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Coherent Raman Interaction
in Gas-Filled Hollow-Core
Photonic Crystal Fibers
Kohärente Raman-Wechselwirkung in gas-gefüllten Hohlkernfasern
Der Naturwissenschaftlichen Fakultät
der Friedrich-Alexander-Universität
Erlangen-Nürnberg
zur
Erlangung des Doktorgrades Dr. rer. nat.
vorgelegt von
Amir Abdolvand
aus Shiraz (Iran)
Als Dissertation genehmigt
von der Naturwissenschaftlichen Fakultät
der Friedrich-Alexander-Universität Erlangen-Nürnberg
Tag der mündlichen Prüfung: 25.07.2011
Vorsitzender der Promotionskommission: Prof. Dr. Rainer Fink
Erstberichterstatter: Prof. Dr. Philip St.J. Russell
Zweitberichterstatter: Prof. Dr. Curtis Menyuk
Drittberichterstatter: Prof. Dr. Florian Marquardt
Abstract
In this thesis I study coherent light-matter interactions via stimulated Raman
scattering (SRS) in a gas-filled hollow-core photonic crystal fiber (HC-PCF). The
HC-PCF constitutes the foundation of the experimental results presented in this thesis,
as without its unique properties realization of these experiments would not have been
possible. These unique properties include tight confinement of laser light and matter
in the small core of the fiber, which leads to extremely high conversion efficiencies in
the SRS, and spectral filtering of unwanted nonlinear waves. This setup creates a
clean system of two optical fields interacting via Raman medium inside the core of
the fiber.
This thesis consists of three parts. The first part (chapter 2) includes a general
overview of the different types of HC-PCF, their fabrication techniques and guidance
mechanisms. Two types of HC-PCF are considered: photonic bandgap HC-PCF and
kagomé-HC-PCF. Guidance mechanisms of these two types of fibers are considered
in detail. Of particular interest to me is the broadband guidance in kagomé-HC-PCF.
Up to now, the mechanism responsible for this broadband guidance is not understood.
A simplified semi-analytical model is derived which accounts for this broadband
guidance in kagomé-HC-PCF and explain many of its guidance properties. In
particular, using the model, the fiber loss as a function of wavelength is reproduced
fairly well. The model also explains an important feature of kagomé-HC-PCF,
namely, the insensitivity of its loss to the number of cladding layers.
In the second part of the thesis (chapter 3), I set the basic theoretical formalism of
SRS to be used in the rest of the thesis. The aim of the chapter is to derive the coupled
wave equations that govern the evolution of the pump, Stokes and coherence fields
using classical as well as semi-classical approaches.
The third part of the thesis (chapters 4, 5, 6 and 7) discusses the results of
experimental and theoretical investigations of my studies on SRS in gas-filled HC-
PCF. I start with an overview of the already know results about SRS in HC-PCF and
explain the mechanism of phase-locking for efficient generation of anti-Stokes SRS.
This work clarifies high conversion efficiencies to anti-Stokes frequencies observed in
early experiments. Most of these early experiments are focused on lowering the
threshold for SRS generation and are done at relatively high pressures. In chapters 5
and 6, I use some of the unique properties of the HC-PCF to explore the coherent
light-matter SRS interaction regimes not previously accessible. This includes the first
experimental observation of backward superluminal solitary waves and self-similar
solutions of SRS coupled-wave equations. These results represent a significant
advance in the study of coherent effects and point to a new generation of highly
engineerable gas cells for studying complex nonlinear phenomena.
In chapter 7, I briefly overview the results presented in previous chapters and
conclude my thesis by explaining possible improvements as well as new interesting
directions in studying coherent light-matter interactions in gas-filled HC-PCF. In
particular, I introduce a simple scheme for generating (purely rotational) broadband
coherent frequency comb at low pump pulse energies using a gas-filled HC-PCF.
Generation of such a frequency comb is quite interesting for synthesizing ultrashort
pulses as well as a broad coherent tunable light source with high spectral brightness.
Zusammenfassung
Die vorliegende Arbeit befasst sich mit kohärenter Licht-Materie Wechselwirkung via
stimulierter Ramanstreuung (SRS) in gas gefüllten hollow-core photonic crystal fibres
(Hohlkernglasfasern/ HC-PCF). Die HC-PCF legt dabei den Grundstein für die
erzielten Ergebnisse, da ohne deren einzigartige Eigenschaften die durchgeführten
Experimente unmöglich wären. Die Eigenschaften beinhalten den räumlich stark
begrenzten Einschluss von Laserlicht und Materie im kleinen Kern der Faser und der
spektralen Filterung von nichtgewünschten Frequenzen. Durch den starken
räumlichen Einschluss werden hohe Konversionseffizienzen erzielt und aufgrund der
spektralen Filterung wird ein klar definiertes Modellsystem realisiert, das nur zwei
optische Frequenzen beinhaltet, die im Faserkern durch Ramanstreuung am Medium
wechselwirken.
Diese Arbeit besteht aus drei Teilen. Der erste (Kapitel 2) beinhaltet einen Überblick
über die verschiedenen Arten von HC-PCF, deren Herstellungstechniken und
zugrunde liegenden Leitungsmechanismen. Dabei werden zwei Arten von HC-PCFs,
photonische Bandlücken HC-PCFs und Kagome HC-PCF, genauer betrachtet.
Spezielles Augenmerk wird dabei auf den breitbandigen Leitungsmechanismus der
Kagome HC-PCF gelegt, welcher bis heute eine noch ungelöste Frage darstellt.
Jedoch kann mit dem in dieser Arbeit beschriebenen halbanalytischen Verfahren
neues Licht auf die Antwort zu dieser Frage geworfen werden. Mit diesem
vereinfachten Modell kann die breitbandige Transmission der Kagome HC-PCF und
weitere Eigenschaften erklärt werden. Vor allem wird der wellenlängenabhängige
Verlust dieses Fasertyps und der Zusammenhang zwischen der Anzahl Ringe der den
Hohlkern
umhüllenden
Mikrostuktur
und
des
Verlusts
im
Einklang
mit
experimentellen Daten beschrieben.
Im zweiten Teil der Arbeit (Kapitel 3) werden die theoretischen Grundlagen zur
Beschreibung von SRS gelegt, welche im weiteren Verlauf wieder aufgegriffen
werden.
Das
Ziel
dieses
Kapitels
ist
die
Herleitung
der
gekoppelten
Propagationsgleichungen für die beteiligten Felder. Diese setzen sich aus dem Pump-
Puls, dem Stokes-Puls und der Materialanregung zusammen und werden sowohl
klassisch als auch semi-klassisch behandelt.
Der dritte Teil (Kapitel 4,5,6,7) beschäftigt sich mit den experimentellen Ergebnissen
und deren theoretischen Beschreibung meiner Studien auf dem Gebiet der SRS in gas
gefüllten HC-PCFs. Ausgehend von einem Überblick über die bereits bekannten
Ergebnisse von SRS in HC-PCFs wird der Mechanismus des sogenannten PhasenLockings für die effiziente Erzeugung von Anti-Stokes SRS erklärt. Dieser beschreibt
die hohe Konversionseffizienz zu Anti-Stokes Frequenzen welche in den ersten
Experimenten beobachtet worden sind. Das Ziel dieser Experimente war die
Reduzierung der Schwelle für SRS, so dass diese bei relativ hohen Drücken
durchgeführt wurden. In den Kapiteln 5 und 6 werden die einzigartigen Eigenschaften
von HC-PCFs ausgenützt um kohärente Licht-Materie Wechselwirkung durch SRS in
zuvor nicht experimentell zugänglichen Regimen zu untersuchen. Dies beinhaltet die
ersten experimentellen Nachweise von “backward superluminal solitary waves” und
selbstähnlichen Lösungen der gekoppelten SRS-Gleichungen. Diese Ergebnisse
bedeuten einen beachtlichen Fortschritt in der Untersuchung von Kohärenzeffekten
und ebnen gleichzeitig den Weg zu einer neuen Generation von präzise
kontrollierbaren Gaszellen zur Untersuchung komplexer nichtlinearer Phänomene.
In Kapitel 7 werden die erzielten Ergebnisse aus den vorangegangenen Kapiteln
zusammengefasst und sowohl
vielversprechende
Wege
zur
mögliche Verbesserungen als auch neuartige
Untersuchung
von
kohärenter
Licht-Materie
Wechselwirkung in gas gefüllten HC-PCFs aufgezeigt. Im speziellen wird ein
einfaches Experiment vorgestellt, welches auf einen breitbandigen kohärenten
Frequenzkamm bei kleinen Pumpenergien in gas gefüllten HCPCFs abzielt. Diese
Frequenzkämme sind sehr interessant sowohl für die Erzeugung ultrakurzer Pulse als
auch als durchstimmbare breitbandige kohärente Lichtquelle mit hoher spektraler
Brillanz.
Acknowledgements
The classic system of education is mainly based on the ability of deductive reasoning
and knowledge of the classics. Quite often, in the course of learning that, we lose the
joy and excitement of thinking. Scientific research, on the other hand, is not just about
knowing and reasoning, but also thinking, creativity and courage of realizing new
ideas. For that, I am hugely indebted to my supervisor, Prof. Philip Russell. Philip,
thank you for teaching me the divergent way of thinking, to see many solutions to one
problem, for giving me the courage to express my ideas and for providing excellent
facilities and environment to realize them.
I would also like to thank Prof. Curtis Menyuk for very nice discussions and
comments about my thesis and specially the work he has done on self-similarity. Your
deep insight and understanding of the physics and the beauty of mathematics behind
it, is truly amazing and I really hope that in future I would have the opportunity to
learn more from you.
Many thanks go to Dr. Johannes Nold for who he is and all he has done for me.
Thank you Johannes for all the scientific and non-scientific discussions we had
together, for all the time we spent in the fiber drawing tower (especially thank you for
making that ugly, massive, fiber holder out of my nice idea), for teaching me about
different thread sizes, for IT support, for giving me that coffee filter, … and not to
forget, for your magic pockets. Vielen Dank Johannes!
I am very much indebted to two Alexander, one from Minsk and one from
Moscow: Dr. Alexander Podlipensky for being such a good friend. Thank you Alex
for your support, for all the discussions we had together, for helping me to join our
group in Erlangen and also settling down in Germany. And Dr. Alexander Nazarkin
for his support and his deep insight in physics and nonlinear optics. Thank you Alex
for showing me that nonlinear optics is more than nonlinear Schrödinger equation – it
doesn’t necessarily need to be a soliton, it can be self-similar!
Большое спасибо Алекс!
Going from Russia to Poland, I am very grateful to my office and lab mate, future
Dr. Marta Ziemienczuk. Thank you Marta for all your “nice” comments about the
awful color scheme of my presentations, for checking my English, for taking my
oscilloscope and power meter and claiming them afterwards, for making me addicted
to watch the Big Bang Theory, for all your “tales”  about Polish food, for telling me
about “the first rule” and having a messy desk. Although my desk is messier, still it is
good not to be alone. Dziękuję bardzo Marta!
Merci beaucoup Professeur Nicolas Joly! Thank you for being such a good friend,
for showing me the tricks of fiber fabrication, for all the scientific discussions, for
being such an amazing cook, for showing Paris to me and Sara, and for all the
moments which you made me feel good.
Thank you very much Dr. John Travers. Although it is not a long time that you
have joined our group, I really appreciate your friendship and all the scientific and
non-scientific discussions we have at work and also at Havana (although it is very
difficult to remember them).
Liebe Philipp Hölzer, Anna Butsch, Dr. Christine Kreuzer and Dr. Sebastian
Stark! Thank you for being so friendly and helpful. Thank you for all the fun time we
have had together. I would also like to express my gratitude to Dr. Michael Scharrer
and Silke Rammler for their guidance and help with fiber fabrication and Dr. Andreas
Walser for all nice discussions about Raman theory and for careful proofreading of
my thesis.
Although now an experimentalist, I started my carrier as a theoretician, working
on the thermodynamics of complex non-equilibrium systems. I would like to thank
my friend and former supervisor, Dr. Afshin Montakhab for introducing me to the
amazing world of thermodynamics, non-equilibrium statistical mechanics, complexity
and fractals. It is truly fascinating to see how probabilities create a world and how
order emerges out of fluctuations and chaos.
Sometimes it is amazing how far away one’s life brings him, so that the memories
of the past look faint and dim. However, the best memories of your childhood with
your family and friends are always bright. I would like to thank my parents, my
brother, Amin and my cousin Vahid, for all those good memories.
And Sara! Writing just a single paragraph in the acknowledgement will never
express how much I am grateful and indebted to you, for tolerating me and my busy
and sometimes stressful times during PhD, for being here with me far away from our
families, for your support and your non-stop efforts to make a calm and relaxed
atmosphere for me to focus on my research. For all you have done for me, I would
like to dedicate this thesis to you.
To Sara
Table of contents
1
1
INTRODUCTION
1
1.1
THESIS OUTLINE
6
2
11
LINEAR OPTICAL PROPERTIES OF HOLLOW-CORE PHOTONIC
CRYSTAL FIBERS
11
2.1 GUIDANCE VIA PHOTONIC BANDGAP
2.1.1 Photonic crystal cladding
2.1.2 Fabrication
2.1.3 Numerical analysis
2.2 GUIDANCE IN THE ABSENCE OF PHOTONIC BANDGAP
2.2.1
2.2.2
2.2.3
2.2.4
2.2.5
Analytical model for guidance in hollow-core Kagomé-lattice
photonic crystal fibers
Model
Reflection coefficients from ML stacks
Calculation of loss
Comparison with experiments
2.3 DISCUSSION AND CONCLUSION
13
13
15
16
19
19
20
27
29
33
36
3
39
WAVE PROPAGATION AND COUPLED WAVE EQUATIONS
IN STIMULATED RAMAN SCATTERING
39
3.1 WAVE EQUATION
41
3.2 STIMULATED RAMAN SCATTERING (CLASSICAL APPROACH) 43
3.3
3.2.1 Mechanism of the Raman effect
3.2.2 Optical phonons and material excitation
3.2.3 Material excitation as a damped oscillator
3.2.4 Basic differential equations
43
45
46
48
SEMICLASSICAL THEORY OF
STIMULATED RAMAN SCATTERING
51
3.3.1 Density matrix approach
3.3.2 Material excitation revisited
51
56
3.4 SUMMARY
59
4
61
OPTIMIZING ANTI-STOKES RAMAN SCATTERING
IN GAS-FILLED HOLLOW-CORE PHOTONIC CRYSTAL FIBERS
61
4.1 PHASE LOCKING
63
4.2
68
NUMERICAL SIMULATION
4.3 OPTIMIZATION SCHEME FOR EFFICIENT ANTI-STOKES
GENERATION
69
4.4 SUMMARY AND CONCLUSION
72
5
73
SOLITARY PULSE GENERATION BY BACKWARD STIMULATED
RAMAN SCATTERING IN HYDROGEN-FILLED HC-PCF
73
5.1 MOTIVATION FOR THE EXPERIMENT
74
5.2
EXPERIMENTAL RESULTS
77
5.3 THEORETICAL ANALYSIS
80
5.4 ANALYTICAL CONSIDERATIONS
83
6
87
OBSERVATION OF SELF-SIMILAR SOLUTIONS OF SINE-GORDON
EQUATION BY TRANSIENT STIMULATED RAMAN SCATTERING
87
6.1 STIMULATED RAMAN SCATTERING
AS A STUDY MODEL FOR SGE
89
6.2 EXPERIMENTAL CONSIDERATIONS
93
6.3
EXPERIMENTAL RESULTS
95
6.3.1
96
Self-similarity of the late-stage oscillations
7
101
CONCLUSION AND OUTLOOK
101
7.1 DIFFRACTIONLESS GUIDANCE OF LIGHT IN VACUUM
101
7.2 SRS IN GAS-FILLED HC-PCF
102
7.2.1 Control of the nonlinearity
7.2.2 Control of the dispersion and phase-matching
102
102
7.3 BACKWARD SRS IN GAS-FILLED HC-PCF
103
7.4
104
SELF-SIMILARITY IN SRS
7.5 GENERATION OF COHERENT BROADBAND
FREQUENCY COMBS
105
APPENDIX A
109
APPENDIX B
113
BIBLIOGRAPHY
117
CURRICULIM VITAE
126
1
Introduction
The propagation of electromagnetic radiation through a transparent material is always
accompanied by various scattering processes. In some materials, the scattering
process can be inelastic, in which case the incident photon is scattered to another
photon of lower or higher frequency. One of the most important inelastic scattering
processes with widespread use in spectroscopy is Raman scattering; named after its
discoverer C. V. Raman [Raman, 1928; Landsberg, 1928]. The Raman process is an
inherently quantum mechanical scattering process in which an incident photon of
energy  ωP is scattered into a photon of energy  ωS , while the difference in energy
 (ω p − ωs ) =Ω
 is absorbed by the Raman active material, shown in Fig. (1.1a).
While the material excitation may be purely electronic, which involves a resonant
transition via state 3 , the excitation may occur far from resonance, mediated by
vibrational or rotational excitations of the molecules, shown schematically by the
horizontal dashed line in Fig. (1.1)*.
The photon which is generated in this process is called the Stokes photon and has
a lower frequency compared to the pump photon. If now the process starts from an
already excited level, say 2 , and is followed by a transition to the ground state via
*
A more informative representation of the Raman process which also takes into account the role played
phase-matching and material excitation in the stimulated case is presented later in chapter 3.
1
Figure 1.1: Schematic energy diagram for (a) Stokes and (b) anti-Stokes Raman scattering.
the Raman process, the scattered photon will have a higher energy. In this case, the
scattered photon is called the anti-Stokes photon, and the difference in energy is given
ωas  (ω p + Ω) , as shown in Fig. (1.1b). In
by the energy conservation law, so that =
thermal equilibrium, the occupancy of a molecular level, say
n
follows the
Boltzmann distribution law, i.e. n ∝ exp(− E / k BT ) where E is the energy of the
k B 1.38 × 10-23 [J/K] is the Boltzmann factor. So, the
level, T is the temperature and =
population of level 2 is smaller than the population of level 1 by a factor of
exp(− Ω / k BT ) . As a result, the anti-Stokes lines are typically weaker than the Stokes
lines.
The spontaneous Raman process accounts for the inelastic scattering of a very
small portion of the incident photons, more specifically one part per million is
scattered. However, as the flux of the incident photons increases, for example by
focusing a coherent laser beam to a small spot in the medium to reach intensities as
high as 107 - 109 W/cm2, the (Raman) scattering process enters the stimulated regime.
As a result of such high intensities, the previously transparent medium becomes
opaque to the incident radiation and a large fraction of the incident photons are
(inelastically) scattered. In this sense, stimulated Raman scattering is a nonlinear
optical process because the probability of inducing a Raman transition depends on the
intensity of the incident light [Delone et al., 1988]. Under suitable conditions, this
process leads to the generation of intense laser radiation at new frequencies, ωn
2
spaced equally with respect to the pump frequency at multiple integers of Ω , given
by ωn = ωP ± m Ω, where m = 1, 2,... is an integer.
The stimulated Raman scattering (SRS) can occur in a variety of systems,
including gases, solids, liquids or plasma and has, compared to the spontaneous case,
several unique and interesting properties:
(i)
It is generated in narrow cones in the forward and backward directions with
respect to the pump laser, in contrast to dipole radiation in the spontaneous
case.
(ii)
It is highly efficient. Indeed under suitable conditions more that 90% of the
pump energy can be transferred to the Stokes frequency.
(iii)
As a result of a higher gain at the center of the Raman line-width, there is a
distinct line-narrowing compare to the line-width of spontaneous Raman
emission.
Theses properties make SRS an invaluable tool with widespread uses in highresolution spectroscopy, generation of intense, ultrashort laser pulses, frequency
conversion and tunable laser sources [Excellent reviews of the subject can be found
in: Bloembergen, 1965 and 1967; Kaiser et al., 1972; Wang, 1975; Penzkofer et al.,
1979; Shen, 1965 and 1984; White, 1987; Raymer, 1990; Reintjes, 1995; Boyd,
2008].
Soon after its accidental discovery by Woodbury and Ng [Woodbury, 1962], it
was realized that the SRS process is accompanied by the generation of spatially and
temporally coherent excitation of optical phonons [Garmire et al., 1963; Bloembergen
et al., 1964]. These are in-phase, non-acoustic excitation of the material internal
degrees-of-freedom, such as molecular vibrations or rotations in the case of gases and
liquids. With SRS serving as a generating source of a coherent material field, the
system of coupled equations describing the phenomenon shows a rich spectrum of
solutions and behaviour. The type of behaviour one should expect from these
equations, i.e. the physics of the solutions, depends strongly on the coherence lifetime,
or T2, of these optical phonons. In mathematical terms, their lifetime determines the
3
form of the equations describing SRS. The coherence or dephasing time is a time
window within which the interaction between the pump, Stokes and material
excitation fields is in-phase or coherent. For a short dephasing time, that is when the
pump pulse duration, τ p , is long compared to T2, τ p >> T2 , the response of the
medium to the laser electric field is instantaneous (steady state regime). In this
regime, the interaction is simple and can be well-explained by rate equations for
intensities (photon number) of the pump and the scattered waves [Hellwarth, 1963].
Due to its nonlinear nature, SRS is observed when the light intensity exceeds some
threshold value. In the steady state regime, in which material excitation only depends
on the electric fields of pump and Stokes at the same time, by increasing the intensity
of the pump, for example by reducing its duration while keeping its energy constant,
the total Stokes output increases. The reason is that the Raman gain in this case
depends on the peak power of the pump [White, 1987]. However, the situation is
completely different when the interaction happens within the coherence time of the
optical phonons, τ P << T2 (transient regime). In this case the response of the medium
to the electric field of the pump laser is not instantaneous. Hence, at any time in the
time window given by the pump pulse duration, the system retains the “memory” of
earlier excitations created in the medium by the leading edge of the pump pulse. A
characteristic of the transient regime is a reduction in the gain seen by the Stokes
wave as compared to the gain in the steady state. The reason for this is that in the
transient regime the gain depends on the integrated energy of the pump pulse [Duncan
et al., 1988; Carman et al., 1970; Akhmanov et al., 1971]. So by reducing the pulse
duration, while entering the transient regime of SRS, the transient Raman gain is
reduced. These requirements entail the use of high energy, high peak power lasers
(typically nanosecond pulses of 1 MW peak power) which make the physical picture
of SRS complex. This complexity involves a number of competing nonlinear
processes such as higher-order SRS, backward SRS, self-focusing and self-phase
modulation that make control and optimization of the process difficult. One way of
tackling this problem is by tightly focusing the laser beam to a small spot in the
medium to reach the required level of nonlinearity for initiating SRS. However, the
high intensity created at the focus of the laser beam could also easily initiate the
aforementioned competing and unwanted nonlinear effects. An additional problem
with the focused-beam geometry originates from a basic limitation of focusing a
4
Figure 1.2: Rayleigh length of a focused Gaussian beam as a function of its waist for two different
wavelengths, 500 nm (red dashed line) and 1000 nm (blue solid line). Note that for a beam waist
of 5 μm (corresponding to the beam diameter of 10 μm), the Rayleigh length is less than 1 mm.
Gaussian beam. As one tries to focus an ideal Gaussian beam to a smaller spot size for
enhancing the nonlinearity in the medium, the beam diverges faster afterward. In
other words, the effective length of interaction of the beam with the medium, the
Rayleigh length, gets shorter as one tries to enhance the nonlinearity in the medium
by focusing the beam more tightly, see Fig. (1.2).
Thus it was a major advance when, using a hollow-core photonic crystal fiber
(HC-PCF) filled with a Raman active gas, researchers in the university of Bath,
United Kingdom demonstrated the generation of an SRS signal with a pump power
threshold one million times lower than previously reported in literature, with photon
conversion efficiencies of more that 90% [Benabid et al., 2002 and 2004].
In an HC-PCF, light is guided in a small hollow core by means of a twodimensional, out-of-plane, full photonic bandgap (see chapter 2). Created by a
suitably designed photonic crystal cladding, as shown in Fig. (1.3), the photonic
bandgap prevents the coupling of the core-guided mode to the cladding. In these novel
guiding structures, the unprecedentedly low energy thresholds reported for highly
efficient Stokes conversion immediately eliminates the need for high power lasers.
Moreover, the long interaction lengths (tens of meters) between the laser light and
matter, offered by the tight confinement of laser beam and gas in the small core of the
fibre, greatly relaxes the complications of using a focused beam geometry or multi5
Figure 1.3: An electron micrograph of the cross section of a hollow-core photonic bandgap fibre.
Here the dark regions show the air holes and the gray regions indicate silica. Note the perfect
crystalline arrangement of air holes in the cladding around the core of the fibre.
pass gas cells, as is common in studies of SRS. In fact, with implementation of HCPCFs, effective control of SRS and many other nonlinear processes in the gas phase
has become possible [Abdolvand et al., 2009, Nazarkin et al., 2010; Nold et al, 2010;
Hoelzer et al., 2010].
In the forthcoming chapters of this thesis, the results of my research over the past
few years, in exploiting the potential of HC-PCF for detailed exploration of the
coherent material excitation and “memory” effects in the transient SRS will be
presented. In summary, I will show that the gas-filled HC-PCF provides an
exceptionally clean, easy-to-use system for exploring and controlling coherent lightmatter interaction.
1.1 Thesis outline
The structure of the thesis is as follows: In chapter 2, after a short general overview of
the different types of hollow-core photonic crystal fibers and their applications, we
focus our attention on their guidance properties. Two types of HC-PCF will be
considered: hollow-core photonic bandgap PCF (HC-PBG-PBG) characterized by its
low propagation loss (~ 1 dB/km) and spectral filtering property and, large pitch HCPCF, characterized by its broadband guidance and relatively high propagation loss (1
dB/m). Typical to this family of HC-PCF is the broadband optical guidance in the
6
absence of full photonic bandgaps. As a prototype of this category, we will present a
semi-analytical model for propagation and guidance of light in kagomé-HC-PCF.
Chapter 3 sets the basic theoretical formalism of SRS to be used in the rest of the
thesis. The aim of the chapter is to derive the main coupled wave equations governing
the evolution of pump, Stokes and material excitation fields using classical as well as
semi-classical approaches. Comparison between sets of classical and semi-classical
system of equations creates a direct link between the two approaches via material
properties.
Chapter 4 tries to shed some light on the mechanism of anti-Stokes generation in
HC-PCF. Due to its parametric character, the efficient generation of anti-Stokes
strongly depends on the phase mismatch between the pump and scattered waves, ∆k .
In free space, the value of ∆k is tuned via non-collinear propagation of pump, Stokes
and anti-Stokes waves. This tuning is obviously not possible in HC-PCF where pump
and scattered waves propagate collinearly along the length of the fiber. Surprisingly,
experiments have shown that conversion to anti-Stokes radiation can be high (about
3%) even in the presence of significant wave mismatch and the collinear propagation
of pump and anti-Stokes in an HC-PCF [Benabid et al., 2002]. In chapter 4, we
analyze the specific features of this process in HC-PCF and show that the main
mechanism behind such efficient energy transfer is the phase-locking of the pump and
scattered waves. Moreover, we show that the unique properties of these fibers allow
for anti-Stokes conversion efficiencies close to the theoretical maximum of 50%.
In chapter 5, we consider another configuration for SRS where the laser pump and
the Stokes pulse are counter-propagating. The backward scheme of SRS amplification
is fundamentally different from the forward case. The intensity of the forward Stokes
pulse can never exceed the intensity of the initial pump since the Stokes and pump
pulse travel with approximately the same velocity. By contrast, the backward
traveling Stokes wave always sees fresh, undepleted pump photons [Maier et al., 1966
and 1969]. As a result, the leading edge of the Stokes pulse is reshaped; the Stokes
pulse becomes shorter and is amplified to intensities much higher than the incoming
pump intensity [Murray et al., 1979]. In chapter 5 we take the advantage of the long
interaction length offered by HC-PCF to show that in the coherent interaction regime
7
when the Stokes pulse is shorter than T2 , the Stokes pulse intensity profile asymptotes
to a soliton hyperbolic secant shape with the peak of the pulse traveling with a
“superluminal” velocity.
In chapter 6 we focus our attention on the forward SRS process in a gas-filled HCPCF. Theoretical studies show that the long distance spatiotemporal evolution of the
nonsolitonic solutions of the forward SRS is governed by self-similar solutions, i.e.
solutions invariant under certain transformations involving dilation in time and
(propagation) length [Menyuk et al., 1992]. This behavior occurs only if the lasermatter interaction is coherent. Chapter 6 reports on the first experimental observation
of clear self-similar behavior in transient stimulated Raman scattering. We obtained
this result by carrying out a detailed study of transient SRS over long interaction
lengths using the unique characteristics of gas-filled HC-PCF.
The work described in Chapters 4-6 has been published in the following journal
papers:
•
Nazarkin, A., Abdolvand, A. and Russell, P. St.J., 2009, “Optimizing antiStokes Raman scattering in gas-filled hollow-core photonic crystal fibers,”
Phys. Rev. A, 79, 031805(R).
•
Abdolvand, A., Nazarkin, A., Chugreev, A.V., Kaminski, C. F. and Russell,
P. St.J., 2009, “Solitary pulse generation by backward Raman scattering in H2filled photonic crystal fibers,” Phys. Rev. Lett. 103, 183902.
•
Nazarkin, A., Abdolvand, A., Chugreev, A. V. and Russell, P. St.J., 2010,
“Direct observation of self-similarity in evolution of transient stimulated
Raman scattering in gas-filled photonic crystal fibers,” Phys. Rev. Lett. 105,
173902.
The results regarding the semi-analytical model of broadband guidance in kagoméHC-PCF, presented in chapter 2, is a work in progress and the results will be
8
published soon*. In addition to the work presented in this thesis, the author has
contributed to the following journal publications and conference presentations:
•
Chugreev, A. V. , Nazarkin, A., Abdolvand, A., Nold, J., Podlipensky, A. and
Russell, P. St.J., 2009, “Manipulation of coherent Stokes light by transient
stimulated Raman scattering in gas filled hollow-core PCF,” Optics Express
17, 8822-8829.
•
Euser, T. G., Whyte, G., Scharrer, M., Chen, J. S. Y., Abdolvand, A., Nold,
J., Kaminski, C. F. and Russell, P. St.J., 2008, “Dynamic control of higherorder modes in hollow-core photonic crystal fibers,” Opt. Express 16, 1797217981.
•
Abdolvand, A., Chugreev, A. V. , Nazarkin, A. and Russell, P. St.J., 2009,
“Generation of sub-nanosecond solitary pulses by backward stimulated Raman
scattering in H2-filled photonic crystal fiber,” EF3.1, CLEO Europe.
•
Nazarkin, A., Abdolvand, A. and Russell, P.St.J., 2010, “Raman amplifiers
without quantum-defect heating,” Optical Communication (ECOC), 36th
European Conference and Exhibition on, Torino, Italy, Tu.4.E.4.
•
Abdolvand, A., Podlipensky, A., Nazarkin, A. and Russell, P. St.J., 2011,
“Coherent multi-order stimulated Raman generated by two-frequency
pumping of hydrogen-filled hollow core PCF,” CLEO-Europe/EQEC,
Munich, Germany, paper EG.P.1.
•
Ziemienczuk, M., Walser, A. M., Abdolvand, A., Nazarkin, A., Kaminski, C.
F. and Russell, P. St.J., 2011, “Three-wave stimulated Raman scattering in
hydrogen-filled photonic crystal fiber,” CD8.1., CLEO Europe.
•
Jiang, X., Euser, T., Abdolvand, A., Babic, F., Joly, N. and Russell, P. St.J.,
2011, “SF6 glass hollow-core photonic crystal fibre,” CE4.2, CLEO Europe.
*
Abdolvand, A., Joly, N., Euser, T. and Russell, P. St.J., “Semi-analytical Model for Guidance in
Hollow-Core Kagomé-Lattice Photonic Crystal Fibers,” (In preparation).
9
10
2
Linear optical properties of hollow-core
photonic crystal fibers (HC-PCF)
A considerable part of my research activities during my PhD studies has been devoted to the
fabrication, design and development of conventional as well as new types of hollow-core photonic
crystal fibers. This chapter presents an overview of some of the technical details of the fabrication of a
HC-PCF and recent advances in our understanding of the guidance mechanism in large pitch kagoméHC-PCF.
Hollow-core photonic crystal fibers, Fig. (2.1), first proposed by Russell in 1991
[Russell, 2003, 2006, 2007], bring together in an elegant way the physics of
waveguides [Snyder and Love, 2010] and photonic bandgap materials [John, 1987;
Yablonovitch, 1987]. By creating an out-of-plane photonic bandgap, i.e. ranges of
frequencies and propagation constants for which the coupling of light to the periodic
cladding of the fiber is inhibited for any direction and polarization state, these novel
waveguides confine and guide light in vacuum over distances much longer than
accessible with diffractive optics. Upon filling the hollow core of the fiber with an
appropriate gas, HC-PCF proves to be an excellent vehicle for gas-based nonlinear
optical experiments [Chugreev et al., 2009; Abdolvand et al., 2009; Couny et al.,
2007; Ghosh et al., 2005; Bhagwat et al., 2008]. Indeed, low propagation loss
( ~ 1 dB/km ) as well as high intensities inside the small core of the fiber ( ~ 10 μm in
diameter), creates a favorable situation for efficient light-matter interaction. However,
if these fibers are to be successfully designed and used in specific experiments, for
example, if low-loss and/or broad-band windows of transmission are required a
reliable understanding of their guidance mechanisms is crucial.
11
Figure 2.1: Scanning electron micrograph (SEM) of (a) PBG-HC-PCF and (b) kagomé-lattice
HC-PCF. Here the core and periodic cladding of the fibers are indicated by dashed arrows. Part
(c) and (d) show the same fibers under white-light illumination. The green color in the core of
PBG-HC-PCF, part (c), is the result of filtering out unguided wavelengths. The multi-color
pattern of the cladding in kagomé HC-PCF, part (d), is due to the size distribution of hexagonal
holes in the fiber cladding which defines the optical resonance condition for each individual hole.
Hollow-core PCF comes in two main varieties: photonic bandgap (PBG) and
kagomé lattice, see Figs. (2.1a) and (b) respectively. Among these two, HC-PBG-PCF
is the only one which guides light based on a true photonic bandgap created via its
periodic cladding structure. It provides low loss (~1 dB/km at 1550 nm in the best
case*) within restricted bands of wavelengths. In a typical experiment, white light
launched into the core of the fiber emerges after propagation with a distinct color –
the result of filtering out of unguided wavelengths [Fig. (2.1c)]. Thus it came as a
surprise when it was discovered that hollow-core PCF with a kagomé-lattice cladding
guides white light, although with much higher loss than in PBG-PCF (typically 1
dB/m), see Fig. (2.1d) [Benabid et al., 2002]. Measurements showed that the
transmission spectrum was fairly flat, interspersed with narrow bands of high loss.
These characteristics make hollow-core kagomé-PCF invaluable for applications
where broad-band single-mode guidance over few-m lengths is required, such as
*
1 dB/km corresponds to 20% loss of intensity after propagation of light over one kilometer of the
fiber.
12
Figure 2.2: Propagation diagram for (a) homogenous slabs of two dielectric materials (inset of the
figure). (b) Triangular lattice of air holes imbedded in silica (inset of the figure, top). The bottom
picture of the inset shows the concept of wave guiding via PBG.
nonlinear spectral broadening in gases [Nold et al., 2010]. Although a number of
papers go some way towards explaining some of the features of guidance in kagoméPCF, the precise nature of the guidance mechanism remains unclear.
In this chapter, after a short introduction to the guidance properties of PBG-HCPCF, we focus our attention on the guidance mechanism of kagomé-PCF. We show
that guidance in this fiber can be best understood by viewing it not as an imperfect
PBG-HC-PCF, but rather as an imperfect Bragg fiber. We develop a simple model
that qualitatively reproduces the loss spectrum of both empty and liquid-filled
kagomé-PCF. Based on this model we gain a clear insight into the mechanism
underlying broad-band guidance in kagomé-PCF.
2.1 Guidance via photonic bandgap
2.1.1 Photonic crystal cladding
The basic idea behind the design of a photonic crystal cladding is to trap light in the
hollow core of a fiber via a photonic bandgap created by a periodic array of microchannels which run along the entire length of the fiber, Figs. (2.1). Quite generally, an
electromagnetic excitation of frequency ω propagating through an isotropic
13
homogeneous medium (glass or air) with refractive index n = n(ω ) has an axial
wavevector β ≤ n k . Any excitation with β > n k will be evanescent in any subregion of the medium. For example, for an air-silica interface and β < k , light is free
to propagate in air and silica, regions 1 and 2 in Fig. (2.2a). However, for k < β < ng k
light is confined in glass substrate via total internal reflection (TIR), region 2, and is
cut off from both air and glass for β > ng , region 3. Now upon introducing a periodic
array of air-holes in the glass substrate, we will have a photonic crystal (PC) structure.
We can consider PC as a composite material with its own dispersion, nPC = nPC (ω ) ,
where 1 < nPC < ng due to the presence of air holes. Light incident from air on this
structure is free to propagate in any sub-region of the PC if (i) β < k , region 1 in Fig.
(2.2b), and (ii) if it is outside the photonic bandgap of the PC, region 5 in Fig. (2.2b).
For k < β < ng k light would be trapped in the glass region and is evanescent in the
hollow channels, regions 2 and 4, and for β > ng k it would be cut off from air, glass
and PC, region 3 in Fig. (2.2b).
Let us consider a situation where for a particular frequency and axial wavevector,
light is prohibited from propagation in a properly designed PC via the photonic
bandgap (PBG), i.e. an electromagnetic excitation incident on the PC cannot find any
resonance to couple with in the PC. In this case, the electromagnetic excitation would
be totally reflected. Now if the PBG is properly positioned so that it crosses the lightline, denoted by βΛ = k Λ in Fig. (2.2), the electromagnetic excitation can actually
propagate in a medium with lower refractive index than the PC, i.e. n − nPC < 0 .
Indeed one can use this situation to create a guide with negative core-cladding index
difference. The position of such a mode in Fig. (2.2) is shown by a white circle as an
air-guided mode. The inset shows the concept of a negative core-cladding index
difference where a hollow channel is sandwiched between two PC layers.
14
Figure 2.3: Different stages of fabrication of HC-PCF using stack and draw technique, including
(a) preparation of the preform and (b) rescaling of the preform. Part (c) shows overall scaling
factors during different stages of fabrication until one reaches the desired fiber structure.
2.1.2 Fabrication
Photonic crystal fibers come into different varieties, e.g. solid-core or hollow-core,
and materials, e.g. silica, soft-glass and polymer-based PCF [Large et al., 2006;
Kumar et al., 2003; Argyros, 2009]. Fabrication methods differ depending on the
material and type of the fiber, with the so-called stack and draw technique being the
most common one for fabrication of silica HC-PCF. The first step in this technique is
to make a preform by horizontally stacking high-purity silica capillaries together
(normally 1 m long with an outer diameter of 1 mm), see Fig. (2.3a). The preform or
stack should be more or less an exact macroscopic version of the final fiber design,
taking into account the reproducible deformations of the silica capillaries which
happen during different stages of the fabrication process. A structural defect is
introduced in the preform by removing several capillaries from the original stack. Fig.
(2.3a) shows a 7-cell structural defect which would finally construct the core of the
HC-PCF. The preform is usually drawn into fiber in a two-stage process, Fig. (2.3c).
Drawing is done by slowly feeding the preform into a furnace (~2000°C) and pulling
the softened glass below the furnace at constant velocity, Fig. (2.3b). During the
drawing process the parameters of the fiber, i.e. fiber diameter, inter-hole spacing
(pitch), thickness of the silica webs, core size, etc. should be accurately tuned to the
15
desired values. This is done by controlling the feed rate, pulling speed, temperature
and inner pressure of the preform. Careful adjustment of the aforementioned
parameters results in an accurate down-scaling of the original macroscopic preform to
length scales on the order of the wavelength of light in the optical frequency domain.
2.1.3 Numerical analysis
Often modal analysis of the photonic crystal cladding of a HC-PCF is challenging.
The reason for this can be traced back to the complicated topology of the cladding as
well as abrupt and strong spatial variation of the refractive index at the material
interfaces. As a result, Maxwell’s equations must be solved numerically using
different well-developed techniques [Birks et al., 1995; Mogilevtsev et al., 1999;
Ferrando et al., 1999; Roberts et al., 2001; McPhedran et al., 1999; White et al., 2002]
Due to the cylindrical symmetry of the PC cladding along the axis normal to its
transverse plane, usually taken as the z-axis, the electromagnetic excitations (modes)
of the structure can be classified according to their axial wavevector β = k . zˆ .
Writing
field
patterns
E E ⊥ (r⊥ )e[i ( β z − ω t )]
formally =
as
and
=
H H ⊥ (r⊥ )e[i ( β z − ω t )] , it is often useful to reformulate Maxwell’s equations as an
eigenvalue problem in β while keeping angular frequency ω constant,
(∇
2
+ k 2ε ( r⊥ ) ) H ⊥ + ∇ ln ε ( r⊥ ) × ∇ × H ⊥ = β 2 H ( r⊥ ) .
(2.1)
The form of Eq. (2.1) allows for the material dispersion to be easily included in the
calculations. The second term on the left hand side of Eq. (2.1) accounts for the
spatial variation of refractive index and is responsible for the coupling between the
vector components of the field.
A quite informative way of representing the solutions of Eq. (2.1) is by plotting
the density of photonic states or simply the density of states (DOS) of the photonic
crystal cladding. The DOS shows the enhancement or reduction of the number of
possible photonic states (modes) of the PC at a fixed frequency relative to vacuum, so
that the regions of zero DOS correspond to photonic bandgaps of the PC. Formally it
16
is calculated via a sum over the Brillouin zone on the number of modes found in the
interval ( β , β + d β ) for Bloch wavevector K B and at a fixed normalized frequency
kΛ , i.e.
−1
=
ρ ( βΛ, k Λ ) ρ vac
∑ ∑ δ (β − β ),
KB
(i )
KB
i
(2.2)
where second summation, (i ) goes over β values found at Bloch wavevector K B
and ρ vac , used as the normalization factor, is the vacuum density of states under the
definition (2.2). It is easy to show that for a triangular lattice ρ vac ( βΛ=
)
A
The real space cell of pitch Λ has an area of =
3β Λ /(2π ) .
3Λ 2 / 2 . Thus the corresponding
π ) / A 8π 2 /( 3Λ 2 ) . If kT Λ is the magnitude
reciprocal space cell has an area of ( 2=
2
of the normalized real-space transverse wavevector, the states in the range kT Λ to
kT Λ + d ( kT Λ ) are contained within a circular shell of area 2π kT Λ d ( kT Λ ) , Fig.
(2.4b), and the number of states must be:
ρ vac ( kT Λ ) d ( kT Λ ) = 2 ×
3kT Λ
d ( kT Λ ) .
4π
(2.3)
where factor of 2 has been added for two different polarization states. From the
vacuum dispersion relation ( βΛ )2 + ( kT Λ )2 = ( k Λ )2 it is clear that the DOS in any kT
range
must
be
the
same
as
the
DOS
in
any
β
range
so
that
ρ vac ( kT Λ ) d ( kT=
Λ ) ρ vac ( β Λ ) d ( β Λ ) . Substituting the relation kT Λd ( kT Λ ) = β Λd ( β Λ )
into Eq. (2.3), we arrive at the desired result.
17
Figure 2.4: (a) Density of photonic states (DOS) calculated via plane-wave expansion method for
the cladding structure of a conventional HC-PBG-PCF. Here the red region corresponds to the
zero DOS for a range of normalized frequencies kΛ . The white dot shows the operating region
of interest for wave guiding in air based on PBG. (b) Schematic of reciprocal space for a
triangular lattice (see the text).
Based on this, one can obtain a detailed map of the photonic band structure of the
PC cladding. Figure (2.4a) shows such a map for the cladding structure of a
conventional HC-PBG-PCF obtained by solving Eq. (2.1) using fixed-frequency plane
wave method [Meade et al., 1993; Pottage et al., 2003; Pearce et al., 2005]. Here red
regions show where the DOS is zero. The color code is chosen so that dark (black)
regions indicate low DOS and bright (white) regions indicate increase of DOS. The
horizontal dashed line shows the light-line where β = n k , n being the refractive index
of the material filling the holes of the cladding. The operating region of interest in
Fig. (2.4a) is below the light-line well within the PBG, shown by a white dot. As
mentioned before, this ensures us that light is able to propagate freely in air (vacuum)
while being prevented from coupling to the PC cladding due to the PBG. This is only
possible if some of the core resonances coincide with the PBG. In practice this is done
by slightly changing the core size (without distorting the cladding structure) during
the fabrication process.
18
2.2 Guidance in the absence of photonic bandgap
2.2.1 Analytical model for guidance in hollow-core Kagomé-lattice
photonic crystal fibers
While guidance mechanism based on PBG in HC-PCF is well understood, the actual
mechanism behind broad transmission in large-pitch kagomé-HC-PCF is still an open
problem. Numerical modeling of the kagomé lattice indicates that, while it has a low
density of photonic states close to the air line ( β < k where β is the modal or axial
wave-vector of the core mode and k the vacuum wave-vector), a full two-dimensional
PBG does not appear. These regions of low DOS are interspersed with narrow bands
of high DOS. Positions of these narrow bands correspond to the loss windows in the
transmission of kagomé-HC-PCF. Measurements showed that these narrow bands of
high loss, which are not wide enough to produce significant coloration in the
transmitted white light, are caused by phase-matching between the core mode and
Mie-like resonances in the glass webs in the cladding. This broad band guidance in
the absence of PBG is in contrast to the behavior of HC-PBG-PCF, indicating that a
guidance mechanism other than photonic bandgap is involved. Although a number of
papers go some way towards explaining some of the features of guidance in kagoméPCF, the precise nature of the guidance mechanism remains unclear. It is evident that
some effect inhibits leakage of core light into the cladding. Indeed, “inhibited
coupling” between core and cladding has itself been suggested as a possible
mechanism [Couny et al., 2007], perhaps because the core-cladding field overlap is
very small [Argyros et al., 2007]. Another suggestion is that the low density of
cladding states slows down leakage through some version of Fermi’s golden rule
[Russell, 2006; Hedley et al., 2003], and yet another sees the effect as a form of antiresonant reflection waveguiding [Février et al, 2009].
The kagomé lattice has some resemblance to so-called Bragg fibers, which consist
of a series of concentric circular rings of low and high refractive index, resulting in a
radial stop-band that prevents leakage of light from a central core, allowing guided
modes to form [Melekin et al., 1968; Yeh et al., 1976]. Although hollow-core Bragg
fibers are available that guide 10 μm light with losses of ~1 dB/m, versions operating
19
in the near-IR and visible have proved elusive. Since Bragg fibers do not possess a
PBG, light is prevented from escaping from the core only if it is incident on the
cladding rings within certain ranges of conical and azimuthal angle. These angular
ranges grow in width as the inter-ring refractive index difference increases. Outside
these ranges, light is free to propagate through the cladding. The natural low-loss
modes of a Bragg fiber are thus azimuthally or radially polarized TE01 and TM01
modes, for these are constructed from rays that propagate conically outwards from the
core center, encountering the cladding boundary with a polarization state and
direction that are identical, relative to the local boundary normal, at all azimuthal
angles. Here we show that guidance in kagomé-PCF can be best understood by
viewing it not as an imperfect PBG-PCF, but rather as an imperfect Bragg fiber. We
develop a simple model that qualitatively reproduces the loss spectrum of both empty
and liquid-filled kagomé-PCF. Our simple model can accurately determine the
position, width and shape of the transmission windows, but the level of the loss
calculated is somewhat higher than the experimentally measured values. Nevertheless,
our simplified model provides us with insight into the mechanism of broadband
guidance in kagomé-HC-PCF, something which we have not been able to extract from
the exact numerical approaches.
2.2.2 Model
We start with the observation that the fundamental core mode can be viewed as
arising from the interference of outward- and inward-going conical waves,
propagating at a fixed angle with respect to the fiber axis, and bouncing to and fro
between the core boundaries. The modal field distribution across the core results from
interference of these two waves, taking the form of a Bessel function (we will address
the issue of polarization state later). The kagomé lattice is constructed from three
overlapping planar multilayer (ML) stacks, each of which will possess stop-bands
over certain ranges of incident angle (see Fig. 2.7). Our model runs as follows. Each
ML stack is viewed as acting independently (this approximation is explored in the
next section) over a 60° range of azimuthal angle. Thus, each 60° section of the
outward-going conical wave is viewed as being reflected by a single ML stack. Using
Fourier decomposition into spectral plane waves, the reflected phase and amplitude
20
distribution is calculated for each 60° conical wave section. This reflected distribution
will then be a distorted version of a perfect inward-going conical wave, so that only a
certain proportion of the reflected power will flow back into the correct conical mode.
This proportion is calculated by evaluating the overlap between the distorted and the
ideal inward-going field and the resulting loss calculated.
Assuming that the kagomé-PCF is perfectly invariant along its axis and is made
from absorption-free materials, the transmission loss will be a combination of leakage
(light that propagates through the cladding) and tunneling (through the stop-bands of
the ML stacks), which we can qualitatively express as:
α=
1
1
1
=
+
,
Lloss Lleak LML (N )
LML ~ tanh 2 ( µ N )
where μ is a numerical factor proportional to the width of the stop-band, N the number
of periods in the ML stack, LML the loss length due to tunneling through the ML stack
and Lleak the loss length due to imperfect reflection back into the core mode. In silicaair kagomé-PCF the leakage mechanism typically becomes dominant after only two
cladding periods (except within narrow pass-bands of high loss), which has the
consequence that additional periods do not reduce the loss. This contrasts with true
PBG-PCF, where reflected rays that do not form part of the guided mode become
evanescent, the leakage length is infinite and the tunneling mechanism dominates; as
a consequence, the loss falls rapidly as the number of cladding layers is increased.
This behavior is in agreement both with previously published numerical study on
kagomé-PCF using a finite-element method [Pearce et al., 2007] and with
experimental studies of polymer-based kagomé HC-PCF [van Eijkelenborg et al.,
2008]. Kagomé-PCF may be thus viewed as occupying a position midway between
hollow core PBG-PCF and Bragg fibre.
21
2.2.2.1 Fourier space analysis
To explore the degree to which each ML stack acts independently, we take the Fourier
transform of the kagomé structure to obtain the dielectric constant profiles of the
individual stacks. Fig. (2.5a) shows the cladding structure of the kagomé HC-PCF,
with its unit cell marked by dashed lines; three ML stacks cross each other at 60° to
create a tiled star-of-David pattern. The dielectric constant of the cladding
ε (r) = n2 (r) can be expanded as follows:
ε (r ) =ε (r + A lm ) =∑ ε pq exp(iG pq ⋅ r )
(2.4)
p,q
where G pq = 2π ( p ĝ1 + q ĝ 2 ) / Λ is the reciprocal lattice vector defining a triangular
lattice (see Fig. 2.5) and p and q are integers. The corresponding real-space lattice is
also triangular, with lattice vector A lm =
Λ (l aˆ 1 + m aˆ 2 ) rotated 30° relative to the
reciprocal space lattice, where Λ is the period and l and m are integers.
Fig. (2.5b) shows the Fourier transform of the real space refractive index

distribution of the cladding, n 2 (r ) . The Fourier transform of the cladding shows a

weak background of refractive index coefficients, n 2 (G ) superimposed by a strong
refractive index modulation lying in three principal directions defined by setting p =
0, q = 0 and p = ±q . Upon comparison with Fig. (2.5a), one can immediately
recognize that the presence of these dominant coefficients is directly related to the
three sets of planes in real space, normal to these directions. This can also be easily
verified by deliberately omitting some of these planes in real space which results in
the absence of the corresponding modulation in the reciprocal space; see Fig. (2.5c)
and its corresponding Fourier transform Fig. (2.5d).
A close inspection of the Fourier transform of the kagomé cladding reveals more
interesting properties regarding the coupling between these principal planes. In
general, one expects that any possible coupling between the three sets of glass struts
in the cladding should happen via their crossing points, i.e., the glass nodes. Indeed,
using Fourier analysis, it is possible to show that such coupling reveals itself as the
22
Figure 2.5: (a) Sketch of kagomé lattice, with the unit cell marked. The period of the triangular
lattice and the ML stacks are respectively Λ and Λ ′ =Λ cos(π / 6) . (b) Discrete Fourier transform
of a unit cell of a full kagomé lattice for a relative membrane thickness h / Λ =0.02 . For clarity
the constant background has been subtracted. The reciprocal lattice coordinates are in units of
2π / Λ . (c) Unit cell with one ML stack removed. (d) Fourier transform of (c). Note the absence
of one pair of “spokes”. (e) Inverse Fourier transform of the weak background shown in (f). The
crossing points of the ML stacks are observed. (f) Weak background spectral distribution in (a)
with the “spokes” removed.
aforementioned weak background modulation of refractive index coefficients in the
reciprocal space. To show that, we deliberately omit the strongest coefficients of the

Fourier transform of n 2 ( r ) and try to retrieve the original structure via an inverse
Fourier transform of these background coefficients, shown in Fig. (2.5f). The result of
such filtering is seen in Fig. (2.5e); upon taking the inverse Fourier transform of the
background part, one can only retrieve the crossing points of the glass struts. So, to
the first order approximation, one can treat the kagomé cladding as if it is made up of
three sets of individual isolated Bragg reflectors.
In the next sections, we show that by proper choice of the fibre core diameter, one
can utilize the stop-bands created by these Bragg reflectors in order to effectively
confine and guide the light with low propagation loss.
23
2.2.2.2 Plane-wave reflection coefficients
Assuming that a mode exists in the core, and that its propagation constant along the
axis is β, permitted wave-vectors will lie on a circle of radius kT, its plane oriented
perpendicular to the fiber axis and displaced from the origin of reciprocal space by β,
kT2 =τ 2 + p 2 =k02 nco2 − β 2 ,
where τ and p are the transverse wave-vector components tangential and
perpendicular to one of the ML stacks in the kagomé cladding, pointing along the
local axes ( x, y , z ) (see Fig. 2.6). The full wave-vector component parallel to the
selected ML stack is then given by q = τ 2 + β 2 . The dispersion relation of Bloch
waves in the ML stack can be obtained analytically using the transfer matrix
approach. Following the formalism in [Russell et al., 1995 & 2003b] we obtain,
pξ
1 pξ
cos( kB Λ′) = A = cos( p1h1 )cos( p2 h2 ) −  1 1 + 2 2
2  p2ξ 2 p1ξ1

 sin( p1h1 )sin( p2 h2 ),

(2.5)
where kB is the Bloch wavevector normal to the ML stack and the subscripts i = 1
and 2 refer to the layers of glass and low index material (air or water in the
experiments reported later). In layers made from the ith material, the wavevector
component normal to the stack is
pi = k 2 ni2 − q 2
, hi is the layer thickness ( h1 + h2 =
Λ′ ),
and ξi = 1 for TE and ξi = ni−2 for TM polarization. Stop-bands occur when kB is
imaginary, in which case light is unable to propagate through the ML stack (although
it can tunnel through if the number of layers is small), which in turn occurs when
A > 1 or A < −1 , since the stop-band edges appear at A = ±1 . Fig. (2.7) shows the
band structure and the map of the stop-bands for a ML stack, expressed via
neff = β / k , as the normalized frequency kΛ is varied for both TE [Fig. (2.7a)] and
TM [Fig. (2.7b)] polarizations. The parameters are chosen to be representative of the
ML stacks in a typical kagomé PCF. There the color map indicates the decay rate of
the intensity of light in the ML stack, i.e. it is proportional to the imaginary part of the
24
Figure 2.6: (a) Constant frequency dispersion sphere for the core material (index nco). The guided
core mode is formed from wavevectors lying on the circular intersection between the k z = β
plane and the dispersion sphere. (c) Intersection circle and its orientation relative to the ML
stack. The components of wavevector normal and parallel to the ML stack are p and q = τ 2 + β 2
respectively. Here ( x, y, z ) defines an orthogonal local axes with
planes and
y being normal to the ML
z showing the propagation direction.
Bloch k-vector, Im{k B } . The dark blue shows regions where Im{k B } = 0 ( A < 1 in
Eq. 2.5) corresponding to the transmission windows of the ML stack, i.e. the loss
windows of the fibre.
A guided mode will appear when a core resonance, expressed as a cylindrical standing
wave, coincides in frequency and effective index with the position of a stop-band in the ML
stacks. The single-lobed fundamental core resonance will occur at an effective index given
approximately by:
=
neff
(2.6)
nco2 − ( z01 / k a ) 2
where a is the core radius and z01 = 2.4048 is the first zero of the Bessel function
J0. The core radius is somewhat uncertain since the core-surround is not circular,
which makes it difficult to predict a precise value for neff . The dispersion of the core
mode, Eq. (2.6), is shown in Fig. (2.7) with white solid lines for an air-filled core,
nco = 1 and different values of the core radius.
25
Figure 2.7: Band structure and map of stop-bands for a ML stack in a typical kagomé fibre
shown as a function of neff and for different vales of normalized frequency, kΛ for (a) TE
polarized and (b) TM polarized states. The color indicates the strength of the intensity decay of
light, Im{k B } . Red color indicates stronger decay, while the dark blue shows loss windows in
which light can easily tunnel through the ML stack. In both graphs white solid lines show the
dispersion of the core mode as approximated by Eq. (2.6) in order of increasing radii from left to
right. By changing the core size, one can position the core resonance near the stop bands of the
cladding ML stack of the fibre.
Among all the stop-bands present in Fig. (2.7), the one closest to neff = nco is of
particular interest since the Bloch wave decay rate is the highest, because one is
operating close to glancing incidence. As can be seen from Fig. (2.7), in the ideal
case, by proper choice of the core size, the propagation constant of the core mode can
be well inside the last stop-band created by the cladding of the fibre resulting in a low
propagation loss. However, one should note that in a real Kagomé fibre, the fibre core
has a hexagonal shape that is rotated by 30° with respect to the cladding. This rotation
results in an uncertainty in the definition of the core radius and consequently limits
the accuracy of Eq. (2.6) as an approximation for the dispersion of a hexagonal core
defect. To take the aforementioned points into account, we replace a in Eq. (2.6) by
aeff which is defined through the relation aeff = γ a, where 0 < γ ≤ 1.
26
Figure 2.8: Schematic of a ML stack consisting of N lattice periods and an infinitely thick glass
substrate. The distance between the glass webs is Λ′ = h1 + h2 , where h1 indicates their thickness
and h2 is the thickness of the air gap between them. Here, the variable hs determines the distance
of the substrate from the last glass web.
2.2.3 Reflection coefficients from ML stacks
As mentioned in the previous section, the Bloch mode may have a traveling or an
evanescent nature upon incidence on the ML stack for different values of incidence
angle α inc = cos −1 (neff / nco ) , with the strength of the reflection as well as the shape of the
transmission window being controlled by the number of layers and the properties of
the substrate in the ML stack. To obtain a better understanding of the aforementioned
point we will first study the reflection characteristics of a ML stack as a function of
effective refractive index of the incident beam, neff. The ML is shown in Fig. (2.8). It
consists of N uniform period of repeating glass-air layers with corresponding
refractive indices n1 and n2 = 1 and thicknesses of h1 and h2, followed by a layer of air
with variable thickness hs, and an infinitely extended glass substrate. The later
represents the supporting silica jacket in a real kagomé fibre. Using the standard
scattering matrix approach it is quite straightforward to calculate the phase and the
reflectivity, | r |2 , of the ML stack for different incident angles.
As the first step we will investigate the role played by the substrate layer by
simply varying the thickness of the final air layer, hs. This is shown in Fig. (2.9a) and
hs 0.98 Λ′ and =
hs 0.96 Λ′ . In both cases we
(2.9b) for N = 2 and two values of=
27
Figure 2.9: Amplitude and the phase of the reflection coefficient of the ML stack for (a)
hs 0.98Λ′ and (b)=
hs 0.96Λ′ .
=
h1 0.02 Λ′ , =
h2 0.98 Λ′ and
have chosen the following parameters for ML stack:=
k0 Λ′ =20 π , where Λ′ = h1 + h2 . The refractive index of the glass is chosen to be
n1 = 1.46 in all cases. As can be seen from the figure, the ML stack shows a loss
window near neff = 0.9985 for both TE and TM polarization states, with the TM
resonance being broader in both cases. However, while the reflectivity of the ML
stack outside the loss window is not sensitive to the changes in the substrate, the
shape of the loss window is heavily affected by the properties of the substrate. This
sensitivity can be attributed to the increased penetration depth of the light in the ML
stack near the resonance. Due to the relatively narrow width of the stop bands and
high values of kΛ , the reflectivity of the ML stack is quite sensitive to its geometrical
properties, such as the pitch and web thickness.
Next, we will consider the affect of the number of layers. This is shown in Fig.
(2.10) as the number of periods in the ML stack is varied form N = 0 (a single glass
hs 0.96 Λ′ from the substrate) to N = 2 . In contrast to the
layer with distance =
substrate the increase in the number of layers affects both the shape of the loss
window and the reflectivity of the ML stack. The increase in the reflectivity by adding
more layers in the ML stack may lead one to naively expect the reduction in the
leakage loss of the kagomé fibre simply by increasing the number of layers in the
cladding of the fibre, as is for example the case in Bragg fibres. However, as will be
shown in the next section, the leakage loss in the kagomé fibre is dominated by the
28
Figure 2.10: Amplitude and the phase of the reflection coefficient of the ML stack as a function of
the number of lattice periods for TE and TM polarization states. In both figures, the right and
left insets show a magnified view of the high reflectivity regions of the ML stack. As expected, an
increase in the number of layers will result in an increased reflectivity of the ML.
imperfect back reflection of the core mode from the ML stack, so that increase in the
number of cladding layers does not help in reduction of the leakage loss of the fibre.
2.2.4 Calculation of loss
Having understood the behaviour of the ML stack in different situations, we move
forward to calculate the transmission loss in a hollow waveguide which is surrounded
by a ML stack with six-fold symmetry, as in the case of the kagomé fibre. Using the
LP approximation for the light in the core – valid since the rays travel almost parallel
to the axis, so that the z-component of the fields is negligible – the electric field of the
LP01 mode may be written,
z ) yˆ
=
E yˆ J 0 (k T ⋅ rT ) exp(i β=
1 (1)
 H 0 (k T ⋅ rT ) + H (2)

0 (k T ⋅ rT )  exp(i β z ),
2
29
(2.7)
Figure 2.11: Schematic of the fibre cross section. The axes are chosen so that the x and y axes
shows the directions parallel and perpendicular to the ML stack, respectively. Here
k T indicates
the transverse k-vector of the incident plane wave to the ML stack and ϕm measures the angle
made by the electric field vector of the plane wave (in this case y-polarized) and the normal to the
m-th ML stack.
where k T =
k 2 nco2 − β 2 ( xˆ cos φ + yˆ sin φ ) . The outward going field, the first term on the
RHS in Eq. (2.7), is Fourier-transformed into a plane-wave spectrum and the
reflection coefficient of each plane wave calculated. If the outward surface normal to
the m-th ML stack, n m , points at an angle φm to the y-axis, the electric field E of one
of these plane-waves can be resolved into TE and TM field components relative to the
ML stack,
TM
TE
EPWout
=
cos φm ⋅ APW exp(i β z ), EPWout
=
sin φm ⋅ APW exp(i β z ),
(2.8)
The reflected field amplitudes are calculated independently for these two field
components and the total reflected field obtained by summing them and evaluating the
field component parallel to the y-axis (the orthogonal field component does not couple
back into the core mode, and has a different phase in each 60° section, so is lost). The
resulting y-component of the reflected field is,
ref
EPW
= ( rTE (τ , β )sin 2 φm + rTM (τ , β )cos 2 φm ) APW exp(i β z ) ,
30
(2.9)
Defining the dimensionless quantities χ = kT x , η = kT y and ζ = kT z , the transverse
=
η k=
η0 of
part of the outward-going y-polarized field at the flat core boundary
T y0
the m-th ML stack takes the form (see Appendix A),
=
Eout
1 (1)
H 0 [ χ 2 + η02 ].
2
(2.10)
For any fixed value of η , it is possible to express the outward-going field, Eq. (2.10)
as a Fourier integral of the conjugate variable χ . This integral serves as the plane
wave expansion of the outward-going field, given by the relation,
Eout =
1
2π
∫
+1
−1
F0 (τ n ,η0 ) exp(i τ n χ ) dτ n ,
(2.11)
where τ n = τ / kT is the normalized, dimensionless wavevector component parallel to
the interface and evanescent waves (for which τ n > 1 ) are neglected – in any case
these do not contribute to the guided core mode.
Making the change of variable τ n = cos(u ) , where u ∈ [ 0, π ] , we may write the
full reflected field amplitude from the m-th ML stack as (see Appendix A, Eqs. A9
and A10),
Emref ( χ=
,η0 )
1
π
u =π
− iη sin( u ) i χ cos ( u )
2
2
e
du.
∫ ( rTE (u, β ) sin φm + rTM (u, β ) cos φm ) e
0
(2.12)
u =0
The field distribution around the entire periphery of the core is finally compared with
a perfect inward-going conical wave (Eq. A11), and the power reflection coefficient
calculated for the entire mode,
31
6
Γ2
2
η0 / 3
∑ ∫
m =1
=
Emref ( χ ,η0 ) Emin* ( χ ,η0 ) d χ
χ = −η0 / 3
6
η0 / 3
∑ ∫
m =1
.
(2.13)
2
Emin ( χ ,η0 ) d χ
x = −η0 / 3
As mentioned before, the core of a real kagomé has a hexagonal shape. In the ray
optic approximation the long lived rays making up the fundamental core mode in a
hollow waveguide with hexagonal cross section are the ones that hit the core wall at
30° with respect to the normal to the wall in the transverse plane. This assures a
closed path (after six bounces) and the same optical path length along the fibre, z B for
all parallel rays making up the mode given by,
zBhex =
6 3 aeff
2
nco2 / neff
−1
(2.14)
,
leading to the following formula for the propagation loss of the fibre in dB/m,
=
α dB/m
−5 × 106 z01 λ
log10 Γ,
2
3 3 π neff aeff
(2.15)
where both wavelength, λ , and radius, aeff, are given in microns. If we assume
log10 Γ neff to be a slowly varying function of core radius, then the loss shows an
inverse quadratic dependence on the core radius, α dB / m ∝ λ / a 2 , while in the case of
hollow dielectric waveguides this dependency is cubic, i.e. α dB / m ∝ λ 2 / a 3 . This
results partly explains the good performance of the kagomé fibre, α dB / m ~ 1 , despite
having a relatively small core size, a ~ 10 - 15 μm, as compared to that of a hollow
dielectric waveguides, a ~ 50 μm.
32
Figure 2.12: The scaling behaviour of loss as calculated using our model for (a) a hollow
dielectric waveguide with a hexagonal core as a function of (b) core radius and (c) wavelength.
2.2.5 Comparison with experiments
Before calculating the loss spectrum of the fibre in detail, we would like to use our
analysis to test its qualitative predictions. To this end, we first apply our model to a
hollow hexagonal waveguide of infinite substrate thickness, as shown in Fig. (2.12a).
Based on analytical calculations [Marcatili et al., 1964], in such a case, one would
expect the loss to scale as α dB / m ∝ λ 2 / a 3 . The result of calculation based on our
method described in previous section is presented in Figs. (2.12b) and (2.12c). In both
graphs, the empty circles show the result of our model and the solid lines show
numerical fits of the form λ 2 , Fig. (2.12b) and 1/ a 3 , Fig. (2.12c). As can be seen, the
calculated result based on our model nicely produces the expected scaling of the loss
in a hollow waveguide with respect to the wavelength and core radius.
To check our model versus experimental results, we fabricated a deliberately highloss kagomé with a single-cell core and only two lattice periods – see Fig. (2.13). As
can be seen in Fig. (2.13b) the fibre supports two transmission windows centered near
525 nm and 725 nm with an average loss of roughly 5 dB/m (experimental data are
shown with full black circles). Transmission windows are separated by resonances of
the glass webs. For a layer of thickness h1, these resonances occur at wavelengths
33
Figure 2.13: (a) Scanning electron micrograph of the fabricated kagomé fibre with only two
Bragg layers. Parts (b) and (c) show experimentally measured (full black circles) and
theoretically estimated (empty circles) loss of the fundamental mode for evacuated and H2Ofilled fibers, respectively. The measured parameters of the fibre are the pitch of Λ' = 14 μm, wall
thickness of h1 = 0.85 μm and core radius of a = 13.86 μm. The parameter γ in both cases is
chosen to be 0.725.
=
λ m 2 h1 n 2 − 1 / m for integer m [Pearce et al., 2007]. Using n = 1.45
values given by
and web thickness of 850 nm, as measured from scanning electron micrograph of the
fibre, these resonances occur at λ = 890, 600 and 450 nm. As seen from Fig. (2.13b),
the positions of these loss windows are correctly reproduced by our model (shown by
empty circles). In addition, although the glass struts in the fabricated fiber are rather
thick (850 nm), the good quantitative agreement between our calculation and the
experimentally measured loss spectrum shows the validity of the assumption of
Section 2.2.2, i.e., to a first order approximation, one can ignore the coupling between
the Bragg layers in the kagomé cladding. To measure the loss experimentally, several
cutback measurements have been done on a constant radius of curvature 7.5 cm
(commercially available fiber winding drum). We mention in passing that the fiber
was sensitive to bending and mechanical tension, which we attribute to the small
number of cladding layers.
To verify the validity of the model in an even more demanding situation, we
performed the loss measurement on an H2O-filled fiber. In this case we would expect
the shift of the transmission (loss) windows due to the scaling properties of Maxwell’s
34
Figure 2.14: Calculated loss versus number of lattice periods. Note the insensitivity of the loss as
well as narrowing of the ML resonances as the number of periods increase from one to four.
equations. The result of this measurement as well as the calculated loss as a function
of wavelength is shown in Fig. (2.13c). Calculations have been done by changing the
refractive index of the base material of the fiber, i.e. nbase = nH 2O , and using otherwise
exactly similar parameters used for the case of unfilled fiber. As can be seen from this
figure, one again finds a good quantitative agreement between the experimental data
and the calculated loss.
One of the still open problems regarding the guidance mechanism in kagomé fibre
is the independence of the loss from the number of lattice periods in the fibre
cladding, as shown by both careful theoretical and experimental studies. After all, if
the guidance mechanism is solely based on Bragg reflection, one would expect an
appreciable reduction of the loss as the number of lattice periods is increased.
However, to the contrary, adding more lattice periods does not reduce the loss beyond
two layers. Here, we would like to tackle this problem using our model. Simple
comparison between Fig. (2.12c) and (2.13b) shows the drastic change in the loss
landscape as one makes a transition from a simple hollow waveguide to a kagomé
fibre, so that the existence of at least one lattice period is crucial to have broadband,
low-loss propagation. However, as mentioned previously, the leakage loss in kagomé
fibre is dominated by the imperfect matching of the reflected core mode, i.e. further
increase in the number of lattice periods does not help reducing the leakage loss. This
35
is shown in Fig. (2.14), as the number of periods is increased from one to four. No
noticeable change in the loss value is observed – a result which is in good agreement
with exact finite-element calculations.
2.3 Discussion and conclusion
In this chapter, after a short discussion of guidance in the presence of a photonic
bandgap, we turned our attention to the case of broadband guidance in HC-PCF in the
absence of any PBG. A semi-analytical model based on the Fourier space analysis of
the cladding structure and plane-wave expansion of cylindrical functions at a planar
(core-cladding) interface was derived to successfully explain the guidance mechanism
and main loss features of the large-pitch kagomé-HC-PCF. Our analysis shows the
existence of several broad transmission windows with a loss of approximately several
dB/m, separated by high loss windows. The positions of these loss windows
correspond perfectly with the position of the resonances of the glass struts.
Using our model, we dealt with the question of the importance of number of
layers in the cladding of kagomé fiber on the loss value of the fiber. As it is shown in
Sec. 2.2.5, after adding a second layer, the loss of the fiber reaches a plateau due to
the inexact matching of the reflected and a perfect inward-going core mode. Based on
this insensitivity of the loss on the number of layers, it has been argued that the
presence of only one layer, i.e. core surround itself, might be sufficient for efficient
confinement of light in the core [Février et al., 2010]. However, a one ring structure
fabricated does not exhibit broadband guidance, although it shows a relatively low
loss over a narrow transmission window. This result is in agreement with our
calculation, shown in Fig. (2.14). Although the addition of more than one layer does
not reduce the loss, it does help narrow the cladding resonances which in turn results
in a broader transmission window. So, we strongly believe that the type of behaviour
common in large pitch HC-PCFs, i.e. a broad transmission window with a relatively
low loss, is a result of having at least more than one Bragg layer surrounding the core
of the fiber, as is the case in a functional kagomé fiber.
36
Using the Fourier space analysis, it is indeed possible to generalize the discussion
presented here to the large pitch square and honeycomb lattices.
While such
generalization is straightforward in the case of a large-pitch square-lattice HC-PCF, it
is not trivial in the case of a honeycomb lattice. The reason is that in this case, due to
the complexity of the cladding geometry, one cannot simply ignore the coupling
between the glass struts, and the approximation of isolated Bragg layers is only
justified for very large pitch, so that the side length of each individual hexagon is
much larger than the wavelength of the light. In fact, a strut thickness of several
hundred nanometers and a pitch of tens of microns, guarantees a big aspect ratio for
each side of the cladding hexagons, i.e. Λ / t >> 1 , so that each of them can be
considered as a planar reflector. That is why the broad band guidance in a honeycomb
lattice happens for pitch sizes larger than that of the kagomé lattice [Beaudou et al.,
2008]. Despite the successful application to the guidance mechanism in the kagomé
HC-PCF, our simple model suffers from several drawbacks:
•
In our analysis, we have not considered diffraction, i.e. we have assumed that
the core resonance faces the core-cladding boundary in the far-field. While
this assumption may well be justified for short wavelengths λ << Λ , due to
the large dimensions of the fibre, it is not the case for the long wavelength
range of the spectrum, λ ≤ Λ . Particularly one should note that the effective
transverse wavelength of the core mode is comparable to the pitch of the fiber.
•
Equation (2.6) is certainly an approximation to the real effective refractive
index of the core mode. In particular, this equation does not capture the exact
behaviour of neff near the resonances of the cladding.
•
Exact figure of the loss depends on the distance that rays which constitute the
fundamental core mode travel along the fiber per reflection from the corecladding interface. Under the assumption of circular core, the traveled distance
is shorter than the one calculated in Eq. (2.14). This will increase the value of
loss from the one presented in Fig. (2.13).
However, I hope that these methods will be useful in clarifying the basic physical
principle behind the guidance of these types of large pitch HC-PCF.
37
38
3
Wave propagation and coupled wave equations in
stimulated Raman scattering
The passage of electromagnetic radiation through matter induces a polarization in the
medium [Boyd, 2008; Reintjes, 1984]. If the intensity of the laser beam is sufficiently
high, the induced polarization shows a nonlinear dependence on the electric field
strength. This nonlinear interaction will result in the mutual interaction between the
electric fields and matter. Since in the stimulated scattering process, we are dealing in
general with a large number of photons, the electromagnetic fields can be treated
classically using Maxwell’s equations. The coupling between the electric field and the
matter is then treated by including the nonlinear polarization induced in the medium
as a source term in Maxwell’s equations.
In dealing with the solutions of Maxwell’s equations in a waveguide such as HCPCF, one should always bear in mind that any electromagnetic excitation in HC-PCF
must be a combination of well-defined sets of modes. These are solutions of
Maxwell’s equations for the electromagnetically coupled system of the fiber’s core
and cladding that satisfy the boundary conditions imposed by the waveguide
geometry. Every mode in the fiber is defined by a unique propagation constant, β ,
spatial intensity distribution, and dispersion, β = β (ω ) , in contrast to free
39
Figure 3.1: Phase-matching diagram for the stimulated Raman scattering in a HC-PCF. The blue
curve shows the dispersion of a typical kagomé-HC-PCF.
space (bulk) interaction where, for a given spatial distribution, the propagation
constants of the interacting fields can be tuned with respect to each other simply by
choosing their relative angles. As a result, information provided by an energy diagram
such as the one shown in Fig. (1.1) is not sufficient to fully describe the interaction in
HC-PCF. A proper way of representing the situation in HC-PCF is depicted in Fig.
(3.1). The figure shows the dispersion of the fundamental core mode in a typical
kagomé-HC-PCF. Here ω p , s , as and β p , s ,as are the frequencies and propagation
constants of pump, Stokes and anti-Stokes waves respectively. The first scattering
process involves generation of Stokes wave at frequency ωs . This process is
automatically phase-matched and leads to the generation of a low-frequency reservoir
of optical phonons, OP1 in Fig. (3.1). The dispersion of these optical phonons is
relatively flat and is shown by the thick horizontal line in Fig. (3.1). In general
efficient generation of any higher order scattering processes require phase-matching.
Only under phase-matched conditions can energy be optimally exchanged between
the interacting waves and the material excitation. Figure (3.1) demonstrates the
situation for the generation of anti-Stokes radiation. In this case the phase mismatch
can be written as Δk ⋅ zˆ=
( OP1 − OP2 ) ⋅ zˆ= β as + β s − 2β p . From the figure it is clear
that this wave mismatch will never be zero for anti-Stokes radiation generated in the
fundamental mode. However, this process might be phase-matched if the anti-Stokes
is generated in a higher-order mode with different dispersion profile. This interesting
point of view opens up the possibility of creating phase-matching simply by using
higher order modes in an HC-PCF [Ziemienczuk et al., 2011].
40
3.1
Wave equation
Our starting point for the derivation of the wave equations for the space-time
evolution of the electric fields of the pump and scattered waves, are the following
equations in SI units, derived directly from Maxwell’s equations for a homogeneous
medium [Boyd, 2008],
∇×E = −
∂B
∂t
(3.1a)
=
∇×B µ
∂D(1)
∂P NL
+µ
+ εµ σ E
∂t
∂t
(3.1b)
Here E and B are the electric and magnetic field vectors, respectively, ε and µ are
the electric permittivity and magnetic permeability of the medium*, c is the velocity
of light†, D(1) is the linear displacement vector, P NL is the nonlinear polarization and
σ is the power absorption coefficient of the medium. The linear part of the
polarization P (1) is related to the linear part of the dielectric displacement vector
(1)
D(1) via the relation D=
ε 0 E + P (1) . For a homogeneous medium, this relation can
be expressed via the electric permeability of the medium as D(1) = ε E . By taking the
curl of Eq. (3.1a) and inserting it into Eq. (3.1b) we arrive at the following wave
equation for the light field in the medium,
∂ 2 D(1)
∂E
∂ 2 P NL
µ
∇ E−µ
− εµ σ
=
.
∂ t2
∂t
∂ t2
2
(3.2a)
For a lossless, nonmagnetic material µ = µ0 and σ = 0 , so that Eq. (3.2a) can be
written as
*
The relation between ε ( µ ) of the medium and ε 0 ( µ0 ) of the vacuum can be expressed via the
relative values of medium electric permittivity and magnetic permeability, i.e. ε = ε r ε 0 ( µ = µ r µ0 ). For
nonmagnetic material µr ≅ 1
The relation between c and the magnetic permeability µ0 and electric permeability ε 0 of vacuum is
given by µ 0ε 0 c 2 = 1
†
41
∇ 2E −
1 ∂ 2 D(1)
1 ∂ 2 P NL
.
=
ε 0c2 ∂ t 2
ε 0c2 ∂ t 2
(3.2b)
In deriving (3.2a) we have used the vector identity ∇ × ∇ × E = ∇(∇. E) − ∇ 2 E and the
0 . This is identically true for transverse electromagnetic plane
assumption that ∇. E =
waves*. Assuming the plane wave propagation of linearly polarized, quasimonochromatic fields with slowly varying amplitudes and phases, along the ± z
direction, with “+” indicating the forward and “–” indicating the backward
propagation, we write the electric field and the nonlinear polarization in the following
form,
=
E ( z, t )
1 N
∑{E j exp[i( k j z + ω j t )] + c.c.},
2 j =1
1 N
{PjNL exp(iω j t ) + c.c.},
=
P ( z, t )
∑
2 j =1
NL
(3.3a)
(3.3b)
where c.c. represents the complex conjugate of the first term. In Eqs. (3.3) ω j and
k j = (n j / c) ω j are the carrier frequency and k-vector of the jth plane wave component
of the field, respectively, n j = ε j µ /(ε 0 µ0 ) is the refractive index of the medium at
the carrier frequency ω j , and N is the total number of waves present. Here E j and
PjNL are the complex, time and space dependent envelope functions. Substituting the
relations in (3.3) into (3.2), after some algebra we arrive at the following equation
describing the space-time evolution of the electric field components in the presence of
the nonlinear polarization,
iω 2j
1
±
+ σ jEj =
−
µ PjNL exp(±ik j z ).
2kj
c ∂t
∂z 2
n j ∂E j
∂E j
*
(3.4)
In general, one should care that the validity of this assumption may break down, as for example in the
case of wave propagation in an anisotropic medium.
42
In deriving Eq. (3.4) we have used the slowly varying approximation for the envelope
functions,
i.e.
∂ 2 E j / ∂ z 2 << k j ∂E j / ∂ z ,
∂ 2 E j / ∂ t 2 << ω j ∂E j / ∂ t ,
and
∂ 2 PjNL / ∂t 2 ≅ −ω 2j PjNL . The latter condition is justified if the induced change in the
nonlinear polarization occurs on a time scale that is much longer than ω −j 1 . Equation
(3.4) will serve as our starting point for investigating the propagation effects in the
process of stimulated Raman scattering.
3.2 Stimulated Raman scattering (classical approach)
3.2.1 Mechanism of the Raman effect
The Raman effect results from the interaction of vibrational and rotational motions of
molecules with an electromagnetic field. By contrast, Brillouin scattering involves the
translational motion of molecules in liquids and solids in the form of accoustic waves.
The effect of the electric field on a molecule is to perturb the electron cloud and
polarize the electron distribution. Thus, a dipole moment is induced in the molecule,
which can quite generally be written as,
μ [ C m ] = ε C m -1V -1  α[m 3 ] : E  Vm -1 
(3.5)
where μ is the induced dipole moment, E is the electric field and α is the
polarizability tensor of the molecule. In an ensemble of randomly oriented diatomic
gas molecules such as H2 (molecular hydrogen), the polarizability tensor averaged
over different (random) molecular orientation with respect to the direction of applied
electric field, is symmetric and we can consider it to be a scalar quantity, so that,
μ =εαE
(3.6)
Under the influence of the electric field E = E0 eiωt of the incident optical field,
the induced dipole in the electronic cloud of the molecule starts to oscillate with
43
frequency ω modulated with the natural vibrational frequency of the molecule itself,
Ω << ω . If we assume a harmonic oscillation of the molecule, then Ω represents the
frequency of the vibration of the inter-nuclear distance, qv = qv0 ei Ω t where qv0 is the
amplitude of the vibration. Now the basic assumption about the polarizability, first
introduced by Placzek [Placzek, 1934], is that it can be expanded as a Taylor series in
the nuclear coordinate qv , that is,
 ∂α 
α (t ) =
α0 + 

 ∂qv qv =0
qv (t ) + ...
(3.7)
Keeping just the first two terms in the above expansion (small atomic displacements)
and entering the explicit form of qv into (3.7), we see that the polarizability of the
molecule oscillates with the natural frequency of the molecular vibration,
 ∂α  0
α (t ) =
α0 + 
 qv cos(Ωt ).
(3.8)
 ∂qv 0
Substitution of Eq. (3.8) into Eq. (3.6) leads to
 ∂α  0
 qv cos(ωt ) cos(Ωt )
q
∂
 v 0
µ=
α 0ε E0 cos(ωt ) + ε E0 
 ∂α  0
1
= α 0ε E0 cos(ωt ) + ε E0 
 qv [ cos(ω − Ω)t ) + cos(ω + Ω)t )] .
2
 ∂qv 0
(3.9)
The frequency content of the field that is radiated by the molecule is given by the
Fourier transform of the motion of the electric dipole, µ . From Eq. (3.9), we see that
there exists a central line of frequency ω (Rayleigh scattering) and two shifted lines
(side bands) of frequency ω − Ω (Raman Stokes scattering) and ω + Ω (Raman antiStokes scattering). It is interesting to note that elastic scattering occurs via α 0 .
44
3.2.2 Optical phonons and material excitation
As can be seen from the previous discussion, the process of Raman scattering is an
inelastic scattering process in which the photon energy changes as a result of
scattering. In the case of Stokes scattering material plays the role of a thermal bath in
which it absorbs the excess quanta of energy (  Ω ) generated during this process.
Classically speaking, the material excitation created in this way will be characterized
by its amplitude and phase. In the case of spontaneous Raman scattering the phases
of the molecular vibrations/rotations are random, i.e. there is no correlation between
individual molecular vibrations/rotations, see Fig. (3.2a). However, as the intensity of
the incident laser beam increases, the scattering process becomes stimulated, in which
case, due to the high flux of the incident photons, the scattered (Stokes) photons
stimulate the scattering of even more (Stokes) photons. In this case, the scattered
photons are coherent and the scattering process induces an in-phase material
excitation in the medium. This coherent material excitation is known as an optical
phonon, in analogy to collective motion of phonons in solid-state material under
excitation by a short pulse [Cheng et al., 1990 and 1991; Zeiger et al., 1992; Zijlstra et
al., 2006]. In this case one can talk about the material coherence wave with frequency
Ω and propagation vector k0 = 2π / λ0 , as shown schematically in Fig. (3.2b).
In this language, the amplitude of the material excitation wave can be simply
expressed by the following plane wave expansion,
=
qv (t )
1
{q exp[i (−k0 z + Ωt )] + c.c.},
2
(3.10)
where q = q (t ) is the slowly varying normal mode amplitude of the material
excitation per molecule. In the presence of molecular collisions, this quantity decays
on the time scale of T2 , the dephasing time of the molecular excitations. As a side
note, the transient regime of stimulated Raman scattering happens before the
relaxation processes (collisions, etc.) destroy the mutual coherence of the
vibrating/rotating molecules interacting with an optical field of duration τ P , that is
when τ P < T2 . As will be shown later, this term is introduced as a phenomenological
45
Figure 3.2: Induced vibrational material excitation in the medium in the case of (a) spontaneous
Raman and (b) stimulated Raman scattering. Here the λ0 shows the effective wavelength of the
material coherence wave (optical phonon).
damping constant, Γ =1/ T2 in the differential equation governing the evolution of the
material excitation amplitude. Moreover, Γ is equal to the half line-width (FWHM)
Γ/2 =
1/(2T2 ) . The dephasing time is the time
of the Raman transition, i.e. δωR =
required for destruction of the mutual correlation between molecular oscillations and
not the actual de-excitation time of the molecular vibrations/rotations. The latter is
governed by the damping constant Γ′ corresponding to 1/ T1 , the inverse lifetime of
the vibrational/rotational state. However, since typically T2 << T1 , the damping of the
molecular excitation, Γ + Γ′ , is entirely controlled by Γ =1/ T2 .
3.2.3 Material excitation as a damped oscillator
Let us consider the medium as an ensemble of oscillators with reduced mass m, and
let us also consider, for simplicity, just one vibrational mode of angular frequency Ω
and amplitude qv . The Hamiltonian for the light field coupled to the molecular
vibrations is given by
H = H radiation + H vibration + H interaction ,
(3.11)
46
where the Hamiltonian* for the radiation field, molecular vibrations and the lightmatter interaction are given by,
=
H
radiation
=
H vibration
1
1
2
2
B ),
(ε E +
µ0
2
(3.12)
1
Nm{qv2 (ri ) + Ω 2 qv2 (ri )},
2
(3.13)
1
H interaction =
− Nε α E ⋅ E
2
1
1
=− N ε α 0 E ⋅ E − N ε (∂α / ∂qv ) qv E ⋅ E,
2
2
(3.14)
respectively. Here, we let N denote the number density of molecules. The differential
equation governing the evolution of the material excitation then follows from the
Hamiltonian equation of motion,
p v = −
∂H
,
∂qv
(3.15a)
qv =
∂H
,
∂pv
(3.15b)
where pv = mqv is the generalized momentum of qv . Substituting from (3.6), (3.7),
and (3.12 – 3.14) into Eq. (3.15a) we arrive at the following differential equation
describing the amplitude of molecular excitation
d2
d
ε  ∂α 
q + 2Γ =
qv + Ω 2 qv

 E⋅E
2 v
2m  ∂qv 0
dt
dt
(3.16)
(2 / T2 )(dqv dt ) has been added to
The phenomenological damping term 2Γ(dqv dt ) =
Eq. (3.16) in order to take into account the dephasing of the molecular excitation.
*
Due to the classical nature of our treatment here, we use the classical definition of the Hamiltonian as
the total energy of the system.
47
Based on Eq. (3.16), the material excitation can be viewed as a damped, forced
t ) (ε / 2)(∂α / ∂qv )0 E2 (t ) .
oscillation with the time dependent force given by F (=
3.2.4 Basic differential equations
Before presenting a more sophisticated semiclassical approach, based on density
matrix and Bloch equations, to the derivation of the system of differential equations
governing the evolution of fields and material coherence in SRS, we consider a
classical approach. That is done under the assumption of no population inversion,
which allows one to decouple the equations for material excitation and population
transfer between ground and excited state (see section 3.3.1). This assumption is
generally true, as in most cases the laser and Raman intensities are not high enough to
create a considerable population difference between the ground and first excited
Raman state. In other words, most of the molecules are in their ground state*.
Let us consider the simple case of forward Stokes scattering. In this case, the
electric field is given by the summation of the electric fields of the pump and Stokes
fields,
=
E
1
{E p exp[i( −k p z + ω p t )] + Es exp[i (−ks z + ωs t )] + c.c.}
2
(3.17)
and the material coherence is given by Eq. (1.10). Here we consider the resonant case,
Ω , and the wave vectors obey the relation, k p − k s =
k 0 . The basic
where ω p − ωs =
differential equation for the slowly-varying amplitude of the material coherence q ,
can be easily obtained by inserting Eq. (3.10) into Eq. (3.16). Using the slowlyvarying amplitude approximation, we arrive at the following equation, that describes
the evolution of the amplitude of material coherence,
*
Here we consider the medium to be in thermal equilibrium. Assuming that the Raman shift is much
larger than the thermal energy (Ω << kBT ) , the population difference between the ground and excited
state reaches the value one, i.e. nGround − nExcited ≈ 1 . Here k B is the Boltzmann factor and T is the
temperature of the medium.
48
i ε  ∂α 
∂q 1
*
+ q =
−

 E p Es
4m Ω  ∂qv 0
∂t T2
(3.18)
In deriving Eq. (3.18), besides making the slowly varying wave approximation for the
material excitation, we have assumed that the Raman transition bandwidth is much
narrower than the Raman frequency shift, i.e. δωR / Ω << 1 . The E p Es* term in (3.18)
represents the beat note of the pump and Stokes fields that drives the material
Ω.
excitation at the resonance frequency ω p − ωs =
The equations for the electric field evolution are obtained by inserting Eq. (3.17)
into the nonlinear wave equation (3.4). To do so we need an explicit form for the
nonlinear polarization induced in the medium via the SRS process. That is simply
given by the following relation, which makes use of the Taylor expansion of the
molecular polarizability, Eq. (3.7),
 ∂α 
 ∂α 
1
P NL N=
Nε 
=
ε
 qv E
 {q exp[i ( −k0 z + Ωt )] + c.c.}
4
 ∂qv 0
 ∂qv 0
× {E p exp[i (−k p z + ω pt )] + Es exp[i(−ks z + ωs t )] + c.c.}
(3.19a)
The nonlinear polarization in (3.19) contains several different frequency components
oscillating at different frequencies. We can rewrite (3.19a) in the form of Eq. (3.3b)
P NL
1
Nε
2
 ∂α  1 *
1
i ( − k z +ω t )
i ( − k s z + ωs t )
+ q Es e p p +c.c.}

 { q E p e
2
 ∂qv 0 2
(3.19b)
The component that oscillates at frequency ωs is the Stokes nonlinear polarization
and its amplitude is given by,
PsNL =
 ∂α  *
1
− iks z
Nε 
 q E p e
2
 ∂qav 0
(3.20)
The component that oscillates at frequency ω p is the pump nonlinear polarization and
its amplitude is given by,
49
1
Nε
2
PpNL =
 ∂α 
− ik P z

 q Es e .
q
∂
 av 0
(3.21)
Inserting Eqs. (3.20) and (3.21) in the right-hand-side of nonlinear wave equation, Eq.
(3.4) we arrive at the following equations for the electric fields of the forward Stokes
and pump,
 1 ∂ ∂ 
iN ω p  ∂α 
1
+  E p ( z, t ) =
−


 q Es − σ p E p ,

4n p c  ∂qav 0
2
 v p ∂t ∂z 
(3.20)
1 ∂ ∂ 
iN ωs  ∂α  *
1
+  Es ( z , t ) =
−


 q E p − σ s Es ,
4ns c  ∂qav 0
2
 vs ∂t ∂z 
(3.21)
=
v j c=
/ n j , j p, s is the phase velocity of the pump and Stokes waves.
where
In some cases, the material excitation is quickly damped over the time scale of the
pump pulse duration. This may happen for instance in high pressure gases due to
intermolecular collisions with a typical value of around T2 ~ 10−12 s . In such case if the
pump is several nanoseconds long then the interaction would be in the steady-state
regime. In this regime, the damping of molecular excitation happens over time scales
much shorter than the pump pulse duration and we can neglect the term ∂q / ∂t in Eq.
2
=
cn jε 0 / 2) E j , j p, s we arrive at
(3.18). By introducing the laser
intensities I j (=
the following rate equations for the SRS in the steady-state interaction regime,
∂I p
∂z
+
1 ∂I p
=
−g p I p Is − σ p I p ,
v p ∂t
∂I s 1 ∂I s
+
=
∂z vs ∂t
(3.22)
gs I p Is − σ s Is ,
where g j , j = p, s is the gain factor for pump and Stokes and defined as
g j [m/W] = N ω jT2κ12 /(4ε 0 c 2 n 2j ) .
50
3.3 Semiclassical theory of stimulated Raman scattering*
One of the main limitations of the classical approach is that it is incapable of fully
treating the coherent interaction between field and molecules. This is of particular
importance if the laser pulse duration becomes comparable to, or shorter than, the
Raman dephasing time T2 . In this case the width of the Raman transition is inversely
proportional to T2 and plays an important role in the precise description of the lasermatter interaction. The semiclassical theory to be presented here gives a detailed
picture of the physical situation. In this approach we treat the molecules quantum
mechanically using the powerful technique of a density matrix. We will derive the
equation of motion for the density matrix operator and use it for the derivation of the
propagation equations for the laser and Stokes field as well as the material excitation.
The equations obtained differ slightly from the previous equations. In the absence of
population difference, however, they converge to the same classical form. In addition,
using the density matrix approach, the time evolution of the coherence in the medium
is properly taken into account.
3.3.1 Density matrix approach
In quantum mechanical terms wave function describing the molecular state is
modified each time a collision between molecules of a gas happens, the. If the
collisions are elastic the collision leads to an overall phase shift in the molecular wave
function. Even if ideally the initial molecular wave function is known to a high
precision, it would be computationally infeasible to keep track of the phase of each
molecule in the gas. In this case the precise state of the system is unknown and one
should turn to statistical description of the system in order to correctly take into
account the effect of this lack of knowledge of quantum mechanical state of the
system. The situation is quite similar to statistical description of the thermodynamic
state of a system with many degrees of freedom, where having the complete
knowledge of the phase state of the system, i.e. positions and velocities of all the
*
The formalism presented here is partially adopted from [Raymer et al., 1990].
51
molecules, is unfeasible. In this case one uses the Boltzmann velocity distribution to
describe the system in a statistical sense [Reif, 1965].
In Dirac notation, the density matrix operator is defined as
=
ρˆ
1
Nt
Nt
∑Ψ
i =1
i
(3.23)
Ψi ,
where Ψ i determines the wavefunction of the system and the summation goes over
all possible states available to the system. The summation in (3.23) is a statistical
summation. If we define the probability of the system being in state i by pi , this
summation can be written as =
ρˆ
Nt
∑p
i =1
i
Ψi Ψi
. The probability distribution p is used in
order to correctly take into account our ignorance of the exact state of the system.
In the Schrödinger picture of interaction, in the presence of an external
perturbation, the density operator of the system evolves in time according to the
following equation of motion,
d
i
ρˆ =  ρˆ , Hˆ  ,
dt

(3.24)
where the Hamiltonian is defined as the sum of the kinetic and potential energy
operators and in the dipole approximation is given by
Hˆ = Hˆ 0 − μˆ ⋅ E .
(3.25)
Here Ĥ 0 is the Hamiltonian of the system in the absence of any external perturbation
and μ̂ is the electric dipole operator. The wavefunction of the system can be
expanded as a function of the unperturbed eigenstates of the unperturbed Hamiltonian
ϕn
which
form
a
complete
orthonormal
52
set,
i.e.
ϕm | ϕn = δ mn
and

H 0 ϕn = ωn ϕn . ωn is the frequency associated to the molecular state (level) n.
This expansion can be formally written as
Ψi =
∑ ani ϕn ,
(3.26)
n
where ani
2
gives the quantum mechanical probability of finding the system in the
state ϕn . Inserting Eq. (3.25) into (3.24) and assuming the electric dipole operator to
lie in the same direction as the linearly polarized electric field we have
d
i
i
 ρˆ , Hˆ 0  − [ ρˆ , µˆ ] E .
=
ρˆ




dt
(3.27)
In order to obtain the equation of motion for the density matrix elements
ρˆ mn = ϕm | ρˆ | ϕn we multiply the left and right hand side of Eq. (3.27) by ϕm and
ϕn respectively. Rewriting Eq. (3.27) by matrix elements we have,
d
i
i
ρmn =
i ( ωn − ωm ) ρmn + ∑ µml ρln E − ∑ ρml µln E ,
 l
 l
dt
(3.28)
where the summation goes over all the levels. Now consider the scheme depicted in
Fig. (1.1) where only two levels 1 and 2 are resonantly driven and the rest of the
levels are just weakly excited. If we show all these intermediate levels by m′ and
ignore possible excitation of them, we can separate the contribution from these levels
and the resonantly driven levels and rewrite Eq. (3.28) in the following form,
d
i 2
i
ρmn =iωnm ρ mn + ∑ ( µ mi ρin − ρ mi µin ) E + ∑ ( µmm′ ρ m′n − ρmm′ µ m′n ) E
dt
 i =1
 m′
(3.29)
= ωn − ωm and the second summation is over all off-resonantly driven,
where ωnm
intermediate states, m′ .
53
If we assume that the dipole transition between eigenstates 1
and 2
is
forbidden and also that these states have a definite parity, then the off diagonal
*
µ=
0 and the diagonal elements of the electric dipole matrix would
element µ=
12
21
be identically zero, =
i.e. µmm
=
m | µˆ | m 0 . Using these relations we arrive at the
following set of equations of motion for the relevant density matrix elements,
d
i
i
ρ1m′ = iωm′1 ρ1m′ − µ1m′ ( ρ11 − ρ m′m′ ) E − µ 2 m′ ρ12 E ,
dt


(3.30a)
d
i
i
ρ m′ 2= iω2 m′ ρm′2 − µ m′ 2 ( ρ m′m′ − ρ 22 ) E + µ m′1ρ12 E ,
dt


(3.30b)
d
i
ρ12 =
iω21 ρ12 + ∑ ( µ1m′ ρ m′2 − µm′ 2 ρ1m′ ) E ,
dt
 m′
(3.30c)
d
i
ρ22 =
− ∑ ( µ m ′ 2 ρ 2 m′ − µ 2 m ′ ρ m ′ 2 ) E .
 m′
dt
(3.30d)
In order to eliminate explicit appearance of the irrelevant off-diagonal density matrix
elements, ρim′ , i = 1, 2 from the equations (3.30), we have to formally integrate the
equations (3.30a) and (3.30b). To do so, we use the adiabatic approximation in order
to separate the time dependence of ρi m′ into a slowly and a fast varying part, i.e.
ρi m′ (t ) = σ i m′ (t ) exp(iωm′i t ) . This separation is justified because the intermediate levels
are driven by a far-off resonance electric field, so that they follow adiabatically the
temporal changes in the field. By substituting the adiabatic form of the ρi m′ in
equations (3.30a) and (3.30b) and formally integrating, we obtain,
ρ1m′ (t ) = −
t
i
iω ( t −t ′ )
∫ dt ′ e m′1
 −∞
(3.31)
× {µ1m′ [ ρ11 (t ′) − ρ m′m′ (t ′)] + µ2 m′ ρ12 (t ′)} E (t ′),
ρ m′ 2 (t ) = −
t
i
dt ′ eiω2 m′ (t −t ′)
∫
 −∞
(3.32)
× {µ m′ 2 [ ρ m′m′ (t ′) − ρ 22 (t ′)] − µm′1 ρ12 (t ′)} E (t ′).
54
The diagonal elements of the density matrix, ρ mm give the probability of state m
being populated. Since the intermediate levels are driven far-off resonance, they will
not be populated and ρ m′m′ = 0 . Defining the slowly varying amplitude Q(t ) , so that
ρ12 (t ) = Q(t ) e
i (ω p −ωs ) t − i ( k p − ks ) z
and using the Eq. (3.17) for the total electric field, we
can formally perform the integrals in Eqs. (3.31) and (3.32) and eliminate ρ1m′ and
ρ m′ 2 from explicitly appearing in Eqs. (3.30c) and (3.30d)*. If exact Raman resonance
ω21 =
Ω . By inserting the expressions for ρ1m′ and ρ m′ 2
is assumed then ω p − ωs =
into Eq. (3.30c), we arrive at the following differential equation governing the time
evolution of ρ12 ,
d
ρ12 (t )= i ( Ω + δ ) ρ12 (t ) + (iκ1* / 4) E p ( z, t ) Es* ( z, t )
dt
×e
iΩt −i ( k p − k s ) z
(3.33)
[ ρ 22 (t ) − ρ11 (t )],
where κ1 is given by
=
κ1
1
2


1
1
+
,
 ωm′1 − ω p ωm′1 + ωs 
∑ µ2m′ µm′1 
m′
(3.34)
and δ= δ1 − δ 2 is the Stark shift in the levels’ frequencies, and δ i , i = 1, 2 is given by
δi
1
2
∑µ
m′
2
m′i
 2 

1
1
+

 Ep 
 ωm′i + ω p ωm′i − ω p 

+ Es
2

 
1
1
+

 .
 ωm′i + ωs ωm′i − ωs  
(3.35)
In what follows we neglect the Stark shift, i.e. δ = 0 . In SI units the coefficient κ1
has the units of [κ1 ] =  mJ2 Cs  . In terms of dimensionless off-diagonal element of the
2
*
2
Note that the definition for the slowly varying amplitude Q is not unique and may vary in literature,
see for example [Mostowski et al., 1981; Raymer et al., 1981].
55
density matrix Q(t ) and the population inversion =
n(t ) ρ 22 (t ) − ρ11 (t ) , the equation
of motion (3.33) is
∂
1
iκ1*
Q+ Q =
E p Es* n(t ) .
4
∂t
T2
(3.36)
Repeating the same line of calculation for ρ 22 (t ) and ρ11 (t ) we arrive at the
following equation of motion for the population inversion n(t )
∂
1
1
n(t ) = iκ1 E *p Es Q − iκ1* E p Es*Q* =Im{κ1 Es* E pQ*} .
2
2
∂t
(3.37)
As mentioned earlier in most of the cases of interest, the population inversion is
negligible, i.e. ∂n / ∂t =0 and ρ 22 (t ) − ρ11 (t ) ≈ −1 . In this case Eq. (3.36) would be
decoupled from Eq. (3.37) and takes the form of its classical counterpart.
3.3.2 Material excitation revisited
As mentioned earlier, the interaction of the pump and Stokes field create a coherence
wave Q in the Raman active medium. By definition, the field Q is proportional to the
off-diagonal elements of the density matrix, ρ12 and it corresponds to the spatial
correlation of the molecular excitations. This field is attenuated on the time scale T2 ,
which enters Eq. (3.36) as a phenomenological damping term, so that on this time
scale the collisions have already destroyed the mutual spatial correlation of the
molecular excitation, although the molecules may still be excited.
In order to make an exact connection between Q(t ) , which is the off-diagonal
density matrix element and the amplitude of the material excitation wave, qv (t ) , we
evaluate the expectation value of the material excitation amplitude operator using the
density matrix approach. From the definition of ρ̂ , given in Eq.
56
(3.23), it is
straightforward to show that the expectation value of any observable quantity can be
evaluated via the relation
Aˆ Tr
=
=
( ρˆ Aˆ )
∑∑
k
k | ρˆ | k ′ k ′ | Aˆ | k ,
(3.38)
k′
where  is the Hermitian operator associated to A . Using Eq. (3.38) we have
qv
=
qˆ Tr ( ρˆ=
qˆ )
=
∑∑
k
k | ρˆ | k ′ k ′ | qˆ | k .
(3.39)
k′
In order to calculate the sum in Eq. (3.29) we note that in our case the population is
residing primarily in the ground state 1 with vibrational quantum number υ = 0 and
the excited state 2 with vibrational quantum number υ = 1 , so that the summation in
Eq. (3.39) simplifies to
qˆ =
1| ρˆ | 2 2 | qˆ |1 + 2 | ρˆ |1 1| qˆ | 2 =
q21 ρ12 + q12 ρ 21 .
(3.40)
In Dirac’s treatment of a quantum mechanical harmonic oscillator one expresses the
position operator q̂ in terms of the non-Hermitian annihilation and creation operators,
a and a † as
1
  2
†
=
qˆ 
 (a + a ) .
 2mΩ 
(3.41)
The operators a and a † have the following properties
a=
n,υ
υ n − 1,υ ,
(3.42a)
a † n,υ = υ + 1 n + 1,υ ,
(3.42b)
where n is the electronic quantum number. Using Eqs. (3.41) and (3.42) we have
57
1
q21=
  2
2,υ= 1| qˆ |1,υ= 0= 
 ,
 2mΩ 
(3.43)
Thus Eq. (3.40) becomes
1
=
qv
  2
=
qˆ 
 ( ρ12 + ρ 21 ) ,
 2mΩ 
(3.44)
or based on the definition of Q(t ) ,
1
qv = {(2 / mΩ)1/ 2 QeiΩt −ik0 z + c.c.} .
2
(3.45)
By comparing equations (3.45) and (3.10) we can easily relate Q(t ) to the slowly
q (t ) (2 / mΩ)1/ 2 Q(t ) . Using this
varying amplitude of the material excitation =
relation we can rewrite Eq. (3.36) for q as
1
∂
1
i  2  2 *
*
q + q =
− 
 κ1 E p Es
∂t
T2
4  mΩ 
(3.46)
n ρ 22 − ρ11 ≅ −1 . Now by comparing Eq. (3.46)
In writing (3.46) we have assumed =
with its classical counterpart, Eq. (3.18) we see that the coefficient κ1 is related to the
molecular polarizability via
1
 1  2  ∂α 
κ =ε
 .
 
 2mΩ   ∂qv 0
*
1
(3.47)
58
3.4 Summary
In order to summarize this chapter we now list the fundamental equations
describing stimulated Raman scattering using the two-photon matrix elements κ1
Coherence:
∂
1
iκ *
Q + Q = i ∆Ω Q + 1 E p Es* n(t ) ,
4
∂t
T2
(3.48a)
Pump field:
∂ 1 ∂
 ω pvp 
1
−i 
 +
 E p =
 κ 2 Q Es − σ p E p ,
2
 ωs vs 
 ∂z v p ∂t 
(3.48b)
Stokes field:
 ∂ 1 ∂
1
−iκ 2 Q* E p − σ s Es ,
± +
 Es =
2
 ∂z vs ∂t 
(3.48c)
Population difference:
∂
n(t ) − n
=
n(t ) +
Im{κ1* E p Es*Q*} .
∂t
T1
(3.48d)
where κ 2 = N ωs vsκ1* /(2ε 0 c 2 ) and ± refers to forward (plus) and backward (minus)
SRS scattering . In Eqs. (3.48) ∆Ω = Ω − (ω p − ωs ) is the frequency shift from exact
Raman resonance, Ω , and n is the probability of finding molecules in the ground
state under thermal equilibrium condition, so that N = N n is the number density of
molecules in the ground state in thermal equilibrium. T1 is the time scale for deexcitation of the molecular excitation.
In the forthcoming chapters we will apply these equations to the various regimes
of stimulated Raman scattering and show how they provide a detailed explanation of
the coupled system of laser field (pump and the scattered Stokes) and material
excitation (coherence field).
59
60
4
Optimizing anti-Stokes Raman scattering in gas-filled
hollow-core photonic crystal fibers*
Stimulated Raman scattering (SRS) is a very efficient tool for generating high-power
laser radiation at multiple wavelengths in an extended spectral region, ranging from
the vacuum ultraviolet (VUV) to the far infrared (FIR) [Loree et al., 1979; Aniolek et
al., 1997; Fischer et al., 1997; Sentrayan et al., 1992 and 1996]. In particular, the
stimulated version of anti-Stokes Raman scattering (ASRS) process is an important
method for frequency up-conversion to the ultraviolet and vacuum ultraviolet regions.
It also constitutes the basis of a powerful spectroscopic technique - coherent antiStokes Raman scattering (CARS) [Eesley, 1981]. ASRS generates an optical field at
frequency ωa = 2ω p − ωs > ω p , s where ω p and ωs are the pump and the Stokes field
frequencies whose difference is resonant to a Raman transition frequency,
ω p − ωs =
Ω , see Fig. (4.1). Because of the parametric character of this process, the
key factor responsible for efficient generation of anti-Stokes is the wave mismatch.
Indeed, the dynamics of ASRS is known to depend strongly on the wavevector
mismatch of the pump, Stokes and anti-Stokes fields: ∆k= 2k p − k s − k a . In media
0 can only be fulfilled for non-collinear
with dispersion, the condition ∆k =
interactions. This can be achieved, for example, by focusing the laser beam in the
*
Published in Nazarkin, A., Abdolvand A. and Russell, P. St.J., 2009, Phys. Rev. A 79, 031805(R).
61
Figure 4.1: Schematic energy diagram of stimulated anti-Stokes Raman scattering.
medium. As a result, in focused beams, ASRS is typically generated off axis, with the
emission angles of the Stokes and anti-Stokes signals lying on a cone close to the
phase matching angle [Ho et al., 2000]. In HC-PCF, where the interaction is collinear,
the three waves have no freedom to choose the preferred direction, and one might
expect much lower interaction efficiency. Surprisingly, experiments have shown that
conversion can be high (about 3%) even in the presence of significant phase mismatch
[Benabid et al., 2002].
In what follows we show that the features observed in the experiments are caused
by the establishment of phase locking between the interacting fields, independently of
the optical path, which leads to higher efficiencies. Moreover we show that, due to the
waveguide dispersion of HC-PCF, the ASRS process can be phase matched and that
by properly adjusting the gas pressure along the fiber, the efficiency can be brought to
its theoretical maximum. These results suggest that gas-filled HC-PCF might be used
as an efficient nonlinear frequency shifter to the UV frequency region. Note that other
collinear SRS techniques, different from the resonant SRS in gas-filled HC-PCFs
discussed here (e.g., SRS with adiabatic molecule preparation [Harris et al., 1998;
Sokolov et al., 2001] and in an impulsively excited medium [Nazarkin et al., 1999 and
2002]) have recently been reported. We also note that phase locking was first
proposed in [Butylkin et al., 1976] to explain the features of ASRS in focused beams.
62
4.1 Phase locking
To analyze ASRS in a gas-filled HC-PCF, we use the semi-classical model developed
in chapter 3 where the dynamics of a Raman transition interacting with fields at ω i ,
i = p, s, a are described by a density matrix equation, and the propagation of the fields
=
Ei ( z , t ) (1/ 2) Ei exp[i(ωi t − ki z )] + c.c. , are modeled by the wave equation. Here E
denotes the fast varying field in time. For laser pulse durations long compared to the
phase relaxation time T2 of the medium, and intensities much lower than the
saturation intensity (i.e., for ρ 22 << 1 , ρ11 ≈ 1 where ρ mn are the density matrix
elements of the transition), the resonant response of the system obeys the following
differential equation (see Appendix B)
dQ 1
+ Q=
−i (κ s Ap As* + κ a Aa A*p ei ∆k z )
dt T2
(4.1)
Here ∆k= Δk ⋅ zˆ is the phase mismatch in the direction of propagation, and κ s and κ a
are the two-photon matrix elements associated with the Stokes and anti-Stokes
processes and are determined by
1
µ µ [(ωm′1 − ω p ) −1 + (ωm′1 + ωs ) −1 ],
2 ∑ 1m ′ m ′ 2
4 m ′
1
=
κa
∑ µ1m′ µm′2 [(ωm′1 − ωa )−1 + (ωm′1 + ω p )−1 ].
4 2 m ′
=
κs
Equation (4.1) should be compared with Eq. (3.48a). The only difference here is the
contribution from anti-Stokes wave to the nonlinear polarization of the medium (the
second term in the bracket). In the steady state when dQ / dt ≈ 0 , e.g. at high gas
pressures, the nonlinear polarization can be written as
Q=
−iT2 (κ s Ap As* + κ a Aa A*p ei ∆k z )
(4.2)
63
The presence of anti-Stokes radiation would also modify the equation for the pump
field evolution, appearing as an additional nonlinear source term on the right hand
side of Eq. (3.48b),
∂ 1 ∂
N
ω p v p (κ s QAs + κ aQ* Aa ei ∆k z )
−i
 +
 Ap =
2
∂
∂
2
ε
z
v
t
c
p
0


(4.3)
The evolution of the anti-Stokes field is governed by
∂ 1 ∂
N
−i
ωa vaκ a QAp e− i ∆k z
 +
 Aa =
2
z
v
t
2
c
∂
∂
ε
a


0
(4.4)
iϕ
Using Eq. (4.2) and introducing amplitudes and phases for the fields, E j = a j e j , and
v=
va , we arrive at the following set of
assuming equal group velocities v=
p
s
equations for the steady-state:
da p
1
=
ω p a p (q 2 aa2 − as2 ) − σ p a p
dz
2
(4.5a)
das
1
=
ωs a 2p (as + qaa cos θ ) − σ s as
dz
2
(4.5b)
daa
1
=
−ωa qa 2p (qaa + as cos θ ) − σ a aa
dz
2
(4.5c)
a
a
dθ
= qa 2p (ωa s − ωs a ) sin θ + ∆k
dz
aa
as
(4.5d)
In Eqs. (4.5) the field frequencies are normalized to the Raman transition frequency,
ω j → ω j / Ω , the field amplitudes are a j → a j / a0 , where a0 is the normalization
L0 [ N ΩT2 v p a02κ s2 /(2ε 0c 2 )]−1 , where N
amplitude and the distance z is in units of =
is the molecular concentration. The coefficients σ j describe linear loss and the
parameter q = κ a / κ s . The function θ= ϕ s + ϕa − 2ϕ p + ∆k z is the phase difference of
the fields in which ∆k is normalized to ∆k L0 . Here, we show the derivation of Eq.
(4.5c) for the anti-Stokes component. For simplicity, we neglect attenuation, setting
64
σ j = 0 . Derivation of the rest of equations is similar. In steady state, when the time
variation of the wave envelopes is slow compared to the dephasing time T2 , we put
∂Ea
≈ 0 and Eq. (4.4) simplifies to
∂t
∂ϕa  iϕa
N vaT2κ s2  κ a  2
 ∂aa
+
=
−
i
a
e
ωa   a p
a

∂z
2ε 0 c 2
 ∂z

 κs 
(
× as e
i (2ϕ p −ϕs −∆k z )
+ (κ a / κ s ) aa e
iϕ a
(4.6)
).
in our original unnormalized variables. Multiplying both sides of Eq. (4.6) by e− iϕa ,
applying our normalization, and equating the real and imaginary parts, we arrive at
following equations for the anti-Stokes field phase and amplitude,
The
daa
=
−ωa qa 2p (qaa + as cos θ ) ,
dz
(4.7a)
dϕ a
a
= qa 2pωa s sin θ .
dz
aa
(4.7b)
dispersion
of
the
propagation
constant
of
gas-filled
HCPCF,
k (ω j ) = β (ω j ) + (ω j / c)∆n(ω j ) , contains the contributions of the HC-PCF [ β (ω j )]
and the Raman gas [∆n(ω j )] . It is assumed that the Stokes field develops from
= as 0 << a p 0 , while the anti-Stokes signal is generated
quantum noise, i.e., as (0)
parametrically starting from zero intensity, aa (0) = 0 [Ottusch et al., 1991].
Before discussing the results of a numerical study of Eqs. (4.5), some important
points can be clarified analytically. Let us consider the initial stage of ASRS, when
the Stokes and the anti-Stokes fields are still small compared to the pump field
(aa , as << a p ) . We assume that the pump is not depleted, i.e. a p ≈ a p 0 . At the
beginning of parametric ASRS aa << as , and the term in brackets in Eq. (4.5d)
associated with a resonant contribution to the phase difference is positive and large.
Under these conditions, the phase difference θ ( z ) evolves toward a stable value
65
θ =
π + sin −1[∆k aa /( qωa as a 2p )] , which is independent of the initial phase θ0 and lies
within the range π / 2 + 2π m < θ < 3π / 2 + 2π m [Butylkin et al., 1976]. This value
remains constant during the interaction because the linear mismatch ∆k on the r.h.s of
Eq. (4.5d) is exactly cancelled by the nonlinear one. Assuming in Eq. (4.5d) that
dθ / dz = 0 , we find from Eqs. (4.5b) and (4.5c) that the amplitudes as and aa
increase with z exponentially at the rate
g = g 0 {−(η − 1) / 2 + (η − 1) 2 / 4 + η sin 2 θ }
(4.8)
where g 0 = ωs a 2p 0 is the Stokes gain (the gain in the absence of the anti-Stokes field)
and η = (ωa / ωs ) q 2 . To derive Eq. (4.8) let us define ξ = aa / as . Inserting this in Eq.
(4.5d) and assuming dθ / dz = 0 we have
ξ=
δ
2q sin θ
+
δ2
4q sin θ
2
2
+
ωa
ωs
(4.9)
where δ = ∆k / g0 . From Eq. (4.9) it can be seen that ξ is independent of propagation
distance along the fiber, z . From Eq. (4.5c) we have
daa
da
ω
=
ξ s =
− a g 0 qas (qξ + cos θ )
ωs
dz
dz
(4.10)
Using Eq. (4.5b) we have
ω
ωs
g 0 asξ (1 + qξ cos θ ) =
− a g 0 as q (qξ + cos θ )
(4.11)
From Eq. (4.11) we obtain a quadratic equation which gives ξ in terms of
parameters η , q and θ
66
1
ξ=
q cos θ
  1 + η 
(1 − η ) 2
+ η sin 2 θ
− 
±
4
  2 



(4.12)
The gain for Stokes g defined through Eq. (4.5b) is given by
g
1 −η
(1 − η ) 2
=
1 + q cos θξ =
+
+ η sin 2 θ
g0
2
4
(4.13)
where plus (+) sign is chosen corresponding to gain for Stokes. The value of the
locked phase θ in Eq. (4.13) is found from
2
=
sin θ
2
 (η − 1)2 + δ 2  δ 2  (η − 1)2 + δ 2 

 + 4η − 
,
8η
8η




(4.14)
It is easy to see that the dynamics of the locked fields are described by the
expressions,
as ( z ) = as 0 e g z ,
aa ( z ) = −
η cos θ
q( g / g0 + η )
(4.15a)
as 0 e g z ,
(4.15b)
showing that a small Stokes signal as 0 gives rise to the generation of a coupled
Stokes-anti-Stokes wave. The most interesting consequence of solutions (4.13)-(4.15)
is that the coupled wave exhibits exponential growth even in the presence of a
nonzero wave mismatch ∆k . When ∆k is large ( δ >> 1 ), it follows from Eqs. (4.5d)
and (4.14) that θ → π / 2 for ∆k < 0 and θ → 3π / 2 for ∆k > 0 , and the gain takes
its maximum value g = g 0 . However, according to Eq. (4.15b), the amplitude of the
anti-Stokes field in the coupled wave is much smaller than that of the Stokes field,
and thus Stokes generation dominates. In the opposite case of small ∆k (or δ << 1 ),
the phase difference θ → π , and growth of the Stokes and anti-Stokes fields is very
slow due to parametric gain suppression ( g → 0 ) [Shen et al., 1965;
67
Figure 4.2: The initial stage of a phase-locked resonant ASRS. The amplification of the antiStokes field intensity
difference
θ
I a = aa2 [the lines labeled with (a)-(c)] and the corresponding locked phase
as a function of the normalized wavevecotr mismatch
∆k L0 for increasing values
of pump intensity: (a) a 2p / a02 = 0.8 , (b) a 2p / a02 = 1.6 and (c) a 2p / a02 = 2.7 . The corresponding gain
factors are: (a) g 0 z = 5 , (b) g 0 z = 10 and (c) g0 z = 18 .
Bloembergen, 1967]. This is caused by destructive interference of the contributions of
0 and
the Stokes and anti-Stokes SRS to polarization, Eq. (4.2) for ∆k =
as ( z ) ≈ qaa ( z ) . As a result, the maximum growth rate of the anti-Stokes field occurs
for intermediate values of ∆k , see Fig. (4.2). A further increase in the pump intensity
leads to a shifting and broadening of this maximum.
4.2 Numerical simulation
These considerations are completely supported by the results of a computer study of
Eqs. (4.5), presented in Figs. (4.3a) and (4.3b). In our calculations we used parameters
from the SRS experiments [Benabid et al., 2002], where a 6 ns long laser pulse at
λ p = 532 nm was used to generate first Stokes and anti-Stokes fields at λs = 683 nm
and λa = 435 nm in a HC-PCF filled with hydrogen. The pump pulse intensity was
approximately 300 MW/cm 2 , resulting in a Stokes gain g 0 ≈ 0.5 cm -1 . Figure (4.3a)
shows the dynamics of the pump, Stokes, and anti-Stokes intensity (in inset) as a
68
Figure 4.3: Vibrational ASRS in a Kagome-type HC-PCF filled with hydrogen at 17 atm. (a)
Intensities of the pump, Stokes and anti-Stokes field (in inset) vs. fiber length (the circles,
triangles, and diamonds show experimental data from [Benabid et al., 2002]; (b) evolution of the
phase difference (solid line). For comparison, the anti-Stokes field (inset) is also shown, calculated
neglecting the nonlinear term in Eq. (4.5d) that gives rise to the phase-locking effect.
function of fiber length z . As can be seen, initially the Stokes and anti-Stokes fields
grow nearly exponentially with z . Efficient ASRS generation appears to be possible
because the phase difference θ , Fig. (4.3b), stays constant over a long interaction
length (about 40 cm). The phase locking breaks down only when the pump field
becomes exhausted due to conversion to the Stokes and anti-Stokes waves. Here the
phase θ differs from its optimum value θ to such extent that the anti-Stokes field
generation and its conversion back to the pump field balance each other, leading to
saturation in the ASRS.
4.3 Optimization scheme for efficient anti-Stokes generation
In what follows we show that the properties of PCFs make the optimization of ASRS
with conversion efficiencies close to the theoretical maximum possible. For a Raman
0 can be approximated as Ω 2∂ 2 k / ∂ω 2 =0 . To see
shift Ω << ω j , the condition ∆k =
this, we expand k (ω ) as a Taylor expansion around ω p ,
k (ω p +
=
Ω) k (ω p ) + Ω
∂k Ω 2 ∂ 2 k
+
+ O (Ω 2 ) ,
2
∂ω 2 ∂ω
(4.16a)
k (ω p −
=
Ω) k (ω p ) − Ω
∂k Ω 2 ∂ 2 k
+
+ O (Ω 2 ) .
2
∂ω 2 ∂ω
(4.16b)
69
Using the definition of the phase mismatch=
∆k 2k (ω p ) − k (ωs ) − k (ωa ) and Eqs.
0 implies operation at zero
(4.16) we obtain ∆k =Ω 2 ∂ 2 k / ∂ω 2 . The relation ∆k =
group velocity dispersion – a regime easily achievable in HC-PCF [Russell, 2006;
Zheltikov, 2006]. In addition, for a collision-broadened Raman line, the density
matrix element ρ12 does not depend on the molecular concentration [Bloembergen,
1967; Shen et al., 1965]. Hereby, that by varying the gas pressure along the fiber, one
0 is a
can setup the wave mismatch ∆k ( z ) in an optimum way. Although ∆k =
necessary condition for efficient ASRS, it does not automatically lead to higher
efficiency [Duncan et al., 1986]. When ∆k → 0 , the exponential gain goes to zero
because a coupled wave with as ( z ) ≈ qaa ( z ) is generated. Once such a wave is
formed, the Raman process terminates. As follows from Eq. (4.5), this structure
Las [(ωa − ωs )a 2p 0 ]−1 which can be treated as the conversion
develops at a length z ~ =
length for phase-matched ASRS. When ASRS starts from a weak Stokes signal
(quantum noise), the anti-Stokes field produced at the length Las is weak too, and the
pump field remains almost undepleted. However, if the Stokes-to-pump ratio at the
input exceeds some critical value, full conversion of the pump to Stokes and antiStokes signals becomes possible, as was shown in [Ottusch et al., 1991]. This result
suggests that the optimum configuration for efficient ASRS should consist of two
interaction regions.
In the first region (where a Stokes signal is generated), the gas pressure must be
high enough to provide a sufficient level of spontaneous noise at the Stokes frequency
and, on the other hand, a large mismatch ( ∆k >> g 0 ) to maximize the Stokes and
minimize the anti-Stokes conversion, Fig. (4.4a). The length of this region is chosen
so that it provides the optimum relation between the input pump and Stokes field in
0 is set by
the second region where phase-matched ASRS occurs. The condition ∆k =
adjusting the gas pressure. By solving Eqs. (4.5a) and (4.5b) in the first region (in
which the anti-Stokes process can be neglected), we find the field amplitudes (using
physical units) as follows:
βz
1+ K
2
2
2 (1 + K ) e
=
a 2p ( z ) a=
,
a
(
z
)
a
,
p0
s
s0
1 + Ke β z
1 + Ke β z
70
(4.17)
Figure 4.4: An optimized scheme for ASRS: (a) Gas pressure varies along a PCF to provide the
optimum
∆k ( z ) . The inset is a scanning electron micrograph of a typical Kagome PCF (core
diameter 26 μm). (b)-(d) The results of optimization of ASRS in a kagomé type HC-PCF filled
with hydrogen. The experimental input pump field intensity is shown [Benabid et al., 2004]. The
dotted, dashed, and solid line shows the results for different lengths (L1=18, 23.5, and 21 cm) of
the high pressure region. The solid line corresponds to the optimum regime of ASRS with
conversion efficiency of 27%.
2 (as20 / a 2p 0 + ωs / ω p ) / L0 . The length L1 of the
where K = (ω p / ωs ) as20 / a 2p=
0 and β
first interaction region is determined from the inequality,
−1
2
s0
2
p0
(a / a ) e
2ωs L1 /(ω p L0 )


4ωa2
≥
− 1 .
2
2
 (ωa q − ωs )

(4.18)
Maximum conversion to the anti-Stokes field is achieved at a distance L1 = L1opt that
satisfies Eq. (4.18). The optimum anti-Stokes field is then given by
−1
a 2p ( L1opt )  (ωa q 2 − ωs ) 2 
a (∞ )
=
1 −
 ,
2q 2 
4ωa2

2
a
71
(4.19)
where a 2p ( L1opt ) is pump intensity Eq. (4.17) corresponding to the optimum length
L1opt . Solution (4.19) gives an asymptotic value of the anti-Stokes field for a semi-
infinite interaction region L → ∞ . In fact for phase-matched ASRS the length L2
should be larger than the above introduced conversion length
Las , i.e.,
L2 > Las =[4πΩN κ s2 a 2p ( L1opt ) /(c)]−1 . The results of optimization of ASRS using the
proposed scheme are presented in Fig. (4.4). As can be seen, there exists a range of
lengths L1 for region one (with ∆L < 3 cm ) which provides efficient conversion in
region two. The optimum efficiency (27%) is one order of magnitude higher than that
reported in experiments [Benabid et al., 2002] and corresponds to a loss of about 3
dB/m. Reduction in the loss below 1 dB/m would lead to an efficiency close to the
theoretical maximum (50% quantum efficiency) predicted by Eq. (4.19).
4.4 Summary and conclusion
In summary, we have discussed the physics of ASRS in gas-filled HC-PCF and
clarified the important role of phase-locking in SRS experiments. We have further
proposed that the ASRS process can be optimized by adjusting the gas pressure along
the fiber. The results suggest that gas-filled HC-PCFs offer a broad spectrum of
possibilities for controlling nonlinear interactions based on stimulated Raman
scattering in the gas phase and may be promising as compact and efficient frequency
shifters to the UV and VUV spectral regions [Couny et al., 2007].
As mentioned in the introduction, another possibility offered by HC-PCF is to
phase match different nonlinear processes such as SRS and third-harmonic generation
via dispersion properties of the HC-PCF [Nold et al., 2010; Ziemienczuk et al., 2011].
In general, the dispersion of a mode in HC-PCF has contributions from waveguide as
well as material dispersion. The waveguide dispersion depends on the specific design
of the HC-PCF (core and cladding design). Material dispersion in gases can be tuned
by adjusting the gas pressure. By accurately adjusting these parameters one should be
able to find optimum condition for phase matching between different nonlinear
waves.
72
5
Solitary pulse generation by backward stimulated
Raman scattering in hydrogen-filled HC-PCF
Stimulated Raman scattering can occur in several geometries, with forward scattering
being the most common. In this configuration, both pump and Stokes signal copropagate along the same direction in the Raman medium. The process can be seeded
or it can start from quantum noise fluctuations, see Fig. (5.1a). However, Raman
amplification can also be achieved with the arrangement shown in Fig. (5.1b). In this
geometry a Stokes wave, which is already generated in a separate Raman generator, is
supplied along with the pump pulse in backward direction*. When SRS occurs in a
counter-propagating geometry (backward stimulated Raman scattering or BSRS), the
Raman process exhibits particularly interesting spatiotemporal dynamics. Indeed the
backward scheme of SRS amplification is fundamentally different from the forward
case. The intensity of the forward Stokes pulse can never exceed the intensity of the
initial pump since Stokes and pump pulse travel with approximately the same
velocity. As a result, the Stokes pulse only has access to the pump energy stored in a
volume traveling with the pulse. In contrast, the backward traveling Stokes wave
always sees fresh, undepleted pump photons and can extract energy stored throughout
the whole amplifying region, see Fig. (5.2) [Maier et al., 1966 and 1969]. As a result,
*
Note that the backward SRS can also be generated starting from noise. However its threshold is much
higher than the forward SRS [Maier et al., 1966 and 1969].
73
Figure 5.1: Schematic diagram of (a) forward and (b) backward Raman interaction geometries.
Note that both forward and backward SRS can start from quantum noise or be seeded via an
external source at the Stokes frequency.
the leading edge of the Stokes pulse is reshaped; the Stokes pulse becomes shorter and
is amplified to intensities much higher than the incoming pump intensity, a
mechanism that has found important applications in high energy, short pulse laser
physics [Murray et al., 1979].
Experimental studies of BSRS performed in free space generally face two major
problems: limited length of the interaction zone and generation of higher order SRS
components. The latter happens because of the backward signal reaching such a high
level of intensity that it produces its own forward signal. In this chapter, I show how
the unique characteristics of gas-filled hollow-core photonic crystal fiber enable us to
overcome these difficulties and make a detailed study of BSRS.
5.1 Motivation for the experiment
Assuming Stokes pulse amplification by an infinitely long counter-propagating pump
wave one can find, from the exact equations of BSRS [Maier et al., 1969], that in the
presence of linear loss the asymptotic solution is a steady-state pulse:
 t − z / vs 
I s ( z , t ) ∝ sech 2 
,
 τs 
(5.1)
74
Figure 5.2: Output temporal traces of the pump in the absence (solid black line) and presence
(red shaded region) of the backward Stokes seed signal. Note that a large fraction of the pump
intensity is transferred to the backward Stokes.
moving with the velocity of light vs and having a duration τ s = T2γ s / Gs , where γ s is
the linear loss coefficient and Gs is the steady-state Raman gain. According to Eq.
(5.1), for a sufficiently high Raman gain (or low loss), so that the condition
γ s / Gs << 1 is fulfilled, the Stokes pulse duration can be much shorter than T2 . This
result suggests that, intrinsically, the mechanism of pulse shortening by BSRS is not
limited by the buildup time of the molecular response of the Raman medium.
However, in early BSRS experiments it was observed that the generated Stokes pulses
could have durations on the order of T2 , the phase relaxation time of the molecular
vibrations (or rotations) [Maier et al., 1969]. This result raises the interesting question
of whether nonlinear pulse shortening is possible in the highly transient regime
[Carman et al., 1970;Duncan et al., 1988], where the pulse duration is much shorter
than T2 . So far, a detailed experimental study of transient effects in SRS has been
difficult. In focused beam geometry, the interaction length needed to observe the late
stage evolution of BSRS signal is limited by beam diffraction, and to observe SRS at
a sub- T2 time scale one needs to pump at multi-gigawatt powers. This then leads to
beam self-focusing, self-phase modulation and the generation of additional SRS
components [Duncan et al., 1988; Koprinkov et al., 2000; Ye et al., 2003].
Here I make use of the unique characteristics of gas-filled hollow-core photonic
crystal fiber (HC-PCF) in a detailed study of BSRS. By eliminating beam diffraction,
75
0Figure 5.3: Experimentally measured loss of the fiber used in the experiment (the dark blue
solid line). Arrows show the wavelengths of the pump laser (1064 nm), first (1134 nm) and second
(1215 nm) rotational Stokes in hydrogen. The narrow transmission band of the fiber allows only
the pump and first Stokes fields to propagate in the gas-filled core.
these novel optical guiding systems offer interaction lengths many times longer than
the Rayleigh length of a focused laser beam, while keeping the laser beam tightly
confined in a single mode. As a result, the threshold power for SRS can be
dramatically reduced [Benabid et al., 2002] below the threshold for deleterious
competing nonlinear effects. Moreover, using HC-PCF with a specially engineered
guidance band, the Raman process can be isolated from competing SRS processes.
That may be done by properly tuning the position and bandwidth of the transmission
window in a HC-PCF. Figure (5.3) shows the transmission window of the fiber used
in our experiment. The arrows in the figure indicate the wavelength of the pump and
the first and second rotational Stokes in hydrogen. As can be seen from the figure the
second rotational Stokes lies well outside the transmission window of the fiber and is
not guided by the HC-PCF. As a result, the Raman signal at this wavelength will
never reach the stimulated regime. This result is in complete contrast to free space
propagation in which the Stokes component generates a higher order Stokes signal, as
soon as it reaches to the required intensity level. So our system would act as an ideal
model for the interaction of two frequencies via a Raman medium. By means of this
approach, we are able to gain deeper insight into the different stages of Stokes
amplification by BSRS. As a short overview, I demonstrate pulse amplification and
shortening below T2 . Well before the pulse reaches its asymptotic shape Eq. (5.1), the
amplification saturates due to formation of a reshaped pulse envelope propagating at a
superluminal velocity. This reshaping occurs as a result of the combined action of
76
Figure 5.4: Schematic of the experimental setup for the study of pulse amplification by backward
rotational SRS in a hydrogen filled HC-PCF. The setup consists of two stages: seed generation
and preliminary amplification (red dashed box) and backward amplification of the seed (black
dashed box).
nonlinear amplification at the pulse leading edge and nonlinear absorption at its
trailing edge - an effect similar to 2π − pulse dynamics in laser amplifiers [Kryukov et
al., 1970; Oraevsky, 1998]. The results represent a significant advance in the study of
coherent effects [Carman et al., 1970; Duncan et al., 1988; Bonifacio et al., 1975;
Harvey et al., 1989], and point to a new generation of highly engineerable optical gas
cells for studying complex nonlinear phenomena.
5.2 Experimental results
Figure (5.4) shows the setup used, comprising a narrow linewidth pump laser emitting
50 μJ pulses of 12 ns duration at 1.06 μm. The seed Stokes pulses were generated by
forward SRS in a 1.5 m long band gap guiding HC-PCF filled with hydrogen (stage I,
shown with a dashed rectangle in the figure). In order to be able to control the seed
pulse energy and the steepness of its front, a part of the pump pulse was delayed and
used to amplify the forward signal generated. The BSRS process was studied in a
77
second stage, consisting of an additional length of hydrogen-filled fiber (shown by a
black dashed rectangle). The Raman-active transition between the rotational levels
J = 1 and J = 3 of molecular hydrogen (Raman shift = 587 cm-1 [Herzberg, 1989])
was chosen to study transient BSRS. As explained before, the transmission window of
the HC-PCF was designed to feature low loss transmission only for the pump and the
first Stokes frequencies. This means that the BSRS process was completely decoupled
from the competing vibrational and higher order rotational SRS processes which are
typically present in focused beam geometry.
The BSRS gain profile of hydrogen exhibits both Doppler and collisional
broadening [Murray, 1972; Owyoung, 1978]. The line broadening due to molecular
50 MHz/bar ),
collisions (MC) is pressure dependent (for H2 the broadening is ∆ν MC =
so that to observe coherent effects with a few ns pulse one should operate at relatively
low pressures ( < 3 bar). In this regime, the phase relaxation is mainly due to
200 MHz ), resulting
collisions with the sidewalls (corresponding linewidth ∆ν wall coll =
in an effective phase relaxation time at 3 bar of T2 = 3.5 ns . In this pressure regime,
the energy relaxation time is significantly longer ( T1 > 15 ns ) [Grasyuk et al., 1982].
The forward and backward pump intensities and gas pressure in the first fiber were
optimized so that the steepness of the leading edge of the generated Stokes pulse was
maximized (the importance of this adjustment is discussed below). The energy of the
output pulse was ≈ 4 μJ with duration of 7 ns (approximately twice the value of T2
in the subsequent amplification stage). The length and the pressure in the second fiber
were chosen to maximize the gain factor for the seed pulse while ensuring that the
pump pulse energy remained below the threshold for forward SRS.
Figure (5.5) shows the evolution of the temporal structure of the BSRS Stokes
pulse (second stage) for increasing pump pulse energies at a pressure of 3 bar .
Amplification of the Stokes field occurs predominantly at the leading edge of the
pulse, giving rise to the formation of an intense spike, the field growth at the trailing
edge of the pulse being strongly saturated. The most interesting feature is that the
spike can reach a duration well below the dephasing time T2 = 3.5 ns . As the pump
78
Figure 5.5: Temporal structure of amplified Stokes pulses measured after propagation along 1.5
m of HC-PCF filled with H2 at 3 bar for pump energies of 4, 8, 12, 16, and 20 μJ. The seed Stokes
pulse is shown with a dashed line, magnified by a factor of 50. The inset shows a single shot
measurement of the amplified Stokes pulse envelope at a pressure of 1.5 bar.
power increases, the spike duration falls while its energy increases. It reaches a
remarkably symmetric form, close to the sech 2 ( x) shape in Eq. (5.1). This is not, as
might be thought, the result of averaging over many shots, but is characteristic of
every single shot, as shown in the inset in Fig. (5.5). Above a certain level of pump
energy, however, the temporal compression saturates to a minimum pulse duration of
1.3 ns , which is significantly shorter than T2 . This compression cannot be attributed
to the transient character of the Raman gain, because even shorter BSRS pulses were
observed when Stokes seed pulses with steeper pulse fronts were used.
The origin of this behavior can be understood by reference to Figs. (5.5) and
(5.6a). It is seen that, for pump energies in the range from 5 to 15 μJ, amplification of
the spike is nearly uniform across its width, the peak of the spike remaining in
approximately the same position. For pump energies > 15 μJ , however, there is a
noticeable shift of the peak to earlier times. In fact, the amplification is not uniform
anymore, which translates into an apparent acceleration of the pulse, with a time
advance that grows monotonically with the pump energy [Fig. 5.6(b)]. Interestingly,
for large enough pump energy the shape of the output Stokes pulse becomes quite
symmetric, remaining more or less unchanged as the pulse energy is further increased.
One finds that the duration of the output pulse is insensitive to linear loss, while
79
Figure 5.6: (a) Duration and temporal position of the peak of the amplified Stokes pulses as a
function of pump energy (HC-PCF is filled with hydrogen at 3 bar). (b) Experimental output
Stokes pulse shapes (see Fig. 5.5) normalized to their peak intensity for increasing values of pump
energy from right to left: 8, 12, 16, 20, 24 μJ.
strongly depending on the steepness of the leading edge of the seed pulse. In this
connection, it is worth noting that the amplification factor for the Stokes pulse is at
least one order of magnitude higher than the linear attenuation factor, indicating that
the observed stabilization of the Stokes pulse cannot be attributed to formation of a
dissipative soliton [Eq. (5.1)], in which amplification is balanced by linear loss.
5.3 Theoretical analysis
To describe the dynamics of pulse amplification in this regime, I consider pump and
Stokes waves propagating, respectively, in the + z and − z directions, and represented
by the fields
=
E p,s ( z, t )
1
Ap , s ( z , t ) exp ( iω p , s t ± ik p , s z ) + c.c. ,
2
{
}
(5.2)
Here Ap , s are the complex amplitudes and ω p , s the carrier frequencies of the fields,
Ω . The
which are two-photon resonant with the Raman transition, i.e., ω p − ωs =
propagation constants of the guided modes in the fiber are k p , s = k p , s (ω p , s ) . Nonlinear
80
propagation of the fields is described by the coupled wave equations [Eqs. 3.48(b) and
(c)]:
−
∂Ap
1 ∂Ap
1
N κ1
=
−i
ω p As Q − σ p Ap ,
2ε 0 cnr
2
v0 ∂t
(5.2a)
∂As 1 ∂As
N κ 1
1
+
=
−i
ωs Ap Q* − σ s As .
∂z v0 ∂t
2ε 0 cnr
2
(5.2b)
∂z
+
In writing Eqs. 5.2(a) and (b) I have assumed that group velocities of the Stokes and
pump pulse and their refractive indices are the same, i.e. v p ≈ vs ≈ v0 and
n p ≈ ns ≈ nr . Doppler broadening of the molecular frequency ∆Ω = Ω − (ω p − ωs )
from the Raman resonance causes the two fields to form an inhomogeneous line shape
described by the normalized function g (∆Ω) , where
∫ g (∆Ω) d (∆Ω) = 1 .
The
macroscopic medium response [i.e., the nonlinear source terms on the right-hand side
of Eq. (5.2)] is calculated from Q - the Raman coherence averaged over g (∆Ω) .
The dynamics of the Raman transition, driven by pump and Stokes fields, are
described by the slowly varying amplitude of the density matrix elements
ρij (i, j = 1, 2) :
∂
1
iκ *
Q + Q = i ∆Ω Q − 1 Ap As* n(t ) ,
4
∂t
T2
(5.3a)
∂
n(t ) − n
=
n(t ) +
Im{κ1 As A*pQ} ,
∂t
T1
(5.3b)
n ρ11 − ρ 22 as the population
where Q is the Raman coherence and I have defined =
difference between the lower and upper levels, normalized to the number density of
the gas molecules. n is the equilibrium value of population difference in the absence
of laser fields.
The modeling shows that the evolution of the Stokes pulse separates into two
phases (Fig. 5.7). In the initial stages of the interaction, amplification is most effective
at the leading edge of the seed pulse, causing the pulse front to steepen and the
81
Figure 5.7: The intensity envelopes of the pump (shaded in gray) and Stokes (red line) pulses
shown at different times in a reference frame moving at the Stokes velocity vs. Temporal
compression of the Stokes pulse (t < 3 ns) is followed by the formation of a quasi-soliton pulse
traveling faster than the velocity of light vs (the vertical dashed line shows the position of Stokes
light moving at exactly vs). The inset shows the amplitudes of the pump and the quasi-soliton
pulse. Also shown is the long-lived Raman coherence at Doppler line center (slanted line fill).
effective pulse duration to fall. This temporal narrowing does not stop even when the
pulse duration is shorter than T2 . This is explained by the fact that when the Stokes
pulse becomes sufficiently strong (Fig. 5.7, t = 2 ns ), the counter-propagating pump
wave is completely depleted before it reaches the trailing edge of the Stokes pulse. As
a result, growth of the Stokes field [which is proportional to the source term in the
right-hand side of Eq. (5.3a)] saturates at the trailing edge. One might expect,
therefore, that a further increase in Stokes intensity would lead to even shorter pulse
durations. The computer simulations show, however, that the Stokes pulse stabilizes
to a nearly symmetric solitary structure with an envelope propagating faster than the
velocity of light (Fig. 5.7, t > 2 ns ). This reshaping effect does not contravene
relativity, but is instead the consequence of pulse reshaping through (a) nonlinear
amplification at the leading edge and (b) nonlinear absorption at the trailing edge by
energy back conversion to the pump frequency. It is the long-lived Raman coherence
which is responsible for this reshaping process. We also note some similarity between
this process and the propagation of coherent 2π − pulses in laser amplifiers [Kryukov
82
et al., 1970]. The formation of a pulse moving faster than the speed of light is caused
by the presence of the long exponential leading edge of the seed pulse. After
propagation over a certain amplification length, the pulse peak finally approaches the
‘‘earliest point’’ on the leading edge of the seed pulse. From this moment on, the
pulse would be further amplified, its duration falling as it approaches the asymptotic
form described by Eq. (5.1). Under our experimental conditions, however, the pulse
peak reaches the end of the amplifying medium before approaching the ‘‘earliest
point’’.
5.4 Analytical considerations
Some basic properties of the BSRS equations can be extracted by considering
solutions of Eqs. (5.2) and (5.3) in the form of an invariant pulse profile that depends
on the variable τ = t − z / v where v > vs . I assume that the Stokes pulse is amplified
by an infinitely long pump wave with constant amplitude
A0 . Neglecting
inhomogeneous line-broadening and linear loss in the system, after some
manipulation of Eqs. (5.2) and (5.3), I find that the coherence is purely imaginary,
Q = −i ρ , obeying the differential equation:
∂ 2 Ψ 1 ∂Ψ
+
= β 2 sin Ψ
2
∂τ
T2 ∂τ
(5.4)
where I have introduced the function
τ
Ψ (τ ) =
2α ∫ ρ (τ ′ ) dτ ′
(5.5)
−∞
N κ 1
and the parameter α =
2ε 0 cnr
1/ 2
 ω pω s 
 −2 −2  .
 v0 − v 
To arrive at Eq. (5.4) I rewrite Eqs. (5.2) using the new variables, i.e the retarded
time τ and ρ ,
83
∂Ap
1/ 2
=
−α ω p ( v − v0 ) / ωs ( v + v0 ) 
∂τ
ρ As ,
(5.6a)
1/ 2
∂As
= α ωs ( v + v0 ) / ω p ( v − v0 )  ρ Ap .
∂τ
(5.6b)
Solution to Eqs. (5.6) can formally be written as,
Ap = A0 cos (αψ ) ,
(5.7a)
1/ 2
As =A0 ωs ( v + v0 ) / ω p ( v − v0 ) 
sin (αψ ) ,
where I have introduced the function ψ (τ =
) (2α )−1 Ψ (τ =)
(5.7b)
∫
τ
−∞
ρ (τ ′ ) dτ ′ . Assuming
that population inversion to the excited state is negligible, so that n ≈ n , the equation
for Raman coherence, Eq. (5.3a) can be written as,
∂ρ 1
n κ1
+ ρ=
Ap As .
∂τ T2
4
(5.8)
Using solutions (5.7) and definition (5.5), I arrive at Eq. (5.4) for the variable Ψ ,
where β 2 =
N n  ωsκ12
A02 .
−1
−1
8ε 0 cnr (v0 − v )
I seek solutions of Eq. (5.4) with initial conditions Ψ ( −∞ ) = Ψ′ ( −∞ ) = 0 . The
behavior of such solutions depends on the dimensionless parameter β T2 and
describes nonlinear oscillations asymptotically approaching the equilibrium value
Ψ ( +∞ ) = π . This can be seen by ignoring the time derivative of Raman coherence in
2 arctan(e β T2τ ) . However, in the limit of long
Eq. (5.8), corresponding to Ψ (τ ) =
2
dephasing times, β T2 >> 1 , the relaxation term in Eq. (5.4) can be neglected, and Eq.
(5.5) has the 2π − pulse solution,
Ψ (τ ) =
4 arctan[exp(τ / τ 0 )] .
(5.9)
84
Inserting solution (5.9) into Eqs. (5.7), I find that this solution is associated with a
solitary pulse of the coupled Stokes and pump fields:
1/ 2
As (τ ) =
A0 ωs ( v + v0 ) / ω p ( v − v0 ) 
sech (τ /τ 0 ) ,
Ap (τ ) = − A0 tanh (τ /τ 0 ) .
(5.10a)
(5.10b)
The characteristic duration τ 0 is given by the expression:
N n  ωsκ12
τ = β=
A02 .
−1
−1
8ε 0 cnr (v0 − v )
−2
0
2
(5.11)
It follows from Eqs. (5.10) that, at the point where the Stokes field reaches its
maximum, the pump field goes through a zero, in direct agreement with the results of
the numerical simulations (see inset in Fig. 5.7). We also note that, unlike dissipative
solitons in amplifying media [Maier et al., 1969; Bonifacio et al., 1975; Harvey et al.,
1989; Picholle et al., 1991], the pulse duration in Eqs. (5.10) - (5.11) is not fixed, i.e.,
it is a free parameter. Therefore, a seed Stokes pulse having an exponential leading
edge (with characteristic time τ 0 ) will evolve towards a sech profile of the same
duration traveling at a velocity v(τ 0 ) determined by the dispersion relation (5.11).
The velocity of such a solitary Stokes pulse increases roughly linearly with the pump
intensity. These considerations qualitatively explain the experimental observations
(Figs. 5.5 and 5.6) and support the numerically modeled pulse dynamics presented in
Fig. (5.7). Finally, we note that by minimizing loss in a specially engineered fiber and
optimizing the experimental configuration, pulse compression factors much greater
than 20 should be possible.
85
86
6
Observation of self-similar solutions of sine-Gordon
equation in transient stimulated Raman scattering*
When resonant light-matter interactions occur on a time-scale shorter than the
characteristic relaxation times of the medium, i.e.,  p  T1 , T2 (  p is the pulse
duration, and T1 and T2 are the population and the coherence lifetimes of an atomic or
molecular transition), the evolution of the optical field becomes particularly
interesting. The reason is that the laser pulse is able to significantly excite the medium
before relaxation comes into play. On the other hand, the response to the field is
highly dispersive and nonlinear since it depends on the history of the field phase and
amplitude from the moment the excitation starts, the response is affected by the
“coherent memory”. Propagation of a laser field in an atomic, molecular, or solid-state
medium with coherent memory is known to lead to a number of specific (coherent)
optical phenomena [Allen et al., 1975]. Many of these phenomena obey the
fundamental equation of light-matter interactions, the sine-Gordon equation (SGE±):
2
  sin  ,
 
(6.1)
*
Published in Nazarkin, A., Abdolvand, A., Chugreev, A. V. and Russell, P. St.J., 2010, “Direct
observation of self-similarity in evolution of transient stimulated Raman scattering in gas-filled
photonic crystal fibers,” Phys. Rev. Lett. 105, 173902.
87
which describes changes in the rotation angle  of the Bloch spins with respect to
the propagation coordinate  and the retarded time   t  z / v . This reduction is
possible because under certain conditions the Maxwell-Bloch equations describing
these processes are integrable and, in particular, can be reduced to the sine-Gordon
equation, Eq. (6.1) (SGE) for the evolution of Bloch spins [Allen et al., 1975].
Solitonic solutions of the SGE, Eq. (6.1) with a minus sign, are associated with
2  pulses in self-induced transparency and higher order solitons in absorbing media,
processes which have been observed and studied experimentally in great detail
[McCall et al., 1969; Lamb, 1971]. In contrast, markedly different behavior is
predicted for non-solitonic solutions of the SGE, Eq. (6.1) with plus sign, which are
associated with coherent pulse amplification [Manakov, 1982; Hope et al., 1969],
superradiant decay [Allen et al., 1975; Gabitov et al., 1984], and transient stimulated
Raman scattering [Elgin et al., 1979; Menyuk et al., 1992; Menyuk, 1993].
A fundamental hypothesis is that at long interaction lengths, and irrespective of
the initial conditions, the spatiotemporal evolution of these non-solitonic solutions
should be self-similar; i.e., at each point in the medium the system should go through
the same phases of temporal evolution but within a different time. This behavior is
only expected if the laser-matter interaction is coherent. Although some features of
the predicted dynamics have been observed before (i.e.,   pulse shortening in laser
amplifiers [Varnavskii et al., 1984; Harvey et al., 1989] and time modulation of the
fields in SRS [Duncan et al., 1988; Carman et al., 1970]), no clear signature of selfsimilarity of this process was established in early works [Duncan et al., 1988;
MacPherson et al., 1989]. This is mainly due to limitations imposed both by the
interaction geometry and by competing nonlinear processes, particularly generation of
higher order Stokes components, so that the amplification length was not sufficiently
long to observe the temporal pulse reshaping characteristic of self-similar dynamics.
In this chapter, I demonstrate the observation of clear self-similar behavior in
transient stimulated Raman scattering by carrying out a detailed study of transient
SRS over long interaction lengths. In order to do this, I make use of the unique
characteristics of gas-filled hollow-core photonic crystal fiber (HC-PCF) [Russell,
2006; Abdolvand et al., 2009]. As mentioned in previous chapters, these novel optical
88
guiding systems offer interaction lengths many times longer than the Rayleigh length
of a focused laser beam, while eliminating diffraction by keeping the beam tightly
confined in a single mode. Moreover, by designing a HC-PCF with a restricted
guidance bandwidth, the Raman process can be completely isolated from competing
higher-order SRS processes, see chapter 5. In this way, we are able to make detailed
measurements of (and gain insight into) late-stage transient SRS when quantum
conversion to the Stokes is close to unity.
6.1 Stimulated Raman scattering as a study model for SGE
The relationship between SRS and the SGE can be established by considering the
transient limit of the semi-classical equations of SRS, i.e when  p  T2 [Elgin et al.,
1979; Menyuk et al., Menyuk et al., 1992; Menyuk, 1993; Carman et al., 1970].
Consider pump and Stokes waves propagating in a Raman medium in the  z
direction and represented by,
1
i  t  k z 
E p , s ( z , t )  { Ap , s ( z , t ) e p ,s p ,s  c.c.}
2
(6.2)
where Ap , s ( z, t ) are the complex electric field amplitudes, k p , s ( p , s ) are the field
propagation constants, and  p , s are the carrier frequencies of the fields, which are
resonant with the Raman transition frequency, i.e.,    p  s . In the regime of
weak excitation of the Raman transition ( n  n  1 ), the interaction dynamics is
described by the equation for the slowly varying amplitude of the off-diagonal
density-matrix element Q(t ) , while the spatiotemporal evolution of the fields obeys
the wave equations,
i *

1
Q  Q   1 Ap As*
T2
4
t
 1 
  pvp
 
 Ap  i 
 s vs
 z v p t 
(6.3a)

  2 Q As

(6.3b)
89
 1 
*
 
 As  i 2 Q Ap


z
v
t
s


(6.3c)
Defining the dimensionless distance   z / LSRS , by normalizing the distance z to
LSRS  2 0 c [ N 1 (s p )1/ 2 (n p ns ) 1/ 2 ]1 , where N is the number density of the
molecules, and changing the variable t    t  z / v (retarded time), and assuming
equal group velocities for both pump and Stokes, v p  vs  v , Eqs. (6.3) can be
rewritten in the following form,
*
Q
Q
 i 1 Ap As*  ,
4

T2
Ap

(6.4a)
 i QAs ,
(6.4b)
As
 i 1Q* Ap .

(6.4c)
Here, we let    p ns /(s n p ) and 1 is the matrix element characterizing the
coupling of the fields to the Raman transition [See chapter 3 on the theoretical basis of
SRS]. In the current analysis it is assumed that the fields have finite energy, so that no
permanent soliton-like structures can exist in the system [Menyuk et al., 1992;
Menyuk, 1993]. Below we consider a special solution of Eqs. (6.4) where the fields
are taken as real valued quantities and the coherence of the Raman transition is purely
imaginary, i.e. Q  i  with Im{}  0 . These special solutions are most significant
from a physical viewpoint because they correspond to the case of maximum Raman
gain. Using the new definitions, the coupled equations of pump, Stokes, and material
excitation, Eqs (6.4), can be rewritten as,
 1

 Ap As  ,

4
T2
Ap

(6.5a)
  As ,
(6.5b)
90
As
  1  Ap .

(6.5c)
The conservation of photon number (Manley-Rowe relation) following from Eqs.
(6.5b) and (6.5c) can be written as
Ap2  , 
p

As2  , 
s

Ap2  0, 
p

As2  0, 
(6.6)
s
where Ap (0, ) and As (0, ) are the temporal shapes of the fields at the input   0 .
The relationship Eq. (6.6) allows one to rewrite the amplitudes using the new variable
 ( ,  ) in the form
Ap  ,    
1/ 2
A0   cos   / 2  ,
(6.7)
As  ,     1/ 2 A0   sin   / 2  ,
where A20     1 A2p     A2s   . By inserting representation (6.7) into Eqs. (6.5),
and assuming the transient interaction regime (  p  T2 ), the three equations in (6.5)
are reduced to only one equation for the function   ,   :
 2  1 2
 A0   sin .
 4
(6.8)
After introducing a new time variable T   
1
4


A20 ( ) d  , one arrives at
SGE:
 2
 sin .
 T
(6.9)
91
Below we consider a special set of solutions of Eq. (6.9), namely self-similar
solutions. These are solutions of (6.9) that depend only on a single variable,  which
can be expressed via a combination of  and T . By doing such a self-similar
transformation, Eq. (6.9) reduces from a partial differential equation in variables 
and T to an ordinary differential equation in the similarity variable  . Let us
consider one such similarity transformation given by   2  T . Upon inserting this
transformation into Eq. (6.9), its self-similar solutions,  (  ) obey the following
differential equation,
 2 1 

 sin  0
 2  
(6.10)
This equation is satisfied by self-similar solutions of the sine-Gordon equation
determined by the proper initial conditions; these solutions can be expressed in terms
of the Painlevé transcendents (P). Indeed, by replacing the independent variable with
2  1/ 2 , Eq. (6.10) can be reduced to one of the standard forms of PIII equation [Elgin et
al., 1979]. Physically interesting solutions of Eq. (6.10) are defined by the boundary
conditions at   0 :  (0)   0 and  (0)  0 . These boundary conditions suggest
that before the fields arrive there is no excitation of the Raman medium, see Eqs. (6.7)
[Elgin et al., 1979; Menyuk et al., 1992; Menyuk, 1993]. The universal function
 (  ) describes all the system dynamics and does not depend on the (initial) forms of
the fields. The region 0    1 corresponds to early-stage Stokes generation when
 (  )  I0 (  ) [ I0 (  ) is a Bessel function of the second kind] and the Stokes field has
a smooth temporal profile. For   1 (late stage),  (  ) oscillates with decreasing
amplitude, asymptotically approaching  , which corresponds to complete conversion
of photons to the Stokes field, Eqs. (6.7). Figure (6.1) shows the behaviour of  (  )
as obtained by numerically solving Eq. (6.10) for two different initial values of  0 .
92
Figure 6.1: Behaviour of the universal function obeying Eq. (6.10) with the initial conditions
 (0)  0 and (0)   0 ; (1)
For small arguments
 0  0.01
(solid blue curve) and (2)
 0  0.02
(solid red curve).
 (  )  I0 (  ) , where I0 (  ) is zero-order Bessel function of the second
kind (solid green and red lines). For larger values of
 ,  ( )
asymptotically tends to the value
.
6.2 Experimental considerations
To observe the self-similar behavior predicted by Eqs. (6.7) and (6.10), we carried out
SRS experiments in a gas-filled HC-PCF [see Fig. (6.2)]. We used a narrow linewidth
laser delivering 10 ns pulses of a 100 μJ energy at  p  1.064 μm . The pump pulses
were launched into a photonic band gap HC-PCF (core diameter = 12 μm) filled with
hydrogen. The PCF had a low-loss transmission window between 1030 and 1150 nm,
which meant that only the pump and first Stokes bands, interacting with the J  1 and
J  3 rotational transition (frequency shift ~ 600 cm-1) could propagate in the fiber
[Abdolvand et al., 2009]. As a result, the competing vibrational and higher-order
rotational SRS typically present in focused beam geometry were completely
suppressed. The gain line of forward SRS in hydrogen is predominantly collision
broadened (for H2 the value is 50 MHz/bar [Abdolvand et al., 2009]). Note that
inhomogeneous (Doppler) broadening is important at pressures much lower than the
working pressures we used ( p  1 2 bar). In this pressure region the Raman line is
93
Figure 6.2: Schematic of the set-up used. Nanosecond laser pulses are launched into a low-loss
band-gap HC-PCF filled with hydrogen, exciting the rotational transition J = 1 to J = 3. The
narrow transmission band of the fiber allows only the pump and Stokes fields to propagate in the
gas-filled core. The inset shows a scanning electron micrograph of the HC-PCF microstructure.
The output traces of the pump and Stokes are separated via dichroic mirror (DM) and suitable
interference filters and are detected using fast photodiodes (PD).
additionally broadened by 200 MHz through collisions with the core sidewalls,
resulting in a net phase relaxation time T2  5 ns at 1 bar. The length of the HC-PCF
was L  200 cm . At p  1 bar the effective Raman length was LSRS  0.1 cm , and the
gain G was as high as 0.15 cm -1 for pump energies of ~ 20 μJ , bringing the
interaction length and gain product, G  L up to a value of 30 , the threshold for SRS
generation. As a result, we were able to operate in a regime where quantum
conversion to the Stokes was close to 100% , see Fig. (6.3a). For the 10 ns pump
pulses used in our experiment the effect of self-phase modulation could be ignored. In
fact, with I p ~ 1 GW/cm 2 , spectral broadening due to self-phase modulation (SPM) in
H 2 was negligibly small (the estimated nonlinear phase shift  SPM  105 ) [Agrawal,
2006]. Moreover, although the pump pulse duration was somewhat greater than T2 ,
94
Figure 6.3: (a) Typical output intensity distributions of the pump and Stokes fields in the regime
of long-path-length transient SRS in a hydrogen-filled HC-PCF. The input pump pulse energy is
60 μJ, the gas pressure 1 bar and the intensity distributions are averaged over 500 shots. The
development of a characteristic oscillating structure (“ringing”) indicating the asymptotic
behaviour of the fields is clearly seen. (b) The results of computer modelling show that the
coherence 12 (shaded gray region) does not follow the instantaneous value of the fields, i.e., it
exhibits “coherent memory”.
the transient dynamics were very clear. This is explained by the sharply increasing
SRS gain at the front of the pump pulse, leading to a Stokes field rise time much less
than T2 .
6.3 Experimental results
Experimental plots of the temporal structure of the pump and Stokes pulses (averaged
over 500 shots) for different values of pump pulse energy are given in Fig. (6.4). For
relatively low pump energies, i.e. E p  30 μJ , the pulse shapes are quite smooth. At
higher input pump energies, however, well-pronounced oscillations appear, with a
period that shortens as the pump energy increases. The time scale of these oscillations
(or ‘‘ringing’’)* is ~ 1 ns , which is shorter than T2  5 ns , suggesting that the Raman
scattering is strongly coherent and the nonlinear interaction is late-stage. Their origin
can be explained as follows. The growth of the Stokes field leads to significant pump
*
These oscillations, if self-similar, are also known as “accordions” [Menyuk et al., 1992; Menyuk,
1993].
95
Figure 6.4: Measured output traces of pump and Stokes pulses (each one is averaged over 500
shots) for input pump energies from 25 to 65 μJ. All input pump pulses have the same Gaussian
form with FWHM duration 10 ns (an input pump pulse of a 65 μJ energy is shown with a dashed
line (bottom figure).
depletion, while the molecular coherence 12 still remains nonzero due to the
presence of ‘‘coherent memory.’’ As a result, a new field at the pump frequency is
generated at the expense of the Stokes field. Because the phase of this new field is
shifted by  , the coherence goes through zero and changes its sign, see Fig. (6.3b).
This leads to oscillating field dynamics and molecular response.
6.3.1 Self-similarity of the late-stage oscillations
The most interesting fact established by our study is that the output pulse shapes are
self-similar. To demonstrate this and to show how we can extract the self-similar
behaviour of the experimental measurements, we express the field intensities as
functions of   ,  , which is supposed to behave in a self-similar way. Because
Stokes generation starts from a very small (spontaneous) signal, at the input we have
As2 ( )  Ap2 ( ) , leading from Eq. (6.7) to A02 ( )  s n p /( p ns ) Ap2 ( ) . Considering
now data sets resulting from dividing the intensity distributions of the output pump
96
I p    L ,  and Stokes I s    L ,  pulses by the input pump pulse shape
I p   0,   I p 0 ( ) (where  L  L / LSRS is the normalized fiber length), we can
write
I p(i ) ( L, )
I p(i0) ( )
 cos 2
 (i )
2
(6.11a)
,
(i )
I s(i ) ( L, ) s
2

sin
,
I p(i0) ( )
p
2
(6.11b)
where i  1, 2, ; n labels the output pump and Stokes intensities in Fig. (6.4)
corresponding to the ith input pump pulse intensity I p2 0 ( ) . If the behavior of the
measured output intensity distributions are self-similar, then the ratios on the left-hand
sides of Eqs. (6.11) should depend, through the functions  (i ) ( L, )   (  ( i ) ) , only on
the self-similarity variable:

 (i )  2 LT(i )  2  LI 0(i )  f p(i ) ( ) d  ,

(6.12)
where we have introduced the form factor f p(i ) ( )  I p( i0) / I 0(i ) for the input pulses, I 0( i )
being the peak intensity of the ith pulse and   2 1 /(n p c ) . Since Eq. (6.5) in the
transient regime ( T2   ) are invariant under a translation in  , there is a possibility
(i )
(i )
, so that one can define the similarity variable as  (i )   (i )   off
for an offset,  off
(i )
[Menyuk, 1993]. Here we have considered  off
being vanishingly small. This point is
supported by the experimental measurements. However, any conclusion about the
reason for the absence of any offset needs more careful experimental and theoretical
investigation. Equation (6.12) suggests that changing the input pulse energy will
simply alter the range of variation of T(i ) and  (i ) . As a result the ratios
I p( i ) ( L, ) / I p(i0) ( )
and
I s( i ) ( L, ) / I p(i0) ( ) , when plotted as a function of T(i )
(proportional to the integrated input pump intensity), should exhibit the same
(universal) behavior for every input pump pulse. In Fig. (6.5) we plot the ratios in
97
Figure 6.5: Dependence of the ratios I P( i ) / I P( i0) and I S( i ) / I P( i0) on the integrated intensity T( i ) of the
input pump pulse for different pump energies (35, 45, 55, and 65 μJ), indicating the behaviour of
squared sine and cosine functions of the universal self-similarity function

. The damped
oscillations of the ratios clearly show the underlying self-similar oscillatory behaviour of the
system for different power levels.
Eqs. (6.11) as a function of T(i ) for pump energies of 35, 45, 55, and 65 μJ . The
curves almost completely agree (within experimental error), suggesting that the latestage dynamics of the SRS process are indeed self-similar. The somewhat smaller
modulation depth for 35 μJ can be explained by the fact that a weaker pump pulse
generates slower temporal modulations [see Fig. (6.4)] which are more strongly
affected by the relaxation process. The behaviour in Fig. (6.5) is associated with the
functions sin 2 ( / 2) and cos 2 ( / 2) where the universal function  completely
characterizes the evolution of the system. As we have already established, this
function is independent of the shape of the input pulse and depends only on the
parameters of the Raman medium.
Since the similarity variable depends on the product of L and I 0( i ) , increasing the
input pulse energy will have the same effect as lengthening the fiber, allowing us to
reconstruct from our data the pump and Stokes pulse shapes at different positions
along the fiber. For example, the pulse shapes at position L / 2 for a launched pulse of
energy of 65 μJ will be identical to those at L for a pulse energy of 32.5 μJ .
98
Figure 6.6: Reconstruction of pump and Stokes pulse shapes for a launched pulse energy of 65
μJ. Experiment: Snapshots of the pulse shapes at different positions along the fiber, obtained
using the equivalence of length and pulse energy (see text). Theory: Pulse shapes (input pump
energy 65 μJ) calculated with the exact model in Eqs. (6.4).
Applying this procedure to the data in Fig. (6.4), we were able to reconstruct the pulse
shapes at several positions along the fiber for a launched energy of 65 μJ . The results
are shown in Fig. (6.6). We also present the results of numerical modeling of SRS
inside the fiber based on solutions of the exact equations, assuming input parameters
(pulse energy and shape, gas pressure, phase relaxation time) close to the
experimental values. Good agreement is obtained between theory and the
experimental data, confirming the self-similar character of the transient SRS process.
In fact, the experimental results presented here for the first time extend the field of
self-similarity [Barenblatt, 1996] to the broad class of nonlinear systems described by
the SGE.
99
100
7
Conclusion and outlook
Throughout the chapters of this thesis I tried to summarize the results of my research
on coherent light-matter interaction in gas-filled hollow-core photonic crystal fibers.
The conclusions drawn from this work and possible interesting directions of future
research on this subject are described here.
7.1 Diffractionless guidance of light in vacuum
In chapter 2 we have seen how a crystalline array of micro channels creates a cage for
light, making it propagate in a diffractionless, single mode manner over long distances
in vacuum. Indeed, many applications of HC-PCF in gas phase exploit the possibility
of filling the hollow micron size core of the HC-PCF, thus creating a strong
interaction between the laser light and gas. The high intensities created in the core of
the fiber immediately eliminates the need for using high power, high energy lasers in
studying SRS. This hugely reduces the complications which arise when using high
peak power, ultrashort laser pulses, such self phase modulation, self focusing, spectral
broadening and continuum generation. Moreover, a long interaction length between
laser and gas results in photon conversion efficiencies more than 90%.
101
7.2 SRS in gas-filled HC-PCF
In chapter 3, on the theoretical bases of SRS, we showed how generation of Stokes
wave is accompanied by the generation of a coherent field of optical phonons. Indeed
the coherence properties of this molecular grating hold a huge potential for different
applications such as manipulation of coherent Stokes light [Chugreev et al., 2009] or
generation of multi-octave frequency comb for low pulp powers [Couny et al., 2007;
Abdolvand et al., 2011].
7.2.1 Control of the nonlinearity
The possibility of controlling the nonlinearity in HC-PCF simply by changing the gas
pressure or creating well defined gas pressure profiles, gives HC-PCF a unique
position when it comes to coherent gas-laser interaction. Indeed in chapter 4 we
showed how using this possibility one can create an optimal condition for efficiently
generating anti-Stokes wave and coherently shift the pump laser to higher frequencies.
Generation of anti-Stokes components is interesting as it not only coherently shifts the
spectrum of the pump to the blue side of the spectral region, but also provides a
mechanism for optically cooling the system [Nazarkin et al., 2010b]. The reason for
this is that during the generation of an anti-Stokes photon, a quantum of energy
(optical phonon) is removed from the system, resulting in a net cooling effect.
7.2.2 Control of the dispersion and phase-matching
Efficient generation of anti-Stokes radiation is of interest in CARS spectroscopy. The
signal generated in CARS is normally weak, limited by the need for short interaction
length over which phase-matching is possible. HC-PCFs may open up new
possibilities for increasing the efficiency of this powerful spectroscopic technique.
Although the efficient generation of anti-Stokes in the early experiments is attributed
to the phase-locking of anti-Stokes field to the pump, the possibility of phase
matching via dispersion properties of HC-PCF offers another route to tackle this
102
Figure 7.1: A comparison between the input and the output backward Stokes pulses showing the
effect of reshaping and pulse shortening de to nonlinear amplification of the seed pulse.
subject; an approach which yet has to be studied in more details [Ziemienczuk et al.,
2011].
7.3 Backward SRS in gas-filled HC-PCF
In chapter 5, I presented a detailed study of backward SRS where pump and Stokes
pulse interact with each other in a head on collision geometry. Observing these
solitary pulses is quite interesting as they represent special solutions of backward SRS
equations. In this case, energy from the counter-propagating pump pulse constantly
flows into the backward Stokes pulse. The nonlinear gain profile along the Stokes
pulse leads to its reshaping and shortening, increasing its peak intensity to values
much higher than the pump pulse, see Fig. (7.1). However, the presence of a long
lived polarization in the medium opens a channel for outflow of energy from the
trailing edge of the Stokes pulse back to the pump. The balance between the inflow
and outflow of the energy into the backward Stokes pulse results in the appearance of
a stable solitary wave which advances in time as the energy of the pump pulse is
increased, a direct consequence of the pulse reshaping. In this situation one would
expect that the recovered energy of the pump after the passage of the first pulse, be
absorbed by the photons at the trailing edge of the Stokes pulse. This should result in
a sequence of solitary pulses in a time window defined by the dephasing time of the
molecular coherence, T2, see Fig. (7.2). While our experimental conditions presented
103
Figure 7.2: Solitary waveforms generated by backward stimulated Raman scattering in gas-filled
HC-PCF. Once the first structure reaches an stable structure (see the text), a secondary structure
should appear at the trailing edge of the leading pulse.
in chapter 5 allowed us to observe an early stage of the formation of a second pulse,
investigating the late-stage evolution and stabilization of the second pulse proved to
be difficult. The complications were mainly due to pump pulse duration and low
threshold for the generation of the forward Stokes which result in less efficient
amplification of the backward Stokes. A possible solution to this problem is to use a
longer pump pulse, together with a short backward Stokes seed pulse. While a long
pump pulse increases the threshold for the generation of forward Stokes, the short
backward Stokes ensures us that the interaction happens with the dephasing time of
molecular excitation. In this way we should be able to observe a train of solitary
pulses following the original pulse. This observation would be a direct experimental
proof of the “transparency” of the leading pulse with respect to the pump pulse.
7.4 Self-similarity in SRS
In chapter 6, we studied the behaviour of the solutions of SRS equations in the
forward configuration, in which pump and Stokes propagate in the same direction. I
showed how in the coherent interaction regime, these equations are reduced to sineGordon equation with plus sign (SGE+) – an equation which appears in many different
physical situations [Allen et al., 1975]. Interesting point is that showed that the SGE+
104
equation describes the long-distance dynamics of nonlinear pulses in SRS. Indeed this
stage is governed by self-similar solutions, that is any pump and Stokes input pulse
shapes launched into a Raman medium tend toward a self-similar solution of SGE+
[Menyuk, 1994]. The situation is similar to the case of solitons where Hasegawa and
Tappert showed that the nonlinear Schrödinger equation describes the long-distance
dynamics of nonlinear pulses in optical fibers [Hasegawa et al., 1973a and 1973b]. In
other words any intense enough pulse launched into an optical fiber in the anomalous
dispersion regime, so that its evolution is subject to the nonlinear Schrödinger
equation, breaks up into solitons plus background radiation [Zakharov et al., 1972].
7.5 Generation of coherent broadband frequency combs
Crucial to the observation of self-similarity in SRS is the absence of any higher orders
Raman components, thanks to the spectral filtering of HC-PBG-PCF. However, as
mentioned in chapter 2, HC-PCF comes into two different types: HC-PBG-PCF, with
narrow transmission window and low propagation loss, and kagomé-type HC-PCF,
with broad transmission window and higher optical loss compared to HC-PBG-PCF.
Recently, kagomé-HC-PCF has shown a large potential in the generation of a multioctave Raman frequency comb [Couny et al., 2007; Abdolvand et al., 2011]. If
coherent, such frequency combs may find a wide spectrum of applications, ranging
from sub-fs pulse synthesis to optical atomic clocks and carrier-envelope control
[Sokolov et al., 2003]. For practical use, a possibility to control the generated SRS
spectrum and the coherence properties of the Raman components would be of great
importance.
Here I describe an efficient way of generating a broadband Raman side-band in a
hydrogen-filled HC-PCF [Abdolvand et al., 2011]. In this technique, with a
combination of a pump (the output of a microchip laser delivering pulses of 80 μJ
energy and 2 ns duration at 1064 nm) and a seed Stokes pulse at 1134 nm, generated
separately from the pump via SRS in a hydrogen-filled HC-PBG-PCF, we drive
resonantly the first rotational Raman level of hydrogen. We achieved this result by
simultaneously coupling the pump and the rotational Stokes seed in a 1 m long
kagomé-HC-PCF with a broad transmission window ranging approximately from
105
Figure 7.3: Schematic of the procedure used for generating broad Raman frequency comb in a
hydrogen-filled kagomé-HC-PCF. Here one drives resonantly the rotational Raman transition in
hydrogen.
800 nm to 1750 nm, see Fig. (7.3). Due to the tight confinement of light and hydrogen
gas along the long length of the fibre, one can observe the generation of a broad,
purely rotational frequency comb of Stokes and anti-Stokes components even at low
input energies.
Figure (7.4a) shows a typical comb produced using this method. We must
determine the extent to which generated comb is coherent. To check this point I
collimate and focus the output of the fiber into a 5 mm thick BBO crystal. Fig. (7.4b)
shows the second-harmonic of the comb after frequency doubling for different phasematching angles of the crystal. A close inspection of the spectral domain reveals the
fact that the generated visible comb consists of the second-harmonics as well as sumfrequencies of the original frequency comb components. The efficient generation of
second-harmonics and sum-frequency lines indicates the presence of mutual
coherence among individual components of the frequency comb. This result can be
explained by noting that each frequency component centered at (ν i +ν i +1 ) / 2 has
2, 3,  , and any phase variation
simultaneous contributions from (ν i − j +1 + ν i + j ) / 2, j =
along the comb would lead to these components generating their own sum-frequency
components with a different phase, leading to a destructive interference between
generated components and less efficiency. Now combined with extremely high
106
Figure 7.4: (a) Pure rotational coherent comb generated via SRS in a gas-filled HC-PCF and (b)
its second-harmonic and sum-frequency comb generated via frequency-doubling in a BBO
crystal as a function of the phase-matching angle of crystal.
Raman gain in a gas-filled HC-PCF, one could imagine performing the same sort of
experiment using CW laser light. It is also possible to shift the whole frequency comb
to the visible spectral region, by for example using a green pump laser.
107
108
Appendix A
We are interested in the Fourier plane-wave spectrum of an outward-going Hankel
function at the flat boundary η = η0 ,
H 0(1) ( ρ ) =
1
2π
∫
+∞
−∞
F0 (τ n ,η0 ) exp(i τ n χ ) dτ n ,
(A1)
= η=
ρ sin(α ) and =
ρ ( χ 2 + η02 )1/ 2 as shown in Fig. (A1).
where χ = ρ cos(α ) , η
0
Here, τ n is the normalized, dimensionless component of the wavewector parallel to
the ML stack.
Our starting point would be the Sommerfeld integral representation of the nth
order Hankel function,
H m(1) ( ρ ) =
( −i ) m
π
∫
C
ei ρ cos ( w) +i m w dw,
(A2)
where n is an integer and the integration should be carried out in the complex plane
w= u + i v with C as the integration path [Sommerfeld, 1964]. Integration of Eq. (A2)
and rewriting the resulting expression in the form of Eq. (A1) will give the following
plane wave expansion coefficients, Fm (τ n ,η0 ) for H m(1) ( ρ ) [Cincotti et al., 1993],
109
Figure A1: Local orthogonal coordinate system used for calculating Eq. A1. Here
η
and
χ
show respectively the directions normal and parallel to the ML stack in the transverse plane,
with η
= η0 representing the core-cladding boundary.
(2 i ( m−1) / 2 / π )1/ 2 (τ 2 − 1) −1/ 2 ( τ 2 − 1 − τ ) m e −η0 τ n2 −1
n
n
n

2
−1

α
m/2
1/ 2
2 −1/ 2 iη0 1−τ n +im cos (τ n )
Fm (τ n ,η
= η0 ) e−im=
(2 (−i) / π ) (1 − τ n ) e

( m +1) / 2
/ π )1/ 2 (τ n2 − 1) −1/ 2 ( τ n2 − 1 + τ n ) − m e −η0
(2 (−i)

− ∞ < τ n < −1
− 1 < τ n < 1 (A3)
τ n2 −1
1<τn < ∞
For m = 0 the relations in Eq. (A3) take the following simple forms,
(2 i −1/ 2 / π )1/ 2 (τ 2 − 1) −1/ 2 e −η0 τ n2 −1
n

2

F0 (τ n ,η= η0 )= (2 / π )1/ 2 (1 − τ n2 ) −1/ 2 eiη0 1−τ n

2
1/ 2
1/ 2
2
−1/ 2 −η τ −1
(2 (−i ) / π ) (τ n − 1) e 0 n

− ∞ < τ n < −1
−1 < τ n < 1
(A4)
1< τn < ∞
Eq. (A4) decomposes H 0(1) ( ρ ) to contributions from propagating, τ n < 1 and
evanescent waves, τ n > 1 . Since we are only interested in the propagating waves
making up the mode of the fibre, we ignore the contributions from evanescent part
which does not contribute to the core mode, i.e.
H 0(1) ( ρ ) =
1
2π
∫
+1
−1
F0 (τ n ,η0 ) exp(i τ n χ ) dτ n .
(A5)
It is easy to show that Eq. (A5) is equivalent to decomposition of the core mode
electric field, J 0 (k T .rT ) , to positively and negatively travelling plane waves along
110
η -axis,
)
(
exp ±iη 1 − τ n2 . Fig. (A2) shows the fundamental core mode intensity
distribution J 0 (k T .rT ) 2 as obtained using Eqs. (A4) and (A5).
The plane wave spectrum of athe perfectly inward-going cylindrical wave,
H 0(2) ( ρ ) can be obtained from Eqs. (A4) and (A5) via the following identity,
=
H 0(2) ( ρ ) [ =
H 0(1) ( ρ )]*
1
2π
∫
+1
−1
F0* (τ n ,η0 ) exp( −i τ n χ ) dτ n .
(A6)
Making the change of variable τ n → −τ n and using the fact that F0 (τ n ,η ) is an even
function of τ n , Eq. (A6) can be rewritten in the following form,
H 0(2) ( ρ ) =
1
2π
∫
+1
−1
F0* (τ n ,η0 ) exp(i τ n χ ) dτ n .
(A7)
Using the plane wave spectrum of H 0(1) ( ρ ) and H 0(2) ( ρ ) along the planar interface
η = η0 , Eqs. (A1), (A4) and (A6), we can write the reflected y-polarized field
distribution, Emref ( χ ,η0 ) , as well as the field distribution for a perfectly inward-going
cylindrical wave as,
ref
E=
m ( χ ,η 0 )
Eminw=
( χ ,η0 )
1
τ n = +1
πτ
1
∫
r (τ n , β ) (1 − τ n2 )−1/ 2 e
− iη0 1−τ n2
eiτ n χ dτ n ,
(A8)
n = −1
τ n = +1
πτ
∫
(1 − τ n2 ) −1/ 2 e
− iη0 1−τ n2
eiτ n χ dτ n ,
(A9)
n = −1
where the reflection coefficient for mth ML stack, rm is given by,
rm (τ n , β ) = rTE (τ n , β ) sin 2 φm + rTM (τ n , β ) cos 2 φm .
(A10)
Making the change of variable τ n = cos(u ) , where u ∈ [ 0, π ] , we can rewrite Eqs.
(A8) and (A9) in the following form, which is more suitable for integration,
111
Figure A2: Normalized fundamental core mode intensity distribution
J 0 (k T .rT )
2
calculated
using the plane wave spectrum of cylindrical waves, Eq. A4. The white solid line circle shows the
first zero of the Bessel function z01
u =π
1
=
Emref ( χ ,η0 )
∫
π
Eminw ( χ ,η0 ) =
= 2.4048 .
rm (u , β ) e −iη0 sin(u ) eiχ cos (u ) du ,
(A11)
e− iη0 sin(u ) eiχ cos (u ) du.
(A12)
u =0
1
π
u =π
∫
u =0
Having the plane-wave spectrum of the reflected and the perfect inward-going files in
hand, we are in the position to evaluate the overlap integral between these two fields
given by the Eq. (2.13), i.e.,
6
Γ
2
η0 / 3
∑ ∫
m =1
Emref
( χ ,η
in*
0 ) Em (
χ ,η0 ) d χ
χ = −η0 / 3
2
=
η0 / 3
6
∑ ∫
m =1
6 π
∑ ∫ rm (u, β ) e−i η
m =1 0
6 π
∑∫e
2
Emin ( χ ,η0 ) d χ
0 sin ( u )
− i η0 sin ( u )
I (u ) du
, (A13)
I (u ) du
m =1 0
x = −η0 / 3
where I (u ) is defined through the following relation,
=
I (u )
π
∫e
i η0 sin ( u ′ )
sinc(η0 (cos(u ) − cos(u′)) / 3) du′.
0
112
(A14)
Appendix B
Here I present the calculations for the Raman coupling constant, κ1 , Eq. (3.34) as
well as the differential equation governing evolution of the Raman coherence, Q ( t ) ,
Eqs. (3.36) and (4.1).
Consider a situation where the electric field has contributions from three
interacting fields of pump, Stokes and anti-Stokes,
=
E ( z, t )
1
{E p exp[i (ω p t − k p z )] + Es exp[i (ωs t − ks z )] + Ea exp[i (ωa t − k a z )] + c.c.}, (B1)
2
where Ei , i = p, s, a is the slowly varying field amplitude of pump, Stokes and anti-
ω p − ωs =
Ω , the Raman frequency shift. In order to derive the
Stokes and ωa − ω p =
differential equation for Q ( t ) we start from the Eq. (3.30c),
d
i
ρ12 = iω21ρ12 + (TI + TII ),
dt

(B2)
where,
TI = −∑ µ m′ 2 ρ1m′ E ( z , t ),
(B3)
TII = ∑ µ1m′ ρ m′ 2 E ( z , t ),
(B4)
m′
m′
113
and the summations go over the intermediate states of the transition, i.e. far-from
resonance virtual levels, see Fig. (B1). Substituting from Eqs. (3.31) and (3.32) for
ρ1m′ and ρ m′ 2 in Eqs. (B3) and (B4) we have,
TI
t
i
µ m′ 2 ∫ dt ′ eiωm′1 (t −t ′) {µ1m′ [ ρ11 (t ′) − ρm′m′ (t ′)] + µ2 m′ ρ12 (t ′)}E ( z , t ′) × E ( z , t ),
∑
 m′
−∞
(B5)
t
i
TII =
− ∑ µ1m′ ∫ dt ′ eiω2 m′ (t −t ′) {µm′2 [ ρ m′m′ (t ′) − ρ 22 (t ′)] − µ m′1 ρ12 (t ′)}E ( z, t ′) × E ( z, t ). (B6)
 m′
−∞
In what follows we neglect the Stark shift of the levels, i.e. we ignore the terms
including ρ12 (t ′) in Eqs. (B5) and (B6). Inserting the Eq. (B1) into Eq. (B5) and
performing the integration explicitly we have,
i (ω t − k z )
1
e p p
e i (ω s t − k s z )
e i (ωa t − k a z )
TI =
− ρ11 (t )∑ µ1m′ µ m′ 2{E p
+ Es
+ Ea
4
ωm′1 − ω p
ωm′1 − ωs
ωm′1 − ωa
m′
+E
*
p
e
− i (ω pt − k p z )
ωm′1 + ω p
× {E p e
i (ω p t − k p z )
+ Es*
e − i ( ωs t − k s z )
e − i (ωa t − k a z )
}
+ Ea*
ωm′1 + ωs
ωm′1 + ωa
(B7)
+ Es ei (ωst − ks z ) + Ea ei (ωat − ka z ) + c.c.}.
In deriving Eq. (B7) we have assumed that the intermediate levels are not populated,
i.e. ρ m′m′ = 0 , and that the ground state population is a slowly varying function of
time. If now in Eq. (B7) we keep only terms which oscillates at the Raman frequency
Ω , we have,
ρ (t ) 
iΩt − i ( k p − ks ) z
− 11  E p Es*e
TI =
µ1m′ µm′ 2 [(ωm′1 − ω p )−1 + (ωm′1 + ωs )−1 ]
∑
4 
m′
* iΩt −i ( ka − k p ) z
p
+ Ea E e

µ1m′ µm′ 2 [(ωm′1 − ωa ) + (ωm′1 + ω p ) ] .
∑
m′

−1
The calculation of TII follows a similar line,
114
−1
(B8)
i (ω t − k z )
1
e p p
ei (ω s t − k s z )
e i ( ωa t − k a z )
TII =
− ρ 22 (t )∑ µ1m′ µm′2 {− E p
− Es
− Ea
4
ω p − ω2 m ′
ω s − ω2 m ′
ωa − ω2 m ′
m′
− i (ω pt − k p z )
− i ( ωa t − k a z )
e − i (ω s t − k s z )
* e
}
+E
+E
+ Ea
ω2 m′ + ω p
ω2 m ′ + ω s
ω2 m′ + ωa
*
p
e
× {E p e
i (ω p t − k p z )
*
s
(B9)
+ Es ei (ωst − ks z ) + Ea ei (ωat − ka z ) + c.c.}.
Keeping only terms oscillating at the Raman frequency, we have,
ρ (t ) 
iΩt − i ( k p − ks ) z
TII =
µ1m′ µm′2 [(ω2 m′ + ωs ) −1 − (ω p − ω2 m′ )−1 ]
− 22  E p Es*e
∑
4 
m′
* iΩt − i ( ka − k p ) z
p
+ Ea E e

µ1m′ µm′2 [(ω2 m′ + ω p ) −1 − (ωa − ω2 m′ ) −1 ] .
∑
m′

(B10)
ω2 − ω1 =
Ω it is easy to show that Eq.
Noting that ωm′2 = −ω2 m′ and that ωm′1 − ωm′ 2 =
(B10) can be rewritten in the following form, similar to Eq. (B8),
TII
ρ 22 (t ) 
* iΩt − i ( k p − ks ) z
 E p Es e
4 
∑µ
1m′
m′
* iΩt −i ( ka − k p ) z
p
+ Ea E e
µm′2 [(ωm′1 − ω p )−1 + (ωm′1 + ωs )−1 ]

µ1m′ µm′ 2 [(ωm′1 − ωa ) + (ωm′1 + ω p ) ] .
∑
m′

−1
(B11)
−1
Defining the coupling coefficients κ s and κ a as,
1
µ µ [(ωm′1 − ω p )−1 + (ωm′1 + ωs )−1 ],
2 ∑ 1m ′ m ′ 2
4  m′
1
=
κa
∑ µ1m′ µm′2 [(ωm′1 − ωa )−1 + (ωm′1 + ω p )−1 ],
4  2 m′
=
κs
We can rewrite Eq. (B8) and (B11) in the following form,
{
(t ) {κ E E e
iΩt −i ( k − k ) z
p
s
TI =
− ρ11 (t ) κ s E p Es*e
+ κ a Ea E *p e
=
TII ρ 22
s
p
* iΩt −i ( k p − ks ) z
s
+ κ a Ea E *p e
115
},
}.
iΩt −i ( ka − k p ) z
iΩt −i ( ka − k p ) z
(B12)
(B13)
Defining the slowly varying amplitude Q ( t ) for the molecular coherence and
inserting from Eq. (B12) and (B13) into the Eq. (B2), we have,
d
=
Q ( t ) i κ s E p Es* + κ a Ea E *p exp i (2k p − ka − k s ) z  [ ρ 22 (t ) − ρ11 (t ) ] .
dt
(
)
(B14)
Introducing the phenomenological damping constant T2 for the material coherence,
Eq. (B14) can be written in the following form,
d
1
Q (t ) + =
Q ( t ) i κ s E p Es* + κ a Ea E *p exp i (2k p − k a − k s ) z  [ ρ 22 (t ) − ρ11 (t ) ]. (B15)
dt
T2
(
For
vanishingly
small
)
population
difference,
i.e.
ρ 22 (t ) − ρ11 (t ) ≈ −1
and
∆k= 2k p − ka − ks , Eq. (B15) reduces to Eq. (4.1), that is,
d
1
−i (κ s E p Es* + κ a Ea E *p exp [i ∆k z ]) .
Q (t ) + Q (t ) =
dt
T2
116
(B16)
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Curriculum Vitae
Personal data:
First name: Amir
Last name: Abdolvand
Born: March 21th, 1979 in Shiraz, Iran
Educational history:
2007 - 2011:
PhD student under the supervision of Prof. Dr. Philip
Russell at the Max Planck institute for the science of
light, Erlangen, Germany.
2005 - 2007:
Researcher at optics group, physics department,
Martin-Luther University, Halle-Wittenberg, Halle
(Saale) Germany.
2002 - Sep 2004:
MSc. with first ranked honor, Shiraz University,
Shiraz, Iran. Major: Atomic Physics.
1997 - 2002:
BSc, Shiraz University, Shiraz, Iran. Major: Physics.
1993 - 1997:
High school, Tohid Gymnasium, Shiraz, Iran. Major:
Mathematics & Physics.
126