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 Phasen-

Lockings 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 mögliche Verbesserungen als auch neuartige

vielversprechende Wege zur 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 13 2.1.1 Photonic crystal cladding 13 2.1.2 Fabrication 15 2.1.3 Numerical analysis 16 2.2 GUIDANCE IN THE ABSENCE OF PHOTONIC BANDGAP 19 2.2.1 Analytical model for guidance in hollow-core Kagomé-lattice

photonic crystal fibers 19 2.2.2 Model 20 2.2.3 Reflection coefficients from ML stacks 27 2.2.4 Calculation of loss 29 2.2.5 Comparison with experiments 33 2.3 DISCUSSION AND CONCLUSION 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 3.2.1 Mechanism of the Raman effect 43 3.2.2 Optical phonons and material excitation 45 3.2.3 Material excitation as a damped oscillator 46 3.2.4 Basic differential equations 48 3.3 SEMICLASSICAL THEORY OF

STIMULATED RAMAN SCATTERING 51 3.3.1 Density matrix approach 51 3.3.2 Material excitation revisited 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 NUMERICAL SIMULATION 68 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 Self-similarity of the late-stage oscillations 96

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 102 7.2.2 Control of the dispersion and phase-matching 102 7.3 BACKWARD SRS IN GAS-FILLED HC-PCF 103 7.4 SELF-SIMILARITY IN SRS 104 7.5 GENERATION OF COHERENT BROADBAND

FREQUENCY COMBS 105 APPENDIX A 109 APPENDIX B 113 BIBLIOGRAPHY 117 CURRICULIM VITAE 126

1

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 Pω is scattered into a photon of energy Sω , while the difference in energy

( )p sω ω− = 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.

2

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

by the energy conservation law, so that ( )as pω ω= +Ω , as shown in Fig. (1.1b). In

thermal equilibrium, the occupancy of a molecular level, say n follows the

Boltzmann distribution law, i.e. exp( / )Bn E k T∝ − where E is the energy of the

level, T is the temperature and -231.38 10 [J/K]Bk = × is the Boltzmann factor. So, the

population of level 2 is smaller than the population of level 1 by a factor of

exp( / )Bk T− Ω . 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, nω

3

spaced equally with respect to the pump frequency at multiple integers of Ω , given

by ,n P mω ω= ± Ω where 1, 2,...m = 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 high-

resolution 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

4

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, pτ , is long compared to T2, 2p Tτ >> , 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, 2P Tτ << (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

5

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 two-

dimensional, 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 multi-

6

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 HC-

PCFs, 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 light-

matter 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 HC-

PCF, 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

7

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

8

when the Stokes pulse is shorter than 2T , 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 HC-

PCF. 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 laser-

matter 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 anti-

Stokes 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 H2-

filled 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 kagomé-

HC-PCF, presented in chapter 2, is a work in progress and the results will be

9

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 higher-

order modes in hollow-core photonic crystal fibers,” Opt. Express 16, 17972-

17981.

• 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).

10

11

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 kagomé-

HC-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.

12

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.

13

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 kagomé-

PCF, the precise nature of the guidance mechanism remains unclear.

In this chapter, after a short introduction to the guidance properties of PBG-HC-

PCF, 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 micro-

channels which run along the entire length of the fiber, Figs. (2.1). Quite generally, an

electromagnetic excitation of frequency ω propagating through an isotropic

14

homogeneous medium (glass or air) with refractive index ( )n n ω= has an axial

wavevector n kβ ≤ . Any excitation with n kβ > will be evanescent in any sub-

region of the medium. For example, for an air-silica interface and kβ < , light is free

to propagate in air and silica, regions 1 and 2 in Fig. (2.2a). However, for gk n kβ< <

light is confined in glass substrate via total internal reflection (TIR), region 2, and is

cut off from both air and glass for gnβ > , 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, PC PC ( )n n ω= ,

where PC1 gn n< < 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) kβ < , 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 gk n kβ< < light would be trapped in the glass region and is evanescent in the

hollow channels, regions 2 and 4, and for gn kβ > 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 light-

line, denoted by kβΛ = Λ in Fig. (2.2), the electromagnetic excitation can actually

propagate in a medium with lower refractive index than the PC, i.e. PC 0n n− < .

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.

15

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

16

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 ˆβ = k .z .

Writing field patterns formally as ( ) [ ( )]e i z tβ ω⊥ ⊥= −E E r and

( ) [ ( )]e i z tβ ω⊥ ⊥= −H H r , it is often useful to reformulate Maxwell’s equations as an

eigenvalue problem in β while keeping angular frequency ω constant,

( )( ) ( ) ( )2 2 2ln .k ε ε β⊥ ⊥ ⊥ ⊥ ⊥∇ + +∇ ×∇× =r H r H 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

17

is calculated via a sum over the Brillouin zone on the number of modes found in the

interval ( ), dβ β β+ for Bloch wavevector BK and at a fixed normalized frequency

kΛ , i.e.

( ) ( )1 ( )vac, ,i

ikρ β ρ δ β β−Λ Λ = −∑ ∑ B

B

KK

(2.2)

where second summation, ( )i goes over β values found at Bloch wavevector BK

and vacρ , 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 ( )vac 3 /(2 )ρ β β πΛ = Λ .

The real space cell of pitch Λ has an area of 23 / 2A = Λ . Thus the corresponding

reciprocal space cell has an area of ( )2 2 22 / 8 /( 3 )Aπ π= Λ . If Tk Λ is the magnitude

of the normalized real-space transverse wavevector, the states in the range Tk Λ to

( )T Tk d kΛ + Λ are contained within a circular shell of area ( )2 T Tk d kπ Λ Λ , Fig.

(2.4b), and the number of states must be:

( ) ( ) ( )vac32 .4

TT T T

kk d k d kρπΛ

Λ Λ = × Λ (2.3)

where factor of 2 has been added for two different polarization states. From the

vacuum dispersion relation ( ) ( ) ( )2 2 2Tk kβΛ + Λ = Λ it is clear that the DOS in any Tk

range must be the same as the DOS in any β range so that

( ) ( ) ( ) ( )vac vacT Tk d k dρ ρ β βΛ Λ = Λ Λ . Substituting the relation ( ) ( )T Tk d k dβ βΛ Λ = Λ Λ

into Eq. (2.3), we arrive at the desired result.

18

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 n kβ = , 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.

19

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 kagomé-

PCF, 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 anti-

resonant 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

20

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

21

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 =

1Lloss

=1

Lleak

+1

LML (N ), LML ~ tanh2 (µN )

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 silica-

air 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.

22

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:

,

( ) ( ) exp( )lm pq pqp q

iε ε ε= + = ⋅∑r r A G r (2.4)

where G pq = 2π (p g1 + q g2 ) /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 1 2ˆ ˆ( )lm l m= Λ +A a a 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, 2 ( )n r . The Fourier transform of the cladding shows a

weak background of refractive index coefficients, 2 ( )n 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

23

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 cos( / 6)π′Λ = Λ . (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 2 ( )n 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.

24

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,

2 2 2 2 2 2T 0 ,cok p k nτ β= + = −

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,

1 1 2 2B 1 1 2 2 1 1 2 2

2 2 1 1

1cos( ) cos( )cos( ) sin( )sin( ),2

p pk A p h p h p h p hp pξ ξξ ξ

′Λ = = − +

(2.5)

where Bk is the Bloch wavevector normal to the ML stack and the subscripts 1i =

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 2ni

2 − q2 , hi is the layer thickness ( 1 2h h ′+ = Λ ),

and ξi = 1 for TE and ξi = ni−2 for TM polarization. Stop-bands occur when Bk 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

1A > or 1A < − , 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

/effn kβ= , 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

25

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 zk 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 y being normal to the ML

planes and z showing the propagation direction.

Bloch k-vector, Im Bk . The dark blue shows regions where Im 0Bk = ( 1A < 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:

2 201( / )eff con n z k a= − (2.6)

where a is the core radius and 01 2.4048z = 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 effn . The dispersion of the core

mode, Eq. (2.6), is shown in Fig. (2.7) with white solid lines for an air-filled core,

1con = and different values of the core radius.

26

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 Bk . 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 eff con n= 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.

27

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 1 2h h′Λ = + , where 1h indicates their thickness

and 2h is the thickness of the air gap between them. Here, the variable sh 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 1cos ( / )inc eff con nα −= , 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 2 1n = 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, 2| |r , 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

(2.9b) for 2N = and two values of 0.98sh ′= Λ and 0.96sh ′= Λ . In both cases we

28

Figure 2.9: Amplitude and the phase of the reflection coefficient of the ML stack for (a)

0.98sh ′= Λ and (b) 0.96sh ′= Λ .

have chosen the following parameters for ML stack: 1 0.02h ′= Λ , 2 0.98h ′= Λ and

0 20k π′Λ = , where 1 2h h′Λ = + . The refractive index of the glass is chosen to be

1 1.46n = in all cases. As can be seen from the figure, the ML stack shows a loss

window near 0.9985effn = 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 0N = (a single glass

layer with distance 0.96sh ′= Λ from the substrate) to 2N = . In contrast to the

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

29

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,

(1) (2)0 T T 0 T T 0 T T

1ˆ ˆJ ( )exp( ) H ( ) H ( ) exp( ),2

i z i zβ β = ⋅ = ⋅ + ⋅ E y k r y k r k r (2.7)

30

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 Tk indicates

the transverse k-vector of the incident plane wave to the ML stack and mϕ 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 ( )2 2 2T co ˆ ˆcos sink n β φ φ= − +k x y . 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, mn , 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 TEPWout PW PWout PWcos exp( ), sin exp( ),m mE A i z E A i zφ β φ β= ⋅ = ⋅ (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 2 2PW TE TM PW( )sin ( )cos exp( )m mE r r A i zτ β φ τ β φ β= , + , , (2.9)

31

Defining the dimensionless quantities Tk xχ = , Tk yη = and Tk zζ = , the transverse

part of the outward-going y-polarized field at the flat core boundary 0 0Tk yη η= = of

the m-th ML stack takes the form (see Appendix A),

(1) 2 2out 0 0

1 H [ .2

E χ η= + ] (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,

1

0 01

1 ( , ) exp( ) ,2out n n nE F i dτ η τ χ τπ

+

−= ∫ (2.11)

where /n Tkτ τ= is the normalized, dimensionless wavevector component parallel to

the interface and evanescent waves (for which 1nτ > ) are neglected – in any case

these do not contribute to the guided core mode.

Making the change of variable cos( )n uτ = , where [ ]0,u π∈ , we may write the

full reflected field amplitude from the m-th ML stack as (see Appendix A, Eqs. A9

and A10),

( ) 0 sin( )ref 2 2 cos ( )0 TE TM

0

1( , ) ( , sin ( , cos .u

i u i um m m

u

E r u r u e e duπ

η χχ η β φ β φπ

=−

=

= ) + )∫ (2.12)

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,

32

0

0

0

0

2/ 36

ref in*0 0

1 / 32/ 36 2in

01 / 3

( , ) ( , )

.

( , )

m mm

mm x

E E d

E d

η

χ η

η

η

χ η χ η χ

χ η χ

= =−

= =−

Γ =∑ ∫

∑ ∫ (2.13)

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, Bz for

all parallel rays making up the mode given by,

hexB 2 2

6 3,

/ 1eff

co eff

az

n n=

− (2.14)

leading to the following formula for the propagation loss of the fibre in dB/m,

6

01dB/m 102

5 10 log ,3 3 eff eff

zn a

λαπ

− ×= Γ (2.15)

where both wavelength,λ , and radius, aeff, are given in microns. If we assume

10log effnΓ to be a slowly varying function of core radius, then the loss shows an

inverse quadratic dependence on the core radius, 2/ /dB m aα λ∝ , while in the case of

hollow dielectric waveguides this dependency is cubic, i.e. 2 3/ /dB m aα λ∝ . This

results partly explains the good performance of the kagomé fibre, / ~ 1dB mα , despite

having a relatively small core size, a ~ 10 - 15 μm, as compared to that of a hollow

dielectric waveguides, a ~ 50 μm.

33

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 2 3/ /dB m aα λ∝ . 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 2λ , Fig. (2.12b) and 31/ a , 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 high-

loss 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

34

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 H2O-

filled 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.

values given by 212 1 /m h n mλ = − for integer m [Pearce et al., 2007]. Using n = 1.45

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

35

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. 2base H On n= , 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

36

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.

37

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 core-

cladding 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.

38

39

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 HC-

PCF, 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

40

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 , ,p s asω and , ,p s asβ 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 sω . 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 ( )2ˆ ˆ 2as s pβ β β⋅ = − ⋅ = + −1Δk z OP OP z . 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].

41

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],

t∂

∇× = −∂BE (3.1a)

(1)

t tµ µ εµ σ∂ ∂

∇× = + +∂ ∂

NLD PB E (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†, (1)D is the linear displacement vector, NLP is the nonlinear polarization and

σ is the power absorption coefficient of the medium. The linear part of the

polarization (1)P is related to the linear part of the dielectric displacement vector (1)D via the relation (1) (1)

0ε= +D E P . For a homogeneous medium, this relation can

be expressed via the electric permeability of the medium as (1) ε=D 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 (1) 2

22 2t t t

µ εµ σ µ∂ ∂ ∂∇ − − =

∂ ∂ ∂

NLD E PE . (3.2a)

For a lossless, nonmagnetic material 0µ µ= and 0σ = , so that Eq. (3.2a) can be

written as

* The relation between ( )ε µ of the medium and 0 0( )ε µ of the vacuum can be expressed via the relative values of medium electric permittivity and magnetic permeability, i.e. 0rε ε ε= ( 0rµ µ µ= ). For nonmagnetic material 1rµ ≅ † The relation between c and the magnetic permeability

0µ and electric permeability 0ε of vacuum is given by 2

0 0 1cµ ε =

42

2 (1) 2

22 2 2 2

0 0

1 1c t c tε ε

∂ ∂∇ − =

∂ ∂

NLD PE . (3.2b)

In deriving (3.2a) we have used the vector identity 2( . )∇×∇× = ∇ ∇ −∇E E E and the

assumption that . 0∇ =E . This is identically true for transverse electromagnetic plane

waves*. Assuming the plane wave propagation of linearly polarized, quasi-

monochromatic 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,

1

1( , ) exp[ ( )] c.c.,2

N

j j jj

E z t E i k z tω=

= + +∑ (3.3a)

1

1( , ) exp( ) c.c.,2

NNL NL

j jj

P z t P i tω=

= +∑ (3.3b)

where c.c. represents the complex conjugate of the first term. In Eqs. (3.3) jω and

( / )j j jk n c ω= are the carrier frequency and k-vector of the jth plane wave component

of the field, respectively, 0 0/( )j jn ε µ ε µ= is the refractive index of the medium at

the carrier frequency jω , and N is the total number of waves present. Here jE and

NLjP 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,

21 exp( ).

2 2j j j j NL

j j j jj

n E E iE P ik z

c t z kω

σ µ∂ ∂

± + = − ±∂ ∂

(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.

43

In deriving Eq. (3.4) we have used the slowly varying approximation for the envelope

functions, i.e. 2 2/ /j j jE z k E z∂ ∂ << ∂ ∂ , 2 2/ /j j jE t E tω∂ ∂ << ∂ ∂ , and

2 2 2/NL NLj j jP t Pω∂ ∂ ≅ − . The latter condition is justified if the induced change in the

nonlinear polarization occurs on a time scale that is much longer than 1jω− . 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,

[ ] -1 -1 -13C m C m V [m ] Vm:ε= μ α E (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,

ε α=μ E (3.6)

Under the influence of the electric field 0i tE E e ω= of the incident optical field,

the induced dipole in the electronic cloud of the molecule starts to oscillate with

44

frequency ω modulated with the natural vibrational frequency of the molecule itself,

ωΩ << . If we assume a harmonic oscillation of the molecule, then Ω represents the

frequency of the vibration of the inter-nuclear distance, 0 i tv vq q e Ω= where 0

vq 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 vq , that is,

00

( ) ( ) ...v

vv q

t q tqαα α

=

∂= + + ∂

(3.7)

Keeping just the first two terms in the above expansion (small atomic displacements)

and entering the explicit form of vq into (3.7), we see that the polarizability of the

molecule oscillates with the natural frequency of the molecular vibration,

00

0

( ) cos( ).vv

t q tqαα α

∂= + Ω ∂

(3.8)

Substitution of Eq. (3.8) into Eq. (3.6) leads to

[ ]

00 0 0

0

00 0 0

0

cos( ) cos( )cos( )

1cos( ) cos( ) ) cos( ) ) .2

vv

vv

E t E q t tq

E t E q t tq

αµ α ε ω ε ω

αα ε ω ε ω ω

∂= + Ω ∂

∂= + −Ω + +Ω ∂

(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 anti-

Stokes scattering). It is interesting to note that elastic scattering occurs via 0α .

45

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 0 02 /k π λ= , 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,

01( ) exp[ ( )] c.c.,2vq t q i k z t= − +Ω + (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 2T , 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 Pτ , that is

when 2P Tτ < . As will be shown later, this term is introduced as a phenomenological

46

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, 21/TΓ = in the differential equation governing the evolution of the

material excitation amplitude. Moreover, Γ is equal to the half line-width (FWHM)

of the Raman transition, i.e. 2/ 2 1/(2 )R Tδω = Γ = . The dephasing time is the time

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 11/T , the inverse lifetime of

the vibrational/rotational state. However, since typically 2 1T T<< , the damping of the

molecular excitation, ′Γ + Γ , is entirely controlled by 21/TΓ = .

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 vq . The Hamiltonian for the light field coupled to the molecular

vibrations is given by

radiation vibration interaction ,H H H H= + + (3.11)

47

where the Hamiltonian* for the radiation field, molecular vibrations and the light-

matter interaction are given by,

2 2radiation

0

1 1( ),2

H εµ

= +E B (3.12)

2 2 2vibration

1 ( ) ( ),2 v i v iH Nm q q= +Ωr r (3.13)

interaction

0

121 1 ( / ) ,2 2 v v

H N

N N q q

ε α

ε α ε α

= − ⋅

= − ⋅ − ∂ ∂ ⋅

E E

E E E E (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,

vv

Hpq∂

= −∂

, (3.15a)

vv

Hqp∂

=∂

, (3.15b)

where v vp mq= is the generalized momentum of vq . 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

2

22

0

22v v v

v

d dq q qdt dt m q

ε α ∂+ Γ +Ω = ⋅ ∂

E E (3.16)

The phenomenological damping term 22 ( ) (2 / )( )v vdq dt T dq dtΓ = has been added to

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.

48

Based on Eq. (3.16), the material excitation can be viewed as a damped, forced

oscillation with the time dependent force given by 20( ) ( / 2)( / ) ( )vF t q tε α= ∂ ∂ E .

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,

1 exp[ ( )] exp[ ( )] c.c.2 p p p s s sE E i k z t E i k z tω ω= − + + − + + (3.17)

and the material coherence is given by Eq. (1.10). Here we consider the resonant case,

where p sω ω− = Ω , and the wave vectors obey the relation, 0p s− =k k k . The basic

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 slowly-

varying 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 ( )Bk TΩ<< , the population difference between the ground and excited state reaches the value one, i.e. 1Ground Excitedn n− ≈ . Here Bk is the Boltzmann factor and T is the temperature of the medium.

49

*

2 0

14 p s

v

q iq E Et T m q

ε α ∂ ∂+ = − ∂ Ω ∂

(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. / 1Rδω Ω << . The *p sE E term in (3.18)

represents the beat note of the pump and Stokes fields that drives the material

excitation at the resonance frequency p sω ω− = Ω .

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),

00 0

1 exp[ ( )] c.c.4

exp[ ( )] exp[ ( )] c.c.

NLv

v v

p p p s s s

P N q E N q i k z tq q

E i k z t E i k z t

α αε ε

ω ω

∂ ∂= = − +Ω + ∂ ∂

× − + + − + +

(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)

( )( )*

0

1 1 1 +c.c.2 2 2

p ps s i k z ti k z tNLp s

v

P N q E e q E eq

ωωαε − +− + ∂= + ∂

(3.19b)

The component that oscillates at frequency sω is the Stokes nonlinear polarization

and its amplitude is given by,

*

0

12

sik zNLs p

av

P N q E eqαε − ∂

= ∂ (3.20)

The component that oscillates at frequency pω is the pump nonlinear polarization and

its amplitude is given by,

50

0

1 .2

Pik zNLp s

av

P N q E eqαε − ∂

= ∂ (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,

0

1 1( , ) ,4 2

pp s p p

p p av

iNE z t q E E

v t z n c qω α σ

∂ ∂ ∂+ = − − ∂ ∂ ∂

(3.20)

*

0

1 1( , ) ,4 2

ss p s s

s s av

iNE z t q E Ev t z n c q

ω α σ ∂ ∂ ∂

+ = − − ∂ ∂ ∂ (3.21)

where / , ,j jv c n j p s= = is the phase velocity of the pump and Stokes waves.

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 122 ~ 10 sT − . 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.

(3.18). By introducing the laser intensities 2

0( / 2) , ,j j jI cn E j p sε= = we arrive at

the following rate equations for the SRS in the steady-state interaction regime,

1 ,

1 ,

p pp p s p p

p

s ss p s s s

s

I Ig I I I

z v tI I g I I Iz v t

σ

σ

∂ ∂+ = − −

∂ ∂

∂ ∂+ = −

∂ ∂

(3.22)

where , ,jg j p s= is the gain factor for pump and Stokes and defined as

2 2 22 1 0[m/W] /(4 )j j jg N T c nω κ ε= .

51

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 2T . In this case the width of the Raman transition is inversely

proportional to 2T and plays an important role in the precise description of the laser-

matter 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].

52

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

1

1ˆ ,tN

i iitN

ρ=

= Ψ Ψ∑ (3.23)

where iΨ 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 ip , this

summation can be written as 1

ˆtN

i i iipρ

=

= Ψ Ψ∑ . 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,

ˆˆ ˆ ,d i Hdtρ ρ =

, (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

0ˆ ˆ ˆH H= − ⋅μ E . (3.25)

Here 0H is the Hamiltonian of the system in the absence of any external perturbation

and μ is the electric dipole operator. The wavefunction of the system can be

expanded as a function of the unperturbed eigenstates of the unperturbed Hamiltonian

nϕ which form a complete orthonormal set, i.e. |m n mnϕ ϕ δ= and

53

0 n n nH ϕ ω ϕ=

. nω is the frequency associated to the molecular state (level) n.

This expansion can be formally written as

ii n n

na ϕΨ =∑ , (3.26)

where 2i

na gives the quantum mechanical probability of finding the system in the

state nϕ . 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

[ ]0ˆˆ ˆ ˆ ˆ, ,d i iH E

dtρ ρ ρ µ = −

. (3.27)

In order to obtain the equation of motion for the density matrix elements

ˆ ˆ| |mn m nρ ϕ ρ ϕ= we multiply the left and right hand side of Eq. (3.27) by mϕ and

nϕ respectively. Rewriting Eq. (3.27) by matrix elements we have,

( )mn n m mn ml ln ml lnl l

d i ii E Edtρ ω ω ρ µ ρ ρ µ= − + −∑ ∑

, (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,

2

1( ) ( )mn nm mn mi in mi in mm m n mm m n

i m

d i ii E Edtρ ω ρ µ ρ ρ µ µ ρ ρ µ′ ′ ′ ′

′=

= + − + −∑ ∑

(3.29)

where nm n mω ω ω= − and the second summation is over all off-resonantly driven,

intermediate states, m′ .

54

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

element *12 21 0µ µ= = and the diagonal elements of the electric dipole matrix would

be identically zero, i.e. ˆ| | 0mm m mµ µ= = . Using these relations we arrive at the

following set of equations of motion for the relevant density matrix elements,

1 1 1 1 11 2 12( )m m m m m m md i ii E Edtρ ω ρ µ ρ ρ µ ρ′ ′ ′ ′ ′ ′ ′= − − −

, (3.30a)

2 2 2 2 22 1 12( )m m m m mm md i ii E Edtρ ω ρ µ ρ ρ µ ρ′ ′ ′ ′ ′ ′ ′= − − +

, (3.30b)

12 21 12 1 2 2 1( )m m m mm

d ii Edtρ ω ρ µ ρ µ ρ′ ′ ′ ′

′

= + −∑

, (3.30c)

22 2 2 2 2( )m m m mm

d i Edtρ µ ρ µ ρ′ ′ ′ ′

′

= − −∑

. (3.30d)

In order to eliminate explicit appearance of the irrelevant off-diagonal density matrix

elements, , 1,2im iρ ′ = 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 imρ ′ into a slowly and a fast varying part, i.e.

( ) ( ) exp( )im im m it t i tρ σ ω′ ′ ′= . 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 imρ ′ in

equations (3.30a) and (3.30b) and formally integrating, we obtain,

1 ( )1

1 11 2 12

( )

[ ( ) ( )] ( ) ( ),

m

ti t t

m

m m m m

it dt e

t t t E t

ωρ

µ ρ ρ µ ρ

′ ′−′

−∞

′ ′ ′ ′

′= −

′ ′ ′ ′× − +

∫ (3.31)

2 ( )2

2 22 1 12

( )

[ ( ) ( )] ( ) ( ).

m

ti t t

m

m m m m

it dt e

t t t E t

ωρ

µ ρ ρ µ ρ

′ ′−′

−∞

′ ′ ′ ′

′= −

′ ′ ′ ′× − −

∫ (3.32)

55

The diagonal elements of the density matrix, mmρ give the probability of state m

being populated. Since the intermediate levels are driven far-off resonance, they will

not be populated and 0m mρ ′ ′ = . Defining the slowly varying amplitude ( )Q t , so that

( ) ( )12 ( ) ( ) p s p si t i k k zt Q t e ω ωρ − − −= 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 1mρ ′ and

2mρ ′ from explicitly appearing in Eqs. (3.30c) and (3.30d)*. If exact Raman resonance

is assumed then 21p sω ω ω− = = Ω . By inserting the expressions for 1mρ ′ and 2mρ ′

into Eq. (3.30c), we arrive at the following differential equation governing the time

evolution of 12ρ ,

( ) * *

12 12 1

( )22 11

( ) ( ) ( / 4) ( , ) ( , )

[ ( ) ( )],p s

p s

i t i k k z

d t i t i E z t E z tdt

e t t

ρ δ ρ κ

ρ ρΩ − −

= Ω+ +

× − (3.33)

where 1κ is given by

1 2 121 1

1 1 1m m

m m p m s

κ µ µω ω ω ω′ ′

′ ′ ′

= +

− + ∑

, (3.34)

and 1 2δ δ δ= − is the Stark shift in the levels’ frequencies, and , 1,2i iδ = is given by

222

2

1 1 1

1 1 .

i m i pm m i p m i p

sm i s m i s

E

E

δ µω ω ω ω

ω ω ω ω

′′ ′ ′

′ ′

= + + − + + + −

∑

(3.35)

In what follows we neglect the Stark shift, i.e. 0δ = . In SI units the coefficient 1κ

has the units of [ ] 2 2

2m C

1 J sκ = . In terms of dimensionless off-diagonal element of the

* 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].

56

density matrix ( )Q t and the population inversion 22 11( ) ( ) ( )n t t tρ ρ= − , the equation

of motion (3.33) is

*

*1

2

1 ( )4 p siQ Q E E n t

t Tκ∂

+ =∂

. (3.36)

Repeating the same line of calculation for 22 ( )tρ and 11( )tρ we arrive at the

following equation of motion for the population inversion ( )n t

* * * * * *1 1 1

1 1( ) Im 2 2p s p s s pn t i E E Q i E E Q E E Q

tκ κ κ∂

= − =∂

. (3.37)

As mentioned earlier in most of the cases of interest, the population inversion is

negligible, i.e. / 0n t∂ ∂ = and 22 11( ) ( ) 1t tρ ρ− ≈ − . 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, 12ρ and it corresponds to the spatial

correlation of the molecular excitations. This field is attenuated on the time scale 2T ,

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, ( )vq t , we

evaluate the expectation value of the material excitation amplitude operator using the

density matrix approach. From the definition of ρ , given in Eq. (3.23), it is

57

straightforward to show that the expectation value of any observable quantity can be

evaluated via the relation

ˆ ˆ ˆˆ ˆ( ) | | | |k k

A A k k k A kρ ρ′

′ ′= = ∑∑Tr , (3.38)

where A is the Hermitian operator associated to A . Using Eq. (3.38) we have

ˆ ˆˆ ˆ ˆ( ) | | | |vk k

q q q k k k q kρ ρ′

′ ′= = =∑∑Tr . (3.39)

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 0υ = and

the excited state 2 with vibrational quantum number 1υ = , so that the summation in

Eq. (3.39) simplifies to

21 12 12 21ˆ ˆˆ ˆ ˆ1| | 2 2 | |1 2 | |1 1| | 2q q q q qρ ρ ρ ρ= + = + . (3.40)

In Dirac’s treatment of a quantum mechanical harmonic oscillator one expresses the

position operator q in terms of the non-Hermitian annihilation and creation operators,

a and †a as

12 †ˆ ( )

2q a a

m = + Ω

. (3.41)

The operators a and †a have the following properties

, 1,a n nυ υ υ= − , (3.42a)

† , 1 1,a n nυ υ υ= + + , (3.42b)

where n is the electronic quantum number. Using Eqs. (3.41) and (3.42) we have

58

12

21 ˆ2, 1| |1, 02

q qm

υ υ = = = = Ω , (3.43)

Thus Eq. (3.40) becomes

12

12 21ˆ ( )2vq qm

ρ ρ = = + Ω , (3.44)

or based on the definition of ( )Q t ,

01/ 21(2 / ) c.c.2

i t ik zvq m Qe Ω −= Ω + . (3.45)

By comparing equations (3.45) and (3.10) we can easily relate ( )Q t to the slowly

varying amplitude of the material excitation 1/ 2( ) (2 / ) ( )q t m Q t= Ω . Using this

relation we can rewrite Eq. (3.36) for q as

12 * *

12

1 24 p siq q E E

t T mκ∂ + = − ∂ Ω

(3.46)

In writing (3.46) we have assumed 22 11 1n ρ ρ= − ≅ − . Now by comparing Eq. (3.46)

with its classical counterpart, Eq. (3.18) we see that the coefficient 1κ is related to the

molecular polarizability via

12*

10

12 vm q

ακ ε ∂ = Ω ∂

. (3.47)

59

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 1κ

Coherence: *

*1

2

1 ( )4 p siQ Q i Q E E n t

t Tκ∂

+ = ∆Ω +∂

, (3.48a)

Pump field: 21 1

2p p

p s p pp s s

vE i Q E E

z v t vω

κ σω

∂ ∂+ = − − ∂ ∂

, (3.48b)

Stokes field: *2

1 12s p s s

s

E i Q E Ez v t

κ σ ∂ ∂± + = − − ∂ ∂

, (3.48c)

Population difference: * * *1

1

( )( ) Im p sn t nn t E E Q

t Tκ∂ −

+ =∂

. (3.48d)

where * 22 1 0/(2 )s sN v cκ ω κ ε= and ± refers to forward (plus) and backward (minus)

SRS scattering . In Eqs. (3.48) ( )p sω ω∆Ω = Ω− − 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. 1T is the time scale for de-

excitation 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).

60

61

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 anti-

Stokes Raman scattering (CARS) [Eesley, 1981]. ASRS generates an optical field at

frequency ,2a p s p sω ω ω ω= − > where pω and sω are the pump and the Stokes field

frequencies whose difference is resonant to a Raman transition frequency,

p sω ω− = 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: 2 p s a∆ = − −k k k k . In media

with dispersion, the condition 0∆ =k can only be fulfilled for non-collinear

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).

62

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.

63

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 iω ,

, ,i p s a= are described by a density matrix equation, and the propagation of the fields

( , ) (1/ 2) exp[ ( )] c.c.i i i iE z t E i t k zω= − + , 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 22 1ρ << , 11 1ρ ≈ where mnρ are the density matrix

elements of the transition), the resonant response of the system obeys the following

differential equation (see Appendix B)

* *

2

1 ( )i k zs p s a a p

dQ Q i A A A A edt T

κ κ ∆+ = − + (4.1)

Here ˆk∆ = ⋅Δk z is the phase mismatch in the direction of propagation, and sκ and aκ

are the two-photon matrix elements associated with the Stokes and anti-Stokes

processes and are determined by

1 11 2 1 12

1 11 2 1 12

1 [( ) ( ) ],4

1 [( ) ( ) ].4

s m m m p m sm

a m m m a m pm

κ µ µ ω ω ω ω

κ µ µ ω ω ω ω

− −′ ′ ′ ′

′

− −′ ′ ′ ′

′

= − + +

= − + +

∑

∑

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 / 0dQ dt ≈ , e.g. at high gas

pressures, the nonlinear polarization can be written as

* *2 ( )i k z

s p s a a pQ iT A A A A eκ κ ∆= − + (4.2)

64

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),

*2

0

1 ( )2

i k zp p p s s a a

p

NA i v QA Q A ez v t c

ω κ κε

∆ ∂ ∂

+ = − + ∂ ∂

(4.3)

The evolution of the anti-Stokes field is governed by

20

12

i k za a a a p

a

NA i v QA ez v t c

ω κε

− ∆ ∂ ∂+ = − ∂ ∂

(4.4)

Using Eq. (4.2) and introducing amplitudes and phases for the fields, jij jE a e ϕ= , and

assuming equal group velocities p s av v v= = , we arrive at the following set of

equations for the steady-state:

2 2 2 1( )2

pp p a s p p

daa q a a a

dzω σ= − − (4.5a)

2 1( cos )2

ss p s a s s

da a a qa adz

ω θ σ= + − (4.5b)

2 1( cos )2

aa p a s a a

da qa qa a adz

ω θ σ= − + − (4.5c)

2 ( ) sins ap a s

a s

a ad qa kdz a aθ ω ω θ= − + ∆ (4.5d)

In Eqs. (4.5) the field frequencies are normalized to the Raman transition frequency,

/j jω ω→ Ω , the field amplitudes are 0/j ja a a→ , where 0a is the normalization

amplitude and the distance z is in units of 2 2 2 10 2 0 0[ /(2 )]p sL N T v a cκ ε −= Ω , where N

is the molecular concentration. The coefficients jσ describe linear loss and the

parameter /a sq κ κ= . The function 2s a p k zθ ϕ ϕ ϕ= + − + ∆ is the phase difference of

the fields in which k∆ is normalized to 0k L∆ . Here, we show the derivation of Eq.

(4.5c) for the anti-Stokes component. For simplicity, we neglect attenuation, setting

65

0jσ = . 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 2T , we put

0aEt

∂≈

∂ and Eq. (4.4) simplifies to

( )

222

20

(2 )

2

( / ) .

a

p s a

ia a a s aa a p

s

i k z is a s a

a N v Ti a e az z c

a e a e

ϕ

ϕ ϕ ϕ

ϕ κ κωε κ

κ κ− −∆

∂ ∂ + = − ∂ ∂

× +

(4.6)

in our original unnormalized variables. Multiplying both sides of Eq. (4.6) by aie ϕ− ,

applying our normalization, and equating the real and imaginary parts, we arrive at

following equations for the anti-Stokes field phase and amplitude,

2 ( cos )aa p a s

da qa qa adz

ω θ= − + , (4.7a)

2 sina sp a

a

d aqadz aϕ ω θ= . (4.7b)

The dispersion of the propagation constant of gas-filled HCPCF,

( ) ( ) ( / ) ( )j j j jk c nω β ω ω ω= + ∆ , contains the contributions of the HC-PCF [ ( )]jβ ω

and the Raman gas [ ( )]jn ω∆ . It is assumed that the Stokes field develops from

quantum noise, i.e., 0 0(0)s s pa a a= << , while the anti-Stokes signal is generated

parametrically starting from zero intensity, (0) 0aa = [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

( , )a s pa a a<< . We assume that the pump is not depleted, i.e. 0p pa a≈ . At the

beginning of parametric ASRS a sa a<< , 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 ( )zθ evolves toward a stable value

66

1 2sin [ /( )]a a s pk a q a aθ π ω−= + ∆ , which is independent of the initial phase 0θ and lies

within the range / 2 2 3 / 2 2m mπ π θ π π+ < < + [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

/ 0d dzθ = , we find from Eqs. (4.5b) and (4.5c) that the amplitudes sa and aa

increase with z exponentially at the rate

2 20 ( 1) / 2 ( 1) / 4 sin g g η η η θ= − − + − + (4.8)

where 20 0s pg aω= is the Stokes gain (the gain in the absence of the anti-Stokes field)

and 2( / )a s qη ω ω= . To derive Eq. (4.8) let us define /a sa aξ = . Inserting this in Eq.

(4.5d) and assuming / 0d dzθ = we have

2

2 22 sin 4 sina

sq qωδ δξ

θ θ ω= + + (4.9)

where 0/k gδ = ∆ . 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

0 ( cos )a s as

s

da da g qa qdz dz

ωξ ξ θω

= = − + (4.10)

Using Eq. (4.5b) we have

0 0(1 cos ) ( cos )as s

s

g a q g a q qωξ ξ θ ξ θω

+ = − + (4.11)

From Eq. (4.11) we obtain a quadratic equation which gives ξ in terms of

parametersη , q and θ

67

2

21 1 (1 ) sincos 2 4q

η ηξ η θθ

+ − = − ± +

(4.12)

The gain for Stokes g defined through Eq. (4.5b) is given by

2

2

0

1 (1 )1 cos sin2 4

g qg

η ηθξ η θ− −= + = + + (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

22 2 2 2 2

2 ( 1) ( 1)sin8 4 8

η δ δ η δθη η η

− + − += + −

, (4.14)

It is easy to see that the dynamics of the locked fields are described by the

expressions,

0( ) g zs sa z a e= , (4.15a)

00

cos( )( / )

g za sa z a e

q g gη θ

η= −

+, (4.15b)

showing that a small Stokes signal 0sa 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 ( 1δ >> ), it follows from Eqs. (4.5d)

and (4.14) that / 2θ π→ for 0k∆ < and 3 / 2θ π→ for 0k∆ > , and the gain takes

its maximum value 0g g= . 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 1δ << ),

the phase difference θ π→ , and growth of the Stokes and anti-Stokes fields is very

slow due to parametric gain suppression ( 0g → ) [Shen et al., 1965;

68

Figure 4.2: The initial stage of a phase-locked resonant ASRS. The amplification of the anti-

Stokes field intensity 2a aI a= [the lines labeled with (a)-(c)] and the corresponding locked phase

difference θ as a function of the normalized wavevecotr mismatch 0k L∆ for increasing values

of pump intensity: (a) 2 20/ 0.8pa a = , (b) 2 2

0/ 1.6pa a = and (c) 2 20/ 2.7pa a = . The corresponding gain

factors are: (a) 0 5g z = , (b) 0 10g z = and (c) 0 18g z = .

Bloembergen, 1967]. This is caused by destructive interference of the contributions of

the Stokes and anti-Stokes SRS to polarization, Eq. (4.2) for 0k∆ = and

( ) ( )s aa z qa 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

532 nmpλ = was used to generate first Stokes and anti-Stokes fields at 683 nmsλ =

and 435 nmaλ = in a HC-PCF filled with hydrogen. The pump pulse intensity was

approximately 2300 MW/cm , resulting in a Stokes gain -10 0.5 cmg ≈ . Figure (4.3a)

shows the dynamics of the pump, Stokes, and anti-Stokes intensity (in inset) as a

69

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

shift jωΩ << , the condition 0k∆ = can be approximated as 2 2 2/ 0k ωΩ ∂ ∂ = . To see

this, we expand ( )k ω as a Taylor expansion around pω ,

2 2

22( ) ( ) ( )

2p pk kk k Oω ωω ω∂ Ω ∂

+Ω = +Ω + + Ω∂ ∂

, (4.16a)

2 2

22( ) ( ) ( )

2p pk kk k Oω ωω ω∂ Ω ∂

−Ω = −Ω + + Ω∂ ∂

. (4.16b)

70

Using the definition of the phase mismatch 2 ( ) ( ) ( )p s ak k k kω ω ω∆ = − − and Eqs.

(4.16) we obtain 2 2 2/k k ω∆ = Ω ∂ ∂ . The relation 0k∆ = implies operation at zero

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 12ρ does not depend on the molecular concentration [Bloembergen,

1967; Shen et al., 1965]. Hereby, that by varying the gas pressure along the fiber, one

can setup the wave mismatch ( )k z∆ in an optimum way. Although 0k∆ = is a

necessary condition for efficient ASRS, it does not automatically lead to higher

efficiency [Duncan et al., 1986]. When 0k∆ → , the exponential gain goes to zero

because a coupled wave with ( ) ( )s aa z qa z≈ is generated. Once such a wave is

formed, the Raman process terminates. As follows from Eq. (4.5), this structure

develops at a length 2 10~ [( ) ]as a s pz L aω ω −= − which can be treated as the conversion

length for phase-matched ASRS. When ASRS starts from a weak Stokes signal

(quantum noise), the anti-Stokes field produced at the length asL 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 anti-

Stokes 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 ( 0k g∆ >> ) 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

the second region where phase-matched ASRS occurs. The condition 0k∆ = is set by

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:

2 2 2 20 0

1 (1 )( ) , ( ) ,1 1

z

p p s sz zK K ea z a a z a

Ke Ke

β

β β

+ += =

+ + (4.17)

71

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%.

where 2 20 0( / ) /p s s pK a aω ω= and 2 2

0 0 02 ( / / ) /s p s pa a Lβ ω ω= + . The length 1L of the

first interaction region is determined from the inequality,

1 0

122 /( )2 2

0 0 2 2

4( / ) 1( )

s pL L as p

a s

a a eq

ω ω ωω ω

−

≥ − − . (4.18)

Maximum conversion to the anti-Stokes field is achieved at a distance 1 1optL L= that

satisfies Eq. (4.18). The optimum anti-Stokes field is then given by

12 2 2

1opt22 2

( ) ( )( ) 12 4

p a sa

a

a L qaq

ω ωω

− −

∞ = −

, (4.19)

72

where 21opt( )pa L is pump intensity Eq. (4.17) corresponding to the optimum length

1optL . 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 2L

should be larger than the above introduced conversion length asL , i.e.,

2 2 12 1opt[4 ( ) /( )]as s pL L N a L cπ κ −> = Ω . 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 1L for region one (with 3 cmL∆ < ) 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.

73

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 co-

propagate 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].

74

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:

2 /( , ) sech ,ss

s

t z vI z tτ

−∝

(5.1)

75

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 sv and having a duration 2 /s s sT Gτ γ= , where sγ is

the linear loss coefficient and sG is the steady-state Raman gain. According to Eq.

(5.1), for a sufficiently high Raman gain (or low loss), so that the condition

/ 1s sGγ << is fulfilled, the Stokes pulse duration can be much shorter than 2T . 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 2T , 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 2T . 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- 2T 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,

76

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 2T . 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

77

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

78

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

1J = and 3J = 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

collisions (MC) is pressure dependent (for H2 the broadening is MC 50 MHz/barν∆ = ),

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

collisions with the sidewalls (corresponding linewidth wall coll 200 MHzν∆ = ), resulting

in an effective phase relaxation time at 3 bar of 2 3.5 nsT = . In this pressure regime,

the energy relaxation time is significantly longer ( 1 15 nsT > ) [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 2T

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 2 3.5 nsT = . As the pump

79

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 2sech ( )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 2T . 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

80

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

( ) , , , ,1( , ) ( , ) exp c.c. ,2p s p s p s p sE z t A z t i t ik zω= ± + (5.2)

Here ,p sA are the complex amplitudes and ,p sω the carrier frequencies of the fields,

which are two-photon resonant with the Raman transition, i.e., p sω ω− = Ω . The

propagation constants of the guided modes in the fiber are , , ,( )p s p s p sk k ω= . Nonlinear

81

propagation of the fields is described by the coupled wave equations [Eqs. 3.48(b) and

(c)]:

1

0 0

1 12 2

p pp s p p

r

A A Ni A Q Az v t cn

κ ω σε

∂ ∂− + = − −∂ ∂

, (5.2a)

*1

0 0

1 12 2

s ss p s s

r

A A Ni A Q Az v t cn

κ ω σε

∂ ∂+ = − −

∂ ∂ . (5.2b)

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. 0p sv v v≈ ≈ and

p s rn n n≈ ≈ . Doppler broadening of the molecular frequency ( )p sω ω∆Ω = Ω− −

from the Raman resonance causes the two fields to form an inhomogeneous line shape

described by the normalized function ( )g ∆Ω , where ( ) ( ) 1g d∆Ω ∆Ω =∫ . 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

( , 1, 2)ij i jρ = :

*

*1

2

1 ( )4 p siQ Q i Q A A n t

t Tκ∂

+ = ∆Ω −∂

, (5.3a)

*1

1

( )( ) Im s pn t nn t A A Q

t Tκ∂ −

+ =∂

, (5.3b)

where Q is the Raman coherence and I have defined 11 22n ρ ρ= − as the population

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

82

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 2T . This is explained by the fact that when the Stokes

pulse becomes sufficiently strong (Fig. 5.7, 2 nst = ), 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, 2 nst > ). 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

83

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 /t z vτ = − where sv v> . I assume that the Stokes pulse is amplified

by an infinitely long pump wave with constant amplitude 0A . 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 iρ= − , obeying the differential equation:

2

22

2

1 sinT

βτ τ

∂ Ψ ∂Ψ+ = Ψ

∂ ∂ (5.4)

where I have introduced the function

( ) ( )2 dτ

τ α ρ τ τ−∞

′ ′Ψ = ∫ (5.5)

and the parameter 1/ 2

12 2

0 02p s

r

Ncn v v

ω ωκαε − −

= −

.

To arrive at Eq. (5.4) I rewrite Eqs. (5.2) using the new variables, i.e the retarded

time τ and ρ ,

84

( ) ( ) 1/ 2

0 0/pp s s

Av v v v Aα ω ω ρ

τ∂

= − − + ∂, (5.6a)

( ) ( ) 1/ 2

0 0/ss p p

A v v v v Aα ω ω ρτ

∂ = + − ∂. (5.6b)

Solution to Eqs. (5.6) can formally be written as,

( )0 cospA A αψ= , (5.7a)

( ) ( ) ( )1/ 2

0 0 0/ sins s pA A v v v vω ω αψ = + − , (5.7b)

where I have introduced the function ( ) ( ) ( )1(2 ) dτ

ψ τ α τ ρ τ τ−

−∞′ ′= Ψ = ∫ . 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,

1

2

14 p sn A A

Tκρ ρ

τ∂

+ =∂

. (5.8)

Using solutions (5.7) and definition (5.5), I arrive at Eq. (5.4) for the variable Ψ ,

where 2

2 2101 1

0 08 ( )s

r

N n Acn v v

ω κβε − −=

− .

I seek solutions of Eq. (5.4) with initial conditions ( ) ( ) 0′Ψ −∞ = Ψ −∞ = . The

behavior of such solutions depends on the dimensionless parameter 2Tβ and

describes nonlinear oscillations asymptotically approaching the equilibrium value

( ) πΨ +∞ = . This can be seen by ignoring the time derivative of Raman coherence in

Eq. (5.8), corresponding to ( ) 222arctan( )Teβ ττΨ = . However, in the limit of long

dephasing times, 2 1Tβ >> , the relaxation term in Eq. (5.4) can be neglected, and Eq.

(5.5) has the 2π − pulse solution,

( ) 04arctan[exp( / )]τ τ τΨ = . (5.9)

85

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

0 0 0 0/ sech /s s pA A v v v vτ ω ω τ τ = + − , (5.10a)

( ) ( )0 0tanh /pA Aτ τ τ= − . (5.10b)

The characteristic duration 0τ is given by the expression:

2

2 2 210 01 1

0 08 ( )s

r

N n Acn v v

ω κτ βε

−− −= =−

. (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 0τ ) will evolve towards a sech profile of the same

duration traveling at a velocity 0( )v τ 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.

86

87

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., 1 2,p T T ( p is the pulse

duration, and 1T and 2T 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.

88

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 self-

similarity 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

89

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 2p T [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( , ) ( , ) c.c.

2p s p si t k z

p s p sE z t A z t e (6.2)

where , ( , )p sA z t are the complex electric field amplitudes, , ,( )p s p sk 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 ( 1n n ), 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,

*

*1

2

1

4 p s

iQ Q A A

t T

(6.3a)

2

1 p pp s

p s s

vA i Q A

z v t v

(6.3b)

90

*2

1s p

s

A i Q Az v t

(6.3c)

Defining the dimensionless distance SRS/z L , by normalizing the distance z to

1/ 2 1/ 2 1SRS 0 12 [ ( ) ( ) ]s p p sL c N n n , 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, p sv v v , Eqs. (6.3) can be

rewritten in the following form,

*

*1

2

,4 p s

Q Qi A A

T

(6.4a)

,ps

Ai QA

(6.4b)

1 * .sp

Ai Q A

(6.4c)

Here, we let /( )p s s pn n 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

2

,4 p sA A

T

(6.5a)

,ps

AA

(6.5b)

91

1 .sp

AA

(6.5c)

The conservation of photon number (Manley-Rowe relation) following from Eqs.

(6.5b) and (6.5c) can be written as

2 22 2, 0,, 0,p ps s

p s p s

A AA A

(6.6)

where (0, )pA and (0, )sA 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

1/ 20

1/ 20

, cos / 2 ,

, sin / 2 ,

p

s

A A

A A

(6.7)

where 2 1 2 20 p sA A A . By inserting representation (6.7) into Eqs. (6.5),

and assuming the transient interaction regime ( 2p T ), the three equations in (6.5)

are reduced to only one equation for the function , :

2

210 sin .

4A

(6.8)

After introducing a new time variable 210 ( )

4T A d

, one arrives at

SGE:

2

sin .T

(6.9)

92

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

2

1sin 0

(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

1/ 22 , 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

0( ) I ( ) [ 0I ( ) 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 .

93

Figure 6.1: Behaviour of the universal function obeying Eq. (6.10) with the initial conditions

(0) 0 and 0(0) ; (1) 0 0.01 (solid blue curve) and (2) 0 0.02 (solid red curve).

For small arguments 0( ) I ( ) , where 0I ( ) 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 1.064 μmp . 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 1J and

3J 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 ( 1 2p bar). In this pressure region the Raman line is

94

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 2 5 nsT at 1 bar. The length of the HC-PCF

was 200 cmL . At 1 barp the effective Raman length was SRS 0.1 cmL , and the

gain G was as high as -10.15 cm 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 2~ 1 GW/cmpI , spectral broadening due to self-phase modulation (SPM) in

2H was negligibly small (the estimated nonlinear phase shift 510SPM ) [Agrawal,

2006]. Moreover, although the pump pulse duration was somewhat greater than 2T ,

95

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 2T .

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. 30 μJpE , 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 2 5 nsT , 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].

96

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

2 2( ) ( )s pA A , leading from Eq. (6.7) to 2 20 ( ) /( ) ( )s p p s pA n n A . Considering

now data sets resulting from dividing the intensity distributions of the output pump

97

,p LI and Stokes ,s LI pulses by the input pump pulse shape

00, ( )p pI I (where SRS/L L L is the normalized fiber length), we can

write

( ) ( )

2( )

0

( , )cos ,

( ) 2

i ip

ip

I L

I

(6.11a)

( ) ( )

2( )

0

( , )sin ,

( ) 2

i is s

ip p

I L

I

(6.11b)

where 1,2,i ; n labels the output pump and Stokes intensities in Fig. (6.4)

corresponding to the ith input pump pulse intensity 20 ( )pI . 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 iL , only on

the self-similarity variable:

( ) ( ) ( )( ) 02 2 ( ) ,i i ii pLT LI f d

(6.12)

where we have introduced the form factor ( ) ( ) ( )0 0( ) /i i i

p pf I I for the input pulses, ( )0iI

being the peak intensity of the ith pulse and 12 /( )pn c . Since Eq. (6.5) in the

transient regime ( 2T ) are invariant under a translation in , there is a possibility

for an offset, ( )ioff , so that one can define the similarity variable as ( ) ( ) ( )i i i

off

[Menyuk, 1993]. Here we have considered ( )ioff 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 ( )iT and ( )i . As a result the ratios

( ) ( )0( , ) / ( )i i

p pI L I and ( ) ( )0( , ) / ( )i i

s pI L I , when plotted as a function of ( )iT

(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

98

Figure 6.5: Dependence of the ratios ( ) ( )0/i i

P PI I and ( ) ( )0/i i

S PI I on the integrated intensity ( )iT 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 ( )iT for pump energies of 35, 45, 55, and 65 μJ . The

curves almost completely agree (within experimental error), suggesting that the late-

stage 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 2sin ( / 2) and 2cos ( / 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 ( )0iI , 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 / 2L for a launched pulse of

energy of 65 μJ will be identical to those at L for a pulse energy of 32.5 μJ .

99

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.

100

101

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%.

102

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

103

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

104

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 sine-

Gordon 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+

105

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 multi-

octave 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

106

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 phase-

matching 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 sum-

frequencies 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 1( ) / 2i iν ν ++ has

simultaneous contributions from 1( ) / 2, 2, 3,i j i j jν ν− + +

+ = , and any phase variation

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

107

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.

108

109

Appendix A

We are interested in the Fourier plane-wave spectrum of an outward-going Hankel

function at the flat boundary η =η0 ,

(1)0 0 0

1( ) ( , ) exp( ) ,2 n n nH F i dρ τ η τ χ τπ

+∞

−∞= ∫ (A1)

where cos( )χ ρ α= , 0 sin( )η η ρ α= = and 2 2 1/ 20( )ρ χ η= + as shown in Fig. (A1).

Here, nτ 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,

(1) cos( )( )( ) ,m

i w imwm C

iH e dwρρπ

+−= ∫ (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, 0( , )m nF τ η for (1) ( )mH ρ [Cincotti et al., 1993],

110

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 0η η= representing the core-cladding boundary.

20

2 10

20

1( 1) / 2 1/ 2 2 1/ 2 2

1 cos ( )/ 2 1/ 2 2 1/ 20

1( 1) / 2 1/ 2 2 1/ 2 2

(2 / ) ( 1) ( 1 ) 1

( , ) (2( ) / ) (1 ) 1 1

(2( ) / ) ( 1) ( 1 ) 1

n

n n

n

m mn n n n

i imim mm n n n

m mn n n n

i e

F e i e

i e

η τ

η τ τα

η τ

π τ τ τ τ

τ η η π τ τ

π τ τ τ τ

−

− −− −

− +− −

− −+ − −

− − − −∞ < < −= = − − − < <

− − − + < < ∞

(A3)

For 0m = the relations in Eq. (A3) take the following simple forms,

20

20

20

11/ 2 1/ 2 2 1/ 2

11/ 2 2 1/ 20 0

11/ 2 1/ 2 2 1/ 2

(2 / ) ( 1) 1

( , ) (2 / ) (1 ) 1 1

(2( ) / ) ( 1) 1

n

n

n

n n

in n n

n n

i e

F e

i e

η τ

η τ

η τ

π τ τ

τ η η π τ τ

π τ τ

− −− −

−−

− −−

− −∞ < < −= = − − < <

− − < < ∞

(A4)

Eq. (A4) decomposes (1)0 ( )H ρ to contributions from propagating, 1nτ < and

evanescent waves, 1nτ > . 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.

1(1)

0 0 01

1( ) ( , ) exp( ) .2 n n nH F i dρ τ η τ χ τπ

+

−= ∫ (A5)

It is easy to show that Eq. (A5) is equivalent to decomposition of the core mode

electric field, 0 T TJ ( . )k r , to positively and negatively travelling plane waves along

111

η -axis, ( )2exp 1 niη τ± − . Fig. (A2) shows the fundamental core mode intensity

distribution 20 T TJ ( . )k r as obtained using Eqs. (A4) and (A5).

The plane wave spectrum of athe perfectly inward-going cylindrical wave, (2)0 ( )H ρ can be obtained from Eqs. (A4) and (A5) via the following identity,

1(2) (1) * *

0 0 0 01

1( ) [ ( )] ( , ) exp( ) .2 n n nH H F i dρ ρ τ η τ χ τπ

+

−= = −∫ (A6)

Making the change of variable n nτ τ→ − and using the fact that 0 ( , )nF τ η is an even

function of nτ , Eq. (A6) can be rewritten in the following form,

1(2) *

0 0 01

1( ) ( , ) exp( ) .2 n n nH F i dρ τ η τ χ τπ

+

−= ∫ (A7)

Using the plane wave spectrum of (1)0 ( )H ρ and (2)

0 ( )H ρ along the planar interface

0η η= , Eqs. (A1), (A4) and (A6), we can write the reflected y-polarized field

distribution, ref0( , )mE χ η , as well as the field distribution for a perfectly inward-going

cylindrical wave as,

2

0

11ref 2 1/ 2

01

1( , ) ( , (1 ) ,n

n n

n

i im n n nE r e e d

τη τ τ χ

τ

χ η τ β τ τπ

=+− −−

=−

= ) −∫ (A8)

2

0

11inw 2 1/ 2

01

1( , ) (1 ) ,n

n n

n

i im n nE e e d

τη τ τ χ

τ

χ η τ τπ

=+− −−

=−

= −∫ (A9)

where the reflection coefficient for mth ML stack, rm is given by,

2 2TE TM( , ) ( , sin ( , cos .m n n m n mr r rτ β τ β φ τ β φ= ) + ) (A10)

Making the change of variable cos( )n uτ = , where [ ]0,u π∈ , we can rewrite Eqs.

(A8) and (A9) in the following form, which is more suitable for integration,

112

Figure A2: Normalized fundamental core mode intensity distribution 2

0 T TJ ( . )k r calculated

using the plane wave spectrum of cylindrical waves, Eq. A4. The white solid line circle shows the

first zero of the Bessel function 01 2.4048z = .

0 sin( )ref cos ( )0

0

1( , ) ( , ,u

i u i um m

u

E r u e e duπ

η χχ η βπ

=−

=

= )∫ (A11)

0 sin( )inw cos ( )0

0

1( , ) .u

i u i um

u

E e e duπ

η χχ ηπ

=−

=

= ∫ (A12)

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.,

0

0

0

00

0

2/ 36 6ref in* sin ( )

0 01 / 3 12 0

6/ 36 2 sin ( )in0

1 01 / 3

( , ) ( , ) ( , ) ( ),

( )( , )

i um m m

m m

i um

mm x

E E d r u e I u du

e I u duE d

η πη

χ ηπη

η

η

χ η χ η χ β

χ η χ

−

= =− =

−

== =−

Γ = =

∑ ∫ ∑∫

∑∫∑ ∫ (A13)

where ( )I u is defined through the following relation,

0 sin ( )0

0

( ) sinc( (cos( ) cos( )) / 3) .i uI u e u u duπ

η η′ ′ ′= −∫ (A14)

113

Appendix B

Here I present the calculations for the Raman coupling constant, 1κ , 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,

1( , ) exp[ ( )] exp[ ( )] exp[ ( )] c.c.,2 p p p s s s a a aE z t E i t k z E i t k z E i t k zω ω ω= − + − + − + (B1)

where , , ,iE i p s a= is the slowly varying field amplitude of pump, Stokes and anti-

Stokes and a p p sω ω ω ω− = − = Ω , the Raman frequency shift. In order to derive the

differential equation for ( )Q t we start from the Eq. (3.30c),

12 21 12 I II(T T ),d iidtρ ω ρ= + +

(B2)

where,

I 2 1T ( , ),m mm

E z tµ ρ′ ′′

= −∑ (B3)

II 1 2T ( , ),m mm

E z tµ ρ′ ′′

= ∑ (B4)

114

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

1mρ ′ and 2mρ ′ in Eqs. (B3) and (B4) we have,

1 ( )I 2 1 11 2 12T [ ( ) ( )] ( ) ( , ) ( , ),m

ti t t

m m m m mm

i dt e t t t E z t E z tωµ µ ρ ρ µ ρ′ ′−′ ′ ′ ′ ′

′ −∞

′ ′ ′ ′ ′= − + ×∑ ∫ (B5)

2 ( )II 1 2 22 1 12T [ ( ) ( )] ( ) ( , ) ( , ).m

ti t t

m m m m mm

i dt e t t t E z t E z tωµ µ ρ ρ µ ρ′ ′−′ ′ ′ ′ ′

′ −∞

′ ′ ′ ′ ′= − − − ×∑ ∫ (B6)

In what follows we neglect the Stark shift of the levels, i.e. we ignore the terms

including 12 ( )tρ ′ in Eqs. (B5) and (B6). Inserting the Eq. (B1) into Eq. (B5) and

performing the integration explicitly we have,

( ) ( ) ( )

I 11 1 21 1 1

( ) ( ) ( )* * *

1 1 1

1T ( ) 4

p p s s a a

p p s s a a

i t k z i t k z i t k z

m m p s am m p m s m a

i t k z i t k z i t k z

p s am p m s m a

e e et E E E

e e eE E E

ω ω ω

ω ω ω

ρ µ µω ω ω ω ω ω

ω ω ω ω ω ω

− − −

′ ′′ ′ ′ ′

− − − − − −

′ ′ ′

= − + +− − −

+ + ++ + +

∑

( ) ( ) ( ) c.c..p p s s a ai t k z i t k z i t k zp s aE e E e E eω ω ω− − −× + + +

(B7)

In deriving Eq. (B7) we have assumed that the intermediate levels are not populated,

i.e. 0m mρ ′ ′ = , 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,

( )* 1 111I 1 2 1 1

( )* 1 11 2 1 1

( )T [( ) ( ) ]4

[( ) ( ) ] .

p s

a p

i t i k k zp s m m m p m s

m

i t i k k za p m m m a m p

m

t E E e

E E e

ρ µ µ ω ω ω ω

µ µ ω ω ω ω

Ω − − − −′ ′ ′ ′

′

Ω − − − −′ ′ ′ ′

′

= − − + +

+ − + +

∑

∑

(B8)

The calculation of IIT follows a similar line,

115

( ) ( ) ( )

II 22 1 22 2 2

( ) ( ) ( )* * *

2 2 2

1T ( ) 4

p p s s a a

p p s s a a

i t k z i t k z i t k z

m m p s am p m s m a m

i t k z i t k z i t k z

p s am p m s m a

e e et E E E

e e eE E E

ω ω ω

ω ω ω

ρ µ µω ω ω ω ω ω

ω ω ω ω ω ω

− − −

′ ′′ ′ ′ ′

− − − − − −

′ ′ ′

= − − − −− − −

+ + ++ + +

∑

( ) ( ) ( ) c.c..p p s s a ai t k z i t k z i t k zp s aE e E e E eω ω ω− − −× + + +

(B9)

Keeping only terms oscillating at the Raman frequency, we have,

( )* 1 122II 1 2 2 2

( )* 1 11 2 2 2

( )T [( ) ( ) ]4

[( ) ( ) ] .

p s

a p

i t i k k zp s m m m s p m

m

i t i k k za p m m m p a m

m

t E E e

E E e

ρ µ µ ω ω ω ω

µ µ ω ω ω ω

Ω − − − −′ ′ ′ ′

′

Ω − − − −′ ′ ′ ′

′

= − + − −

+ + − −

∑

∑

(B10)

Noting that 2 2m mω ω′ ′= − and that 1 2 2 1m mω ω ω ω′ ′− = − = Ω it is easy to show that Eq.

(B10) can be rewritten in the following form, similar to Eq. (B8),

( )* 1 122II 1 2 1 1

( )* 1 11 2 1 1

( )T [( ) ( ) ]4

[( ) ( ) ] .

p s

a p

i t i k k zp s m m m p m s

m

i t i k k za p m m m a m p

m

t E E e

E E e

ρ µ µ ω ω ω ω

µ µ ω ω ω ω

Ω − − − −′ ′ ′ ′

′

Ω − − − −′ ′ ′ ′

′

= − + +

+ − + +

∑

∑

(B11)

Defining the coupling coefficients sκ and aκ as,

1 1

1 2 1 12

1 11 2 1 12

1 [( ) ( ) ],4

1 [( ) ( ) ],4

s m m m p m sm

a m m m a m pm

κ µ µ ω ω ω ω

κ µ µ ω ω ω ω

− −′ ′ ′ ′

′

− −′ ′ ′ ′

′

= − + +

= − + +

∑

∑

We can rewrite Eq. (B8) and (B11) in the following form,

( ) ( )* *I 11T ( ) ,p s a pi t i k k z i t i k k z

s p s a a pt E E e E E eρ κ κΩ − − Ω − −= − + (B12)

( ) ( )* *II 22T ( ) .p s a pi t i k k z i t i k k z

s p s a a pt E E e E E eρ κ κΩ − − Ω − −= + (B13)

116

Defining the slowly varying amplitude ( )Q t for the molecular coherence and

inserting from Eq. (B12) and (B13) into the Eq. (B2), we have,

( ) ( )[ ]* *22 11exp (2 ) ( ) ( ) .s p s a a p p a s

d Q t i E E E E i k k k z t tdt

κ κ ρ ρ = + − − − (B14)

Introducing the phenomenological damping constant 2T for the material coherence,

Eq. (B14) can be written in the following form,

( ) ( ) ( )[ ]* *22 11

2

1 exp (2 ) ( ) ( ) .s p s a a p p a sd Q t Q t i E E E E i k k k z t tdt T

κ κ ρ ρ + = + − − − (B15)

For vanishingly small population difference, i.e. 22 11( ) ( ) 1t tρ ρ− ≈ − and

2 p a sk k k k∆ = − − , Eq. (B15) reduces to Eq. (4.1), that is,

( ) ( ) [ ]( )* *

2

1 exp .s p s a a pd Q t Q t i E E E E i k zdt T

κ κ+ = − + ∆ (B16)

117

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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.