~50 Mbit/s.
4.2.1 Arbitrary bandwidth by pump modulation
SBS gain resonance has a linear dependence on the pump frequency. It means that if a
polychromatic pump wave is involved in the SBS process, each monochromatic wave of the
pump can generate its own gain resonance at different frequency, corresponding to frequency
matching conditions for stimulated Brillouin scattering [13,14]. The combination of each
gain spectrum represents the effective Brillouin gain spectrum, so that the spectral
distribution of SBS gain can be drastically engineered and shaped by simply modifying the
pump power spectrum. Consequently, the effective power spectrum of SBS gain geff(ν) is
expressed by the convolution of the pump power spectrum and the intrinsic Brillouin gain
spectrum gB(ν) [15,16]:
g (ν eff ) = P (ν ) ⊗ g B (ν ) ,
(4.18)
where ⊗ denotes convolution and P(v) is the normalized pump power spectral density
(integral over spectrum is unity). For a Lorentzian shape of the pump spectrum, the effective
gain curve remains Lorentzian, and its width is given by the sum of the two spectral widths
ΔνB+Δνp, ΔνB being the spectral width of natural Brillouin gain given by Eq.(4.6) and Δνp
being that of the pump power spectrum. Figure 4.2.1 depicts the spectral distribution of the
effective Brillouin gain geff as a result of the convolution, showing the smoothed power
Î
P(v)
gB(v)
geff(v)
Figure 4.2.1: The effective Brillouin gain spectrum geff (ν), resulting from the convolution of the pump power
spectrum and the intrinsic Brillouin gain spectrum.
CHAPTER 4.2. BROADBAND BRILLOUIN SLOW LIGHT
53
spectrum of the pump. Therefore, when the pump wave is adequately modulated to broaden
the pump spectrum and its spectral width is much broader than that of the natural Brillouin
resonance (ΔνB<<Δνp), the spectral shape of the effective Brillouin gain is essentially given
by the pump power spectrum.
Actually, the extension of the effective SBS bandwidth led to a tremendous
improvement on signal bandwidth for Brillouin slow light, and thus Brillouin slow light turns
out to be a suitable solution as a delay line in a multi-Gbits/s transmission. However, it must
be pointed out that the bandwidth extension requires an increase of the pump power to
maintain an equivalent delay. This comes from two facts: first, the peak level of the effective
Brillouin gain is decreased proportionally to the relative spectral broadening as a result of the
convolution. Second, as shown in Eq.(4.10) the amount of delay is inversely proportional to
the bandwidth of the Brillouin resonance.
Broadband SBS slow light was, for the first time, experimentally demonstrated by
Gonzalez-Herráez et al. using a randomly modulated pump source [16], as shown in
(a)
Gain
ΔvB
v
B
ΔvB
Gain
(b)
v
Gain
(c)
ΔvB
gain1
gain2
loss1
vB
loss2
v
Figure 4.2.2: Effective SBS gain spectra induced by a randomly modulated pump source. (a) The first
experimental demonstration of the broadband Brillouin gain resonance, and (b) the maximum achievable SBS
bandwidth limited by the overlap of the SBS loss resonance. The right wing of the gain is canceled out by the
identical left wing of the loss, resulting in the maximum bandwidth ΔνB ≈ νB. (c) However, when introducing
another pump (pump2) the SBS gain2 generated by the pump2 compensates the SBS loss1, which allows the
further extension of the effective bandwidth of the SBS gain1.
CHAPTER 4.2. BROADBAND BRILLOUIN SLOW LIGHT
54
Figure.4.2.2a. By directly modulating the current of the pump laser diode, the Brillouin
bandwidth was effectively enlarged up to 325 MHz at FWHM and produced successful signal
delays up to 1.1-bit delays for 2.7 ns pulse through the Brillouin slow light medium. The
realization of broadband signal delay in Brillouin slow light was thoroughly studied by
Zhu et al., showing the fundamental limitation of the maximum achievable bandwidth [17]. It
is mainly due to the presence of the SBS loss resonance, as shown in Figure.4.2.2b.
Simultaneous growth of the Brillouin loss resonance restricts the Brillouin gain to spread
over in frequency, resulting in the maximum bandwidth up to 14 GHz. However, Song et al.
proposed a solution to open unlimited-bandwidth Brillouin slow light by introducing another
pump (pump2) positioned at frequency +2νB above the pump1 frequency [18]. As in a typical
SBS process, two distinct pumps generate the Brillouin gains and losses at ±νB with respect
to the frequencies of the pumps. Here, it must be pointed out that the spectral characteristics
of the loss1 and gain2 are identical in terms of the bandwidth, the peak amplitude and the
center frequency except the opposite signs as shown in Figure.4.2.2c. In consequence, these
two resonances completely overlap and mutually neutralize, so that it offers the possibility to
overcome the fundamental limitation and to lead to further extension of SBS bandwidth. This
way 25 GHz bandwidth SBS slow light was experimentally demonstrated.
4.2.2 Gain-doublet by a bichromatic pump
As discussed in the previous section, a double-frequency pump generates two separate SBS
gain bands with identical spectra [19-21]. In between the resonances a spectral region with
anomalous dispersion appears with a broad window, as shown in Figure.4.2.3. Therefore, a
broadband fast light can be realized when the signal pulse is spectrally set in the valley
between the two peaks. Moreover, the pump laser can be directly modulated by a noise
Pump
Signal in
EOM
Brillouin shift vB
RF DC
OI
Fast light
Vas
vp
vs
vp
Slow light
Delayed signal out
Figure.4.2.3: Schematic diagram to produce a Brillouin gain/loss doublet using a two-tone pump generated by
external modulation. EOM; electro-optic modulator, OI; optical isolator.
CHAPTER 4.2. BROADBAND BRILLOUIN SLOW LIGHT
55
generator to broaden its spectrum [16], so that the effective spectral profile of the Brillouin
gain doublet can be modified. This way the spectral shape of the gain is precisely chosen so
as to optimize the delay/distortion characteristics of the medium, which gives rise to
maximum pulse advancement with low distortion and small power variation.
A conventional single mode optical fiber with a length of 2 km is used as the SBS
gain medium. The Brillouin characteristics of this fiber were measured, showing a Brillouin
shift of 10.8 GHz and an SBS gain bandwidth of 27 MHz. A commercial DFB laser emitting
at 1532 nm is used to generate the double-frequency pump beam. The laser light is modulated
by means of a Mach-Zehnder electro-optic modulator (EOM) driven with a 100 MHz tone.
The DC bias voltage on the modulator is accurately adjusted, so that a full extinction of the
carrier is achieved. As a consequence, the optical spectrum at the output of the modulator
contains only two sidebands that create two close SBS gain resonances with identical spectra,
showing a separation between central frequencies of 200 MHz with perfect inherent stability.
Moreover, a direct modulation was applied to the pump laser in order to broaden its spectrum.
This way an optimized spectral shape could maximize the pulse advancement with low
distortion in this Brillouin slow light system. The laser current is directly modulated by use of
a noise generator and its spectral width is slightly broadened to 44 MHz. This way a
wideband quasi parabolic spectral distribution was obtained with a width of FWHM
160 MHz across the two peaks. The signal pulse train is produced from a distinct DFB laser
operating at the same wavelength of 1532 nm. The temperature and bias current of the laser
are precisely adjusted to ensure propagation in the dip between the two gain resonances. The
laser is then fast-optically gated using another EOM to generate a pulse train. This pulse in
turn enters into the fiber and counter-propagates with the pump.
A gain doublet is generated after propagating through the 2 km optical fiber as shown
in Figure.4.2.4a. It must be pointed out that the Brillouin spectrum in the case of modulation
is much smoother than in the unmodulated case. The overlapping of the two gain
distributions obtained using the modulated pump gives rise to a perfect linear transition in the
effective refractive index that results in fast light propagation with minimal distortion.
Positioning spectrally the signal in the dip between the two gain peaks offers the further
advantage to give a minimal amplification while fully maintaining the delaying effect. The
FWHM pulse width used in this experiment is 6.6 ns and Figure.4.2.4b shows the nonnormalized time waveforms of the pulse after passing through the fiber. The pulses are
clearly advanced with respect to the pump power.
The scheme is very flexible and can match any signal bandwidth up to several GHz
by simply increasing the frequency doublet separation and the spectral broadening. It makes
possible a fine optimization of the distortion and the advancement by properly setting the
CHAPTER 4.2. BROADBAND BRILLOUIN SLOW LIGHT
1.0
Peaks width
25 MHz
44 MHz
(a)
0.6
0.4
0.2
1.0 ns
(b)
0.25
Amplitude, a.u
Amplitude, a.u
0.8
0.30
56
0.5 ns
Pump power
00 mW
25 mW
50 mW
0.0 ns
0.20
0.15
0.10
0.05
0.00
0.0
-400 -300 -200 -100
0
100
200
300
400
-0.05
-15
Frequency, MHz
-10
-5
0
5
10
15
Time delay, ns
Figure.4.2.4: (a) The SBS doublet generated by the spectrally broadened bichromatic pump source. (b) The
time waveforms of the signal pulses after propagating through the fiber for different pump powers, showing
clear signal advancement.
amount of broadening with respect to the doublet separation. The scheme can also be applied
for slow light generation in the middle of the loss doublet induced by anti-Stokes. This
scheme offers an advantage that signal delays can be produced with a reduced signal
amplitude change. However, a presumable limitation of maximum achievable time delay can
be discussed. As the pump power increases the Brillouin gains in the two resonance peaks
starts to be so large that amplified spontaneous emission present in these peaks leads rapidly
to pump saturation. Consequently, the maximum pump power to generate the delays remains
limited.
4.2.3 Gain-doublet by a concatenated fiber
In general, a double Brillouin gain peak can be created when the pump beam is externally
modulated, and the spectral separation between the two gain/loss resonances is controlled by
the modulation frequency. However, such a double resonance in optical fibers can be
achieved in a passive way with a very simple experimental configuration. This system is
conceptually similar to those developed previously in [19-21], but it has two crucial
advantages: first, it eliminates the need of an external electro-optic modulator, drastically
simplifying and optimizing the operation; second, it makes the operation in the optimum
delay-bandwidth product conditions easily possible. The setup proposed here simply consists
of two appended fiber segments with different Brillouin shifts [22]. This causes naturally and
passively the generation of two spectrally close Brillouin gain resonances using a single
pump and with no need to insert any kind of modulator, as far as the total gain after
propagation through the 2 segments is considered. To optimize the delay-bandwidth product,
a definite pump broadening is introduced by direct current modulation of the pump laser. The
CHAPTER 4.2. BROADBAND BRILLOUIN SLOW LIGHT
57
width of the gain resonances can be adjusted this way to closely match the optimum value
relating the separation of the resonances and their widths.
4.2.3.1 Direct consequence of the linearity of Brillouin amplification
The Brillouin shift is given by νB=2nva/λ where n is the refractive index, λ is the wavelength
of the pump in vacuum and va is the acoustic velocity within the fiber. Since the Brillouin
shift νB is determined by the velocity of the acoustic grating along the fiber, it can be readily
influenced by changing the mechanical properties of the fiber such as the applied strain or the
ambient temperature [10,23]. In addition, the frequency shift is also affected by the doping
concentrations in the core and cladding of the fiber [10]. Let us now consider two spliced
segments of optical fibers with similar length but different core doping concentrations (and
hence different Brillouin shifts) as shown in Figure 4.2.5. From a qualitative point of view, it
can be seen that each fiber segment will have the Brillouin gain curve tuned at a different
position, and hence two gain resonances will appear in the spectrum of the full system. In
these conditions, anomalous dispersion appears in the middle of the gain doublet, thus signal
advancement and fast light can be realized when the signal is spectrally positioned in the
valley between the two peaks.
Pump
Signal in
vB1
VOA
vB2
OI
vas
vp
vs
vp
Fast light
vas
vs
vp
vp
Slow light
Delayed signal out
Figure.4.2.5: Principle of the single-pumped passive configuration to generate a SBS gain or loss doublet. A
partial overlap of the gain spectra can be created by using a spectrally broadened pump, as shown on the
bottom situation.
CHAPTER 4.2. BROADBAND BRILLOUIN SLOW LIGHT
58
The evolution of the probe wave intensity along the fibers will be given by the
following equation:
dI s
= g B ( z ,ν ) I p I s − α I s ,
dz
(4.19)
where Ip and Is are, respectively, the pump and probe intensities, α is the attenuation
coefficient of the fiber and gB(z,ν) is the natural Brillouin gain curve at each position, given
by the usual Lorentzian shape:
g B ( z ,ν ) = g B
1
,
1 − 2 j ⎡⎣(ν −ν B ( z ) ) / Δν B ( z ) ⎤⎦
(4.20)
where ν is the frequency different between the pump and probe waves and ΔνB is the FWHM
width of the Brillouin gain spectrum. Assuming negligible pump depletion, the evolution of
the counter-propagating pump intensity can be written as:
I p ( z ) = I o exp ( − ( L − α ) z ) ,
(4.21)
where Io is the pump intensity launched into the far end of the fiber and L the fiber length.
Inserting Eq.(4.21) into Eq.(4.19), the following explicit expression is obtained for Is at the
output:
(
)
I s ( L ) = I s ( 0 ) exp I 0 ∫ [ g B ( z ,ν ) exp ( −α z ) − α ] dz .
L
0
(4.22)
In the case depicted in Figure 4.2.4 and assuming that the Brillouin linear gain gB is positionindependent in each fiber segment, the signal intensity reads:
(
)
I s ( L ) = I s ( 0 ) exp ( −α ( L1 + L2 ) ) exp I 0 ⎡⎣ g B 2 (ν 2 ) L2 eff + g B1 (ν 1 ) L1eff exp ( −α L2 ) ⎦⎤ ,
(4.23)
where gB1 and gB2 are the Brillouin linear gains of each fiber segment respectively, L1 and L2
are the lengths of the fiber segments, and L1eff and L2eff are their corresponding usual
nonlinear effective lengths [8]. Hence the total gain spectrum observed at the output of fiber2
is the superposition of the gain spectra accumulated along the two fibers, with different
weights depending on the relative ordering of the fiber segments, their length and a possible
different Brillouin linear gain. Assuming a small loss, the two resonances will show nearly
identical strength for equally long fiber segments.
This way two separate gain or loss windows with arbitrary bandwidth can naturally
appear in the Stokes and the anti-Stokes regime. These, in turn, can induce anomalous
dispersion or normal dispersion in the middle of the gain or loss doublets, respectively.
Hence, signal advancement and fast light can be observed when the signal is spectrally placed
at the center of the gain doublet. On the contrary, an equivalent temporal delay or slow light
can be produced as well by simply tuning the signal frequency in between the absorption
CHAPTER 4.2. BROADBAND BRILLOUIN SLOW LIGHT
59
doublet. The optimum delay-bandwidth product for the two-resonance scheme is achieved for
a particular ratio between the separation of the resonances and their width, thereby this
relationship should be approximately (νB1-νB2)/ΔνB ≈ 3. In this experiment, νB1 - νB2 is
approximately 120 MHz, so it was necessary to broaden the spectrum of the Brillouin
interaction from its characteristic value of 27 MHz to approximately 40 MHz, so that it could
match the optimum delay-bandwidth product conditions. This is done simply by introducing
a noise modulation of the current passing through the pump laser, in a similar way to the one
developed in [16] and later extended to extreme bandwidths in [17,18].
4.2.3.2 Broadband window in the center of gain-doublet
Figure.4.2.6 depicts the schematic diagram of the experimental set-up. Two segments of
different single-mode optical fibers with similar length (approximately 2 km) were appended
to be used as the SBS gain medium. These two fibers are both step index single mode fibers,
but constituted with a 20 % difference in core doping concentration and in the core radius.
These conditions lead to a Brillouin shift difference of 120 MHz between the two Brillouin
resonances while showing basically identical Brillouin bandwidth of approximately
27 MHz [10]. Two conventional temperature and current-controlled distributed-feedback
(DFB) lasers operating at a wavelength of 1532 nm are used to generate the pump and probe
beams, respectively. The pump laser is directly modulated using a noise generator in order to
broaden its spectral linewidth through the current-frequency dithering effect. Then the pump
power is amplified using an erbium doped fiber amplifier (EDFA) and the output power of
the EDFA is adjusted by a variable optical attenuator (VOA) before routing via a circulator to
pump the cascaded optical fibers. The spectral width of the pump is simply controlled by
varying the amplitude of the noise signal. When the pump modulation is turned off, two well
separated natural SBS gain/loss resonances are observed with a central frequency spacing of
EOM
Pump
Signal
EDFA
Noise
generator
O.I.
VOA
Fiber1
Fiber2
vB1
vB2
Pulse
generator
ΔvB = 120 MHz
Det
Osc.
Trigger
Figure.4.2.6: Experimental setup to realize fast light propagation with low distortion, by appending two optical
fibers showing different Brillouin shift and a spectrally broadened pump laser. EDFA: erbium doped fiber
amplifier, VOA: variable attenuator; EOM: electro-optic modulator, PC: polarization controller.
CHAPTER 4.2. BROADBAND BRILLOUIN SLOW LIGHT
Spectral width
25 MHz
38 MHz
56 MHz
78 MHz
98 MHz
0.24
0.22
0.20
Amplitude, a.u
60
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
-200
-100
0
100
200
Frequency, MHz
Figure.4.2.7: The spectral profiles of gain-doublets as a function of frequency for different spectral widths of
the pump while the pump power is kept constant.
about 120 MHz (νB1 - νB2). The frequency separation remains perfectly stable in this
experiment since there is no applied strain and/or temperature variations around the fibers. It
must be pointed out that a temperature variation would only cause a global spectral shift of
the 2 resonances, but their frequency difference would remain constant [10].
To observe the shape of the gain doublet, the signal amplitude at the end of the
concatenated fibers was measured while the signal frequency was swept linearly in time. In
the proposed method, SBS gain doublets were clearly observed for different values of the
pump spectral width while the pump power is kept constant, as shown in Figure 4.2.7. It is
seen that the two gain peaks of Brillouin amplification are slightly mismatched mainly as a
result of the pump attenuation after propagation in the first fiber segment, but the
measurement experimentally confirms the adequacy of the expression in Eq.(4.23). It turns
out that the global gain is given by the superposition of the individual gains in each fiber.
Signal advancements in the center of gain doublet
To generate a signal pulse train, a distinct DFB laser is optically gated using a fast external
electro-optic modulator. The signal then enters the compound fiber in opposite direction to
the pump beam. The frequency of the signal is precisely tuned by adjusting the temperature
and current settings applied to the laser, so that the center frequency of the signal is
accurately placed in the middle of the two Brillouin gain resonances. The temporal
advancements and amplitudes of the signal pulses after propagating through the fiber cascade
are measured and displayed on a digital oscilloscope for different values of pump power and
pump linewidth. Optimized results are found when the width of the effective gain curves of
the fiber segments reaches approximately 40 MHz. The FWHM width of signal pulse used in
this experiment is about 25 ns and the pulses have a Gaussian-like intensity profile with
CHAPTER 4.2. BROADBAND BRILLOUIN SLOW LIGHT
61
Pump power
0 mW
30 mW
50 mW
Norm. Amplitude
1.0
0.8
0.6
0.4
0.2
0.0
-40
-20
0
20
40
Delay time, ns
Figure.4.2.8: Normalized temporal traces of the signal pulses after propagating through the concatenated fibers
for different pump powers, showing clear signal advancements.
smooth edges, though showing a somehow longer trailing edge. Figure.4.2.8 shows the
normalized time waveforms of the signal pulses after experiencing the fast light propagation
through concatenated fibers for different pump powers at 0 mW, 30 mW and 50 mW,
respectively. It is clearly observed that signal advancement is achieved with minor signal
distortion, resulting from the spectral filtering (high-pass filtering) in a fast light process. The
largest signal advancement achieved is about 2 ns with 2.7 dB signal amplification at the
pump power of 50 mW. Evaluating the FWHM spectral width of the valley through simple
arithmetics to be approximately 80 MHz, an equivalent delay would be associated with a
6.4 dB gain or loss using a standard single resonance configuration, showing clearly that the
output power variation is substantially reduced.
To accurately determine the amount of signal advancement induced by the proposed
scheme, a sinusoidally modulated light was used to avoid any biasing due to a possible
Advancement, ns
2.0
1.5
1.0
0.5
0.0
0
10
20
30
40
50
Pump power, mW
Figure.4.2.9: Signal advancements for a 1 MHz sine modulated signal as a function of the pump power in the
optimum delay-bandwidth conditions (resonance separation: 120 MHz, effective resonance width 40 MHz).
CHAPTER 4.2. BROADBAND BRILLOUIN SLOW LIGHT
62
distortion. This way it unambiguously measures the real time advancement without any
arbitrary criterion, by measuring the phase shifts of the sine wave. Figure.4.2.9 shows the
achieved time advancement as a function of the pump power, showing that the signal
advancement has a linear dependence along the pump power with a slope efficiency of
approximately 0.04 ns/mW.
Bandwidth limitation
Using this concept, the bandwidth limitation is determined by the spectral distance between
the two Brillouin shifts. It is not difficult to find two fibers whose Brillouin shift is separated
by at least 1 GHz, such as standard and dispersion compensating fibers, extending the data
rate to at least 1 Gbit/s. For a larger separation, one can elongate one of these two fibers in
order to shift the Brillouin frequency by 500 MHz/% [10], resulting in a further extended
bandwidth. Higher resonance separation would require fibers manufactured in different
materials, such as fluoride and chalcogenide glasses showing Brillouin shifts several GHz
lower than silica [24,25].
Signal delays it the center of loss doublet
This technique cascading fibers can also be applied to produce slow light, by simply placing
a signal spectrally in the center of a loss doublet, occurring in the Brillouin loss regime. In
this condition a large normal dispersion is observed in the valley of the two peaks and the
refractive group index is increased with pump power. In this case, the pump source was also
spectrally broadened by means of direct modulation in order to optimize the characteristics of
the slow light medium. This configuration was also experimentally tested and Figure.4.2.10
shows the obtained delays. The insert presents the measured Brillouin loss doublet created in
1.6
Time delay, ns
1.2
0.8
0.4
0.0
0
10
20
30
40
50
Pump power, mW
Figure.4.2.10: Signal delays for 1 MHz sine modulated signal with respect to the pump power after
propagating through two different optical fibers. Insert shows the Brillouin loss doublet created in the
Brillouin loss regime.
CHAPTER 4.2. BROADBAND BRILLOUIN SLOW LIGHT
63
the Brillouin loss regime while the pump propagates through the cascaded fiber segments. As
anticipated, a linear dependence of the time delay on the pump power was observed, showing
a slope efficiency of approximately 0.03 ns/mW.
CHAPTER 4.3. TRANSPARENT BRILLOUIN SLOW LIGHT
64
4.3 Transparent Brillouin slow light
The bandwidth of SBS-based slow light can be made arbitrarily large by actively broadening
the pump spectrum using random direct current modulation of the pump laser [15-18] or by
using SBS gain/loss doublets [20-22]. However, all these techniques still suffer from the
drawback of a significant amplitude change associated with the delaying effect, which may
be highly impairing in a real system. Indeed, the delaying effect in slow & fast light is
intimately related to a narrowband gain or loss process. With typical SBS characteristics in a
standard single-mode fiber, 1-bit delay for a pulse with 30 ns FWHM duration gives rise to a
large 30dB pulse amplitude change [1,2,26].
This problem was soon identified as severe in the early experiments on slow & fast
light using atomic absorptions, the high loss rendering the output signal unobservable for
large delays. Elegant solutions were proposed to open transparency windows in narrow
atomic absorption lines such as coherent population oscillation [7] and electromagneticallyinduced transparency [27], this latter demonstrating large delay with much reduced amplitude
change. In this thesis work, it was demonstrated that the high flexibility of SBS can offer the
possibility to synthesize a gain spectral profile, so that a signal delay or advancement is
achieved with an absolute null amplitude change [28]. This can be obtained by the
combination of gain and loss spectral profiles with identical depth but different width,
resulting in a net zero gain and a differential delaying effect. The possibility to finely tune
independently the depth of each spectral profile results in a perfect compensation of gain and
loss and induces an ideal electromagnetically-induced transparency window in the spectral
profile. This way continuously tunable optical delays with nearly zero amplitude change are
experimentally achieved using slow and fast light.
4.3.1 Principle
In Brillouin slow light, a signal propagating in a medium showing a linear gain G will
experience a net amplitude change by a factor exp(G), together with an extra delay T, due to
the associated group velocity change. For a SBS gain or loss process following a Lorentzian
spectral distribution, linear gain G and delay T are simply given by:
G = g o I p Leff
and
T=
G
2πΔν
,
(4.24)
where Δν is the full width at half maximum of the Lorentzian distribution, go is the peak
value of the Brillouin gain, Leff is the effective length of fiber and Ip is the intensity of the
pump. The delay T thus depends on two parameters: the linear gain G (negative sign for loss)
CHAPTER 4.3. TRANSPARENT BRILLOUIN SLOW LIGHT
65
Gain
vo
Pump 1
vo+vB
VOA2
Signal
VOA1
Pump 2
vo
Gain + Loss
vo-vB
Signal out
vo Loss
Figure.4.3.1: Principle of the experimental configuration to generate transparent spectral resonances where
two distinct optical pumps were used to produce Brillouin gain and loss, respectively.
and the bandwidth Δν of the gain or loss process. Two optical resonances can be generated
by two distinct pump lasers placed at a frequency ±νB above and below the signal frequency
ν, respectively. Then the bandwidth of each pump Δν can be arbitrarily broadened and
controlled using the scheme proposed in [16-18]. A gain is generated using SBS by placing a
pump frequency at frequency +νB above the signal frequency, νB being the Brillouin shift. On
the contrary, an equivalent loss at the same frequency can be also generated simply by
placing the pump frequency at frequency -νB below the signal frequency. Now let superpose
in the frequency domain a SBS gain with linear gain +G1 and a bandwidth Δν1 on a SBS loss
with negative linear gain (thus loss) -G2 and a bandwidth Δν2, as shown in Figure.4.3.1. In
consequence, the resulting linear gain Gnet and the overall delay Tnet must be modified as:
Gnet = G1 – G2
and
Tnet =
G1
2πΔν 1
−
G2
2πΔν 2
.
(4.25a)
If G1 = G2, then:
Gnet = 0
and
Tnet =
G1,2 ⎛ 1
1 ⎞
−
⎜
⎟.
2π ⎝ Δν 1 Δν 2 ⎠
(4.25b)
If the bandwidths of the gain and loss spectra are substantially different, e.g. Δν2 » Δν1, it is
possible to obtain a significant time delay Tnet with nevertheless a zero linear gain Gnet. The
effect is fully comparable to electromagnetically-induced transparency or coherent population
oscillation, in which a transparency window is opened in the middle of an absorption line.
The configuration offers a large flexibility, since a spectral hole can be similarly created in a
gain line by swapping the spectral positions of the broad and narrow pumps, to generate fast
light. In addition the supplementary degree of freedom offered by the possibility to tune the
CHAPTER 4.3. TRANSPARENT BRILLOUIN SLOW LIGHT
66
pumps spectral width makes possible to control to a wide extent the slope between signal
delay Tnet and the pump powers.
4.3.2 Transparency in Brillouin gain resonance
Figure.4.3.2 shows the schematic diagram of the experimental set-up used to demonstrate the
transparent delays. As Brillouin gain medium, a conventional 2-km-long optical fiber was
used, showing a Brillouin shift of 10.8 GHz and a spectral width of 27 MHz. Pump 1 was
kept unmodulated and spectrally placed at a frequency ν+νB above the signal at frequency ν,
so that the frequency difference between Pump 1 and signal is equal to the Brillouin
frequency shift νB. To secure a high stability of the frequency difference between Pump 1 and
signal, these two optical waves were generated through modulation of the CW light from one
commercial DFB laser diode emitting at a wavelength of 1532 nm. This is achieved by
modulating the laser light using an electro-optic Mach-Zehnder intensity modulator (EOM) at
half the Brillouin frequency shift (5.4 GHz). The DC bias voltage on the modulator is
adequately set to achieve a full extinction of the carrier. Therefore, the optical spectrum at the
output of the modulator contains only the two modulation sidebands, exactly separated by the
Brillouin frequency shift of 10.8 GHz. The higher frequency component is then amplified by
a first Erbium-doped fiber amplifier (EDFA) to a maximum power of 17 dBm. On the
contrary, the lower frequency sideband was modulated either sinusoidally or as a signal pulse
train using a second EOM. This technique secures a total absence of spectral drift between
Pump 1 and the signal, together with a perfect centering of the signal in the gain spectrum
[10]. Pump2 was generated from a distinct DFB laser diode, operating at the same 1532 nm
wavelength. This laser was directly modulated by superposing the random noise signal on the
ν
Pump1
EOM
ν
FBG 2
FBG 1
RF DC
Probe
Pump 1
ν
Noise
generator
Pump1
ν
Pulse
generator
EDFA 1
50/50
1W
EDFA 2
VOA 1
VOA 2
Det
Osc.
Figure.4.3.2: Experimental setup to realize transparent slow light via optical fibers, by spectrally superposing a
gain spectrum over a loss spectrum, generated from distinct sources showing different linewidths. VOA:
variable attenuator; BPF: band-pass filter; FBG: fiber Bragg grating; EDFA: Erbium-doped fiber amplifier.
CHAPTER 4.3. TRANSPARENT BRILLOUIN SLOW LIGHT
67
DC bias current in order to substantially broaden its optical spectrum through the current-tofrequency dithering effect. This way the linewidth of pump2 Δν2 could be obtained up to
312 MHz. The center frequency of pump2 was set below the signal frequency, at a frequency
of ν-νB, so that this pump generates a broadband loss through SBS at the signal frequency ν,
following the principle sketched in Figure 4.3.2.
Since the spectral width Δν2 is about 12-fold broader than the natural Brillouin
linewidth Δν1 induced by the pump1 the frequency setting of pump2 is less critical and just
requires a fine tuning by adjusting the DC bias current and temperature applied to the laser.
The broadened pump was strongly boosted using a high power EDFA with a saturation
power of ~1 W since the pump2 power must be larger than the pump1 power by a factor
identical to the broadening factor Δν2 /Δν1 to make the peak gain/loss generated by the two
pumps identical [16]. A variable optical attenuator (VOA1) was used to adjust the pump2
power to perfectly match G2 to G1. This adjustment was carried out by substituting to the
signal the light from a frequency-sweeping laser and by monitoring the signal amplitude at
the fiber output. G1 and G2 are equal when the signal amplitude at the peak of the narrow
band gain is equal to the amplitude of the signal far away from the broadband loss spectrum,
as shown in Figure.4.3.3. It clearly demonstrates that the obtained spectral transmission
profile of the probe after propagating through the test fiber is very similar to an ideal EIT
profile. It must be pointed out that the procedure using the frequency sweeping signal is also
useful to center the gain and loss spectra generated by the two pumps. It illustrates that a
good compensation of gain and loss can be obtained by using two optical pumps via
stimulated Brillouin scattering in optical fibers. The light from the two pumps is combined
through a 3dB directional coupler and their state of polarization is aligned to get a stable and
evenly distributed double SBS interaction along the fiber. The VOA 2 is then used to vary
0.325
Amplitude, a.u.
0.300
0.275
0.250
0.225
0.200
0.175
0.150
-400
-300
-200
-100
0
100
200
300
400
Frequency, M H z
Figure.4.3.3: Variation of the amplitude of the probe signal as a function of frequency after propagation
through a 2-km fiber, showing the achievement of a well-compensated SBS gain/loss profile.
CHAPTER 4.3. TRANSPARENT BRILLOUIN SLOW LIGHT
68
simultaneously and identically the power of the two pumps, so that the gain/loss
compensation is maintained at any pumping level.
4.3.3 Signal delay with small amplitude change
In typical SBS slow light, the temporal delay induced by the slow & fast light effect was
normally measured as a function of the SBS gain/loss, as a consequence of the direct and
simple relationship between these two quantities in Eq.(4.10). However, by essence it turns
out to be conceptually inappropriate in this experimental configuration since only a small
residual gain/loss of the signal is observed as a result of the peak gain compensation. Hence
the delays were rather evaluated as a function of the pump1 power while the pump2 power
varies proportionally. Since the gain G1 is proportional to the pump1 power, a linear relation
must also be observed between the induced delays and this power. It also makes comparisons
possible with the standard technique by simply measuring delays using the same procedure
after turning off the pump2.
Delays and amplitudes of signal pulse after propagation through a 2-km standard
single-mode fiber were precisely measured while incrementing the pump1 power from 0 mW
to 19 mW. The duration of signal pulse used in this experiment was 50 ns FWHM and typical
traces are shown in Figure.4.3.4. It must be pointed out that the signal pulses are shown with
non-normalization using a fixed scope vertical scale, demonstrating a moderate amplitude
change due to a slight mismatch between peak gain and loss. The effect of the mismatch
logically increases for higher pump levels and a perfect balance would experimentally require
a dedicated feedback control circuit that was not implemented for this simple demonstration
of the experimental principle. The effect of a negative mismatch was also observed as
6 ns
0.10
Amplitude, a.u.
0 mW
10 m W
19 m W
12 ns
0.12
0 ns
0.08
0.06
0.04
0.02
0.00
-100
-75
-50
-25
0
25
50
75
100
125
Time, ns
Figure.4.3.4: Time traces of signals after propagation in a fiber with a transparent profile for different pump1
powers, showing a clear delay and a minor amplitude change. Traces are non-normalized and measured in
unmodified experimental conditions. Arrows indicate the pulses peak position.
CHAPTER 4.3. TRANSPARENT BRILLOUIN SLOW LIGHT
69
decreasing pulse amplitudes for increasing pump levels, with no noticeable effect on the
observed delays. An equivalent maximum delay using the non-compensated standard
technique would result in a 12 dB amplitude change using the ~1 ns/dB delay versus gain
relationship commonly observed using SBS in standard single mode fibers [1]. However, the
possible strong noise, caused by the high power erbium doped amplifying process, should be
considered for practical applications. Also, two-fold SBS processes could generate significant
spontaneous noise, which would be imposed onto the signal.
Since the signal experiences non-negligible distortion and pulse broadening, a
sinusoidally modulated light was used to precisely determine the delays induced by the group
velocity change. This way it makes possible to unambiguously obtain the real delays induced
by the group velocity change without applying any arbitrary criterion, by measuring the
difference between the phases of the sine modulation with and without the pump power.
Delays and amplitudes for a 1 MHz sine modulated signal were measured for different pump
levels, as shown in Figure.4.3.5. It clearly demonstrates that the delaying effect is maintained
and fully comparable in magnitude to those obtained using the standard method [1], while
observing a maximum amplitude change of 0.9 dB that can be reasonably considered as a flat
response. The largest induced delay was limited by the onset of the amplified spontaneous
Brillouin scattering generated by the spectrally broadened pump2. The observed threshold is
well below the theoretical expectation, when taking into account the broadening effect. This
is certainly due to a residual coherence in the spectral properties of the pump2, the noisegenerated broadening effect being not perfectly optimized in terms of frequency and
amplitude to result in a fully chaotic linewidth enhancement. Modifications of the noise
generator parameters towards optimization raised significantly Brillouin threshold for the
12
6
10
4
8
2
6
0
4
-2
2
-4
0
-6
-2
0
2
4
6
8
10
12
14
16
18
20
Probe gain, dB
Time delay, ns
pump2 and this strongly supported the validity of the explanation. A full and careful
22
Pump1 pow er, mW
Figure.4.3.5: Delays and amplitudes for a 1 MHz sine modulated signal as a function of the pump1 power in a
transparent slow light configuration. The pump2 power is 12 times larger than the pump1 power.
CHAPTER 4.3. TRANSPARENT BRILLOUIN SLOW LIGHT
70
optimization of the noise-generated broadening would lead to delays comparable to those
obtained with no gain compensation.
The broadband SBS gain induced by the spectrally broadened pump2 can be replaced
by any broadband amplification process based on another type of interactions such as Raman
scattering and parametric amplification, but also using doped fibers. However, the operation
of the delay line would be in this case less convenient since the signal and the SBS pump1
will both be subject to the bidirectional broadband gain. This demonstrates the key role and
the excellent suitability of SBS for the generation of slow & fast light in optical fibers due to
its directivity and its unique narrowband and flexible spectral property.
Effect of spectrally broadened pump on efficiency
The effect of gain compensation results in a decrease of the slope efficiency between the
pump1 power and delay by a factor (Δν2-Δν1) /Δν2 when compared to the standard technique.
The validity of the model could be verified by measuring the slope efficiencies with the
pump2 turned on (gain compensation) and off (no compensation) and comparing with the
expected slope reduction as calculated from the gain linewidths. The slope efficiency
deduced from the linear fit was 0.709 ns/mW when the pump2 is on and rises to
0.776 ns/mW when the pump2 is off. Experimentally the slope is thus reduced by a factor
0.913. Using the measured broadened linewidth Δν2=312 MHz, such a slope reduction would
be obtained after calculation with a gain linewidth Δν1=27 MHz, that is equal to the Brillouin
natural gain linewidth in standard single mode fibers. This shows the perfect adequacy
between the simple model and the experimental results.
Polarization effect on SBS slow light
In Figure 4.3.5, the relationship between delay and pump power is not perfectly linear and
deviates from the linear fit in particular for low pump levels. This effect results from the
residual random birefringence in the fiber that makes the counter-propagating probe and
pump waves mostly interacting with unmatched polarization, thus the efficiency of SBS
decreases. However, it turns out that this penalizing effect is substantially reduced for high
gain since the Brillouin interaction transfers photons from the pump to the probe. These
coherently transferred photons keep the polarization property of the pump, so that they pull
the probe polarization state towards the pump polarization, resulting in a better polarization
matching [29,30].
71
0
8
-1
6
-2
4
-3
2
-4
0
-5
-2
-6
-4
-7
-6
-8
Probe loss, dB
Time delay, ns
CHAPTER 4.3. TRANSPARENT BRILLOUIN SLOW LIGHT
-8
-2
0
2
4
6
8
10
12
14
16
18
20
Pump 1 pow er, mW
Figure.4.3.6: Advancements and amplitudes for a 1 MHz sine modulated signal as a function of pump1 power
in a transparent fast light configuration. Power of Pump2 is 8 times larger than power of Pump 1. The insert
shows the measured SBS gain/loss profile in this configuration.
Signal advancements with negligible amplitude change
The gain compensation technique can also be applied to generate fast light, by simply
swapping the frequency positions of the pump1 and pump2. In such environmental conditions,
the broadened pump generates a gain at the signal frequency and the narrow pump burns a
hole at the center of the gain spectrum. This configuration was also experimentally tested and
resulted in a successful fast light propagation with nearly zero loss, as shown in Figure.4.3.6.
A larger pump1 power could be used through a smaller broadening of the pump2
(Δν2=210 MHz) that resulted in a better coherence reduction. The gain and loss compensation
is nearly perfect in this case, but a substantial slope efficiency reduction was observed when
compared to slow light, by nearly a factor 2 down to -0.414 ns/mW. It can only be partly
explained by a smaller broadening of Pump2 that would result in a 10 % reduction of the
slope efficiency. The explanation certainly lies in the interplay between line broadening
through the random frequency dithering of the pump2 and polarization dragging effect: as a
result of the frequency dithering Brillouin gain - and thus polarization dragging of the probe occurs only at limited and random locations, so that most of the time the probe polarization is
only subject to random birefringence. Matching between pump and probe polarizations is
thus poorly improved even for high gain and the slope efficiency is reduced substantially,
accordingly. This explains the asymmetric behavior between slow and fast light and this
observation is certainly an important contribution for guiding the future design of fiber optics
delay line based on spectrally-broadened SBS.
CHAPTER 4.4. OPTIMIZED SHAPE OF SIGNAL FOR BRILLOUIN SLOW LIGHT
72
4.4 Optimized shape of signal for Brillouin slow light
Since the first demonstration of Brillouin slow light, many solutions towards the objective of
a real implementation solutions were proposed to enlarge the bandwidth of the Brillouin
resonance [16-18] and to optimize its dispersion characteristics [31-33]. All these
experiments modified the Brillouin spectrum by slightly modulating the current applied to the
pump laser, so that the effective gain spectrum is broadened to contain the entire signal
spectrum. Although such SBS slow light systems appear to be very promising timing tools,
they clearly need a set of optimizations to get the maximum benefits for a given bandwidth,
since a broader resonance results in a lower delaying efficiency that can be only compensated
at the expense of a higher pump power.
In this section, an optimization based on the time-domain intensity profile of a signal
will be introduced in order to modify its spectrum for a best fit within the Brillouin gain
bandwidth. It must be noticed that this approach in the context of a delay line is mainly for
tunable timing purposes and not for the transmission of information through a continuous
data stream. In this particular case, it is assumed that a train of isolated pulses is transmitted
to carry the timing information that is extracted when the pulse amplitude crosses a preset
level, typically 50 % of its peak amplitude. Clearly the scope of this study is definitely more
oriented to metrology applications than the telecommunication domain. Since the overlap
between adjacent pulses is inexistent, the issues related to inter-symbol interferences can be
ignored. In addition, the relevant quantities to characterize the pulse for this class of
applications will be the position of its peak (or front edge) and its FWHM temporal and
spectral widths. This method is therefore distinct from a recent complete study [34], in which
the signal shape is optimized for a continuous data stream for communication applications. In
this latter case relevant quantities characterizing the signal are different and in particular
RMS temporal and spectral widths have to be considered. This leads to optimal signal shapes
substantially different in the two studies.
4.4.1 Optimization of the signal spectral width
The relationship between a pulse shape and its optical spectrum can be characterized by its
time-bandwidth product K. This parameter is essentially given by the product of Δν and Δt,
here defined as the full widths at half maximum (FWHM) of the spectral and temporal
distributions, respectively. Table.1 shows the values of K for various pulse shapes in the case
of transform-limited pulses [11]. It turns out that, for a given pulse duration, its spectral width
can be substantially modified by properly shaping the pulse envelope. Thereby a significant
CHAPTER 4.4. OPTIMIZED SHAPE OF SIGNAL FOR BRILLOUIN SLOW LIGHT
Shape
ε(t)
K
Gaussian function
exp[-(t/t0)2/2]
0.441
Exponential function
exp[-(t/t0)/2]
0.140
Rectangular function
u(t+t0/2) - u(t-t0/2)
0.892
Lorentzian function
[1+(t/t0)2]-1
0.142
73
Table 4.1: Values of the FWHM time-bandwidth product K for various pulse shapes. u(t) is the unit step
function at the origin..
narrowing of the effective spectral width in terms of FWHM can be obtained. The signals can
therefore be spectrally well confined in the center of the Brillouin resonance where a perfect
linear transition in the effective refractive index is observed. As a result, larger delays of the
pulse peak or front edge could be achieved through Brillouin slow light systems.
In a simulation test, three different pulse intensity profiles were defined with
identical FWHM duration, showing successively exponential, Gaussian and rectangular
temporal distributions. The spectra of the pulses were numerically obtained through a Fourier
transform of the pulse waveforms, as shown in Figure.4.4.1. It is clearly observed that as
anticipated the different pulses show different spectral widths. Then, to demonstrate the
validity of the proposed approach, the three different pulse shapes were experimentally
generated with identical 14 ns FWHM duration by using an arbitrary waveform generator.
The spectral widths of the differently shaped pulses were measured by the typical delayed
self-homodyne method based on a Mach-Zehnder interferometer in which one arm contains a
delay line to break the coherence of the analyzed beat signal. The measured pulse spectra
show a good agreement with the numerical predictions, clearly illustrating the spectral width
dependence on the pulse time-domain waveform. Unavoidable small deviations remain in the
Norm. Amplitude
Intensity profiles
Time, 100 ns/div
Spectral profiles
Frequency, 100 MHz/div
Figure.4.4.1: Signal pulses under study (exponential, Gaussian and rectangular time distribution from left to
right with corresponding spectra on bottom chart). Numerical and measured results are shown in solid and
dashed lines, respectively.
CHAPTER 4.4. OPTIMIZED SHAPE OF SIGNAL FOR BRILLOUIN SLOW LIGHT
74
spectral width, which are induced by the smoothed rising and falling edges of the pulses.
Among these intensity profiles, the exponential pulse offers the narrowest FWHM spectral
width of 18 MHz, while the Gaussian and Rectangular pulses show a measured spectral width
of 35 MHz and 58 MHz, respectively.
4.4.2 Enhanced signal delay
Figure.4.2.2 depicts the schematic diagram of the experimental setup. As a Brillouin gain
medium a 1-km-long standard single mode fiber was used. The Brillouin characteristics of
this fiber were measured, showing a Brillouin shift of 10.8 GHz and an SBS gain bandwidth
of 27 MHz. The experimental validation has been realized using the natural Brillouin gain
without engineered broadening, since the bandwidth issues are less critical for timing
applications and the gain spectral characteristics are fully stable and accurately known.
However, the approach can be extrapolated to any gain spectral distribution, in particular
when the spectrum is synthetically broadened.
A commercial distributed feedback (DFB) laser diode operating at 1532 nm was used
as a light source and its output was modulated through an electro-optic Mach-Zehnder
intensity modulator at half the Brillouin frequency shift to generate two first-order sidebands.
The DC bias on the modulator was well set, so that the carrier was completely suppressed and
only two sidebands are present at the output. Each sideband was then filtered and directed to
a distinct fiber using a set of 2 fiber Bragg gratings associated with 2 circulators. The higher
frequency sideband was amplified using an Erbium doped fiber amplifier (EDFA) to play the
role of the Brillouin pump and its power was precisely controlled by a variable optical
Laser
EOM
ν
FBG 1
ν
FBG 2
RF DC
Signal
Arbitrary
pulse generator
ν
ν
Pump
EDFA
EOM
VOA
PC
Osc.
Det
Figure.4.4.2: Experimental setup to demonstrate the effect of the signal shape on the time delay. FBG: fiber
Bragg grating, EDFA: erbium doped fiber amplifier, VOA: variable optical attenuator, PC: polarization
controller.
CHAPTER 4.4. OPTIMIZED SHAPE OF SIGNAL FOR BRILLOUIN SLOW LIGHT
75
attenuator. The lower frequency sideband was launched into another external modulator to
properly shape the signal pulse, so that trains of signal pulses with distinct intensity profiles
are generated showing identical 14 ns FWHM pulse duration.
To observe the delaying effect induced by the Brillouin gain, the pump power was
varied from zero to 50 mW. The temporally delayed signals after propagation through the
fiber were measured as a function of the pump power using a fast detector and displayed on a
digital oscilloscope. Figure.4.4.3 shows the normalized time waveforms of the signal pulses
with different pulse envelope for a set of fixed pump levels: 0 mW, 20 mW, 35 mW and
50 mW. In all situations the signals experience more delay for an increasing pump power, as
expected from a SBS slow-light system, but it is clearly observed that the exponentially
shaped pulse achieves the largest delay. When rectangular and Gaussian pulses exit the fiber,
significant distortion is imposed onto the signal pulses and a reduced time delay is observed,
since these two pulses suffer by essence a stronger spectral filtering effect by the narrower
Brillouin resonance.
Increasing pump power
Norm. Amplitude
Exponential
Gaussian
Rectangular
-60
-40
-20
0
20
40
60
80
100
Time, ns
Figure.4.4.3: Normalized time traces of the signal pulses with pump power at 0 mW, 20 mW, 35 mW and 50
mW, showing a clear time-delay dependence on the signal shape.
In previous works [1,2,26], temporal delays induced by SBS were evaluated in terms of
Brillouin gain. But, in this experiment the signal delays are rather plotted as a function of the
pump power for a flat comparison between the time delays obtained in the three different
cases, as shown in Figure 4.4.3, the effective gain value being substantially different for
different pulse shapes. To evaluate the amount of time delay, the peak position of the signal
pulse was used. It is interesting to immediately point out that, for a given pump power and
fixed input pulse duration, the obtained time delay is larger with a reduced distortion when
the pulse shaping results in a smaller FWHM spectral width. However, the signal delaying
CHAPTER 4.4. OPTIMIZED SHAPE OF SIGNAL FOR BRILLOUIN SLOW LIGHT
30
Time delay, ns
25
76
Pulse shape
Exponential
Gaussian
Rectangular
20
15
10
5
0
0
10
20
30
40
50
Pump power, mW
Figure.4.4.4: Comparison of the temporal delays as a function of the pump power for the signal pulses with
three different shapes.
remains unavoidably accompanied by a pulse broadening as a result of the spectral filtering
(low-pass filter) in the slow light process. The largest time delay achieved by the exponential
pulse is about 31 ns (or a fractional delay of 2.1) with a broadening factor of 3. For
comparison it was calculated that in a typical SBS slow light system an equivalent delay for
14 ns FWHM Gaussian pulses would result in a broadening factor of 3.4. The advantage may
look minor in terms of broadening, but the main benefit of the optimized pulse shape turns
out to be the following: the signal delay shows a linear dependence on the pump power with a
slope efficiency of 0.73 ns/mW for the exponential pulse. In the Gaussian case, this slope
efficiency is observed to be 0.28 ns/mW, resulting in a 2.5 times decrease when compared to
the exponential pulse.
It must be noticed that this description is justified when the FWHM widths of the
time and spectral distributions are considered, the product of the RMS time and spectral
widths keeping of course minimized for Gaussian pulses. This approach is justified for timing
applications when a transition threshold is defined to determine the pulse arrival time. It must
also be pointed out that the exponential pulse also offers another spectral optimization in the
sense that its spectral distribution is Lorentzian as the natural Brillouin gain spectrum. This
also indicates that a particular pulse shape is certainly optimal for a given gain spectral shape.
CHAPTER 4.5. REDUCED BROADENING SIGNAL DELAY IN BRILLOUIN SLOW LIGHT 77
4.5 Reduced broadening Signal delay in Brillouin slow light
In order to overcome the trade-off relation between signal delay and signal distortion, several
linear slow light schemes have been theoretically and experimentally investigated [16,22,3139]. In particular, tailoring the shape of the spectral resonance to optimize the dispersive
properties in the material could partially reduce the induced distortion while keeping the
fractional delay. However, this approach could not fully eliminate the strong distortion, thus
the maximum achievable signal delay remains limited to a few pulse durations. In parallel,
the limits of delay and distortion have been explored for several kinds of slow light systems
[40-42], and recent studies seem to indicate that linear slow light systems will never be
candidates for making distortionless (or zero-broadening) slow light systems [43].
Nevertheless, the vast majority of the slow light systems proposed up to now in the literature
can be considered linear in terms of the signal propagation. Therefore, a general scheme to
achieve slow light without distortion for any arbitrary fractional delay is still under
investigation, but the results of these papers seem to indicate that nonlinear elements should
be essential for this task. In a recent work [44], a possible solution to mitigate the broadening
effect was put forward using a depleted (therefore nonlinear) Brillouin amplifier as slow light
system. Even though the signal broadening issue was improved, the compensation of the
broadening was certainly not complete, and the saturation in the amplification had the
unwanted outcome of reducing also the delaying effect.
In this thesis work, a new configuration to compensate the inherent signal broadening
in all slow light systems was proposed and experimentally demonstrated. The demonstration
is based on an all-fiber setup, rendering the system very attractive for future applications in
the field of optical communications. It makes use of the combination of a conventional SBS
linear slow light system and a nonlinear fiber loop mirror (NOLM) as broadening
compensation element. The NOLM plays the role equivalent to a fast saturable absorber. In
the Brillouin slow light segment, the signal pulse experiences both a time delay and the usual
broadening. Then this delayed signal is delivered into the nonlinear fiber loop mirror, where
it experiences a compression as a result of the nonlinear transmission (the peak of the signal
experiences a larger transmission than both tailing parts). In this experiment, a fractional
delay above unity could be effectively achieved with no signal broadening in the SBS slow
light delay line. Additionally, this regeneration element can eliminate most of the background
noise introduced by the Brillouin amplifier when the signal is “off”, therefore improving the
contrast between the ones and the zeros in a transmission system. There is in principle no
limitation to cascade this system and achieve large fractional delays with very minor
distortion.
CHAPTER 4.5. REDUCED BROADENING SIGNAL DELAY IN BRILLOUIN SLOW LIGHT 78
4.5.1 Compensation of signal distortion
Stimulated Brillouin scattering in optical fibers is usually described as a nonlinear interaction
between two counter-propagating waves, a strong pump and a weak probe wave. However, it
can be shown that the effect of the amplification/loss on the signal wave can be treated as a
linear effect (in absence of pump depletion) and therefore modeled through a linear transferfunction approach [43]. Thus, as a linear low-pass system, the signal pulse is broadened and
this limits the maximum delay that can be achieved with a reasonable distortion. To avoid
this effect, one can use a nonlinear regeneration element at the output. In the time domain,
this nonlinear element should sharpen the shape of the pulse. In the frequency domain, the
role of this element will be to broaden the signal spectrum and re-generate the frequencies
that have been filtered out by the slow light element.
The experimental scheme comprises two basic building blocks: an SBS slow light
delay line and a nonlinear optical loop mirror (NOLM) that acts as a regeneration element.
Upon propagation through the SBS slow light line, the signal essentially experiences
distortion. In the proposed configuration, the NOLM acts similarly to a fast saturable
absorber. The transmission of the NOLM is larger for higher input powers and decays rapidly
as the input power is reduced. When a signal enters into the compensation element, the peak
of the signal experiences a larger transmission coefficient than the wings, leading to a
sharpening in its shape. By using only the input power to the NOLM as control variable, one
can also accommodate the sharpening of the signal in the NOLM, and potentially one can
discuss on achieving a complete regeneration of the signal. This way the system can allow
the delayed signal to have effectively no broadening at the output of the nonlinear element for
large delays in the SBS delay line.
The NOLM basically consists of a Sagnac loop, in which an attenuator is placed at
one loop end and thus introduces a large power imbalance between the clock-wise and
counter-clockwise fields while they propagate in the loop. A polarization controller was used
in the loop to match the states of polarization (SOP) so as to maximize the visibility of the
interference and secure a total absence of transmission at low powers. In linear operation, the
Sagnac mirror acts as a perfect mirror. The input signal is split by a directional coupler into
two counter-propagating electric fields, as shown in Figure.4.5.1. The two fields are in turn
re-combined at the coupler after propagation through the optical fiber loop. Since they travel
the same path but opposite direction, the optical path length is identical to both propagating
fields, resulting in the same linear phase shift. As a result, the input signal is totally reflected
into the input port. Therefore, for low input powers the loop acts as a perfect mirror, and no
light exits through the output port. In the high power regime, however, the refractive index of
CHAPTER 4.5. REDUCED BROADENING SIGNAL DELAY IN BRILLOUIN SLOW LIGHT 79
12 Km DSF
α
PC
G
E1
Input pulse
G
E2
Reflection
Output pulse
Transmission
Figure.4.5.1: Schematic diagram of an attenuation imbalanced nonlinear optical fiber loop to compress the
shape of signal. E1 and E2 present clock and counter-clock wise electric fields of pulses, respectively. DSF;
dispersion shifted fiber, PC; polarization controller and α; an attenuation factor.
the fiber depends on the light intensity. This means that the imbalance in the optical power of
the two arms caused by the attenuator will lead to a difference in the effective optical path
length for the clockwise and counter-clockwise signals [45].
In this experiment, the input light was equally split as required for maximum contrast,
but 1-dB attenuation was set for the counter-clock wise field prior to its propagation through
the loop using a variable optical attenuator. In addition, a 12-km dispersion shifted fiber
(DSF) was used as a nonlinear medium over which the two fields experience a differential
phase shift, mainly caused by Kerr effect. The clock and counter-clockwise electric fields E1
and E2 before entering the DSF can be written as:
E1 =
and
1 / 2 Ain
E2 = i α / 2 Ain ,
(4.26)
where Ain presents the amplitude of the input electric field and α is the attenuation factor for
the E2. Upon recombination in the coupler, the signal at the output port will depend on the
input power and the power imbalance between the two arms. In this circumstance, the electric
fields after a single pass via the loop are given by the expressions:
E1′ out =
α
2
⎡ 1 ⎤
Ain exp i ϕ
⎢⎣ 2 ⎥⎦
and
α
⎡ α ⎤
E2′ out = −
Ain exp i ϕ
⎢⎣ 2 ⎥⎦
2
(4.27)
The nonlinear phase shift φ is defined as φ =γPoL where γ is the nonlinear coefficient, Po is
the optical power of the light at the input port and L is the fiber length. Therefore, the
transmission coefficient T can be written as:
T =
(
)⎦⎥
α (1 − α ) 2
α⎡
1−α ⎤
ϕ ϕ
1 − cos
2 ⎣⎢
2
2
16
for φ<<1,
(4.28)
CHAPTER 4.5. REDUCED BROADENING SIGNAL DELAY IN BRILLOUIN SLOW LIGHT 80
and the system turns out to be equivalent to a saturable absorber: the transmission grows for
higher input powers, with a quadratic dependence on the power for small accumulated
nonlinear phase shifts. Since the power in the peak of the signal is larger than in the wings,
the output pulse will be sharper than that at the input, resulting in pulse compression [46].
However, due to the nonlinear response of the phase shift induced by self-phase modulation
across the signal, a negative frequency chirp will be imposed through the signal. It shows
presumably that the delayed signal can be further compressed, probably to a very small extent,
by placing a dispersive medium with anomalous dispersion, such as dispersion compensated
fiber.
4.5.2 Nearly non-broadened signal delay
The schematic diagram of the experimental scheme is depicted in Figure.4.5.2. As a Brillouin
gain medium a 1-km-long standard single-mode fiber was used. The Brillouin characteristics
of this fiber were measured, showing a Brillouin shift of 10.8 GHz and an SBS gain
bandwidth of 27 MHz. A commercial distributed feedback (DFB) laser diode operating at
1532 nm was used as the light source and its output was split using an optical coupler. Then
one branch was amplified using an erbium-doped fiber amplifier (EDFA) to play the role of
Brillouin pump, and its power was precisely controlled by a variable optical attenuator before
entering into the delaying fiber segment. The other branch was modulated through an electrooptic Mach-Zehnder intensity modulator (EOM) at the Brillouin frequency of the fiber so as
to generate two first-order sidebands. The DC bias of the modulator was adequately set to
obtain complete suppression of the carrier. Therefore, only two sidebands were present at the
output of the modulator. Only lower-frequency sideband was then filtered by a fiber Bragg
DFB-LD
EOM
EDFA
VOA
12 Km DSF
RF
Pump
DC
Signal
v
PC
α
v
v
EDFA
FBG
Optical
gate
[ SBS slow light delay line]
VOA
Det.
Osc.
[ Nonlinear optical loop mirror ]
Figure.4.5.2: Experimental setup to produce non-distorted signal delays, by combining a nonlinear generation
element with a typical Brillouin slow light system. EDFA; erbium doped fiber amplifier, EOM; electro-optic
modulator, FBG; fiber Bragg grating, VOA; variable optical attenuator, DSF; dispersion shifted fiber, PC;
polarization controller and α; an attenuation factor.
CHAPTER 4.5. REDUCED BROADENING SIGNAL DELAY IN BRILLOUIN SLOW LIGHT 81
Norm. Amplitude
1.0
(a)
0.5
0.0
1.0
(b)
0.5
0.0
-100
-50
0
50
100
Time, ns
Figure.4.5.3: (a) Normalized waveforms of signals that experienced time delays through SBS slow light and
(b) normalized waveforms of transmitted signals through a saturable absorber, showing noticeable pulse
compression.
grating via a circulator, and this wave was optically gated using a fast external modulator to
generate a signal pulse train. Consequently, a signal pulse train was generated at frequency
downshifted from the pump frequency by the Brillouin frequency. The signal pulse showed
duration of 27 ns FWHM at a repetition rate of 200 kHz.
To produce relative time delays, the signal was sent into the SBS delay line while the
pump power was increased from 0 to 30 mW. Figure.4.5.3a shows the normalized time
waveforms of the signal pulses at the output of the Brillouin delay line. As in any typical
Brillouin slow light system, it is clearly observed that the signal delay increased a function of
pump power. Also, the pulse exiting from the delay line was temporally both delayed and
broadened with respect to the pump power. This way the largest signal delay achieved was
about 36 ns (corresponding to 1.3-bits delay) at the pump power of 30 mW and the delayed
pulse was significantly broadened by a factor of 1.9. After passing through the delay line, the
delayed signal was amplified using another EDFA and delivered into the nonlinear loop. In
practice, the NOLM acts as a saturable absorber as previously described. Consequently, the
shape of the broadened signal was sharpened at the output, as shown in Figure.4.5.3b.
Moreover, it must be pointed out that the unwanted background components imposed onto
the pulse train (mainly amplified spontaneous emission from the EDFAs and the Brillouin
amplifier) were cleaned up since they were rejected for the transmission. It is clearly
observed that the output signal is compressed at the output of the loop, nevertheless fully
preserving the time delays achieved in the previous stage, SBS delay line. Figure.4.5.4 shows
the measured fractional delay and signal broadening as a function of signal gain. This system
allows producing fractional delays of up to 1.3-bits without any broadening. This result is
fully consistent with the prediction for a transmission depending quadratically on the
CHAPTER 4.5. REDUCED BROADENING SIGNAL DELAY IN BRILLOUIN SLOW LIGHT 82
Fractional delay
Pulse broadening
2.0
1.0
1.5
0.5
1.0
0.0
0.5
0
10
20
30
Broadening, B
Fractional delay, bit
1.5
40
Gain, dB
Figure.4.5.4: Factional delays and broadening factors of signal pulses, respectively, with square and star
symbols as a function of signal gain when the nonlinear loop mirror is present (filled symbols) or absent
(opened symbols).
intensity, which gives a compression by a factor 2-½ when applied to a signal with a
Gaussian intensity profile. Of course the non-delayed signal is also compressed, but this
should not be an issue in detection, since the detection system is inherently band-limited.
Another advantage takes into account the signal-to-noise ratio of the setup. It was observed
that the output pulse contained amazingly very clean zero background level, which is rather
unusual in Brillouin amplifier setups. This zero-background level can be very helpful in real
transmission systems to enhance the contrast between the “on” and “off” states in the
detection process.
CHAPTER 4.6. SELF-ADVANCED BRILLOUIN FAST LIGHT
83
4.6 Self-advanced Brillouin fast light
Stimulated Brillouin scattering offers unmatched flexibilities for an all-optical control of the
signal delay in a fiber. However, considering practical issues, Brillouin slow light shows two
actual drawbacks: first, this scheme inevitably requires an external pump source; second, the
frequency separation between the pump and probe lasers has to be constant and precisely
controlled within typically a 1 MHz uncertainty. These requirements force a certain
complexity in the experimental system, which should be preferably avoided in many practical
applications. In this section, key contribution of this thesis will be demonstrated, that is, a
light signal with a sufficient average component can make itself speed up along the fiber
without any external pump source. The working principle is the following: when the signal
power grows beyond a certain critical power - commonly denominated Brillouin threshold - a
significant Stokes component is generated at a frequency down-shifted νB below the signal.
This Stokes wave, in turn, acts as Brillouin pump to create an absorption peak in the
transmission spectrum of the fiber at the signal frequency, hence creating relative
advancement on the signal. In other words, simply controlling the power of the signal
entering the fiber eventually determines its advancement. This configuration is considerably
simpler than all previously reported techniques and may serve as a practical basic concept for
several applications.
4.6.1 Noise-seeded stimulated Brillouin scattering
The process of SBS can be initiated without the need of an external probe wave. The
background energy present in the fiber in ambient conditions causes the presence of
thermally activated acoustic waves that spontaneously scatter the light from the pump. This
noise-scattered seed light initiates the stimulation and is thus gradually amplified if it lies
within the SBS gain spectrum. This can eventually lead to a considerable amount of power
that is back-reflected at the Stokes wavelength νpump-νB. A useful quantity in this case is the
Brillouin critical power Pc, which is conventionally defined as the power of the input CW
light necessary to have an equal amount of power present in the backscattered Stokes wave in
the fictitious case of an absence of depletion. For a uniform fiber, this critical power can be
estimated [47] as Pc=21Aeff/(gBLeff), where gB is the Brillouin gain coefficient, Aeff is the
nonlinear effective area of the fiber and Leff is the nonlinear effective length. In long
conventional optical fibers with L > Leff, this critical power is about 5 mW at a wavelength of
1550 nm and can thus be easily reached using a conventional distributed feedback lasers.
CHAPTER 4.6. SELF-ADVANCED BRILLOUIN FAST LIGHT
Amplitude
t
rg y
Ene
sfe
ran
84
r
vStokes : weak power
vsignal : strong power
v
Narrowband absorption
generated by the Stokes wave
Figure.4.6.1: Principle of the configuration to generate self-advanced fast light. The signal power is high
enough to generate a strong amplified spontaneous Stokes wave, which in turn depletes the signal wave. The
depletion is assimilated to a narrowband loss spectrum.
The basic idea of the self-induced fast light scheme is to avoid using a distinct pump
wave to modify the signal propagation conditions through SBS. This is simply realized by
delivering a sufficiently powerful average signal into the fiber, above the Brillouin critical
power Pc. Seeded by noise, the process of stimulated Brillouin scattering will generate a
substantial Stokes signal, which in turn will induce through depletion a narrowband loss for
the signal, as depicted in Figure.4.6.1. Associated to this narrowband loss, a spectral region
of anomalous dispersion is induced, in which the temporal envelope of the signal will
experience advancement through fast light.
The advantage of this configuration is that the signal is continuously and accurately
centered in the spectrum of the loss resonance created by the spontaneously amplified Stokes
wave. The pump-signal frequency difference automatically compensates for any
environmental and wavelength changes and remains perfectly stable without the need of any
optical component or instrument (such as external modulators, microwave generators, etc.)
typically used in other configurations [1,2,19,28]. Moreover, in case of very large gain as in
the present situation, the polarization state of the Stokes wave is precisely parallel to that of
the signal at the fiber input since the SBS interaction coherently transfers photons from the
pump to the Stokes wave and preserves their states of polarization [30]. Thus, in these
conditions, the polarization of the strongly amplified Stokes wave experiences a pulling
effect and eventually aligns to the pump polarization. This holds for a common standard fiber
with a reasonably low birefringence value. This secures a maximum efficiency and stability
for the interaction, and hence the highest possible advancement for a given input power.
When addressing the delay-bandwidth characteristics and the efficiency of a slow
and fast light scheme, it is important to know the spectral width of the Brillouin resonance.
The bandwidth of the gain/loss process is obtained from the convolution of the intrinsic
CHAPTER 4.6. SELF-ADVANCED BRILLOUIN FAST LIGHT
85
Brillouin spectral distribution with the pump spectrum [16]. In this configuration, it is
important to realize that the Stokes signal is not purely monochromatic, since it builds up
from a noise-seeded SBS process and will therefore present a certain spectral distribution
[48]. For low input power the spontaneous Brillouin noise shows a linewidth close to the
intrinsic Brillouin linewidth ΔνB. However, for higher input power, the linewidth experiences
a dynamic narrowing. This narrowing stabilizes when the critical power is reached and a
significant depletion of the input signal is observed [48]. In this case, significant
advancement of the signal starts to take place when the signal power exceeds the Brillouin
critical power, hence when significant pump depletion starts to occur. In these conditions, the
spectral width of the amplified spontaneous Brillouin emission should remain moderate and
constant for all input powers [48]. Therefore, a power-invariant loss spectral distribution for
the signal must be expected, since it is essentially given by the convolution of the Stokes
wave spectrum with the natural Brillouin gain, both being constant for all the relevant input
powers in the present experimental conditions.
The power of the signal must also be considered as constant when time-averaged
during transit in the fiber, so that an amplified spontaneous Stokes emission showing a
constant power is generated and no time jitter is observed on signal delays. Practically, this
condition requires that the fiber length must be much longer than the typical periodicity of the
signal, or equivalently a large number of symbols (> 100-1000) forming the data pattern must
simultaneously propagate through the fiber. Under this condition each separate symbol in the
data stream taken individually has a negligible impact on the amplitude of the Stokes wave
and therefore experiences a Brillouin loss actually similar to that produced by an external
constant pump. This makes the system behave identically to a standard Brillouin fast light
configuration in terms of distortion and limitation.
4.6.2 Characteristics of spontaneous Stokes wave
The experimental setup realized to demonstrate the self-advanced fast light through SBS is
shown in Figure.4.6.2. A 12-km-long conventional dispersion shifted fiber (DSF) with a
Brillouin shift of 10.6 GHz and a FWHM gain bandwidth of approximately 27 MHz is used
as the SBS gain/loss medium. To generate the signal a commercial distributed feedback laser
diode was used, operating at a wavelength of 1532 nm. The output of the laser is modulated
using an external electro-optic modulator to produce a signal pulse train with a width of 45 ns
(FWHM) at a repetition rate of 5 kHz. With this periodicity, only one pulse is present at a
time over the entire optical fiber, so that any cross-interaction between adjacent pulses during
propagation is avoided at a first stage. The signal includes a definite DC component obtained
CHAPTER 4.6. SELF-ADVANCED BRILLOUIN FAST LIGHT
86
12 km DSF
νΒ
EOM
Signal
νs
1W
EDFA
Pulse
generator
Det
Osc.
Det
ESA
VOA
Amplified spontaneous Stokes
Delay line
νΒ
νs
50/50
50/50
EOM
Figure.4.6.2: Experimental configuration to realize the self-pumped signal advancement based on both
amplified spontaneous and stimulated Brillouin scattering. EOM; electro-optic modulator, EDFA; Erbiumdoped fiber amplifier, VOA; variable optical attenuator, DSF; dispersion shifted fiber.
simply by adjusting the DC bias applied to the EOM. This DC component is essentially
responsible for the generation of the Stokes wave. In a realistic fiber system, a sufficiently
long pulse sequence present in the fiber would equally generate the Stokes component
responsible of the signal advancement. The DC power is approximately 14 % of the peak
power of the pulse, but creates a much larger integrated gain over the fiber length considering
the very low pulse repetition rate. Then this compound signal is strongly boosted using a high
power erbium-doped fiber amplifier (EDFA) with ~30 dBm saturation power before it is
launched into the DSF. The signal power is controlled with a variable optical attenuator
(VOA) after being amplified by the EDFA. The strong DC component present on the signal
generates a strong backward Brillouin Stokes at a frequency downshifted vB below the pulse
signal frequency. This Stokes wave causes an absorption peak in the spectral transmission of
the fiber at the frequency of the input signal, which consequently experiences fast light
conditions. This is observed at the fiber output by measuring the temporal advancement of
the pulse signal for different input signal powers. To perform this measurement the amplitude
of the pulse at the input of the detector was controlled using a variable optical attenuator, so
as to avoid any possible biasing of the trace from an amplitude-dependent time response of
the detector. The higher the input power, the stronger the Stokes wave, the deeper the peak
absorption is and the faster the pulse will travel.
The Stokes power as a function of the input signal power is shown in Figure.4.6.3a. It
is seen that there is no significant Stokes component below the Brillouin critical power, while
an abrupt change is observed over this threshold power. For higher signal power all the light
intensity in excess of the threshold power is transferred to the Stokes waves, making the
output signal power saturated at a constant value. For even higher input power exceeding
CHAPTER 4.6. SELF-ADVANCED BRILLOUIN FAST LIGHT
6
18
(a)
5
15
4
12
3
9
2
6
1
3
0
0
0
4
8
12
16
20
24
28
Signal power
13 dBm
16 dBm
19 dBm
22 dBm
25 dBm
28 dBm
1.0
Norm. Amplitude
Stokes power, a.u
21
Signal output power, a.u
7
87
0.8
(b)
0.6
0.4
0.2
0.0
-60
Average signal power, dBm
-40
-20
0
20
40
60
Frequency, MHz
Figure.4.6.3: (a) Measured optical powers of the Stokes waves and transmitted signals. (b) Linewidths of the
generated Brillouin Stokes waves recorded in the ESA, by use of the delayed homo-heterodyne system.
twice the Brillouin threshold the Stokes waves is powerful enough to generate its own Stokes
wave co-propagating with the signal. This turning point at approximately 24 dBm input
power is observed as an apparently resumed growth of the signal output power. The linewidth
of the Stokes wave with respect to the signal power was characterized. The spectral width of
the Stokes wave was measured by the delayed self-heterodyne method [49]. This method is
an interferometric method, and is based on a Mach-Zehnder interferometer in which one arm
contains a frequency shifter (EOM2) and the other is used as a delay line to break the
coherence of the analyzed beat signal. The beating is recorded using a fast detector connected
to an electrical spectrum analyzer. Figure.4.6.3b shows the spectral profiles of the generated
Stokes wave for different input signal powers, all of them over the SBS critical power. The
spectra show no linewidth change of the Stokes wave for all the relevant input powers, hence
the spectrum of the SBS-induced absorption will remain similar for all input powers,
generated by a lightwave showing a linewidth of approximately 10 MHz.
4.6.3 Self-advanced signal propagation
Self-advancements for isolated signals
The signal delay induced by the SBS effect was typically measured as a function of the
Brillouin gain/loss, as a consequence of the simple linear relationship between these two
quantities. However, in this configuration, it turns out to be conceptually inappropriate since
the signal pulse experiences fast-light propagation with no independent pump source.
Additionally, the signal annihilates part of its power as a result of the Brillouin loss and
saturates to a roughly constant value. Therefore, a direct measurement of advancement as a
function of loss experienced by the signal is of limited interest and can not be easily extracted
from the raw data. A more interesting quantity to plot, however, is the delay as a function of
the input signal power. Figure.4.6.4 shows the measured time waveforms of the signal pulses
CHAPTER 4.6. SELF-ADVANCED BRILLOUIN FAST LIGHT
Signal power
00 dBm
16 dBm
22 dBm
28 dBm
1.0
0.8
Norm. Amplitude
88
0.6
0.4
0.2
0.0
-0.2
-100
-50
0
50
100
Time delay, ns
Figure.4.6.4: Temporal traces of the signal pulse after propagating through the dispersion shifted fiber for
different input signal powers, showing clear advancements.
Advancement, ns
12
10
8
6
4
2
0
0
4
8
12
16
20
24
28
Average signal power, dBm
Figure.4.6.5: Temporal advancements of the signal pulses as a function of the signal average power, showing
logarithmic dependence of delay on the signal power.
for different average powers of the input signal, ranging from 0 dBm to 28 dBm. It is clearly
observed that the pulse experiences more advancement as the input power increases.
Additionally, in all cases, the signal pulse experiences low distortion. The advanced pulses
show a slightly sharper leading edge and a longer trailing edge, consistent with previous
observations in SBS fast light [2]. The largest advancement induced by the proposed scheme
is 12 ns (corresponding to a fractional delay 0.26) at a signal power 28 dBm. The signal
power in this setup was limited by the saturation power of the EDFA. Longer delays could
eventually be realized in this method with an EDFA with higher saturation power or with a
fiber showing a smaller effective area. Figure.4.6.5 shows the advancement observed on the
signal as a function of the input power. To determine the amount of signal advancement the
position of the peak of the pulse was used. Notice that for low power levels, there is no
visible advancement, since there is no significant loss of the signal below the Brillouin
CHAPTER 4.6. SELF-ADVANCED BRILLOUIN FAST LIGHT
89
critical power. When the signal power exceeds this critical value, the pulse train immediately
experiences an observable advancement. In the range of measured values, the advancement
depends logarithmically on the input power, with a slope efficiency of 0.69 ns/dBm.
This logarithmic dependence is actually a direct consequence of the total power
transfer from the signal to the Stokes power above the Brillouin critical power Pc. The
effective total loss A experienced by the signal is defined as
Pout
Pin
=
e− A ,
(4.29)
where Pin and Pout represents the input and output signal power, respectively. Since above the
Brillouin threshold the signal output power Pout saturates to a constant value Psat, the effective
loss simply depends on the input signal power Pin following this relationship:
⎛ Pout ⎞ = − ln ⎛ Psat ⎞ .
⎟
⎜P ⎟
⎝ Pin ⎠
⎝ in ⎠
A( Pin ) = − ln ⎜
(4.30)
The temporal advancement being proportional to the effective total loss A and Psat being
constant, the logarithmic dependence on the input power comes out immediately from this
simple description. It incidentally shows one more advantage of this scheme: the output
signal power remains constantly fixed at the saturation power Psat for any signal advancement,
since all light over the critical level is scattered to build up the Stokes wave.
Self-advancements for a data stream
To demonstrate that this technique can also be used for a data stream with negligible DC
component, the repetition rate of the signal pulse was modified to 20 MHz, so that it
reasonably simulates a real sequence of bits when averaged over the fiber length. In this case,
the pulses in this case have a FWHM of 14.22 ns, hence the duty cycle is 30 % and the DC
component is reduced to a negligible fraction of the pulse peak power. Figure.4.6.6a shows
7
(a)
6
0.8
Advacement, ns
Norm. Amplitude
1.0
0.6
0.4
0.2
0.0
-100
-50
0
50
Time delay, ns
100
(b)
5
4
3
2
1
0
-1
0
5
10
15
20
25
30
Average signal power, dBm
Figure.4.6.6: (a) Temporal traces of data streams for a signal power below the critical power (solid line) and at
maximum signal power realized in our setup (dashed line). (b) Signal advancement as a function of the
average signal power, showing the logarithmic dependence over the Brillouin critical power at 10 dBm.
CHAPTER 4.6. SELF-ADVANCED BRILLOUIN FAST LIGHT
90
the pulse train for 2 different signal powers - one below the critical power and the other at the
maximum possible value using this setup - and the pulse advancement is again clearly visible.
Figure.4.6.6b shows the advancement as a function of the signal power, with a maximal
obtained fractional delay of 0.42. The slope is in this case 0.35 ns/dBm. It must be pointed
out that the power of the data stream time-averaged over the fiber length must remain
constant to avoid any timing jitter at the output, requiring a steady fraction of bits "1" with
respect to the total number of bits.
The main limit for the maximum possible advancement in this configuration is
caused by the onset of the 2nd order SBS amplified Stokes emission. Once the backward
Stokes reaches its own critical power for SBS, there will be a forward Stokes wave
downshifted by 2νB below the frequency of the input signal. This wave will deplete the
backward Stokes wave and hence make the advancement saturated. This places a
fundamental limit to the range of signal power suitable to produce a delaying effect. Other
limitations are related to the requirement of a permanent constant average power in the data
stream to avoid fluctuations in the signal delay and amplitude.
4.6.4 Self-adapted signal bandwidth
A last aspect of self-pumping was investigated related to the capability of the Stokes
amplified emission to adapt its spectral width to the signal bandwidth. For this purpose, the
Stokes emission generated by a pulse train was observed while the pulse averaged power is
well above the Brillouin critical power Pc. The initial pulse width was 50 ns at a repetition
frequency of 4 MHz, corresponding to a normalized repetition rate of 5. Such a signal shows
a measured FWHM bandwidth of 10 MHz that is substantially lower than the Brillouin
natural linewidth. Then the pulse width was reduced while adapting proportionally the
Pulse width
50 ns
40 ns
20 ns
10 ns
(a)
0.8
0.6
0.4
0.2
0.0
-80
-60
-40
-20
0
20
40
60
24
Stokes linewidth, MHz
Norm. Amplitude
1.0
22
(b)
20
18
16
14
12
10
80
Frequency, MHz
8
10
20
30
40
50
Pulse bandwidth, MHz
Figure.4.6.7: (a) Measured spectra of the Stokes emission by the delayed self-homodyne technique, for
different signal widths at a constant normalized repetition rate. (b) Measured linewidth of back-scattered
Stokes wave as a function of the measured signal bandwidth.
CHAPTER 4.6. SELF-ADVANCED BRILLOUIN FAST LIGHT
91
repetition frequency to maintain a constant normalized repetition rate. This way the average
pump power is kept constant and the only modified relevant signal characteristic is its
bandwidth. Figure.4.6.7a shows the measured Stokes spectra for different pulse width. From
a floor value of 10 MHz the Stokes linewidth clearly self-adapts to the incremental
broadening of the signal, as illustrated in Figure.4.6.7b. It must be pointed out that the Stokes
linewidth represents only a fraction of the input signal bandwidth. It could be extrapolated
from the measurements that this fraction corresponds asymptotically for a wideband signal to
about 45 % of its bandwidth. Even after convolution with the natural Brillouin spectrum this
fractional linewidth has certainly a substantial impact on the pulse distortion.
CHAPTER 4.7. DISPERSIVE DELAY LINE BASED ON WAVELENGTH CONVERSION
92
4.7 Dispersive delay line based on wavelength conversion
In this section, another approach for the development of all-optically controlled delay lines
will be discussed that is entirely distinct from a slow light-based delaying scheme. In any
slow light system the maximum time delay that a signal can experience is inherently limited
to a few bits delay. It is mainly caused by the growing noise associated to the amplification
process and the signal distortion, resulting from the frequency dependent Brillouin gain
(spectral filtering effect) and the large dispersion induced by the slow light medium.
Therefore, the scientific challenge to produce a large amount of signal delay with negligible
distortion remains opened. To overcome this issue, an elegant solution was soon proposed,
which makes use of the combination of wavelength conversion and group velocity dispersion
[50-52]. This scheme allows tremendous signal delays for broadband signals. In the work of
this thesis, the conversion-dispersion technique was also studied and experimentally
demonstrated to improve to a large increment the fractional delay. The main progress in the
configuration compared to the previous works is to simply and efficiently realize the
wavelength conversion with a compact and low power concept. The wavelength of the signal
was converted to a desired wavelength using cross gain modulation via a semiconductor
optical amplifier. It must be noticed that this type of conversion offers the key advantage to
be efficient over a much broader wavelength range than parametric processes. This technique
extends the range of delays obtained by the dispersive line and a large range of optical delays
with low distortion could be demonstrated experimentally up to tens of nanosecond for
100 ps optical pulses.
4.7.1 Wavelength conversion using cross gain modulation
Cross gain modulation (XGM) in a semiconductor optical amplifier (SOA) is usually
described as a nonlinear interaction between two co- or counter-propagating beams, a strong
pump wave at νpump and a weak probe wave at νprobe. The principle to realize the wavelength
conversion using XGM is depicted in Figure.4.7.1. Let suppose an intensity modulated pump
light entering into an SOA and a continuous wave (CW) probe light simultaneously injected
counter-directionally into the SOA. Due to the gain saturation, the pump light will modulate
the gain inside the amplifier [53]. In turn XGM in the amplifier will impose the pump
modulation on the probe. Consequently, the intensity of the probe at the target wavelength is
inversely modulated and carries the complementary data pattern, so essentially the same
information as the pump modulation. This way the wavelength of the signal can be converted
with high conversion efficiency and no requirement for strict phase matching conditions,
CHAPTER 4.7. DISPERSIVE DELAY LINE BASED ON WAVELENGTH CONVERSION
Pump
93
Signal in
SOA
Pulse modulation
t
Continuous wave
t
t
Converted wave
Signal out
Figure.4.7.1: Schematic diagram of the principle to generate the XGM wavelength conversion, in which the
pump and signal waves counter-propagate through semiconductor optical amplifier.
which are necessary to obtain parametric amplifications [50-52]. Moreover, this process can
be polarization-independent if an SOA showing a polarization independent gain is employed.
After the first wavelength conversion is accomplished, the probe light exiting from
the SOA is delivered into a highly dispersive optical medium such as long lengths of optical
fiber [50,52] or even a highly dispersive Bragg grating [51]. The propagation velocity of the
probe through the dispersive fiber is continuously varied by simply tuning the wavelength of
the probe due to the wavelength dependence of the group velocity. As a result, the probe
wave can exit the dispersive fiber with relative temporal delays or advancements. The
amount of time delay achieved in the delay line can be simply estimated to first order as the
product of probe wavelength change and the group-velocity dispersion (GVD) of the fiber.
The delayed converted signal in turn experiences a second wavelength conversion through
another SOA back to the original signal wavelength, restoring in the same process the
original modulation pattern.
4.7.2 Continuous control of large fractional delay
The schematic diagram of the experimental setup is depicted in Figure.4.7.2. A commercial
distributed feedback (DFB) laser diode operating at 1571 nm was used as a light source and
its output was split by a 90/10 directional coupler. The higher power branch was optically
gated using a fast electro-optic Mach-Zehnder intensity modulator (EOM) to produce a signal
pulse train with duration of 100 ps FWHM at a repetition rate of 500 MHz. Then the pulse
train was boosted using an erbium-doped fiber amplifier (EDFA) before entering into the first
SOA, so that it could play the role of the pump in the first wavelength conversion. A weak
continuous wave (CW) probe beam at a desired wavelength was generated by a tunable laser
source (TLS) and was also simultaneously launched into the SOA, but in the direction
opposite to the pulse direction. This way the converted probe signal could be simply
CHAPTER 4.7. DISPERSIVE DELAY LINE BASED ON WAVELENGTH CONVERSION
Pump
94
Tunable laser
EOM
SOA
RF DC
t
t
Signal pulse
Continuous wave
t
25-km SMF
t
t
Signal out
SOA
Figure.4.7.2: Experimental setup to generate a wide range of signal delays, using wavelength conversion
through semiconductor optical amplifier and group velocity dispersion in optical fibers.
separated from the input pump pulse without any need of optical filters. According to the
principle described above, XGM in the amplifier induces an intensity modulation on the
probe, resulting in a wavelength converted signal that is inverted compared to the modulation
pattern of the input pulses. The converted signal was then delivered to a dispersive medium
to experience a relative time delays. In our setup, it was simply a 25 km-long single-mode
fiber, showing the group velocity dispersion of ~180 ps/nm. After exiting from the dispersive
fiber, the wavelength converted signal was amplified using another EDFA before entering
into the second SOA. The lower power channel of the initial DFB laser was also launched
into the SOA to play the role of the probe in the second wavelength conversion.
Consequently, the original pattern of the signal pulse is restored and the signal returns to its
original wavelength.
The normalized time waveforms of the signal pulses exiting from the delay line were
measured and displayed on a digital oscilloscope for different wavelength of the TLS, as
TLS wavelength
1550 nm
1552 nm
1554 nm
1556 nm
Norm. Amplitude
1.0
0.8
0.6
0.4
0.2
0.0
-0.5
0.0
0.5
1.0
1.5
2.0
Delay time, ns
Figure.4.7.3: Time waveforms of the delayed signal pulse trains while the TLS wavelength was swept from
1550 nm to 1556 by 2 nm steps, showing clear delays of the signal pulse.
CHAPTER 4.7. DISPERSIVE DELAY LINE BASED ON WAVELENGTH CONVERSION
95
14
Time delay, ns
12
10
8
6
4
2
0
-2
1520 1530 1540 1550 1560 1570 1580 1590 1600
Wavelength, nm
Figure.4.7.4: The relative signal delays as a function of the TLS wavelength and the red curve represents the
result of the 4th order fitting.
shown in Fig.4.7.3. The pulse delays are clearly observed without significant broadening
while the TLS wavelength is increased by 2 nm steps. To determine the amount of the signal
delay the peak point of the pulse, as shown in Figure.4.7.4. The largest time delay achieved
was 14 ns corresponding to a fractional delay of 140 while the TLS wavelength was swept
from 1526 nm to 1596 nm. This ratio can certainly be further improved using a more
dispersive delay line and an even broader wavelength scanning range.
It must be pointed out that the delay lines, based on wavelength conversion and
group velocity dispersion, also suffer the trade-off relations between the maximum
achievable delay and the associated signal distortion expressed by broadening. Since the
chromatic dispersion is a key parameter to control the amount of signal delay, the frequency
components of the signal travel through the medium at slightly different velocities. More
specifically, blue components propagate faster than red components in the anomalous
dispersion regime observed in standard single mode fibers in the minimum attenuation
window. Therefore, the signal must experience a broadening effect after propagating through
the material, limiting the signal bandwidth. This can be alleviated by imposing an extrinsic
frequency chirp across the signal. Recently, Okawachi et al. demonstrated experimentally the
possibility to completely compensate the large induced signal broadening using phase
conjugation [54].
The anticipated signal broadening was calculated as a function of the GVD for the
transform-limited Gaussian pulse with duration of 100 ps FWHM [55] and the wavelength
change required to achieve 1000-bits delay, as shown in Figure.4.7.5. According to the
simulation test, this delay line demands wavelength change of ~550 nm, which is far from
real operational conditions. In an alternative way a larger GVD of the fiber can be replaced to
CHAPTER 4.7. DISPERSIVE DELAY LINE BASED ON WAVELENGTH CONVERSION
Broadening factor, B
500
1.4
400
1.3
300
1.2
200
1.1
100
1.0
0
0
500
1000
1500
2000
2500
Wavelength change Δλ, nm
Pulse broadening
Wavelength change
1.5
96
3000
GVD, ps/nm
Figure.4.7.5: The associated signal broadening to achieve 1000-bits delay for a transform limited Gaussian
pulse with a width of 100 ps FWHM and the required wavelength change as a function of group velocity
dispersion.
compensate the small change of wavelength, but leading to more degradation on the data
stream.
The maximum bit-rate of the data packet through this delay-line was restricted by the
carrier recovery time of the SOAs used in this experiment. However, it must be pointed out
that this delay-line can be applied for high speed networks since error-free wavelength
conversion at 160 Gbit/s has been realized [56]. Moreover, the original signal wavelength is
preserved while producing optical delays. The signal bandwidth can be moderately modified
by the possible chirp effects by gain saturation in SOAs [57]. However, this scheme can be a
promising timing tool for future communication and microwave photonics systems.
CHAPTER 4. BIBLIOGRAPHY
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Chapter 5
Slow light and linear light-matter
interactions
The main target applications of slow light so far have been the achievement of all-optical
timing tools such as all-optical delay lines, all-optical buffers and signal synchronization
elements for future all-optical high capacity networks [1,2]. These target applications benefit
from the group delay changes caused by the slow light element as an indirect consequence of
the modification of the group velocity in the medium. However, the most remarkable feature
of slow light media, in which extreme changes in the group velocity can be possibly achieved,
remains unexploited. Recently, some research effort has been devoted to understanding the
possibilities of slow light for enhancing light-matter interactions. For instance, the role of
slow light for enhancing nonlinear effects has been theoretically investigated [3,4]. This
enhancement is accomplished by two main contributions of slow light: the longer transit time
of light through an optical medium and the higher energy density since the signal tends to be
spatially compressed due to the increased group index. It has also been theoretically and
experimentally proved that the extreme dispersion of slow light can lead to an enhancement
of the spectral sensitivity of interferometers [5,6]. The role of slow light in gyroscopes has
been also theoretically investigated [7]. Besides, recent works [8,9] have theoretically
demonstrated that Beer-Lambert-Bouguer (BLB) absorption can be increased while light
transmits through a special kind of microstructured photonic crystal (PC) cuvette. This
absorption enhancement was attributed to the slow group velocity in the cuvette [8,9]. A
simpler interpretation of these results would be that the PC structure acts as a quasi-cavity. In
such a structural waveguide the incident light bounces back and forth several times, resulting
in a longer interaction length of light with the liquid analyte. The experimental proof of these
CHAPTER 5. SLOW LIGHT AND LINEAR LIGHT-MATTER INTERACTIONS
102
theoretical results seems very challenging when considering the dimension of the
microstructure and the degree of infiltration required.
In this thesis, the relationship between material slow light (slow light in travelingwave media) and light-matter interactions was experimentally investigated and it is clearly
observed that material slow light has no impact on Beer-Lambert absorption. This
experimental demonstration was successfully achieved, using a special kind of gas cell. A
photonic crystal fiber (PCF) with solid core plays the role of the gas cell as the fiber holes are
filled with gaseous acetylene at reduced pressure. A small fraction of the mode field (the
evanescent field) propagates through the holes and thus experiences Beer-Lambert absorption.
The fiber is also used as a slow light medium. Since light mostly propagates through a solid
silica core the group velocity of the light can be controlled by a nonlinear interaction between
optical waves. The absorption can therefore be measured under normal and slow-light
conditions without modifying the experimental implementation. By comparing the two
regimes, one can observe if the slow light has any impact on the absorption by the gas. The
results indicate that Beer-Lambert absorption is not at all affected by slow light. To the best
of our knowledge, this is the first experimental observation of the effect of material slow light
on light-matter interactions.
5.1 Principle
The absorption of a gas analyte is usually expressed by means of the Beer-Lambert (BL) law.
When light with an intensity Io propagates through the gas cell, the light experiences an
exponential decaying and the transmitted intensity is given as I=Ioexp[-α·C·L]. C is the gas
concentration, L is optical path length and α is the molar absorption coefficient, which is a
unique property of the gas, related to the imaginary part of the refractive index. Previous
papers (Ref [8], eq. 3) have suggested that the absorption coefficient is inversely proportional
to the group velocity. However, in this work, it is experimentally shown that α is not related
to the group velocity. To clarify the relationship between BL absorption and slow light, a
solid-core microstructured optical fiber replaces the classical gas cell. There are two main
reasons for this: first, in such fiber, the optical mode is mainly confined to the solid core.
Thus, the velocity of light propagating through the PCF can be controlled by manipulating
the material properties of the core using a narrowband gain process such as stimulated
Brillouin scattering. The intrinsic fast light effect caused by the gas absorption resonance can
be reduced to a negligible amount when compared to the effect of Brillouin gain (weaker
linear absorption and much broader resonance width). Second, with a proper choice of the
CHAPTER 5. SLOW LIGHT AND LINEAR LIGHT-MATTER INTERACTIONS
103
fiber geometry, a non-negligible fraction of the guided light propagates through the air holes
filled with gas, and hence can probe Beer-Lambert absorption. The key advantage of this
setup is that the effects of slow light and absorption are totally de-coupled, so that it is
possible to vary one without having a direct impact on the other. Under these conditions, the
equation of BLB absorption can be slightly modified as:
I = I o exp[ − N ⋅ σ ⋅ f ⋅ L ]
where N is the gas molecular density, σ is the absorption cross section, f is the fraction of
light in holes and L is the optical length.
Figure.5.1a shows the SEM image of the single-mode PCF. Using the source-model
technique (SMT) [10], the mode field distribution along the core cross section of the PCF
was calculated at a wavelength of 1535 nm, as shown in Figure.5.1b. The evanescent part of
the guided field propagating in the gas, representing 2.9 % of the total power, is shown in
Figure.5.1c. The PCF was then filled with acetylene gas and hermetically sealed at a low
pressure, using a normal arc fusion splicer. More details on the development of the gas cell
are referred to the paper [11].
(a)
(b)
(c)
Figure.5.1: (a) SEM image of the solid-core microstructured photonic crystal fiber. (b) Calculated mode field
distribution of the fundamental mode and (c) the small evanescent fraction of the guided field present in air
holes.
5.2 Group velocity change through the fiber gas cell
As a Brillouin gain medium, a 9.18 m-long PCF was used. The Brillouin spectrum of this
fiber was characterized, showing a Brillouin shift of 10.85 GHz and an SBS gain bandwidth
of 38 MHz. Figure.5.2 depicts the experimental setup to clarify the relationship between slow
light and BLB absorption. A commercial distributed feedback (DFB) laser diode, operating at
1535 nm, was used as a light source and its output was split using a directional coupler. One
branch was strongly boosted using a high power erbium-doped fiber amplifier (EDFA) with
CHAPTER 5. SLOW LIGHT AND LINEAR LIGHT-MATTER INTERACTIONS
EDFA
DFB-LD
VOA
104
PC
Det
1W
Pump
Pump
EOM
Ramp
RF
DC
ν
Signal
Microstructured fiber:
Absorption cell
+
Slow light medium
n
Lock-in Amp.
Oscilloscope
ν
Absorption
Signal
ν
FBG
EOM
Figure.5.2: The experimental setup to verify the effect of slow light on BLB absorption. EDFA; erbium doped
fiber amplifier, VOA; variable optical attenuator, PC; polarization controller, EOM; electro-optic modulator,
FBG; fiber Bragg grating.
30 dBm saturation power so as to play the role of Brillouin pump. The output of the EDFA
was then precisely controlled by a variable optical attenuator before entering into the gas cell.
The other branch was modulated at the Brillouin frequency shift of the PCF through an
electro-optic Mach-Zehnder intensity modulator (EOM) to generate two first-order sidebands
and the DC bias of the modulator was adequately set for the complete suppression of the
carrier. In consequence, the output of the modulator contains only two waves at frequencies
of Brillouin Stokes and anti-Stokes. Only lower-frequency sideband was then precisely
filtered by a fiber Bragg grating (FBG) and was launched into the fiber as signal.
The amount of group velocity change was first measured as a function of pump
power. For this measurement, the frequencies of the pump and signal waves were spectrally
placed in a region where absorption resonances are absent. As in typical Brillouin slow light
the signal was clearly delayed with respect to the pump power, as shown in Figure.5.3a. To
precisely determine the SBS-induced group delay, the signal was sinusoidally modulated at
1 MHz by another external EOM. The phase of the sine wave after propagation through the
PCF was measured while the pump power was incremented from 0 to 600 mW by 100 mW
steps. This way the group delay change achieved through the PCF for different pump powers
was accurately determined from the difference of phase shift, as shown in Figure.5.3b.
Additionally, the transit time through the fiber was also determined by making the same
measurement with and without the PCF. The group velocity change is quantified by defining
a slow-down factor S corresponding to the ratio between the transit times through the gas cell
in slow light conditions (with pump) and in normal conditions (without pump), respectively.
The largest time delay achieved through the PCF was 11.7 ns at a pump power of 600 mW,
corresponding to an increment of the effective optical path length by 3.5 m and this time
CHAPTER 5. SLOW LIGHT AND LINEAR LIGHT-MATTER INTERACTIONS
0.8
0.6
0.4
0.2
0.0
-60
-40
-20
0
20
40
60
Time, ns
80
12
1.5
(b)
10
1.4
8
1.3
6
1.2
4
1.1
2
1.0
0
0.9
0
100
200
300
400
500
Slow-down factor, S
Pump power
no pump
200 mW
400 mW
600 mW
(a)
Time delay, ns
Norm. Amplitude
1.0
105
600
Pump power, mW
Figure.5.3: (a) Time waveforms of the signal after propagating through the PCF for different pump powers,
showing clear signal delays and (b) the time delays achieved in this Brillouin delay line as a function of pump
power and the associated slow-down factor.
delay is equivalent to a 26 % reduction of the group velocity.
5.3 Effect of slow light on Beer-Lambert absorption
To observe an absorption resonance of acetylene, the frequency of the signal was swept all
over the absorption line, simply by introducing a slow variation of the current applied to the
initial DFB laser. Since the signal and pump waves were generated from the same DFB laser
source, the pump frequency was also swept with a perfect synchronization to the signal. This
way the spectral distance between signal and pump was kept stable during frequency
sweeping and matched the Brillouin shift. In consequence, the pump constantly generated a
Brillouin gain resonance centered at the signal frequency, even though the signal frequency
was swept across the absorption line. It results in a constant Brillouin gain for the signal
while the frequency is swept, which means that the group velocity of the signal remains
constant and simply controlled by the pump power. The intensity of the signal wave was also
sinusoidally modulated at 100 kHz by an EOM to obtain a clean signal using a lock-in
detection. The signal wave was then amplified using another EDFA, so as to even further
improve the signal-to-noise ratio, still avoiding saturation of the Brillouin amplifier. The
output power was then precisely adjusted before entering into the gas cell using a VOA in
order to avoid any risk of saturation of the atomic transition [12].
The signal amplitudes after propagating through the fiber gas cell were recorded for
different pump powers on a digital oscilloscope while the laser frequency was swept.
Figure.5.4 shows the measured normalized absorption when the signal is centered in the
P(17) absorption line at 1535.453 nm. The pump power was incremented following the same
sequence as during the group delay calibration. Therefore, the curves are fully representative
of the gas absorption while the signal propagates through the gas cell at different group
CHAPTER 5. SLOW LIGHT AND LINEAR LIGHT-MATTER INTERACTIONS
106
velocities. The absorption curves were normalized in logarithmic scale for a better
comparison of the absorption levels. To evaluate the confidence level, measurements were
repeated 5 times for each power levels, so that the mean value and the standard deviation of
the peak attenuation due to the BL absorption could be accurately estimated. These values
representing the absorption at the center of the molecular line are plotted as a function of the
slow-down factor in Figure.5.5. It is clearly observed that BL absorption is totally
independent on the slow-down factor S, hence group velocity change. The same experiment
was also performed for the P(15) and P(19) absorption lines at 1534.158 nm and 1536.771
nm, respectively, resulting in very similar results despite fairly different magnitudes for their
Norm. Absorption, dB
absorption coefficient α. The result of this experimental verification of the relationship
0
-1
-2
Pupmp power
0 mW
100 mW
200 mW
300 mW
400 mW
500 mW
600 mW
-3
-4
-5
-6
-4
-2
0
2
4
6
8
10
12
Frequency, GHz
Figure.5.4: Variation of the signal amplitude in logarithmic scale after propagating through the PCF gas cell
for different pump powers.
Absorption, dB
5.8
Measured absorption
Linear dependence
5.6
5.4
5.2
5.0
4.8
4.6
1.00
1.05
1.10
1.15
1.20
1.25
1.30
Slow-down factor, S
Figure.5.5: Measured optical power loss at the peak attenuation due to the Beer-Lambert absorption as a
function of the slow-down factor. The error bars show the measured standard deviation on the attenuation
measurement and the red line represents the hypothetical response expected for an absorption coefficient
inversely proportional to the group velocity.
CHAPTER 5. SLOW LIGHT AND LINEAR LIGHT-MATTER INTERACTIONS
107
between absorption and group velocity could be at first glance anticipated, since the BL
absorption law is obtained from light-matter interaction by only considering the phase
velocity. The experimental results then unambiguously clarified that the linear interaction
between light and matter is not influenced by group velocity changes, which is induced by
modification of the material properties. However, it could be argued that the time delay
(11.7 ns) induced by slow light is negligible compared to the signal period (10 μs) of the sine
wave used for the measurements, and thus does not expand substantially the interaction time
to lead to an observable effect on the BL absorption. Besides, in this environment, the sine
wave acts as a quasi continuous wave which propagates through the gas cell at the phase
velocity. To decisively clarify this issue, this experiment was repeated using an optical pulse
as signal. The duration of the pulse was 22 ns FWHM, which is shorter than the propagation
1
Absorption, dB
0
-1
-2
Pump power
000 mW
100 mW
200 mW
300 mW
400 mW
500 mW
600 mW
-3
-4
-5
0
100
200
300
400
500
Pump power, mW
Figure.5.6: Amplitude variation of the pulsed signal in logarithmic scale after propagating through the PCF
gas cell for different pump powers.
5.6
5.4
Absorption, dB
5.2
5.0
4.8
4.6
4.4
4.2
1.00
1.05
1.10
1.15
1.20
1.25
1.30
Slow-down factor, S
Figure.5.7: Measurement of pulsed signal power loss at the peak attenuation due to the Beer-Lambert
absorption as a function of the slow-down factor.
CHAPTER 5. SLOW LIGHT AND LINEAR LIGHT-MATTER INTERACTIONS
108
time (45 ns) through the gas fiber cell. This way the interaction time of the signal pulse with
the gas is clearly and noticeably modified by slow-light effect. However, this experiment
resulted in the same flat response, as shown in Figure 5.5 and Figure 5.6. As a result, it
confirms that BL absorption is not influenced by group velocity since photons interacting
with gas atoms can be considered as propagating through the material at phase velocity.
These results decisively lead to the evidence that material slow light does not bring
any benefit for this kind of sensing, and the same conclusion can be extrapolated with a good
confidence to all types of linear light-matter interaction. In fact, in case of material-related
slow light the electric field itself may be not enhanced while its energy density is basically
condensed due to increased group index in a proportional manner. The extra energy is then
transferred into the material to reinforce the acoustic wave, so that the electric field remains
constant. However, it must be pointed out that an entirely different conclusion will be
probably drawn for structural slow light, like that created in coupled resonators or using the
dispersion in a properly designed waveguide structure. In such systems the optical path
length is artificially extended through loop recirculation or multiple reflections, causing
primarily an apparent slowing of light but also a longer interaction length for light-matter
interaction. In that case the apparently enhanced absorption can always be interpreted with
classical arguments, out of the framework of slow light theory.
CHAPTER 5. BIBLIOGRAPHY
109
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[2]
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[7]
M. Terrel, M. J. F. Digonnet, and S. Fan, "Performance comparison of slow-light
coupled-resonator optical gyroscopes," Laser & Photon. Rev., 1-14 (2009).
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absorption," Appl. Phys. Lett., 90, 141108 (2007).
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hydrogen cyanide in hollow-core photonic bandgap fibers," Opt. Express, 13, 1047510482 (2006).
Chapter 6
Conclusions and Perspectives
Brillouin slow and fast light has shown an unprecedented flexibility to offer an all-optically
controlled delaying element for modern optical telecommunication systems based on high
speed networks. However, it has been proved that any slow light system, based on a large
dispersion induced by a spectral resonance, presents a critical deadlock in terms of optical
storage capacity. As already discussed in many studies [1-3], the maximum achievable time
delay in such slow light-delay lines is strictly limited up to a few periods of the signal.
Actually, the storage of more than 5-bits of information remains unrealistic, since it requires
a tremendous pump power in order to achieve such an amount of time delay, and tends to
being subject to a significant noise induced by the amplification process. For this main target
application, slow light based on micro-ring resonators could be implemented to extend the
maximum delay to 10-bits, but remains restricted to a system, in which the wavelength of a
light signal is controlled and very well-defined [4,5]. Also, coherent optical storage seems to
be a better approach for the generation of tunable delaying. Recently, it was reported that
SBS can transform the information encoded by a bit sequence to an acoustic excitation and
retrieve the signal information at a later time, resulting in a considerable delaying effect.
From a practical point of view, dispersive delay lines based on wavelength conversion looks
like more promising solution to obtain a multi-bits delay. This technique has already shown
the capability to delay a full sequence of more than 1000-bits at 10 Gbit/s transmission [6-9].
As a result, although slow light was initially motivated as a scientific challenge to realize an
all-optical buffer for future all-optical routers, in practice, it has been proved to be an
insufficient timing tool when compared to other methods mentioned above. According to this
conclusion the slow-light community started to reconsider potential applications of slow light,
where one can take the maximum benefit of the real slowing of light.
CHAPTER 6. CONCLUSIONS AND PERSPECTIVES
112
Optical storage is just one aspect of slow light applications. Slow light-delay lines
can be implemented as a robust solution for retiming applications in digital communications,
in which only error-free 1-bit delay is required. In a wavelength-division multiplexing
transmission, a light signal in each channel can be independently delayed, so that one channel
can be precisely synchronized to other ones using a single bit delaying [10,11]. Moreover, as
far as analog signals are concerned in communication systems, slow light is possible to
produce tunable delays slightly more than one signal period or 2π phase shift. It turns out that
this tunable range is sufficient for the development of optical switches and modulators [12],
and also for the generation of true time delays for microwave photonic applications such as
photonic microwave filters [13] and phased-array antenna systems [14]. It must be pointed
out that the intrinsic linear response of Brillouin amplification can further support the
applications to analog signals as linearity is another essential requirement in analog systems.
In slow light conditions, light actually propagates slowly through a medium, so that
the transit time turns to be longer. This fascinating fact may potentially lead to a longer
interaction length between light and matter. According to this particular feature of slow light,
a significant enhancement of light-matter nonlinear interactions has been already
demonstrated when light propagates through the optical medium with a reduced group
velocity [15-18] and the enhancement of nonlinear effects must scale as the square of
slowing-down factor. Additionally, it is anticipated that the spectral sensitivity of
interferometers or optical sensors can be considerably improved when a slow light medium is
involved in these systems [19-23]. The key point of such applications is that transformations
and functions based on optical nonlinearities could be observed in miniaturized structures,
showing identical nonlinear efficiency. For instance, if the propagation velocity of light is
reduced by a tenth of its normal group velocity, a 1 m-long nonlinear medium under test
could be replaced by a 1 cm-long one for an equivalent nonlinear effect. However, this point
must be cautiously addressed since particular waveguide structures such as photonic crystal
and coupled resonator lead to a longer transit time of light in the medium by increasing the
effective path length through multiple reflections (or a similar effect), resulting in a longer
interaction time of light with material. A reduced group velocity induced by structural slow
light is not a fundamental cause of the sensitivity or nonlinearity enhancement in these
implementations. Naturally, structural slow light will give rise to such an enhancement since
delaying and enhanced interaction result from the same effect, i.e. an extended effective
optical path length.
Nevertheless, some exotic and unexpected applications as well as all these
anticipated applications will be discovered in coming years through innovative and creative
CHAPTER 6. CONCLUSIONS AND PERSPECTIVES
113
solutions, and such scientific challenges makes slow light yet a very fascinating field of
research.
CHAPTER 6. BIBLIOGRAPHY
114
Bibliography
[1]
R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Slow-light optical buffers:
Capabilities and fundamental limitations," J. Lightwave Technol., 23, 4046-4066
(2005).
[2]
R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A.E. Willner, "Maximum time delay
achievable on propagation through a slow-light medium," Phys. Rev., A, 71, 023801
(2005).
[3]
J. B. Khurgin, "Power dissipation in slow light devices: A comparative analysis," Opt.
Lett., 32, 163-165 (2007).
[4]
Khurgin, J.B., "Optical buffers based on slow light in electromagnetically induced
transparent media and coupled resonator structures: Comparable analysis," J. Opt. Soc.
Am. B, 22, 1062-1074 (2005).
[5]
F. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip,"
Nature Photonics, 1, 65-71 (2007).
[6.
J. E. Sharping, Y. Okawachi, J. V. Howe, C. Xu, Y. Wang, A. E. Willner, and A.L.
Gaeta, "All-optical, wavelength and bandwidth preserving, pulse delay based on
parametric wavelength conversion and dispersion," Opt. Express, 20,7872-7877 (2005).
[7]
M. Fok and C. Shu, "Tunable Pulse Delay using Four-Wave Mixing in a 35-cm
bismuth Oxide Highly Nonlinear Fiber and Dispersion in a Chirped Fiber Bragg
Grating," in Proc. ECOC'06, Cannes, France (2006).
[8]
S. Chin and L. Thévenaz, "Large multi Gbit/s delays generated in an all-optical tunable
delay line preserving wavelength and signal bandwidth," in Slow and Fast Light,
Technical Digest (CD), SMC3 (2008).
[9] Y. Okawachi, M. A. Foster, X. Chen, A. C. Turner-Foster, R. Salem, M. Lipson, C. Xu,
and A.L. Gaeta, "Large tunable delays using parametric mixing and phase conjugation
in Si nanowaveguides," Opt. Express, 16, 10349-10357 (2008).
[10] B. Zhang, L. S. Yang, J. Y. Yang, I. Fazal, and A.E. Willner, "A Single slow-light
element for independent delay control and synchronization on multiple Gb/s data
channels," Photon. Technol. Lett., 19, 1081-1083 (2007).
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Express, 15, 8317-8322 (2007).
[12] D. M. Beggs, T. P. White, L. O'Faolain, and T.F. Krauss, "Ultracompact and lowpower optical switch based on silicon photonic crystals," Opt. Lett., 33, 147-149
(2008).
CHAPTER 6. BIBLIOGRAPHY
115
[13] J. Capmany, B. Ortega, D. Pastor, and S. Sales, "Discret-time optical processing of
microwave signals," J. Lightwave Technol., 23, 702-723 (2005).
[14] J. Capmany and D. Novak, "Microwave photonics combines two worlds," Nature
Photonics, 1, 319-330 (2007).
[15] M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y.
Rostovtsev, E. S. Fry, and M.O. Scully, "Ultraslow group velocity and enhanced
nonlinear optical effects in a coherently driven hot atomic gas," Phys. Rev. Lett., 82,
5229-5232 (1999).
[16. A. B. Matsko, Y. V. Rostovtsev, H. Z. Cummins, and M.O. Scully, "Using slow light
to enhance acoustooptical effects: Application to squeezed light," Phys. Rev. Lett., 84,
5752-5755 (2000).
[17] M. Soljacic, S. G. Johnson, S. H. Fan, M. Ibanescu, E. Ippen, and J.D. Joannopoulos,
"Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc.
Am. B, 19, 2052-2059 (2002).
[18] M. Soljacic and J.D. Joannopoulos, "Enhancement of nonlinear effects using photonic
crystals," Nature Mater., 3, 211-219 (2004).
[19] L. Thévenaz, K. Y. Song, S. Chin, and M. Gonzalez-Herraez, "Light Controlling Light
in an Optical Fibre: From Very Slow to Faster-Than-Light Speed," in IEEE Int.
Symposium Workshop on Intelligent Signal Processing, Madrid, Spain (2007).
[20] M. González-Herráez, O. Esteban, F. B. Naranjo, and L. Thévenaz, "How to play with
the spectral sensitivity of interferometers using slow light concepts and how to do it
practically," in Third European Workshop on Optical Fibre Sensors, Napoli, Italy,
SPIE Proc., 6619, 661937 (2007).
[21] Z. Shi, R. W. Boyd, D. J. Gauthier, and C.C. Dudley, "Enhancing the spectral
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[22] Z. Shi, R. W. Boyd, R. M. Camacho, P. K. Vudyasetu, and J.C. Howell, "Slow-Light
Fourier Transform Interferometer," Phys. Rev. Lett, 99, 240801 (2007).
[23] M. Terrel, M. J. F. Digonnet, and S. Fan, "Performance comparison of slow-light
coupled-resonator optical gyroscopes," Laser & Photon. Rev., 1-14 (2009).
Nomenclature
List of symbols
α
Attenuation coefficient
γe
Electrostrictive constant
εo
Electric permittivity in vacuum
ε
Dielectric constant
Δε
Scalar fluctuations of the dielectric constant
κ
Thermal conductivity
λ
Optical wavelength
μo
Magnetic permittivity in vacuum
μ
Dielectric constant
Δρ
Density variations
σ
Absorption cross section
τp
Average life time of acoustic phonons
νp
Optical frequency of the pump wave
νs
Optical frequency of the signal wave
νB
Brillouin frequency shift
ΔνB
Bandwidth of Brillouin resonance
φ
Phase of optical wave
χj
j-th order susceptibility
ωp
Angular frequency of the pump wave
ωs
Angular frequency of the signal wave
δω
Frequency detuning of the signal wave
Γ
Acoustic damping coefficient
ΓB
Bandwidth of Brillouin scattering
Κ
Bulk modulus, Time-bandwidth product
NOMENCLATURE
Ω
Angular frequency of the acoustic wave
Φ SBS
p,S
Nonlinear phase shift induced by the Brillouin process
c
Light velocity in vacuum
cp
Specific heat at a constant pressure
f
Electrostrictive force
gB(ν)
Brillouin gain spectrum
geff(ν)
Effective Brillouin gain spectrum
kp
Wavevector of the pump wave
kS
Wavevector of the Stokes wave
k1
Group velocity
k2
Group velocity dispersion
k3
Dispersion slope
n
Refractive index
ng
Group index
pst
Electrostrictive pressure
Δp
Pressure fluctuations
q
Wavevector of the acoustic wave
Δs
Entropy fluctuations
to
Pulse duration factor
tin
Pulse FWHM duration
Δu
Energy density fluctuations
vp
Phase velocity
vg
Group velocity
va
Acoustic velocity
A
Amplitude of the acoustic wave
Aeff
Effective area
Cs
Adiabatic compressibility
Ẽ
Fourier transform of the field envelope function
Ep
Field amplitude of the pump wave
ES
Field amplitude of the Stokes wave
G
Linear Brillouin gain
I
Optical intensity
Leff
Effective length
N
Gas molecular density
118
NOMENCLATURE
Pc
Critical power
Pin
Input signal power
Pout
Output signal power
PL
Linear polarization vector
PNL
Nonlinear polarization vector
T
Transfer function
ΔTd
Effective time delay
List of acronyms
BLB
Beer-Lambert-Bouguer
CPO
Coherent population oscillation
CW
Continuous wave
DC
Direct current
DFB
Distributed-feedback
DSF
Dispersion shifted fiber
EIT
Electromagnetically-induced transparency
EDF
Erbium-doped fiber
EDFA
Erbium-doped fiber amplifier
EOM
Electro-optic modulator
FBG
Fiber Bragg grating
FWHM
Full width at half maximum
GVD
Group velocity dispersion
IFT
Inverse Fourier transform
NOLM
Nonlinear optical loop mirror
PC
Photonic crystal, Polarization controller
PCF
Photonic crystal fiber
RMS
Root mean square
SBS
Stimulated Brillouin scattering
SOA
Semiconductor optical amplifier
SOP
State of polarization
TLS
Tunable laser source
VOA
Variable optical attenuator
XGM
Cross gain modulation
119
Sanghoon Chin
Address : 89, Route Neuve, Ecublens VD, Switzerland
Phone : (+41) 021 693 68 77
Email : [email protected]
Civil Status : Married
Nationality : South Korea
Gender: Male
Date of Birth: 27 May 1977
Education
2006.01 onwards
Ph.D Candidate in Electrical and Electronics Engineering
Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland
Group for Fiber Optics, directed by Prof. Luc Thévenaz
Thesis title: Governing the speed of a light signal in optical fibers: Brillouin slow and
fast light
Experimental research for the development of all-optical buffers
Research interests: slow & fast light, optical storage, optical fiber sensors, nonlinear fiber
optics, light source generation
2003.03 ~ 2005.02
Master Degree in Information and Communications
Gwangju Institute of Science and Technology, South Korea
Photonic Device Measurement Lab., by Prof. Dug Young Kim
Thesis title: New method for characterization of nonlinear phase chirp of a laser
pulse by fitting its autocorrelation trace
Experimental research for the generation and characterization of ultrashort optical pulse, generated by Figre-8 EDFL and gain-switched
method.
1996.03 ~ 2003.02
Diploma in Physics
Chonnam National University, South Korea
Major in Optics
Awards and fellowships
2009.06.24
Conference Grant for Early Stage Researchers
ICT domain in COST
2003.03 ~ 2005.02
Scholarship
Korea Ministry of Education
Project involved
2008.09 onwards
European Project (219299)
Governing the Speed of Light (GOSPEL)
2009.01 onwards
Swiss National Science Foundation (200020-121860)
Optical storage
2006.01 ~ 2008.02
Swiss National Science Foundation (200021-109773)
Slow and fast light in optical fibers
2005.05 ~ 2005.10
European Space Agency (17267/03/NL/CH)
Tunable frequency stabilization scheme
Work experience
2006.02 ~ 2009.06
Teaching assistant
Ecole Polytechnique Fédérale de Lausanne, Switzerland
Preparation of experimental demonstrations for teaching lectures.
2005.05 ~ 2005.10
Academic Host,
Ecole Polytechnique Fédérale de Lausanne, Switzerland
Main activities: Relative frequency stabilization of semiconductor diode lasers, using
mode-locking method and spectral resonance
Development of a fiber gas cell: sealing CO2 gas in a photonic crystal
fiber at low pressure
2001.06 ~ 2001.08
Assistant teacher
Ivanhoe grammar school, Melbourne, Australia
Main activities: Teaching assistant for mathematics in a sixth grade class.
1997.04 ~ 1999.06
Military service
South Korea
Administrative soldier
Languages
Mother tongue Korean
Understanding
Speaking
Writing
English: Fluent in listening and reading
Fluent
Fluent
French: Intermediate in listening and reading
Intermediate
Intermediate
Non professional skill
Computer skill Competent with most MS office programmes, Matlab, Origin and
Labview (data acquisition, instrument control and step-motor control),
and some experience with Mathematica.
Sport and activity
football, ski, hiking and travel
Publications
Journal paper
1. S. Chin, Y. J. Kim, H. S. Song and D. Y. Kim, “Complete chirp analysis of a gainswitched pulse using an interferometic two-photon absorption autocorrelation,” Appl.
Opt. 45, 7718-7722 (2006).
2. S. Chin, M. Gonzalez-Herraez and L. Thevenaz, “Zero-gain slow & fast light
propagation in an optical fiber,” Opt. Express, 14, 10684-10692 (2006).
3. S. Chin, M. Gonzalez-Herraez and L. Thevenaz, “Simple technique to achieve fast light
in gain regime using Brillouin scattering,” Opt. Express, 15, 10814-10821 (2007).
4. S. Chin, M. Gonzalez-Herraez and L. Thevenaz, “Self-advanced fast light propagation in
an optical fiber based on Brillouin scattering,” Opt. Express, 16, 12181-12189 (2008).
5. S. Chin, and L. Thevenaz, “Optimized shaping of isolated pulses in Brillouin fiber slow
light systems,” Opt. Lett., 34, 707-709 (2009).
6. A. Zadok, S. Chin, L. Thevenaz, E. Zilka, A. Eyal and M. Tur, “Stimulated Brillouin
scattering induced polarization mode dispersion in slow light setups,” submitted to Opt.
Lett.
7. S. Chin, and L. Thevenaz, “Large multi Gbit/s delays generated in an all-optical tunable
delay line preserving wavelength and signal bandwidth,” Photon. Technol. Lett., under
submission.
8. S. Chin, M. Gonzalez-Herraez and L. Thevenaz, “Complete compensation of pulse
broadening in a linear slow light system using a non-linear regeneration element,” Opt.
Express, under submission.
Invited presentation
1. L. Thévenaz, S. -H. Chin, K. -Y. Song, and M. Gonzalez-Herráez, "Flexible Slow and
Fast Light in Optical Fibers," LEOS 19th Annual Meeting 2006, Montreal QC, October
29 – November 2 2006, Paper MA3, pp. 18-19, Invited.
2. M. Gonzalez-Herráez, K. -Y. Song, S. -H. Chin, and L. Thévenaz, " Progress in Brillouin
Slow Light and Its Impact in Fiber Sensing," 18th International Conference on Optical
Fiber Sensors, Cancún, Mexico, October 23-27 2006, OSA Technical Digest (CD)
(Optical Society of America, 2006), paper TuC1, Invited.
3. L. Thévenaz, M. Gonzalez-Herráez, K. -Y. Song, S. -H. Chin, "La lumière lente et
rapide: une future pièce maîtresse pour la photonique", 25e Journées Nationales
d'Optique Guidée JNOG'06, Metz F, November 7-9, 2006, Invited.
4. L. Thévenaz, K. Y. Song, S. Chin and M. Gonzalez-Herráez, " Light controlling light in
an optical fibre: from very slow to faster-than-light speed", IEEE International
Symposium on Intelligent Signal Processing WISP 2007, Alcala de Henares, Spain,
October 3-5 ( 2007), Plenary talk.
5. S. Chin, K. Y. Song, M. Gonzalez-Herráez and L. Thévenaz, “Slow and fast light in
optical fibers, based on stimulated Brillouin scattering,” Seminar, University of Chosun,
South Korea, August 7, (2008), Invited.
6. L. Thévenaz, M. Gonzalez-Herráez, S. Chin, “Potentialities of show and fast light in
optical fibers,” Proceedings Photonic West, SPIE, 7226, 72260C (2009), Invited.
Presentation in conferences
1. S. Chin, Y. J. Kim, H. S. Song and D. Y. Kim, “Non-linear phase chirp retrieval of a
1.55 um gain switched pulse laser using two photon absorption autocorrelation,” CLEO
Pacific/Rim, CWAB3-P10 (2006)
2. L. Thévenaz, S. H. Chin, M. Gonzalez-Herráez, "Zero-Gain Slow Light in Optical
Fibres", Proceedings of the 32nd European Conference on Optical Communication
ECOC'2006, Sept. 24-28, 2006, Cannes, F, paper Tu1.1.2.
3. L. Thévenaz, S. -H. Chin, K. -Y. Song, and M. Gonzalez-Herráez, "Flexible Slow and
Fast Light Using Tailored Brillouin Spectra in Optical Fibers," in Slow and Fast Light,
Technical Digest (CD) (Optical Society of America, 2006), paper TuB4.
4. R. Matthey, C. Affolderbach, G. Mileti, S. Schilt, D. Werner, S.-H. Chin, L. Abrardi, L.
Thévenaz, “Frequency-stabilized laser reference system for trace-gas sensing applications
from space”, 6th International Conference on space Optics, Noordwijk, Netherlands, June
27-30, 2006.
5. R. Matthey, C. Affolderbach, G. Mileti, S. Schilt, D. Werner, S.-H. Chin, L. Abrardi, L.
Thévenaz, “Water vapour DIAL optical frequency laser reference system”, 23rd
International Laser Radar Conference, Nara, Japan, July 24-28, 2006.
6. S. Chin, M. Gonzalez-Herráez and L. Thévenaz, “Simple scheme for realizing fast light
with low distortion in optical fibers,” Research day & Photonics day, EPFL, Poster
section (2006).
7. S. Chin, M. Gonzalez-Herráez and L. Thévenaz, “Simple scheme for realizing fast light
with low distortion in optical fibers,” European Conference on Lasers and Electro-Optics,
June 17-22, CD9-4-Fri (2007).
8. S. Chin, M. Gonzalez-Herráez and L. Thévenaz, “Self-advanced propagation of light
pulse in an optical fiber based on Brillouin scattering,” in Slow and Fast Light, Technical
Digest (CD), SWC5 (2007).
9. L. Thévenaz, S. Chin and M. Gonzalez-Herráez and, “Low distortion fast light in an
optical fiber using stimulated Brillouin scattering,” in Slow and Fast Light, Technical
Digest (CD), JTuA8 (2007).
10. S. Chin, L. Thévenaz and M. Gonzalez-Herráez, “Self-advanced fast light in an optical
fiber based on Brillouin scattering,” Research day & Photonics day, EPFL, Poster
section (2008).
11. J. P. Dakin, S. Chin, L. Thévenaz, “A multiplexed CW Brillouin system for precise
interrogation of a sensor array made from short discrete sections of optical fiber,” 19th
international conference on Optical Fiber Sensors, 7004, 70046L-4 (2008).
12. S. Chin, and L. Thévenaz, “Enhancement of Brillouin slow-light in optical fibers through
optical pulse shaping,” in Slow and Fast Light, Technical Digest (CD), JMB10 (2008).
13. L. Thévenaz and S. Chin, “Self-pumped optical delay line based on Brillouin fast light in
optical fibers,” in Slow and Fast Light, Technical Digest (CD), STuC5 (2008).
14. S. Chin, and L. Thévenaz, “Large multi Gbit/s delays generated in an all-optical tunable
delay line preserving wavelength and signal bandwidth,” in Slow and Fast Light,
Technical Digest (CD), SMC3 (2008).
15. S. Chin, M. Gonzalez-Herráez and L. Thévenaz, “Complete broadening compensation in
a slow light system using a non-linear regeneration element,” in Slow and Fast Light,
Technical Digest (CD), JWB2 (2009).
16. S. Chin, I. Dicaire, J. C. Beugnot, S. Foaleng-Mafang, M. Gonzalez-Herráez and L.
Thévenaz, “Material slow light does not enhance Beer-Lambert absorption,” in Slow and
Fast Light, Technical Digest (CD), SMA3 (2009).
17. L. Thévenaz, S. Chin, I. Dicaire, J. C. Beugnot, S. Foaleng-Mafang and M. GonzalezHerráez, “Experimental verification of the effect of slow light on molecular absorption,”
20th international conference on optical fiber sensors (2009), accepted.
18. A. Zadok, S. Chin, E. Zilka, A. Eyal, L. Thevenaz and M. Tur, “Polarization dependent
pulse distortion in stimulated Brillouin scattering slow light systems,” submitted to the
post-deadline for Slow and Fast Light, Technical Digest (CD) (2009).

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