Stimulated Raman Scattering in Gas Filled
Hollow-Core Photonic Crystal Fibres
Stimulierte Raman-Streuung in mit Gas gefüllten Hohlkernfasern
June 2013
Der Naturwissenschaftlichen Fakultät
der Friedrich-Alexander-Universität
Erlangen-Nürnberg
Zur
Erlangung des Doktorgrades Dr. rer. nat.
vorgelegt von
Nguyen Manh Thang aus Hanoi, Vietnam
Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät at der Friedrich-Alexander- Universität at
Erlangen- Nürnberg
Tag der mündlichen Prüfung: 26.9.2013 Vorsitzender des Promotionsorgans: Prof.Dr. Johannes Barth Gutachter: Prof.Dr. Philip St. J. Russell Prof.Dr. Maria Chekhova
Abstract
In this thesis I use unique properties of hollow-core photonic crystal fibre (HC-PCF) to
study stimulated Raman scattering (SRS) in gaseous medium. HC-PCF offers excellent
abilities such as tight confinement of light and matter along diffractionless interaction
length in the micron-size core, low loss and adjustable guidance bandwidth. These allow
us to achieve extremely high Raman conversion efficiencies and to optimize optical
processes for a desired frequency range as well as exploring SRS regimes inaccessible in
conventional ways.
I first give an overview of the guidance mechanisms and fabrication techniques of
HC-PCF. There are two main types of HC-PCFs. Hollow-core bandgap fibre (PBG-PCF)
have quite low power loss and narrow guidance bandwidth. Hollow-core photonic crystal
fibres with Kagomé lattice (Kagomé-PCF) provide broadband guidance and higher loss.
The light-matter nonlinear interaction efficiency in HC-PCF has been shown several
orders of magnitude higher than those of previous approaches. Next, the theoretical
background of SRS is described in detail through both classical and quantum mechanical
pictures. Maxwell-Bloch equations governing the spatio-temporal evolution of light-gas
interaction system via SRS are also derived.
For application purposes, I performed a two consecutive stage pulse compression in
H2 gas-filled PBG-PCF by backward stimulated Raman scattering (BSRS). As a result, a
signal pulse 20 times shorter than that of the original pump pulse was efficiently
generated. Moreover, a new dynamical process generating a train of Raman pulses with
flexibly controllable peak intensities have been observed in transient BSRS. We also
have been able to generate a broad, mutually coherent, purely rotational Raman
frequency comb by a relatively simply setup consisting of a micro-chip pump laser
source and two H2 gas-filled HC-PCFs. Lastly, I consider the effect of the collision
between gaseous molecules and the fibre core on the spectral linewidth of forward
stimulated Raman scattering (FSRS) in a low gas pressure range.
Zusammenfassung
In dieser Arbeit nutze ich die einzigartigen Eigenschaften von photonischen
Hohlkernfasern (engl.: hollow-core photonic crystal fibre: HC-PCF), um stimulierte
Raman-Streuung (SRS) in gasförmigen Medien zu untersuchen. HC-PCF bieten
exzellente Möglichkeiten wie beispielsweise den Einschluss von Licht und Materie auf
engstem Raum und über lange Wechselwirkungslängen im mikrometer-großen
Faserkern, sowie geringe Transmissionverluste und einstellbare Transmissionbänder.
Dies erlaubt uns extrem hohe Raman-Konversionseffizienzen zu erreichen und die
optischen Prozesse für die gewünschten Frequenzbereiche zu optimieren. Darüber hinaus
können wir SRS-Bereiche erforschen, die auf herkömmliche Weise nicht zugänglich sind.
Ich werde zunächst einen Überblick über die Leitungsmechanismen und die
Herstellungsverfahren von photonischen Hohlkernfasern geben. Es gibt hauptsächlich
zwei verschiedene HC-PCF-Typen. Bandlücken-Hohlkernfasern (engl.: hollow-core
photonic bandgap fibre: PBG-PCF) haben sehr geringe Leistungsverluste und leiten in
einem schmalen Frequenzband. Hohlkernfasern mit Kagomé-Gitterstruktur (Kagomé-
PCF) erlauben breitbandige Lichtleitung, allerdings bei höheren Verlusten. Es ist
bekannt, dass die Licht-Materie-Wechselwirkungseffizienz für nichtlineare Effekte in
HC-PCF um mehrere Größenordnungen höher ist als mit herkömmlichen Methoden. Im
Anschluss an diese Kapitel wird der theoretische Hintergrund zur SRS im Detail erklärt,
ausgehend von sowohl klassischem als auch quantenmechanischem Bild. Dabei werden
unter anderem die Maxwell-Bloch-Gleichungen hergeleitet, die die raumzeitliche
Ausbreitung der Licht-Gas-Wechselwirkung bei SRS beschreiben.
Zu Anwendungszwecken habe ich durch stimulierte Raman-Rückstreuung (engl.:
backward stimulated Raman scattering: BSRS) zwei aufeinanderfolgende
Pulskompressionen in Wasserstoff-gefüllten PBG-PCF durchgeführt. Damit konnte auf
effiziente Weise ein Signalpuls erzeugt werden, der zwanzigmal kürzer als der
Ausgangpuls der Pumpquelle war. Darüber hinaus konnte ein neuer dynamischer Prozess
bei der transienten BRSR beobachtet werden, welcher einen Raman-Pulszug mit flexibel
kontrollierbarer Spitzenintensität erzeugt. Außerdem gelang es uns einen zugleich breiten
und kohärenten Raman-Frequenzkamm ausschließlich mit Hilfe von
Rotationsübergängen zu erzeugen. Der verhältnismäßig einfache Aufbau besteht
hauptsächlich aus einer Mikrochip-Pumplaserquelle und zwei Wasserstoff-gefüllten HC-
PCF. Abschließend befasse ich mich mit der spektralen Linienbreite der stimulierten
Raman-Vorwärtsstreuung (engl., forward stimulated Raman scattering: FSRS) bei
geringem Gasdruck, die maßgeblich von Zusammenstößen der Gasmoleküle mit der
Wand des Faserkerns beeinflusst wird.
Acknowledgements
Firstly, I am sincerely grateful for my supervisor Prof. Dr. Philip St.J. Russell whose give
me the continuous support during my research course. Thank you for giving me an
invaluable chance to work in such a highly scientific environment.
Secondly, I would like to thank Amir and Andy whose spend a lot of time to explain
clearly for me the dynamical processes in Raman scattering as well as nonlinear optics.
Thank you for your patient reading and corrections to my thesis. You not only help me in
job but also teach me the way to overcome the difficult problems in life. Amir, I learn
much about your careful characteristic. The thesis could not be completed without you.
Andy, thank you for sharing your plentiful knowledge in culture
I have a great time with Azhar, Patrick, Sarah, Nicolai and Xiao Ming. Thank you for
sharing my office and funny stories. Azhar, you are very friendly and thank you for your
help about computer problem and “Taj Mahal tea” gifts. Thank Sarah for translating
thesis abstract into German version.
I also would like to thank my all my colleagues Tran Xuan Truong, Xin Jiang, Martin
Finger, Barbara Trabold, Federico Belli, Michael Schmidberger, Anna Butsch, Oliver
Schmidt, Gordon Wong, Ana Maria Cubillas, Tijmen Euser, Johannes Koehler, Micheal
Frosz, David Novoa, Alessio Stefani, Thomas Weiss, Sebastian Bauerschmidt, Philipp
Hoelzer, Martin Butryn, Stanislaw Doerchner and Fatma Tuemer.
Finally, I would like to thank my parents, my wife and my daughter whose encourage
continuously me to complete my PhD work.
Contents
Chapter 1 Introduction ......................................................................................... 1 Chapter 2 Hollow-core photonic crystal fibres ....................................................... 4
2.1 Conventional fibre ............................................................................................. 4
2.2 Hollow-core photonic crystal fibre .................................................................. 5
2.3 Guidance via photonic bandgaps ...................................................................... 5
2.4 Density of states ............................................................................................ 8
2.5 Fabrication technique ...................................................................................... 10
2.6 Guidance via low density of states ................................................................. 11
2.7 HC-PCF enhances the gas-based nonlinear effect ........................................... 13 Chapter 3 Theoretical background of Raman scattering ................................... 18
3.1 Origin of Raman scattering ........................................................................... 18
3.2 Spontaneous and stimulated Raman scattering ............................................... 20
3.2.1 Spontaneous Raman scattering ................................................................. 20
3.2.2 Spontaneous versus stimulated Raman scattering ...................................... 25
3.3 The coupled wave equations and stimulated Raman scattering .......................... 27
3.3.1 Wave propagation ................................................................ 27
3.3.2 Stimulated Raman scattering ................................................................ 30
3.3.3 SRS in the language of optical phonons ..................................................... 32 3.3.4 Phase-matching diagram ................................................................ 33
3.3.5 The classical description ................................................................ 35
3.3.6 The semi-classical description .................................................................. 40
3.3.6.1 Density matrix formalism .................................................................. 40
3.3.6.1 Schematic of energy levels ................................................................ 43
3.3.6.1 Motion equation of density matrix ....................................................... 44
3.3.6.4 Transient regime in SRS .................................................................... 52 Chapter 4 Backward stimulated Raman scattering in H2 gas-filled PBG-PCF .... 55
4.1 Introduction .................................................................................................... 55
4.2 Backward and forward Raman gain asymmetry ............................................. 56
4.3 Motivation ................................................................................................... 60
4.4 Optical pulse compression via BSRS ............................................................. 61
4.4.1 Experimental setup ............................................................... 61
4.4.2 Results and discussion ............................................................... 63
4.4.3 Dynamical analysis of reverse-pumped Raman pulse ................................ 65
4.5 Generation of like-solitary pulse train .......................................................... 67
4.5.1 Experimental process and results .............................................................. 67
4.6 Conclusion .......................................................... 70 Chapter 5 Phase-coherent frequency comb generation in gas filled HC-PCFs . .71
5.1 Introduction ................................................................................................. 71
5.2 Purely rotational frequency comb generation ................................................. 72
5.3 Stable phase-locking charateristic in comb lines ............................................ 77
5.4 Summary .................................................................................................. 80
Chapter 6 Raman linewidth broadening in gas filled HC-PCF .............................. 81
6.1 Introduction ................................................................................................. 81
6.2 Analysis of Raman linewidth change in gas medium ....................................... 81
6.3 Experimental setup and results ..................................................................... 85
6.4 Conclusion .................................................................................................. 88 Chapter 7 Summary and outlook .............................................................................. 89 References ................................................................................................................... 95 Curriculum Vitae .......................................................................................................... 102
Chapter 1 Introduction
Raman scattering is a result of the interaction of light with the oscillation modes of
molecules constituting the scattering medium. It can be described as the scattering of
light from optical phonons, differing from acoustic phonons in Brillouin scattering [1,2].
Raman scattering is a two-photon inelastic scattering, where the frequency of scattered
photons is different from that of the incident photons, with the down-shifted frequency
referred to as Stokes scattering and the up-shifted frequency referred to as anti-Stokes
scattering. Raman scattering can occur in various media such as solid, liquid, gases and
plasma. It was first discovered in 1928 by C.V. Raman in liquids [3] and by G. Landsberg
in solid [4]. It had long become important for investigating the vibronic structure of
molecules and crystals. However, these initial experiments used sources with low photon-
density resulting in only a spontaneous regime where the scattered light is not coherent,
emitting in every direction and providing a negligible scattered efficiency only few parts
in 105 of the incident radiation [1]. After the coherent light source (laser) was invented in
1960, the first experiment in a stimulated regime was also accidentally observed in 1962
by J. Woodbury [5]. SRS has notably advantageous characteristics: the high conversion
efficiency to scattered frequency, high directionality, definite excitation threshold, quite
narrow linewidth compared with the spontaneous regime [2,6]. These make it an
excellent tool with a wide range of applications in areas such as high-resolution
spectroscopy [7], optical communication, frequency shifter, pulse compression [8], comb
frequency generation as well as ultrashort pulse synthesis [9,10].
Apart from the common research on the SRS in forward direction (FSRS) for frequency
shifting, backward SRS (BSRS) first observed in 1966 [1] is considered as a method for
amplification and generation the signal pulse of highly spatial quality from the pump
beam of poor spatial quality [13,14,15,16,17]. FSRS and BSRS are different in behavior.
The forward-traveling Stokes pulse just has access to the energy stored in the co-
propagating volume element of pump pulse envelop, the forward Stokes intensity is
limited by the initial pump. On the other hand, the backward-travelling Stokes is
amplified by encountering continuously with long pump pulse, resulting in a backward
1
signal intensity can be amplified to a value far in excess of the pump intensity [8]. This
mechanism has a promising potential in generation of powerful ultra-short pulses
[18,19,20].
For low-density media such as gases, the maximization of the SRS efficiency requires
following conditions: high intensity at low power, long effective interaction length and
good quality transverse beam profile. Initially, to reach the Raman threshold, the laser
beam was tightly focused to a small point by lens inside a gas cell. In this simple way, the
effective length of interaction is not longer than a few mm (~Rayleigh length) caused by
the strong diffraction limit of the focused laser beam, which results in the SRS efficiency
only a few percent [21]. Then, for increasing the effective interaction length, the laser
beam was coupled into multi-pass or high-finesse Fabry-Perot cavities [24, 25], or
hollow-core capillaries [22,23]. However, far better conversion efficiencies are obtained
when using HC-PCF as a gas-filled novel guidance system [26]. The light is confined
inside the small core of HC-PCF by means of photonic bandgap of the cladding. These
structures offer unique characteristics: the free-diffraction effective interaction length,
quite low loss attenuation, flexible in designing of guidance bands, small effective area
(~25µm2), single-mode transverse beam profile. These excellent characteristics make
HC-PCF a desired candidate for studying light-matter interactions in low-density media
at very low pump power level. This approach made the Raman threshold energy drop
significantly with only a single-pass interaction, much lower than that of the threshold of
unwanted other nonlinear processes such as self-phase modulation, self-focusing [27].
Choosing the suitable guidance band also allows us to optimize conversion to a desirable
frequency by getting rid of unwanted higher order rotational and vibrational Stokes and
anti-Stokes frequencies. As a result Raman energy threshold could reduce six orders of
lower than previously reported [28]. Moreover, it is possible to gain deeper insight into
the different states of SRS; good overviews can be found in [29,30,31,32,33].
In this thesis, I exploited novel characteristics of HC-PCF for carrying out experimental
studies in both backward SRS and forward SRS regimes. The outline of the thesis is as
following:
2
Chapter 2 gives a short overview on the novel light guidance mechanisms of photonic
crystal fibres (PCFs). The propagation diagram is used to analyze and compare with the
conventional waveguide. Then, we will focus on two HC-PCF types including hollow-
core narrowband guidance fibre (PBG-PCF) and hollow-core broadband guidance fibre
(Kagomé-PCF). Finally, the advanced applications of HC-PCF in nonlinear optical
interactions between the light and low-density media were also introduced.
Chapter 3 introduces a theoretical background of Raman scattering. Initially we explain
the physical origin of this process based on the classical picture. Coupling equations
describing the spatiotemporal evolution of stimulated Raman scattering will be
considered and compared from both classical and quantum viewpoints. The transient SRS
regime (high coherence) important in ultrashort synthesis will also be introduced at the
end of the chapter.
Chapter 4 describes BSRS in H2 gas filled PBG-PCF. Firstly, the gain asymmetry in
backward and forward Raman scattering in H2 gas medium will be analyzed. By using a
two-stage compression scheme, the signal pulse 20 times shorter than the original pulse
was efficiently generated. Interestingly, a train of solitary-like Raman pulses with
flexibly controllable peak intensities has been also observed in transient BSRS regime.
Chapter 5 presents the generation of a broad, phase-coherent, purely rotational-
Raman frequency comb by a microchip pump laser source and two H2 gas-filled
HC-PCFs. Then, the doubled-frequency interferometry was used to consider the
phase characteristic of the generated comb.
Chapter 6 investigates the pressure dependence of the rotational Raman linewidth of
hydrogen confined in the core of a PBG-PCF with a radius of 5.5µm, in which the effect
of the collision between gas molecules and fibre core wall will come into play at the
pressure below 1bar when the molecular mean-free path is of order of the fibre core (a
few µm).
Chapter 7 gives a summary and outlook for future research.
3
Chapter 2 Hollow-core photonic crystal fibres
I will introduce briefly optical properties of two types of HC-PCF, i.e. photonic bandgap
PCF (PBG-PCF) and kagomé-PCF. The reviewed material of this chapter is mainly based
on these references [27,34,37].
2.1 Conventional fibre
In order to distinguish conventional fibre clearly from HC-PCF, firstly we summarize
their guiding mechanism. Conventional “step-index” fibres operate by total internal
reflection (TIR). They consist of a solid core with the refractive index n1 surrounded by
an outer cladding of slightly lower refractive index n2<n1 [34]. Incident light rays are
completely reflected into the fibre core (TIR) if their incident angles (on the core-
cladding boundary) are smaller than that of a critical angle ⎟⎠⎞⎜
⎝⎛=≤ −
1
21cr n
nsinθθ . The
guided rays are illustrated for highly multimode fibres in figure 2.1.
Figure 2.1 Schematic of a highly multimode fibre with core index n1>n2 (cladding
index), green rays are guided when they incident on an acceptance angle. In contrast, red
rays are not guided (leak into fibre’s cladding) because they are outside the acceptance
one [34]. c
Conventional fibre has been developed and used since the 1970s for a range of important
applications such as telecommunications, imaging and high power laser. However, these
fibres have some limitations: waveguide geometry and refractive index deviation of core
and cladding are restricted. Fabrication of single mode fibre becomes more difficult when
4
guided wavelength gets shorter. Furthermore, for specialized applications, which require
hollow core, conventional fibres are impossible because of their dependence on TIR.
. Photonic bandgaps are formed by a periodic
avelength-scale lattice of microscopic air holes running along the entire length of fibre,
plotted illustratively in figure 2.2.
a
2.2 Hollow-core photonic crystal fibres
HC-PCFs are a special class of the photonic crystal fibres (PCFs), which guide light in a
hollow core instead of solid core, as is the case for conventional fibres, first proposed by
Phillip Russell [35]. These low-loss waveguides enable new applications such as studying
matter-light interactions in gas-filled or liquid-filled cores. HC-PCF guides light by
means of 2D-photonic bandgaps
w
Figure 2.2: A structure of PBG-PCF with hexagonal cladding structure. It consists of a
hollow core (diameter~10µm) surrounded by the cladding formed by a periodic array of
ir holes with diameter d~2.8 µm and pitch Λ~2.9µm (the distance between two closest
a
umber of gratings that consist of periodic arrays of glass rods and air holes. These
pagation of light is forbidden completely [36].
a
air-holes), the cladding is created in a glass substrate.
The appearance of photonic bandgaps can be intuitively understood in the form of “stop
bands” caused by Bragg reflections [34]. However, photonic bandgaps are created by
n
gratings add up appropriately so that pro
2.3 Guidance via photonic bandbaps
5
It is well known that when light is incident on any interface between materials, the
component of the wave-vector parallel to the interface is conserved [34]. In the fibre, if
the structure is invariant along its entire length, the interface of core and cladding is
always parallel to the fibre axis, labeled usually as z-axis, conserved vector is called
propagation constant, β . Propagation constant can be obtained by solving the Maxwell
equations (as Eq. (2.1) in section 2.4 below) and gives information on the dispersion of
fibres. Its maximu nk0 (m is 0nkβ ≤ ), with n being the refractive index of the
homogeneous medium and λ2π
0k = is the vacuum wave-vector corresponding to the
wavelengthλ . For a given value of 0nkβ > , light propagation is forbidden. Results in
light being confined in the higher index areas by TIR.
A very useful tool to describe regimes where light is able to propagate or be evanescent is
the propagation diagram, described in figure 2.3. The propagation diagram shows the
relation between propagation constant and light frequencies normalized to the pitch, Λ of
bre cladding. This diagram allows us to present clearly the propagation mechanisms of
light in conventional fibres as well as PCFs.
fi
Figure 2.3 Propagation diagram of a step-index fibre is presented in figure 2.3a. PCFs are
presented in figure 2.3b. Where the horizontal axis shows normalized propagations β Λ,
ormalized frequency is presented by the vertical axis Λ/c. Points A, B and C and n ω
regions 1,2,3,4 are described below (also see [37]).
6
Propagation of step-index fibre composed for example of a Ge-doped silica core
and a pure silica cladding with slightly lower refrac , pr
regimes
tive index esented in figure 2.3a:
light can proRegion 1: pagate in all regions; air refractive index of 1nair0air knβ < ≈ ;
cladding index of 1.45n ≈ and solid core of 1.47n ≈cladding core .
<< li
kn such as point A in
gure 2.3a. This is TIR in conventional fibre.
regimes of PCFs with an average refractive index of micro-structured
an
ight propa
k
as TIR regim in conventional fibre, PCFs
average index of air-glass cladding is always smaller than that of pure glass core
Region 2: n ght can propagate in both fibre cladding and core but not
in air.
0cladding0air knβk
Region 3: 0coreknβ << light only propagate in fibre core0cladding
fi
Region 4: 0reknβ > no propagation with any refractive index of n.
Propagation
co
cladding n d an air-filling fraction of 45% made of pure glass are presented in
figure 2.3b.
glass-air
Region 1: gates freely in all regions of PCF, air, air-glass cladding
and glass-pure core.
0air knβ < l
Region 2: light propagation is allowed in air-glass cladding and
pure-glass core, but not in air.
0glass-air0air knβkn <<
Region 3: 0coreknβn << light guidance is only allowed in solid core (point C) in
figure 2.3b, which is similar to a TIR mechanism in conventional fibres.
Region 4: 0coreknβ > light propagation is forbidden for any refractive index n. The same
with solid core can guide light because
irrespective of distribution structure of air holes, i.e. guidance condition
coreglassair nn <<− 0k/
0glass-air
e
β is satisfied. However, a very interesting feature of this kind of
7
PCF is that its core keeps single mode no matter how short is the wavelength of the
guided light, i.e. it is endlessly single mode (ESM-PCF). Conventional fibres, however,
tend to become multimode for shorter wavelengths [35].
bandgaps unique to PCF. By designing appropriately the cladding with periodic air-hole
arrays in the pure glass substrate, it is possible to form photonic bandgaps where light
propagation is forbidden at certain values of β . Full photonic bandgaps are presented by
black thin “fingers” in figure 2.3b. Photonic bandgaps are possible to appear in regions
1&2 and pass through the air line (diagonal line) to intersect the guided line at point B.
Points such as point B are only possible in
Moreover, PCF also contribute another light guidance mechanism, namely photonic
PCF. Hence, light propagation is possible in
cladding of air holes by mean of photonic
is impossible in conventional fibre, because hollow core has a
ractive index smaller than that of air-glass cladding material which does not satisfy the
resent qualitative
provides the information about the band structure or the range of prohibited wavelengths.
a desired propagation
order to get the DOS plot, Maxwell equations must be solved numerically using some
special methods [38,39]. Maxwell equations can be solved with as lue
y the equation (Eq.) below.
air (hollow-core) but not in the periodic
bandgaps. This mechanism
ref
requirement of TIR.
2.4 Density of states
Whereas the tool of the propagation diagram can be used to rep
information on the the position of photonic bandgaps, density of states (DOS) plot
This gives parameters for the fabrication of PBG-PCF with
wavelength ranges.
In2β eigenva s given
b
( ) ( )( )[ ] T2
TTTT20
2 HβHyx,rlnε)]Hyx,ε(rk[ =×∇×∇++∇ (2.1)
This form allows material di to be easily included.
spersion
8
Here the plane (x, y) is the transverse plane normal to the direction of propagation, z,
( )Trε is the dielectric constant at position rT (x . H s the transverse component
of magnetic field vector H.
,y) denoteT
cωk0 = is the vacuum wave-vector.
The plane-wave solution of (2.1) at fixed frequency shows a range of possible guided
cladding modes in propagation constant from to
ω
Λβ ( )Λdββ + at a particular normalized
frequency of on figure 2.4a. 0Λk
parameters for the cladding structure (2.4b) with pitch Λ =3 µm and d/Λ=0.98 [39].
Here, normalized frequency Λk
Figure 2.4: DOS plot (2.4a) for the micro-structured cladding shows on the right. Design
0 and the propagation constant ( )0nkβΛ − are horizontal
and vertical axe respectively, n is a refractive index of filling material in fibre cladding.
The horizontal blue line shows air-line where 0nkβ 0 =− . Red areas indicate the
is calculated for a cladding structure (fig2.4b) consisting of rounded
bandgaps where photonic density of states is zero. Dark color shows low DOS, and
brighter regions describe increased DOS in cladding. Guidance in hollow core via
cladding’s photonic bandgaps takes place in the red region below the air-line.
The shown DOS plot
hexagonal air-holes arrays (white) in a glass substrate (black strand), similar to a
honeycomb lattice. The position and width of the photonic bandgaps can be controlled by
the cladding structure. The different cladding pitch will result in different locations of
transmission bands.
9
The typical loss level of PBG-PCF is low, narrow guidance band. The best reported
attenuation of PBG-PCF of 1.2dB/km at wavelength 1620nm [35]. With the feature of
light propagation in the empty space, this level has the great potential to be reduced
drastically with the further development of fabrication technology. Transmission window
restricted to the range of guided wavelengths in the photonic bandgaps. Figure 2.5
(left) shows loss spectrum with a transmission bandwidth of 150nm, the lowest loss about
0.13dB/m at 1064nm and microscope image figure 2.5 (right) of PBG-PCF fabricated at
Max Planck institute for the science of light.
is
Figure 2.5 Loss spectrum of a PBG-PCF (left) and its microscope end-face image (right).
as low loss, narrow transmission bandwidth and spectral
ositions are adjustable by the cladding parameters. Hence, PBG-PCF is unique and very
desired micro-structured fiber. It is done by
orizontally stacking pure-silica capillaries (1m long, 1mm in diameter) in a “crystalline”
structure before being inserted into a jacket tube. The preform is about 2cm (fig2.6a) in
The optical characteristics
p
suitable for optimized investigations of light-matter interactions [28,31,32,40].
2.5 Fabrication technique
Although the idea of light guidance in an air core by means of photonic bandgaps in the
cladding came early in 1991, its realization had to wait until 1999 when the first HC-PCF
was fabricated successfully [26]. A widely used technique for fabrication of HC-PCF is
the stack-and-draw technique. It consists of two main stages. The first stage is to build a
preform i.e. a macroscopic version of the
h
10
outer diameter. Functional defects like the hollow core are simply formed by removing
several capillaries from the original stack.
Figure 2.6: Fabrication stages of PBG-PCF use the stack-and-draw technique [35].
The second stage is the fibre drawing, which is usually done in a two-stage drawing
process. For the intermediate stage (fig2.6b from real image), the preform is drawn down
to a cane whose diameter is 10 times smaller (~2mm) inside a furnace with appropriate
temperature (~2000°C for silica HC- PCF). In the next drawing, the cane is continuously
drawn down to the final structure with diameter about 100µm (fig2.6c from real image).
HC-PCF parameters as hole diameter/pitch, core diameter, outer diameter which are
related to the transmission wavelengths and to the fibre loss can be precisely controlled
y the feed rate, drawing speed, temperature and inner pressure of the perform. Careful
ro-structured PCFs with the desired
haracteristics.
structure (fig2.7a) instead of the honeycomb in PBG-PCF (fig2.7b). It is
b
adjustment of these parameters can lead to mic
c
2.6 Guidance in the low density of states regime
Guidance mechanism in a large-pitch HC-PCF such as Kagomé-PCF is rather different
than the guidance in air by means of photonic bandgaps. Guidance mechanism, especially
the influence of cladding structure (pitch, glass thickness) is not understood clearly yet
[41,42]. The Kagomé cladding includes an array of thin glass strands in air forming a
star-like
11
established that the cladding structure does not exhibit any photonic bandgaps. Indeed,
wave guidance of the Kagomé lattice happens in the presence of low density of photonic
states.
Figure 2.7 Scanning electron-microscope images of Kagomé-PCF (fig2.7a) and PBG-
PCF (fig2.7b). Kagomé fibre has a 6 wings-star structure (red) in its cladding and is 30
m in core diameter, pitch of 12 µm. While PBG-PCF has a cladding of honeycomb-like
ax-Planck Institute for Science of Light.
Figure 2.8 shows the attenuation spectrum of a Kagomé-PCF with transmission window
~1100nm, level of lowest loss ~ 2dB/m.
µ
structure (parallelogram unit cell) and a 3 times shorter core diameter about 10 µm, pitch
of 3 µm. These fibres are fabricated at M
Figure 2.8 Low-loss spectrum of a Kagomé-PCF (left) is very broad transmission
window of 1100nm. Its microscope picture (right) is coupled by excitation source.
12
Typically transmission window of Kagomé-PCF is much broader compared to ones
caused by bandgaps of PBG-PCF. Loss level can be down to 0.180dB/m a transmission
and can over 1200nm [41]. This fibre is useful for applications requiring a wide
ency comb generation [33,43],
ltraviolet generation [44].
order to get a feeling of the possible enhancement in the nonlinear light-matter
interaction we get by usin
experiment in free space) we defined a figure of merit M expressed as [27].
b
bandwidth of guided wavelengths such as Raman frequ
u
2.7 HC-PCF enhances the gas based nonlinear effect
In
g HC-PCFs (as compared when one performs the same
effeff A
LM λ= (2.2)
M is a function of Leff, the effective interaction length, λ the vacuum wavelength and
intensity of
SRS. In order to achieve that condition, there are some approaches as following:
2eff rπA ×= , the effective cross-section or area where r is the effective radius associated
to Aeff.
Nonlinear effects require high enough light intensity, for example threshold
13
Figure 2.9 The effective interaction lengths Leff (red color) for different configurations
with the same effective area. Figure 2.9a shows Leff is limited by the Rayleigh length in a
focused free-space laser. Leff in hollow capillary is reduced quickly from the radius of
capillary core (fig.2.9b). Figure 2.9c illustrates the long interaction length (approximate
the fibre length) supported by the 2D photonic bandgap mechanism of cladding which
block almost light (very low loss) [27].
A simple way used commonly in early gas-based nonlinear experiments is the tight
focusing of free-space laser beam by lens into gas-filled cuvette which results in a high
intensity near the focal point as shown in figure 2.9a. For a focused Gaussian beam has
the beam waist of 2r and wavelength λ, the effective interaction length and the effective
cross-sectional area are considered in Eq.(2.3&2.4):
λr2lengthRayleigh 2L
2
eff ×=×= π (2.3)
2
eff rπA ×= (2.4)
14
The figure of merit for the focused Gaussian beam Mfb is written in Eq.(2.5):
2Mfb = (2.5)
It is clear that the effective cross-sectional area is smaller (or higher focused intensity)
Eq.(2.4), results in a shorter effective interaction length Leff Eq.(2.3) so that the two
counterbalance each others effect. Hence, tighter focusing is inefficient in increasing the
effect of matter-light nonlinear interaction.
Another approach to improve the nonlinear effect is the use of dielectric capillaries, or
metal-coated tubes [22,23]. This can increase the effective interaction length. However,
their propagation losses are very high, as illustrated in figure 2.9b.
For a dielectric capillary with an inner radius r, refractive index of glass n=1.5, the loss
rate for fundamental mode [22].
3
2
rλ4246.0 ×=α (2.6)
The effective interaction length is related to the length of capillary Lcapillary:
α1
αe1L
yαLcapillar
eff ≈−
=−
(2.7)
From Eq.(2.2, 2.6&2.7), we obtain the figure of merit for the hollow capillary
(normalized to Mfb) Mhc,
λr0.375Mhc ×≈ (2.8)
From Eq.(2.6) we note that the loss increase hugely (or the effective decrease of the
interaction length) as the inverse radius cubed (loss ~ 3r − ). For metal-coated tubes, loss is
15
even many orders of magnitude higher, particularly at optical frequencies where metals
absorb strongly [35].
An ideal configuration for effective gas-based nonlinear interactions needs to satisfy the
following requirements: diffraction-free, lossless, single-mode waveguide, core diameter
same as focused laser beam waist (~ µm). HC-PCF with the core radius r = 5 µm and an
achievable loss of 1.2dB/km [35] comes to this ideal situation. Hence, the effective
interaction length is approximated by the length of the fibre Lfibre and the normalized
figure of merit of HC-PCF become.
2fibre
2
αL
hcf rπλ
2L
2λ
rπ1
αe1M
fibre
×≈×
×−
=−
(2.9)
Eq. (2.9) shows that the figure of merit of HC-PCF increases quickly with the decrease of
core radius. Figure 2.9c illustrates he effective interaction length without the depth of
focus in HC-PCF. Light is confined tightly (high intensity) along the entire length of the
fibre.
Next, we compare the gas-based nonlinear effect for above approaches. Assume that
propagation wavelength (1µm), Lfibre=3m, refractive index of glass n=1.5, core radius is
changed in a range of 1-20µm. Figure 2.10 shows that figure of merit of HC-PCF with
loss of 195dB/km is about 8 orders of magnitude higher than that of capillary at the core
radius of 5µm (PBG-PCF) and about 4 orders of magnitude at radius of 15 µm (Kagomé-
PCF). Mfb of focused beam is invariant.
HC-PCFs with unique characteristics such as designable transmission window, very high
nonlinear effects are considered as the best candidate to study the light-matter interaction
in the low power regime in general and in gas-based nonlinear interactions in particular.
This thesis exploits these unique features to investigate stimulated Raman scattering in
hydrogen gas filled HC-PCF.
16
Figure 2.10 Comparison of the figure of merit M for different configurations: a focused
free beam (blue line); hollow capillary (pink curve); the red curve was calculated for
PBG-PCF, loss of 195dB/km and black curve was calculated for Kagomé-PCF, loss of
1.4dB/m [27].
17
Chapter 3 Theoretical background of Raman scattering
In this chapter I review the theoretical background behind stimulated Raman scattering.
This is mainly based on the references [1,2,45,46,53].
3.1 Origin of Raman scattering
Raman scattering is the result of the interaction of optical field pE~ [Vm-1] with the
oscillation excitations of the molecules in the Raman active medium. Although the
optical frequency is too high to follow by the nuclei of the molecule, it can cause the
distortion of electron cloud, making each molecule become polarized. On the other hand,
the electron potential depends on the nuclear coordinate. Hence, we can say that the
electronic polarizability α~ [m3] perturbed by the presence of nuclear oscillation. This
section is derived from [1,6,45,46].
Dynamically, the oscillations in diatomic molecules can be rotational, vibrational or
rotational-vibrational depending on the excitation conditions such as the polarization state
of the molecule, the type of the scattering medium. Figure 3.1 illustrates intuitively two
simple motion states in the H2 molecule. The oscillation of atoms under the externally
electric field force are presented by spherical balls (red) bonded each other by the spring.
Figure 3.1 Motion states of the H2 molecule are indicated by the direction of arrows: a)
Vibrational state with the frequency of 125THzΩv = , b) Purely rotational state with the
frequency of . 18THzΩR =
18
The different motions correspond to the frequencies of Raman excited transitions. At the
room temperature, the frequency of Raman excited transition for vibration of
(4155 cm125THzΩv =-1) and for rotation of 18THzΩR = are dominant [6].
In the context of our experiments, only the rotational Raman transition is considered.
However, the formalisms for the description of the Raman scattering used below are valid
for both states.
The induced electric dipole moment [Cm] likes a dipole emitter. Its magnitude is equal
the product of the strength of the applied field of and the Raman polarizability of
, expressed in Eq.(3.1).
μ~
(t)E~
( )tα~
( ) ( ) ( )tΕ~tα~εtμ~ p0= (3.1)
Where, [Cm0ε-1V-1] is the electric permittivity in vacuum.
We let is the motion coordinate or the deviation of the internuclear distance from its
equilibrium. It may either be the linear position in the vibrational motion or the angular
position in rotational motion. Then, can be expressed by the Taylor expansion in
motion coordinate
( )tq~
α(t)
( )tq~ (Placzek model) [1,46].
( ) ...q~qααq~α~
00 +⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+= (3.2)
The term of in Eq.(3.2) is the polarizability of the molecule at the absence of
oscillation, it can be approximated as constant in
0α
( )tq~ and contributes to the Rayleigh
scattering. The first order correction of 0q
α⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂ is interpreted as the coupling strength
between the nuclei and electrons. The higher-order terms are responsible for the multi-
photon processes. The induced dipole moment makes the molecule polarized. The
macroscopic polarization of the scattered medium is obtained by the statistic sum of all
dipole moments per unit volume N[m-3].
19
( ) ( )tμ~NtΡ~ = (3.3)
Where [Cm( )tP~ -2] plays role as a source term in the Maxwell wave propagation
equations (section 3.3.1).
Raman scattering can be split into spontaneous and stimulated Raman scattering (SRS).
The former one is typically a weak excitation process of incoming intensity with the
Stokes transfer efficiency of only being about one millionth of the incident light
radiation. Spontaneous scattering is incoherent and its Stokes radiation can spread in any
directions. The latter is observed when excited with an intense laser beam. This
stimulated process increases the transfer efficiency and the coherence is much higher in
the spontaneous one leading to the emission process in a narrow cone in the backward
and forward direction.
Before the detail description of SRS is done as essential part of this chapter, we consider
some basic properties of the spontaneous and its relationship with the stimulated Raman
scattering.
3.2 Spontaneous and stimulated Raman scattering
3.2.1 Spontaneous Raman scattering
If the incident field is not strong enough and the scattered Stokes photons don’t
affect to the scattering process, then we talk about spontaneous Raman scattering.
According to the classical description, the oscillation can be featured by its amplitude and
phase. We assume the material excitation and the applied field are represented by
monochromatic plane waves propagating in z-direction. The following description is
referred from [1,46]
(t)E~
( ) ( )[ ]( c.cΩt-iexptz,Q21(t)q~ +Φ= ) (3.4)
20
( ) ( )[ ]( c.ctω-zkiexptz,E21(t)E~ pppp += ) (3.5)
Where Q(z,t) is the complex, time and space-dependent envelopes of the internuclear
motion, is the nuclear motion frequency (assumed that the nuclei are not moving
initially), is the phase of the nuclear mode oscillation established by random phases.
E
Ω
Φ
p(z,t) is the complex, time and its space dependent envelopes of the input fields, where
is the carrier frequency and its wavevector pω cωnk PP
P = , nP denotes the refractive
index of medium at the frequency of . C.c indicates as the complex conjugate
component, c is the velocity of light in vacuum.
Pω
Substitute Eq.(3.2,3.4&3.5) into Eq.(3.1), then the dipole moment is calculated as:
( )( )[ ] c.c-tΩ-ω-zkiexpEQqα
2μ~ ppp
*
0
0 +Φ⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=ε
( )[ ]( )c.ctω-zkiexpEα ppp00 ++ ε
( )( )[ ] ...c.ctΩω-zkiexpQEqα
2 ppp0
0 ++Φ++⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+ε (3.6)
The induced dipole moment of μ~ in Eq.(3.6) contains the different components of
frequency with the shifted spacing of . The figure 3.2 illustrates the shift frequencies
for Stokes, Rayleigh and anti-Stokes scatterings.
Ω
21
Figure 3.2 The Raman scattering is expressed in the frequency axis ofω . Here is the
pump frequency and also Rayleigh scattering signal (black line); anti-Stokes (purple line)
at the shifted frequency and
pω
Ωωp + Ωωp − at the Stokes shift (red line).
The term of in Eq.(3.2) contributes to the elastic or Rayleigh scattering (the scattering
frequency is equal to the input frequency). The first order correction of
0α
0qα⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂ contributes
to the first Raman scattering order consisting of the first Stokes and the first anti-Stokes
scattering. This term describes how the polarizability changes with the molecular motion.
Of course, the higher order correction corresponds to the higher order Raman scattering.
Experimentally spontaneous Raman scattering is useful to obtain the Raman cross section
of [mσ 2]. It is explained as the effective area of molecule for removing light of the
incident beam. We assume the signal power Ps[W] is linearly proportional to the intensity
[WmPI -2], 2
p0p Ec2εI = falling onto an individual molecule by
σIP ps = (3.7)
We can rewrite Eq.(3.7) in a different manner by
dΘdσI
dΘdP
ps = (3.8)
Eq.(3.8) describes the power of dPs scattered in some directions in the solid angle
element of . Here dΘdΘdσ is the different cross section. Because the total power of the
scattered radiation of dΘdΘdPP
4
ss ∫=
π
, Raman cross section can be calculated by
22
dΘdΘdσσ
4π∫= (3.9)
We denote ϕ is the angle between the induced dipole moment of molecule and the
direction r which the radiation is scattered shown on figure 3.3.
Figure 3.3 Geometry of Raman scattering from the induced dipole of an individual
molecule [1].
According to the classical electrodynamics, the Stokes power of dPs per the solid angle
unit of is radiated from the above induced dipole [dΘ 46].
( )( ) ϕϕ 22
p2
3
4s02
3
2s sinEΩα~
2πcωεsin
π2c
tμ~
dΘdP
==
ϕ22
p
2
003
4s0 sinE
qα
Ω2m2πcωε
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛=
η (3.10)
Where we use the revised relation of 2
1
0Ω2m0q1 ⎟⎟
⎠
⎞⎜⎜⎝
⎛=
η the polarizability of ( )tμ~ is
defined in Eq.(3.1), the angular brackets ... mean that the time average of the enclosed
quantity is to be taken, c denotes the velocity of light, is the Planck
constant, is a reduced nuclear mass.
[ ]Js106.625 34−×≈η
0m
23
Because the angle dependence of dΘdPs is contained entirely in the quantity of .
Integrating Eq.(3.10), we have the total power emitted from the oscillating dipole
moment [46].
ϕ2sin
2
p
2
003
4s0
4π
ss E
qα
Ω2mπ2cωε
34πdΘ
dΘdPP ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛=== ∫
η (3.11)
From Eq.(3.8) and Eq.(3.10) we have the different cross section of spontaneous Raman
scattering:
ϕ22
004
4s1-
p sinqα
Ω2mc2ω
dΘdP)(I
dΘdσ
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛==
η (3.12)
Here 2
p0p Ec2εI =
Combining Eq.(3.12) with Eq.(3.9) gives the total cross-section
( 2322
004
4s
4π
m10qα
Ω2m3c16πdΘ
dΘdσσ −≅⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛== ∫
ηω ) (3.13)
Equation (3.11) shows that the classical model of the electrodynamics predict the power
of the first Stokes scattering depends on the incident light intensity (~2
pE ) with the scale
of 2
0qα⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂ . Its phase (Eq.3.6) is dependent on the nuclear mode motions. In
equilibrium state, the motions of different molecules are random or the total phase of the
Stokes fields from the dipole emitters is uncorrelated. As a result the total field of the
Raman emission is incoherent in the spontaneous regime.
Φ
24
In order to describe explicitly the spatial evolution of spontaneous Raman process and its
connection with stimulated Raman scattering, it is useful to use the classical photon
occupation number formalism [47].
3.2.2 Spontaneous versus stimulated Raman scattering
At staring point, Sζ is defined as a probability per unit time for emitting a photon into
mode Stokes S and depends on the mean photon per mode in pump ( PN ) and Stokes
( SN ) beam by [1]
( 1NND SPS +=ζ ) (3.14)
Where, D denotes a proportional constant depending on the Raman medium. On the other
hand, the time rate of nS is given by SS
dtNd
ζ= , we can rewrite Eq.(3.14) as following.
( 1NNDdtNd
SPS += ) (3.15)
We assume the Stokes mode (corresponding to a Stokes wave) travels in the positive
direction-z in the medium of the refractive index n, we have the relation tncz = .
Associate this relation with Eq.(3.15) we get by
( 1NNDnc
dzNd
SPS += ) (3.16)
We consider two extreme situations.
• For a spontaneous case 1NS << , Eq.(3.16) becomes
25
PS ND
nc
dzNd
= (3.17)
Assume that the laser field is independent of the travel propagation z, integrate of
Eq.(3.17) we have
zNDnc)0(N)z(N PSS ⎟
⎠⎞
⎜⎝⎛+= (3.18)
The first term of the right side of Eq.(3.17) denotes the Stokes photon occupation number
at the input position of the Raman medium. In this case, (z)NS or Stokes intensity
increases linearly respect to the active medium length- z.
• For a SRS case 1NS >>
The Eq.(3.16) gives
zNDnc
NNd
PS
S ⎟⎠⎞
⎜⎝⎛= (3.19)
Integrate two sides of Eq.(3.19)
( zgexp)0(N)z(N SSS = ) (3.20)
We introduce PS NDncg = in Eq.(3.20) and it is called the gain coefficient of SRS. Here
)0(NS denotes the Stokes photon occupation number at the input of the Raman medium.
Raman process follows the Eq.(3.20) is called stimulated and its Stokes intensity in SRS
actually experiences exponential increase with the medium length-z.
26
The relationship between SRS and spontaneous Raman scattering is expressed by the
Raman gain coefficient of in Eq.(3.20) and the Raman cross-section of in
Eq.(3.13). This relationship is given [1]
Sg σ
P2SP
2S
P23
S IΘσ
ωnωωN~c4Nπg ⎟
⎠⎞
⎜⎝⎛∂∂
Δ=
η (3.21)
Where, nS is a refractive index of the Stokes radiation, ⎟⎠⎞
⎜⎝⎛∂∂Θσ denotes the differential
spectral cross section, where is the total linewidth of the Stokes radiation, is an
element of solid angle. I
Δω Θd
P denotes the pump intensity of P
PPP Vn
N~cωI
η= , where V is the
effective volume of the Raman scattering, nP is the refractive index of the pump laser
wavelength.
3.3 The coupled wave equations and stimulated Raman scattering
The previous section provides an overview picture of the Raman scattering. However, it
can not reveal the information relating to the coherent interaction between the fields and
the molecules. This information becomes especially important when SRS occurs in
highly coherent regime (transient regime) where the pump pulse duration is comparable
or shorter than the relaxation time of the molecular coherence. This section describes
detail the coherent SRS interaction in terms of the coupled propagation approach in a
nonlinear optical media. Because the coherent excitation is dominant in SRS, the applied
electromagnetic fields can be treated suitably as a classical quantity [48].
3.3.1 Wave propagation
We consider a lossless nonlinear optical media with no free charge, no free current and
no magnetization. The travel of light obeys the Maxwell equation is derived from [1].
27
Ρ~tεc
1Ε~tc
1Ε~ 2
2
022
2
2 ∂∂
−=∂∂
+×∇×∇ (3.22)
Where, is the electric permittivity constant and the light velocity c in vacuum. Where 0ε
Ρ~ denotes the nonlinear polarization vector of the nonlinear optical medium depending
nonlinearly on the electric strength vector of the classical field of Ε~ .
The first term in Eq.(3.22) is analyzed as follow:
( ) Ε∇−Ε⋅∇∇=Ε×∇×∇ ~~ 2 (3.23)
Here, we have for most cases interested in nonlinear optics. For example, 0~ ≈Ε⋅∇ Ε~ is a
transversely, infinite plane wave. More general, it often demonstrated to be small for the
case of slowly varying amplitude approximation.
Inserting Eq.(3.23) into Eq.(3.22) we have
Ρ∂∂
=Ε∂∂
−Ε∇ ~1~1~2
2
022
2
22
tctc ε (3.24a)
D~1~2
2
20
2
tc ∂∂
=Ε∇ε
(3.24b)
Where the displacement field vector Ρ+Ε= ~~D~ 0ε
We split Ρ~ into two parts: a linear part of LΡ~ (depend linearly on the field of Ε~ ) and a
nonlinear part PN (depending nonlinearly on Ε~ ).
N1
0NL Ρ~Ε~χεΡ~Ρ~Ρ~ +=+= (3.25)
Here is the linear electric susceptibility 1χ
28
Hence N100 Ρ~Ε~χε~D~ ++Ε= ε (3.26)
We rewrite Eq.(3.26)
N
02 P~E~εnD~ += (3.27)
where 1χ1n += is the refractive index of the medium.
We substitute Eq.(3.27) into Eq.(3.24b) and obtain the general equation of wave
propagation in an isotropic, dispersionless optical nonlinear medium.
N2
2
02
2
2
22 P~
tμΕ~
tcnΕ~
∂∂
=∂∂
−∇ (3.28)
Here NP~ is on the right-hand side and acts as the source term of new components in
nonlinear optical interactions in general and in stimulated Raman scattering in particular.
Where is the magnetic permeability in vacuum. Assume the applied field of
the Raman active medium consists of j linearly polarized monochromatic plane waves
with the carrier frequency . Their respective wavevectors
-10
-20 εcμ =
jω cωnk jj
j = , where nj is the
refractive index corresponding to the jω . The solution of Eq.(3.28) can be written as
( ) ( )[ ](∑ +−±=j
jjj c.ctωzkiexptz,E21E~ ) (3.29)
( ) ( )[ ]( )∑ +−=j
jNj
N c.ctωiexptz,P21P~ (3.30)
Where, , are the temporal spatial complex envelope functions (defined as
Eq.(3.5)). The signs “ ” represent the propagation direction of the incident waves. We
take the plus (+) for forward propagation increasing the distance z, in contrast the minus
(-) for backward propagation reducing the distance z.
( )tz,PNj ( tz,E j )
±
29
Insert Eq.(3.29&3.30) into Eq.(3.28) and apply some slowly varying amplitude
approximations: z
Ek
zE
;t
Eω
tE j
j2j
2j
j2j
2
∂
∂<<
∂
∂
∂
∂<<
∂
∂; N
jj
Nj Pωt
P <<∂
∂
The propagation Eq.(3.28) for the forward and backward directions are given by
( ) ( ) ( ) ( )zikexptz,P2kωiμ
ttz,Ε
cn
ztz,Ε
jNj
j
2j0jjj μ=
∂
∂+
∂
∂± (3.31)
If the attenuation loss is included with loss coefficient [ ]1j mγ − , Eq.(3.31) is modified as
following
( ) ( ) ( ) ( ) ( tz,Ε2γ
zikexptz,P2kωiμ
ttz,Ε
cn
ztz,Ε
jj
jNj
j
2j0jjj −=
∂
∂+
∂
∂± μ ) (3.32)
3.3.2 Stimulated Raman scattering
SRS occurs with high applied intensity and can be understood schematically in terms of
two different regimes in figure 3.4 which is related to the way the Raman transition takes
place. The first regime, Raman transition is addressed with one sufficiently intense laser
field EP (fig.3.4a). In this case, the nuclear motion modulates its refractive index with the
natural frequency of the molecule oscillation of and frequency sidebands are
developed. This scattering process is excited initially with spontaneous emission from the
random noises of the molecular system. It becomes stimulated after passing the given
distance of pump laser with the sufficient Stokes photon number created. Hence, we have
no chance to control the phase of the output signal and result in the high phase and
energy fluctuations between frequency components. This approach have been applied in a
hydrogen filled Kagomé-PCF which can generate multioctave Raman optical frequency
combs [
RΩ
33].
30
Figure 3.4 Schematic of rotational SRS a) Raman transition is addressed by one incident
frequency; b) Raman transition is driven by two incident frequencies.
The second regime, the amplification of the Stokes signal in a manner the molecular
transition is driven resonantly or slightly detuned from Raman resonance by two
incoming fields. The molecular Raman transition driven far-off resonant by two strongly
incident mono-chromatic laser fields can give a very high average coherence of
frequency sidebands. This technique requires an adiabatic preparation of Raman medium,
for example Raman active gas is cooled down to a quite low temperature ~77K [49]. The
molecular transition is driven resonantly with two pump and seed fields. This approach
provides various advantages: the input frequencies and intensities are well defined,
unwanted higher order Stokes and other competing nonlinear processes is eliminated,
high selection of the excited molecule states (Chapter 6).
We assume that the molecule is driven by two monochromatic incident laser fields,
expressed in fig.3.4b. These fields will form a total intensity modulation with beat
frequency of . Then, this modulated intensity correlatively excites the molecule
motion at the resonance frequency of . The oscillation is the strongest when the
frequency difference matches the molecule resonance frequency.
SP ωω −
RΩ
31
3.3.3 SRS in the language of optical phonons
Whereas spontaneous Raman scattering occurs with small number of scattered Stokes
photons and uncorrelated phases Φ of the individual oscillations (excitations), the SRS
has larger number of scattered Stokes photons in the scattered fields and the phases Φ of
the individual excitations are correlated. This collective excitation of the Raman active
medium can be considered as a coherent material excitation wave and material excitation
is called an optical phonon [2]. Optical phonons are analogous with photons and they
describe a special type of motion at the same angular frequency Ω as in the quantum
mechanical description. Each optical phonon has energy of Ωη as excited quanta of the
oscillation mode. The coherent wave of material excitation has no dispersion and an exact
analogy of the classical wave with the determined wavevector K (or wavelength) [50].
Hence, the optical phonon field of in Eq.(3.4) can be rewritten with by q~ KzΦ =
( ) ( )[ ]( c.cΩt-KziexptQ21t)(z,q~ += ) (3.33)
Where, Q(t) is the time dependent complex envelope functions of the optical phonon
amplitude. Like photons, optical phonons can be destroyed in collisions. The molecular
coherent decay is characterized by the rate 2Γ which is the inverse of the relaxation time
of the molecular coherence (duration for the coherence to relax to its
equilibrium). The collisions are mainly between molecules. The collisions with their
container wall may affect to the mutual correlation of excitations in the low pressure
gases filled micro-containers as HC-PCF [
-122 ΓT =
51]. In addition, the population decay from the
excited levels to the ground state also contributes slightly to the reduction of molecular
coherence (see 3.3.6 for more detail). Experimentally, we can obtain the coherent decay
rate by measuring the full width at half maximum of the Raman gain profile (FWHM). In
the next part, by using the language of optical phonons for the coherent material
excitation, we will express the full picture of SRS in the diagram of phase matching.
32
3.3.4 Phase-matching diagram
Every optical mode with the frequency passing a HC-PCF is affected by the dispersion
characterized by the propagation constant of
ω
( )ωβ [m-1]. We consider the applied fields
consisting of the pump and Stokes seed pulse beams passing the Raman active medium
filled HC-PCF and assume that the dispersion relation ( )ωβ is expressed by single mode
dispersion curves in the figure 3.5. These fields can excite SRS in two geometrical
manners: the pump and Stoked seed have the same direction (forward SRS) or in two
opposite directions (backward SRS). The sidebands of Stokes (S) and anti-Stokes (AS)
frequencies are separated equally by the optical phonon frequency Ω and presented in
the frequency (vertical axis). For forward SRS, the propagation constants of pump ( );
Stokes ( ) and anti-Stokes ( ) fields have the same sign and hence the dispersion
curve is presented in the same left side of the frequency axis. For a backward SRS, the
pump ( ) and Stokes ( ) fields are of opposite sign, the dispersion curve for negative
is a mirror image of the dispersion curve for positive flipped at the frequency axis.
Pβ
Sβ Sβ
Pβ Sβ
β
Figure 3.5 Phase-matching schematic for the SRS. The different optical phonons are
expressed in the optical phonon branch: pump-forward Stokes (red vector), pump-
antiStokes (green vector) and pump-backward Stokes (pink vector).
33
In order to get the optimum interaction efficiency for the SRS, the phase-matching
conditions must be satisfied. For comparison of optical phonons created at the different
phase-matching conditions, they are expressed by the different color vectors: pump-
forward Stokes seed (red), pump-antiStokes (green), pump-backward Stokes (pink) and
give the respective group velocities: ( )SPS ββ
Ω−=ϑ , ( )PAS
AS ββΩ
−=ϑ and
( )SPBS ββ
Ω+=ϑ . Because the optical phonons have no dispersion, hence the
characteristic wavelengths of coherent excitation waves for the different SRS are given
by
• For the forward Stokes SRS: ( )SP
phFS ββ
2λ −= π
• For the forward antiStokes SRS: ( )PAS
phAS ββ
2λ −= π
• For the backward Stokes SRS: ( )PS
phBS ββ
2λ += π
The wavelength of a coherent wave backward SRS is small compared to the ones in
forward cases illustrated in figure 3.6.
Figure 3.6 Comparison of the wavelengths of optical phonons created in forward SRS
(pink) and backward SRS (red).
34
In the next sections, we describe mathematically the coupling of a pair of applied pump
and Stokes seed fields with the coherent material excitation waves via a given Raman
active medium. The equations for the description of theses dynamical processes are
derived gradually by the classical and semi-classical approaches.
3.3.5 The classical description
In this approach, the coherent oscillation of the molecule system is approximated as a
classical harmonic oscillator and the dynamical equations for the coupled wave problem
are derived in the formalism of Lagrangian density [2]. The coupling parameter 0q
α⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂ is
given by the classical Placzek model Eq.(3.2). We assume the Lagrangian densities for
the classical fields (Lrad), the oscillation field (Los) and the interaction field (Lint) in the
dilute (negligible dispersion), isotropic medium are given by
intosrad LLLL ++= (3.34)
Where ⎟⎟⎠
⎞⎜⎜⎝
⎛−= 2
0
20rad B~
μ1E~ε
21L (3.35)
( 2220os q~Ωq~Nm
21L −= & ) (3.36)
E~.E~q~qα
2NεE~.E~
2εNαE~.E~
2NεL
0
0000int ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+== α (3.37)
Where and are the electric and magnetic field vectors, N is the number density of
molecules, denotes the reduced nuclear mass.
E~ B~
0m
Inserting Eq.(3.35-3.37) in the motion equation of Lagrangian density given by
35
0q~d
dLq~d
dLdtd
=−⎟⎟⎠
⎞⎜⎜⎝
⎛&
We receive
E~.E~qα
2mεq~Ω
dtq~dΓ2
dtq~d
00
022
2
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=++ (3.38)
Where denotes the phenomenologically added damping constant. Eq.(3.38) is
rewritten to
Γ
( ) ( ) ( )0
22
2
mt)(z,F~tz,q~Ω
dttz,q~dΓ2
dttz,q~d
=++ (3.39)
Where ( ) E~.E~qα
2εtF~
0
0⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
= plays the role of the applied force exerting on the oscillator
with the eigenfrequency of .Assume the applied field consists of two pump Ω E~ PE~ and
Stokes seed SE~ components. The total field can be written as
( ) ( )[ ] ( ) ( )[ ]( )c.cetz,Eetz,E21E~E~E~ tω-zki
Stω-zki
PSLSSPP ++=+= ± (3.40)
According to Eq.(3.39) only the time varying part of the stimulated force. The signs “± ”
represent the forward (+) and backward (-) SRS. below contributes dominantly to
the resonant process.
(t)F~
[ ]( c.ceEEqαε(t)F~ )tω(ω)zk(ki*
SP0
0SPSP +⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
= −−μ ) (3.41)
36
Where the signs “μ ” represent the forward (-) and backward (+) SRS, the beat frequency
is the stimulating frequency. The exchange efficiency becomes optimum
when the stimulated frequency is equal to the resonant frequency Ω .
SPbeat ωωω −=
beatω
We substitute Eq.(3.33&3.41) into Eq.(3.39) and use the slowly varying amplitude
approximation q~Ωdt
q~d2
2
<< and Ω<<Γ . We obtain the temporal evolution equation for
the coherent envelop Q.
*SP
00
0 EEqα
2miεΓQ
dtdQ
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
Ω=+ (3.42)
Here, ( )[ ]( )c.cΩt-Kzi-expq~21t)Q(z, +=
Next, we will consider the temporal-spatial evolution of the applied amplitudes by using
the propagation equations (3.32) for two incoming fields (j=P,S).
From Eq.(3.2&3.3) we can write the macroscopic polarization for the Raman active
medium consisting two components linear (L) and nonlinear (N) parts by
( ) ( ) ( ) ( ) ( )tz,E~tq~qαNεtz,E~Nεαtμ~NtΡ~
0000 ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+== (3.43)
NL P~P~ += (3.44)
We substitute Eq.(3.33 & 3.40) into Eq.(3.43) and receive the nonlinear polarization for
the forward (+) and backward (-) travel of the field pumps.
( ) ( )tz,E~tq~qαNεP~
00
N⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
= (3.45)
37
[ ]( ) ( )[ ] ( )[ ]( )c.ceEeEc.cQeqα
4Nε tω-zki
Stω-zki
Ptzi
0
0 SSPP +++⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
= ±Ω−Κ (3.46)
( )[ ] ( )[ ]
⎭⎬⎫
⎩⎨⎧ ++⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
= ± c.ceQE21eEQ
21
qαNε
21 tω-zki
Stω-zki
P*
00
PPSS (3.47)
NP
NS P~P~ +=
Where Ω=− SP ωω
We used the relation with the sign (-) for the forward SRS and the sign (+)
for backward case. We also assumed the nonlinear polarization does not contain the
frequency components (2
Κ=SP kk μ
Ω−Sωnd Stokes) and Ω+Pω (anti-Stokes). The parts of
Eq.(3.47) oscillating at the Stokes & pump frequencies are
( )[ ]⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
= ± c.ceEQqαNε
41P~ tω-zki
P*
00
NS
SS (3.48)
( )[ ]⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
= c.ceQEqαNε
41P~ tω-zki
S0
0NP
PP (3.49)
Comparing Eq.(3.30) with Eq.(3.48 & 4.49) we have the complex amplitudes
( ) (3.50) zkiP
*
0
0NS
SeEQqα
2NεP ±
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=
( zkiS
0
0NP
PeQEqα
2NεP~ ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
= ) (3.51)
Inserting Eq.(3.40,3.50&3.51) into Eq.(3.32) we obtain
38
For the pump field
PP
S0P
2P00PPP Ε
2γQE
qα
4kωεiNμ
tΕ
cn
zΕ
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=∂∂
+∂∂ (3.52)
For the Stokes field
SS
P*
0S
2S00SSS Ε
2γEQ
qα
4kωεiNμ
tΕ
cn
zΕ
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=∂∂
+∂∂
± (3.53)
Where nP,S are the pump and Stokes refractive index, kP,S are the pump and Stokes
wavevectors, are the loss coefficient of pump and Stokes respectively the set of
coupled-wave equations for SRS are Eq.(3.42,3.52&3.53).
SL γ&γ
PP
S0P
PPPP Ε2γQE
qα
c4niNω
tΕ
cn
zΕ
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=∂∂
+∂∂ (3.54)
SS
P*
0S
SSSS Ε2γEQ
qα
c4niNω
tΕ
cn
zΕ
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=∂∂
+∂∂
± (3.55)
*SP
00
0 EEqα
2miεΓQ
tQ
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
Ω=+
∂∂ (3.56)
The classically harmonic model gives a qualitative description of the coherent Raman
state. However, it does not provide the quantized nature of optical phonons and hence the
lack of the contribution of the population occupation in the coupled-wave equations.
39
3.3.6 The semi-classical description
In this model the molecules are treated quantum mechanically using the density operator
formalism which has the ability of including the quantum mechanic characteristics of the
molecules. This section is adapted from [1,53].
3.3.6.1 Density matrix formalism
We review this formalism from the basic laws of quantum mechanics. According to
quantum mechanics, we can describe all the physical properties of a quantum system
(such as a molecule) in terms of a wave function ( )tr,Ψs of a known particular state s and
which obeys the Schrodinger equation [1].
( ) ( )tr,ΨHt
tr,Ψi ss
=∂
∂η (3.57)
VHH 0 += (3.58)
Where, is the Hamiltonian operator of the system consisting for a free operator
and interaction operator of . In order to determine how the wave function evolve in
time, it is often represented in the superposition of the eigenstates of the
Hamiltonian operator of .
H 0H
V
( )ru n
0H
( ) (r(t)uCtr,Ψ nn
ns ∑= ) (3.59)
Where are assumed to be orthonormal by the relation ( )ru n
( ) ( ) ( ) ( ) ⎢⎣
⎡≠=
=== ∫ mn 0mn 1
δdrrurururu mn3
n*mnm (3.60)
40
Where is the probability amplitude of the eigenstate of n. The expectation value of
any operator can be calculated by
( )tCn
A
( ) ( )tr,ΨAtr,ΨA ss= (3.61)
The angular brackets denote a quantum-mechanical average, the wave functions are
written in Dirac notation. The matrix element Amn of the operator is given by A
( ) ( )ruAruA nmmn = (3.62)
If the Hamiltonian operator and the initial state of the quantum system are known, the
time evolution of quantum system and its observable properties are described completely.
However, there are situations under which the state of system is not known precisely, for
example a collection of gas molecules where molecules can interact with each other by
means of collisions. Each collision, the wave function is modified. If the collision is
sufficiently weak, the modification may only relate to the change of the total phase of the
wave function. Therefore, the calculation for keeping track of the phase of each molecule
is impossible and the state of each molecule is unknown. Under such situations, the
density matrix formalism is adequate to present the system in a statistical manner. The
density matrix operator is defined by the relation
H
( ) ( )∑=s
sss tr,Ψtr,Ψpρ (3.63)
Where the index s runs over all of the possible states of the system, the quantity p(s) is
nonnegative and understood as a classical probability of that system which reflects the
lack of our knowledge about the actual quantum state. We can write p(s) under the
normalized fashion
41
(3.64) 1ps
s =∑
It is useful to determine the elements of the density matrix. Multiplying two sides of
Eq.(3.63) with ( )ru n and ( ) ru n , then we get the elements of the density matrix by
using Eq.(3.59&3.60).
sn
s
s*msmn CCpρ ∑= (3.65)
The indices of m,n run over all the energy eigenstates of the system and . The
elements of the density matrix have the following physical meanings:
*nmmn ρρ =
• The diagonal elements nnρ give the probability of the molecule being in energy
eigenstate n.
• The off-diagonal elements ( )nmρmn ≠ are generally complex numbers and
contain a phase, interpreted to be the coherence between levels m and n. It is only
nonzero when the coherent superposition of energy eigenstates m, n occurs and
proportional to the induced electric dipole moment of the molecule of ( )tμ .
In the case of the exact state of system is unknown, the expectation value of any
observable quantity A is given by the trace of the product matrix ( )Aρ
( ) ( )AρTrAρAn
nn ≡=∑ (3.66)
The evolution of the density matrix under the action of Hamiltonian operator in
Eq.(3.58) is given by
ρ H
[ ] nmnmnmnm ρH,ρitρ
Γ−=∂
∂η
(3.67)
42
Where, the damping terms are added phenomenologically. We also assumed that the
molecular coherence is zero in the thermal equilibrium state. We rewrite Eq.(3.67)
nmΓ
[ ] nmnmnmnmnmnm ρV,ρiρiωtρ
Γ−+=∂
∂η
(3.68)
We can rewrite Eq.(3.68) being more specific
( ) mn ρΓE~ρμμρiρiωtρ
nmnmν
νmνnmννnnmnmnm ≠−−+=∂
∂ ∑η (3.69)
( ) mn ρΓE~ρμμρitρ
nnnnν
νmνnmννnnn =−−=
∂∂ ∑η (3.70)
Here, we used the matrix representation : 0H nmnnm0, δEH = and η
mnnm
EEω −= denotes
the transition frequency between the energy eigenstates. Here for the off-
diagonal elements of density matrix is the damping rate for the coherence and
provides the decay rate of population. We also assumed that the energy interaction
operator is defined adequately by the electric dipole approximation
nmΓ
nnΓ
E~.μ-V = . Where
mnμnμm = and denotes the matrix elements of the dipole moment. Here we
used the electric dipole moment
*mnmn μμ =
( )tμ rather than the molecular polarizability ( )tα~ in
previous description.
3.3.6.2 Schematic of energy levels
We consider a collection of identical molecules. Each molecule begins in its ground state
1 may absorb a pump laser photon (red arrow) at the frequency and scatter a Stokes
photon (green arrow) with the frequency
Pω
21P ωω − which is near or equal the Stoke
43
frequency and leaving it from the virtual level to the excited (final) stateSω 2 . Where
denotes the transition frequency 21ω 12 − of the molecule. In general, we assume the
molecule has an arbitrary number of intermediate states m . The Raman scattering is
expressed in the figure 3.7.
Figure 3.7 Energy level schematic for SRS. The molecule is initially in the ground state
1 driven by a strong laser field of the frequency (red arrow), the molecule is moved
to the virtual state and leaving to the excited state after scattering a photon with Stokes
frequency (green arrow). The relaxation terms
Pω
Sω nmΓ in Eq.(3.68) is illustrated
schematically in the ground state 1 and excited state 2 , where is the
dephasing rate of the molecular coherence,
212 Γ=Γ
111 Γ=Γ denotes the decay rate of the
population between the levels 12 − or the inverse of the life time of the level 2
( ) [111 ΓT −= 49,52].
3.3.6.3 Motion equation of density matrix
For simplicity, we neglect the relaxation terms for the moment. Using the Eq.(3.69&3.70)
for the energy level scheme of SRS, we receive the matrix density equations of motion
[53, 54].
44
( E~ρμμρiρ iωtρ
m2m1mm2m12112
21 ∑ −−=∂∂
η) (3.71)
( E~ρμμρitρ
m2mm2
*m2
*2m
22 ∑ −=∂∂
η) (3.72)
( E~ρμμρitρ
mm1m1
*m1
*m1
11 ∑ −=∂∂
η) (3.73)
( ) E~ρμiE~ρρμiiωtρ
212m11mm1mm1m1
ηη−−+=
∂∂ (3.74)
( ) E~ρμiE~ρρμiiωtρ
21m1mm22m22m2m
ηη+−−=
∂∂ (3.75)
We have assumed that the states 1 and 2 have a definite parity so that the diagonal
matrix elements of μ vanish (ˆ 0μμ 2211 == ). The excitation of the intermediate states
m is negligible. The total applied field SP E~E~E~ += , the index m runs over all of
possible intermediate states.
Integrating in time on both sides of Eq.(3.74&3.75) gives us
( ) ( ) ( )( ) ( ) ( ) ( )[ ] ( )ttiω212m11mm1m
t
0
tiωm1m1
m1m1 etE~tρμtE~tρtρμdtie0ρρ ′−′′−′′−′+= ∫η (3.76)
( ) ( ) ( )( ) ( ) ( ) ( )[ ] ( ) (3.77) ttiω21m1mm22m2
t
0
tiω2m2m
2m2m etE~tρμtE~tρtρμdtie0ρρ ′−′′−′′−′−= ∫η
Inserting Eq.(3.76&3.77) into Eq.(3.71), we have
45
+=∂∂
211221 ρ iωtρ
( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ′−
⎥⎥⎦
⎤
⎢⎢⎣
⎡′′−′′−′′+
m
ttiω
**
212
m2
*
11mm1mm2
t
02
m1etρtE~tE~μtE~tE~tρtρμμtd1444 3444 21444444 3444444 21η
( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ′−
⎥⎥⎦
⎤
⎢⎢⎣
⎡′′+′′−′′+
m
ttiω
**
212
1m
*
mm221mm2
t
02
2metρtE~tE~μtE~tE~tρtρμμtd1444 3444 21444444 3444444 21η
(3.78)
It is useful to express E and as below ~21ρ
( ) ( ) ( ) ( ) ( )( c.cetEetE21tE~tE~tE~ tiω
Stiω
PSPSP ++=+= −− ) 3.79a)
( ) ( ) ( ) ( ) ( )( c.cetEetE21tE~tE~tE~ tiω
Stiω
PSPSP +′+′=′+′=′ ′−′− ) (3.79b)
( ) ( )( c.cet21tρ tiω
2121 +′ℜ=′ ′− ) (3.79c)
Substituting Eq.(3.79) into Eq.(3.78) and assuming the exact Raman resonance occurs at
, then only those terms in Eq.(3.78) oscillating near the frequency are
kept. It is also noted that only two of 16 terms in (*) arising from
21PS ωωω −= 21ω
( ) ( )tE~tE~ ′ contribute to
the beat frequency 21PS ωωω =−
( ) ( ) ( ) ( ) tiω*S
t-iωP
tiω*S
t-iωP
SPSP etEetEetEetE ′′ ′+′
The same as in (**) only two components are retained
46
( ) ( ) ( ) ( ) tiω*S
t-iωS
tiω*P
t-iωP
SSPP etEetEetEetE ′′ ′+′
Then integrating in t´ and eliminating the frequency components containing the index m,
we have
+≈∂∂
211221 ρ iωtρ
( ) ( )( ) ( ) ( ) ( ) ( ) tiω-2tω-ωi*SP11mm
*1
21SP et8χietEtEtρtρiκ
41
ℜ−−+ −
( ) ( )( ) ( ) ( ) ( ) ( ) tiω-1tω-ωi*SPmm22
*1
21SP et8χietEtEtρtρiκ
41
ℜ+−+ − (3.80)
( ) ( )( ) ( ) ( ) ( ) ( ) ( ) tiω21tω-ωi*SP1122
*12112
21SP et8χχietEtEtρtρiκ
41ρ iω −− ℜ
−+−+= (3.81)
Where ωωω 21SP =−
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛+
+−
=m Sm1Pm1
1mm221 ωω1
ωω1μμ1κ
η
∑ ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
+−
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
+−
=m Sm1Sm1
2S
Pm1Pm1
2P
21m21 ωω
1ωω
1Eωω
1ωω
1Eμ1χη
∑ ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
+−
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
+−
=m Sm2Sm2
2S
Pm2Pm2
2P
22m22 ωω
1ωω
1Eωω
1ωω
1Eμ1χη
We rewrite Eq.(3.81)
47
( ) tiωtiω*SP1122
*12112
21 2121 e8ΔieEEρρiκ
41ρ iω
tρ −− ℜ+−+=∂∂ (3.81)
Where the Stark shift is expressed by 21 χχ −=Δ
Subtracting Eq.(3.72) from Eq.(3.73)
( ) ( )E~ρμρμμρμρitρρ
m2mm2m1m1
*m1
*m1
*m2
*2m
1122 ∑ −+−=∂−∂
η (3.82)
Replacing Eq.(3.76&3.77) into Eq.(3.82) and do some calculation similar to , we have 21ρ
( ) **SP
*1S
*P1
1122 EEiκ21EEiκ
21
tρρ
ℜ−ℜ=∂−∂ (3.83)
The population difference between the ground level 1 and excited level 2 to be
and write Eq.(3.81&3.83) under the slowly varying amplitude
functions.
( ) ( ) ( )tρtρtn 1122 −=
*SP
*1 EnEiκ
21
4Δi
t+ℜ=
∂∂ℜ (3.84)
**SP
*1S
*P1 EEiκ
21EEiκ
21
tn
ℜ−ℜ=∂∂ (3.85)
Where ( ) ( ) ( )( )c.cet21tρ ΩtKzi
21 +ℜ= −
21ω=Ω ; SP kkK −=
The equations (3.84&3.85) provide an adequate description of the resonant Raman
process where relaxation processes can be neglected, such as pump pulse duration is
much shorter than the relaxation time of material T
Pτ
2. They will be modified in the
48
presence of relaxation processes by adding the decay rate of the molecular coherence
formed and the decay rate of occupation photon numbers from the 2 level to 1
level respectively. We also assume the population difference
2Γ 1Γ
( ) 0ntn = in thermal
equilibrium [1].
( 01**
SP*1S
*P1 nnΓEEiκ
21EEiκ
21 )
tn
−−ℜ−ℜ=∂∂ (3.86)
ℜ−+ℜ=∂∂ℜ
2*SP
*1 ΓEnEiκ
21
4Δi
t (3.87)
In order to consider the meaning of the decay constants, we will examine the nature of
the solutions to these equations in the absence of applied fields [1].
From the Eq.(3.86), the solution for population difference n with E=0
[ ]t
T1
001en-n(0)nn(t)
−
+= (3.88)
Where -111 ΓT =
The Eq.(3.88) shows that the population inversion n relaxes from its initial value n(0) to
its thermal equilibrium value n0 in a time of the order of T1. Hence, T1 is called the
population relaxation time.
From the Eq.(3.87), the solution for coherence ℜ with E=0
[ ] tT1
tiω2121
tT1iω
221221
e(0)eρ(t)ρor (0)e(t)⎟⎟⎠
⎞⎜⎜⎝
⎛−
−⎟⎟⎠
⎞⎜⎜⎝
⎛+−
=ℜ=ℜ (3.89)
Where and assume -122 ΓT = 0=Δ
The solution (3.89) shows the coherence oscillates at the molecular transition frequency
and decays to zero in the characteristic time T21ω=Ω 2. It also can be explained more
49
directly by considering the expectation of the induced dipole moment oscillates near
the resonance frequency for E=0. Using Eq.(3.76, 3.77&3.89), we have the trace with the
density operator.
( )tμ
( ) ( ) [ ] tT1
tiω2121
m2m2mm1m1
221 ec.ce(t)(0)ρμρμρμtμ⎟⎟⎠
⎞⎜⎜⎝
⎛−
− +=+= ∑ (3.90)
This result shows that the induced dipole moment also oscillates at the frequency
and decays in a time of the order T21ω=Ω 2 as the coherence. So, it has the meaning of
the dipole dephasing time.
Using formula (3.66), the macroscopic polarization is calculated for wave propagation
equation (3.32)
( c.cρμρμN)μρNTr(Pm
2m2mm1m1N ++== ∑ ) (3.91)
Inserting Eq.(3.76&3.77) into Eq.(3.91) and only keep the terms oscillating near the
resonance frequency 21ω
( )[ ] ( )[ ] ⎟⎠⎞
⎜⎝⎛ +ℜ+ℜ= ± c.ceE
21eE
21
2NκP tω-zki
Stω-zki*
P
*1N PPSS
η (3.92)
The complex amplitude functions
( )zki*P
*1N
SSeE
2NκP ±ℜ=η (3.93a)
( )zkiS
*1N
PPeE
2NκP ℜ=η (3.93b)
Substituting Eq.(3.93) into Eq.(3.32)
50
For a pump field
PP
SS
P2
P
P
P Ε2γE
vvκ
tΕ
v1
zΕ
−ℜ⎟⎟⎠
⎞⎜⎜⎝
⎛=
∂∂
+∂∂
S
Piωω (3.94a)
For a Stokes field
SS
P*
2S
S
S Ε2γEκ
tΕ
v1
zΕ
−ℜ=∂∂
+∂∂
± i (3.94b)
Where, S
*1S0
2 4nκωcNμκ η
= and SP,
SP, ncv = denotes group velocities
The Eq.(3.86,3.87&3.94) describes detail the temporal-spatial evolution and the
interaction between the applied fields and the coherent excitation of material under the
quantum nature. Let us consider the connection the quantum mechanic model and the
classical mechanic model by the relation of the oscillated amplitude q and the molecular
coherence [( )tρ22 53].
The expectation value of the harmonically oscillated operator ( )tq is given by a trace
with the density operator ( )tρ
∑=nm
mnnmqρq (3.95)
In the case of classical harmonic oscillator, it corresponds to the ground 1 and the
excited 2 energy levels in quantum mechanic. Hence, the indexes m,n only get values 1
and 2.
( ) ( )12212112122121nm
mnnm ρρqqρqρqρq +=+== ∑ (3.96)
We let ,...2,1,0=ν are the oscillated quantum numbers. According to the quantum
harmonic oscillator [54], the elements of oscillated operator are given by.
51
( )⎟⎟⎠
⎞⎜⎛
+=η
⎜⎝
+ 1νω2m
q210
ν1,ν (3.97a)
Where 1,0=ν corresponds to the 1 and 2 levels in Eq.(3.7a). Hence Eq.(3.97a)
become s
( ) c.cρ2m
ρρqq 210
122121 +Ω
=+=η (3.97b)
Where
Comparison Eq.(3.54-3.56) with Eq.(3.86,3.87&3.94) and using the revised relation
we hav some conclusions:
ter,
21ω=Ω
(3.97a), e
*1
21
0Ω2mα ⎞⎛⎞⎛ ∂• The Raman coupling parame 2
00
κεq ⎟⎟
⎠⎜⎜⎝
=⎟⎟⎠
⎜⎜⎝ ∂
.
en the Stark shift, population inversion
effects are negligible
• Classical model is relatively adequate wh
( )0n0,Δ == .
ficie• The Raman gain coef nt, 2
2P21 Eκκ2
g =Γ
(3.98)
teristic of the created Stokes fields in SRS strongly
epends on the dephasing rate of coherenc
3.3.6.4 Transient regime in SRS
The temporal and spectral charac
d e 2Γ , the effective interaction length z, the
m
Raman gain g given in Eq.(3.98) as well as the pump duration Pτ . Hence, it is convenient
to determine the scattering regime of the syste by comparing the pump pulse duration
Pτ with the characteristic times 1τ & 2τ defined in Eq(3.99). This section is mainly
derived from [30, 52].
Γgz1τ = &
Γgτ =
z2 (3.99) 1
52
Spontaneous Raman scattering: The pump pulse duration is smaller than the
haracteristic time ). It is too short to provide a coherent emission from the
Stokes intensity
c 1τ ( 1P ττ <
generated Stokes photons.
( ) gz2SΓτz,I ≈ , where c
z-tτ = is the local time. The Stokes
the pulse pump duration
Steady state regime: The pump pulse duration is longer than the characteristic
ti between the
amplification is very low and independent from Pτ .
time 2τ ( 2P ττ > ). This regime causes the loss of the mutual phase correla on
Stokes and the initial pump fields by the effect of the collisional dephasing rate 2Γ .
Stokes intensity ( )πgz2
Γeτz,Igz
S ≈ , the Stokes intensity is also independent on romf the
tion as the
higher.
nt regime
uration lies in the range of the characteristic times, . In
hich the pulse pump duration is short enough that the dephasing rate of coherence
pulse pump dura spontaneous one, but the Raman amplification is much
Transie
Pτ
The pulse pump d 2P1 τττ <<
w 2Γ
has minimum effect during the generation time of the Stokes pulse but enough long that
the number of pump photons is sufficiently large to trigger a SRS process. The transient
regime is mostly interested by the Stokes field generated can retains the high molecular
coherence with the initial pump field which is important in coherent generation of
cascaded Raman scattering for ultrashort compression in a gas-filled capillary [55,56].
From Eq.(3.99), it is clear that the required range of pump duration for the transient
regime depend strongly on the net Raman gain gz. The higher net Raman gain, the wider
the range of pump pulse width and vice versa.
The evolution of Stokes intensity
53
( )πτ
τ
8eτz,I
gz2
S
Γ
≈ (3.100)
The Raman amplification in this regime IS depends on the pulse pump duration , the
longer gives the higher amplification.
Pτ
Pτ
The temporal and spatial evolution of Stokes intensity IS as a function of the normalized
interaction time as well as its respectively scattering regimes are illustrated in fig.3.8 Γτ
Figure 3.8 Evolution of the Stokes intensity IS (red curve) in the normalized interaction
time (log scale). The operation regimes are respectively expressed: spontaneous
(white), transient (blue shaded) and steady state (shaded yellow) regimes [52].
Γτ
54
Chapter 4 Backward stimulated Raman scattering in
H2-filled PBG-PCF
This chapter focuses on the application of BSRS for pulse shortening and generation of a
pulse train using this technique. During my studies I have built the setup for this
experiment and performed experimental measurements for backward Raman processes.
4.1 Introduction
The most popular SRS interaction geometry is the forward SRS geometry (FSRS) where
the signal Stokes (seeded or generated from noise) has the same propagation direction as
the pump beam (fig.4.1a). Another SRS geometry is the backward SRS (BSRS) in which
the signal Stokes and pump beams travel in two opposite directions (BSRS), as shown in
fig.4.1b. In BSRS, the Stokes signal is usually provided by an external seed because the
backward to forward Raman gain ratio is very small [8].
Figure 4.1 Stimulated Raman scattering geometries a) FSRS the pump of (red) and
signal Stokes of (from noise) travel in the same direction; b) BSRS the pump and seed
(green) travel in opposite directions [8].
Pω
Sω
In principle FSRS and BSRS are different in the amplification scheme of the Stokes
signal. The former one, the forward-traveling Stokes pulse extracts just the energy stored
55
in a small co-propagating volume element of the pump pulse envelope. Therefore, the
Stokes intensity is limited by the initial pump intensity. In the latter case, the backward
Stokes signal continuously encounters the undepleted pump pulse. As a result the leading
edge of the backward pulse is amplified and its intensity can reach to a value far in excess
of the pump intensity. Beside the high amplification, the backward pulse is also
sharpened and steepened [8,12].
4.2 Backward and forward Raman gain asymmetry
The difference in the interaction geometry of forward and backward SRS causes the
difference of Raman gain. SRS gain coefficients are given by [15]
( )PBF,2
S
BF,BF,2SBF, I
ΔωπndΘ
dσΔN2λg = (4.1)
Where the indices (F,B) denote the backward and forward Raman scattering cross-
sections, is the scattered wavelength. In general, the asymmetry of backward and
forward Raman gain is caused by the differences in the following factors: Raman
linewidth including the Doppler-linewidth broadening of the Raman scatters
Sλ
BF,Δω DΓ
which is different for forward and backward scattering; the differential spectral cross
section ( ) BF,
dΘdσ ; the pump linewidth ; the initial and final state population
difference [
PΓ
BF,ΔN 8,15].
For simplicity, we assume that the differential spectral cross section and the initial and
final state population difference are equal for the two cases and that the influence of the
pump linewidth is neglected. Then, the Raman gain asymmetry by the Doppler-linewidth
broadening in H2 gas medium will be considered below.
At low gas pressures, assuming the velocity vector v of a molecule has a component vz in
the same direction as a pump photon (moving with the light velocity c) [8]. It will
56
provide a Doppler shift of cvω z
P− on the pump wave, cvω z
S− on the scattered wave in
forward direction, and the shift of cvω z
S+ on the scattered wave in backward direction.
Therefore, the net shift from Raman resonance in the forward Stokes wave of
( )cvωω z
SP −− and in the backward Stokes wave of ( )cvωω z
SP +− . Two scattering
processes are illustrated in figure 4.2. A pump photon (the blue-arrow curve) is
scattered by a hydrogen diatomic molecule (a black dumbbell), resulting in the emission
of a Stokes photon (the red-arrow curve) in the same direction as pump photon for
forward SRS (fig.4.2a) and in opposite direction with pump photon for backward SRS
(fig.4.2b).
Pω
sω
Figure 4.2 Motion diagram of Raman scattering: a) forward SRS; b) backward SRS
Using Maxwell-Boltzmann velocity distribution for the mean thermal velocity of
mT2kB , where T is the temperature in Kelvin (here we assume T= 298K) and m is the
H2 mass in the atomic units, kB is a Boltzmann constant. The Doppler line width in
forward and backward Raman scatterings are given as following [8,57]
For forward SRS
( )m
T2ln2kcωω2Γ BSPF
D−
= (4.2)
57
For backward SRS
( )m
T2ln2kcωω2Γ BSPF
D+
= (4.3)
It is clear that the Doppler-linewidth broadening depends strongly on the type of
scattering and is much larger in the backward type than forward SRS. The Doppler
broadening backward and forward linewidth ratio is 30 in the rotational Raman transition
at pump wavelength 1064nm. Doppler-linewidth broadening is also conversely
proportional to the mass of the scatters. Hence, the larger the scatters mass, the smaller
the effect of Doppler broadening to Raman gain.
In the limit of high gas pressure, the linewidth of Raman medium at the room temperature
is given by [57]
ρB=RΓ (4.4)
Where ρ is a gas number density per volume unit (amagat), B (MHz/amagat) is a
broadening constant of Raman linewidth. For rotational Raman transition, B=110
(MHz/amagat).
If we ignore the effect of collisional narrowing at relatively low pressures, the total
Raman linewidth has contributions from Doppler-broadening effect at low pressure and
pressure-broadening effect at high pressure.
The asymmetry ratio R between the Raman gain of the forward and backward SRS
caused by the Doppler-broadening effect is given by.
FDR
BDR
F
B
B
F
ΓΓ
ΔωΔω
ggR
+Γ+Γ
=≈= (4.5)
58
We also neglected that the effect of molecular collisions with the fibre’s core-wall in the
relation (4.5).
Inserting Eq.(4.2-4.4) into Eq.(4.5), the dependency of R on the pressure p (bar) is
described in the figure 4.3. Where the parameters are calculated for the rotational Raman
transition ,( )MHz1017.4ωω 6SP ×=− ( )MHz10547ωω 6
SP ×=+ , m=2(amu), T=298(K).
Figure 4.3 shows the high gain asymmetry curve (red) R of forward and backward
rotational SRS in H2 at low gas pressure, which is due to the larger Doppler effect in
backward scattering. When the pressure increases, this ratio R decreases and achieves the
values 5.5, 2, 1.5 at the pressure 7bar, 50bar, 100bar respectively. In reality, these ratios
R are even higher by other contributions to the backward scattering such as the pump
linewidth, intensity fluctuations, differential Raman cross section not included in our
calculation.
Figure 4.3 Dependence of forward/backward rotational SRS gain ratio on the H2 gas
pressure at the pump wavelength of 1064nm [15].
59
4.3 Motivation
The first backward Raman scattering was observed in 1966 [11]. The experimental
demonstration of 20 times amplification by backward Raman in liquid CS2 has been
performed impressively [12]. BSRS has several advantages over the forward case, which
results in its use in different applications such as high power amplification, pulse
shortening and wavefront conservation [12,58], converting poor quality pump beam to
high quality output beams [14,16,58] and high power ultraviolet excimer lasers [8].
Recently BSRS technique is emerging as a promising candidate in generating of powerful
ultra-short pulses in plasma. By pulse amplification in plasma it is possible to overcome
the limitation of the thermal damage encountered in the more traditional chirped pulse
amplification method [59,60,61].
SRS in gases offers many advantages over solids and liquids: high damage threshold, a
low linear absorption, less self-focusing. However, SRS in gaseous medium requires high
pump power for initiating stimulated Raman operation. In previous experiments done in
focused-beam geometry or capillaries, the pump power requirement is up to several MW
with a conversion efficiency of only a few percent from the initial pump energy [62,63].
The threshold is even higher for BSRS caused by the low backward Raman gain [64].
Moreover, pulse shortening becomes more difficult for light molecule such as H2 caused
by the large Doppler linewidth broadening (see Eq.(4.3)).
HC-PCF as a microcell offers an excellent guiding structure: diffraction-free long
interaction length allows light to be tightly confined during its propagation while the
flexibility in designing the position of the guidance band helps us to remove unwanted
higher order Raman components [28]. These excellent features make it become an
excellent candidate for the investigation of light interaction with gases or vapor filled into
its hollow-core. The very long interaction length of HC-PCF makes its net Raman gain
increase hugely. As a result the Raman threshold energy is quite low (six orders of
60
magnitude smaller compared with previous setups). Such low energy levels are
significantly below the threshold for other competing nonlinear processes. Because of the
high net Raman gain of HC-PCF, SRS transient regime (coherent interaction regime) can
be achieved in a very large range of pump pulse durations [30] and allows us to gain
deeper into different schemes in the stimulated Raman scattering [31,32,65].
4.4 Optical pulse compression via BSRS
4.4.1 Experimental setup
The schematic for the experimental setup is presented in figure 4.4. A laser source
1064nm delivering 40ns pulses, with a maximum energy of 100µJ is split into three parts
for Generator, Amplifier 1 and Amplifier 2.
Generator provides pump energy for generating the seed Stokes by 2.5m PBG-PCF filled
with H2 gas at pressure 4bar. The rotational Raman transition between the levels j=1 and
j=3 ( ) of the hydrogen molecule is chosen by its dominance in room
temperature.
18THzΩR =
61
Figure 4.4 Setup diagram for two-consecutive stage pulse compression using BSRS in H2
filled PBG-PCF. Generator provides the seed Stokes 21 ns. The first compression stage-
Amplifier 1 gave the signal duration of 3.6ns at the signal detection SD1. The second
compression stage-Amplifier 2 gave the signal duration of 2ns at the signal detection
SD2.
The transmission wavelength window of this PBG-PCF is shown in figure 2.5 which only
guides the pump wavelength λP=1064nm and Stokes seed wavelength λS=1134nm. The
higher order Stokes, anti-Stokes as well as vibrational transition frequencies are not
guided caused by the high loss. Notch filter 1064 nm is placed at the output window to
filter the seed pulse 1134 nm having a duration of 21ns which is about 2 times shorter
than that of pump. The BSRS is studied in two consecutive stages Amplifier 1 and
Amplifier 2. Amplifier 1 is used for the first compression stage, where the fibre length
3m and pressure-9bar are chosen to increase a backward-forward Raman gain ratio while
ensuring that forward Raman threshold is not quite low. These conditions require the
pump range from to 12psτ1 = 75nsτ2 = for transient regime of rotational BSRS in HC-
PCF, which the experimental pump duration ns04τP = is suitable. It is noted that the
backward/forward Raman gain ratio is reduced at low pressure by the Doppler linewidth
strong broadening in backward direction. In contrast, it increases with the gas pressure
increasing (see fig.4.3). In this stage, FSRS threshold is measured to be 6.2µJ. The pump
pulse beam propagates through a quarter-wave plate (λ/4) that alters the polarization to a
circular state. The delay line (R1) is used to optimize the BSRS performance. The pump
energy is easily changed by a polarizer (GL) and half-lambda plate (λ/2). Dichroic mirror
(DM) with high transmission for seed wavelength 1134nm and high reflection for pump
at 1064nm is used to combine the Stokes and pump. The seed Stokes pulse with energy
of 1µJ meets the pump pulse at the entrance of the fibre. Timing is so that the seed Stokes
enters the fibre exactly when the leading edge of the pump pulse is leaving the fibre. The
pump energy is varied from 0 to 7µJ. This energy range is high enough to get enough
amplification by BSRS effect but also small enough so that forward Stokes is not
dominant. In order to maximize the BSRS performance, the seed is injected into the fibre
window at the time the leading edge of the pump pulse begins to exit through that
62
window. The seed and pump polarizations are optimized to be circular in opposite sense.
Output signals of Amplifier 1 are received by signal detector (SD1) by the beam splitter
(BS) and recorded with one channel of oscilloscope.
The output pulse of the Amplifier 1 also enters the Amplifier 2 (the second compression
stage). We used the same parameters as the first stage with the fibre length of 3m,
pressure 9bar. The energy is only changed in the range of 0-3µJ. The delay line (R2),
dichroic mirror (DM) and quarter-lambda plate (λ/4) is also used for optimization. Output
signal of Amplifier 2 are received by signal detector (SD2) and recorded with another
channel of the oscilloscope.
4.4.2 Results and discussion
The measurements are presented in figure 4.5: the seed Stokes with the duration of 21ns,
energy-1µJ and rising time about 7ns is coupled into the Amplifier 1. The leading edge of
the seed pulse always encounters undiminished pump pulse and continues to grow rapidly
until it leaves the end of the Amplifier 1 (the entrance window of the pump pulse). This
results the shortening and sharpening of the seed pulse duration and front during its
propagation in fibre. At the pump energy 7 µJ and the end of the Amplifier 1, the signal
1 pulse is shortened down to 3.6 ns with its rising time 2ns and energy of 4 µJ.
63
Figure 4.5 Time and intensity evolution of backward Stokes in Amplifier 1 from seed
Stokes with 21ns, 1µJ is compressed and sharpened gradually to Stokes pulse of signal 1
with 3.76ns, 4µJ when pump energy is increased from 0 to 7 µJ.
The dependence of the pump energy on the backward amplification is also showed in
figure 4.5 which is divided into two stages: The linear stage (pink->green)-the pump
energy is changed from 0 to 2.5µJ, the amplification of the Stokes seed is nearly uniform
through the entire pulse. The nonlinear stage (blue->red) the pump energy is increased
from 2.5µJ to 7µJ, the sharpening and shortening occurs strongly, resulting in increase in
its rise time and decrease in the duration of Stokes pulse from 21ns to 3.6ns. Moreover,
the position of pulse peaks is moved to earlier positions in time, which is caused by
reshaping of the Stokes pulse due to its amplification. For this work, the trailing edge is
not critical to our consideration of the Raman pulse compression ability. We are able to
eliminate seed pulse’s trailing edge by the electro-optic technique.
The output signal of Amplifier 1 (3.6ns, 4µJ) keeps compressing by the Amplifier 2. The
Stokes seed energy used for the Amplifier 2 is 0.3 µJ. The measured result is shown in
figure 4.6. The signal 1 with 3.6ns, 0.3µJ is converted into the signal 2 with 2ns and 1µJ
energy.
64
Figure 4.6 Time and intensity evolution of backward Stokes in Amplifier 2, which the
Stokes pulse of signal 1 with 3.6ns, 0.3 µJ is compressed and sharpened gradually to
Stokes pulse of signal 2 with 2ns, 1µJ when pump energy is increased from 0 to 3 µJ.
4.4.3 Dynamical analysis of reverse-pumped Raman pulse
To discuss the dynamics of pulse compression by the BSRS formalism, we consider the
pump field ( )[ ]( c.ctωzkiexpE21E~ PPPP +−= ) moves in the direction +z and the Stokes
seed field ( )[ ]( c.ctωzk-iexpE21E~ SSPS +−= ) moves in the opposite direction –z inside
PBG-PCF filled with Raman active medium (hydrogen gas). Here ( )SP,SP,SP, ωβk = are
the propagation constants of the guided modes pump and Stokes seed respectively
in the fibre. Spatial and temporal evolution of pulse propagation in the Raman active
medium are represented by field equations
Pω Sω
PP
SS
PP2
P
P
P Ε2γE
vvκ
tΕ
v1
zΕ
−ℜ⎟⎟⎠
⎞⎜⎜⎝
⎛=
∂∂
+∂∂
S
iωω (4.6)
65
SS
P*
2S
S
S Ε2γEκ
tΕ
v1
zΕ
−ℜ=∂∂
+∂∂
− i (4.7)
Where is the loss coefficients at the frequency of . Two incoming fields pump
and Stokes drives the transition dynamics of the ground and excited states in the H
SP,γ SP,ω
2
molecule by following equations
ℜ−=∂∂ℜ
2*SP
*1 ΓEnEiκ
41
t (4.8)
**SP
*1S
*P1 EEiκ
21EEiκ
21
tn
ℜ−ℜ=∂∂ (4.9)
Where, denotes the amplitude of Raman coherence oscillating at the rotational
transition frequency of
ℜ
SPR ωωΩ −= , 1122 ρρn −= is the population inversion of the
excited and ground states. We also neglected the contribution of Stark-shift ( 0Δ = ).
Equations (4.8&4.9) present the coupling of input fields with the molecule [31].
66
Figure 4.7 Evolution of Stokes pulse (red) shortening by the reverse-Raman pumping
(blue). The coherence (black curve) is formed in backward SRS.
The coherence is driven by a term depending on the product of the inversion n with a
term ( ), and the inversion n is driven by a term depending on the product of the
coherence with a term ( ). The process of pulse compression in BSRS is
described numerically in the figure 4.7. The long pump pulse (blue curve) and the shorter
Stokes seed (red curve) move in opposite directions in PBG-PCF and they collide each
other at time t=0 at the entrance of Stokes seed. The pulse front of Stokes seed is
amplified according to Eq.(4.7) and concurrently create the coherence in the amplifier.
As the Stokes wave continues to propagate at next times t=2ns, 3ns, … its leading edge
keeps encountering and extracting the energy from the pump pulse, consequently the
Stokes pulse continues to be amplified. When the leading edge of the Stokes pulse grows
rapidly enough then the pulse will be narrowed and has the “shark fin” shape shown in
fig.4.7. This process continues until a high-power short pulse is obtained at the output of
the amplifier.
*SPEE
ℜ *SPEE
4.5 Generation of solitary-like pulse train
In this work, we present a newly dynamical process generating a train of solitary-like
pulses observed in backward stimulated Raman scattering via H2 gas filled PBG-PCF.
4.5.1 Experimental process and results
Experimental setup is shown in figure 4.8. It is the same as the figure 4.4 except the
amplifier 2. However, some experimental parameters have been changed for amplifying
other peaks (oscillations): Generator’s fibre length is 1.6m filled with H2 gas at pressure
5bar. The duration of Stokes seed is 5 times shorter than the pump duration (40ns). For
the amplifier stage, fibre length of 1.4m was chosen to limit the effect of forward Raman
scattering and enhance the pump energy for other peaks at the first pulse’s trailing edge
67
but also obtain reasonable high amplification. The pressure is kept at 5bar. Pump pulse
energy is in the range of 0-18µJ with a FSRS threshold at 12 µJ.
Figure 4.8 Setup diagram for generating a solitary-like pulse train in transient BSRS by
H2 filled PBG-PCF. Generator provides the seed Stokes 1134nm, pulse width 7ns.
Amplifier: the seed Stokes is counter-propagated by the pump wave 1064nm; pulse
duration 40ns; pressure gas is 5bar
The experimental result shows the appearance and evolution of backward peaks at the
seed’s trailing edge when the pump energy is increased in the range 0-18µJ, presented on
figure 4.9. In the absence of the pump (0 µJ), the seed pulse energy (dash-black curve) is
small, duration of 7ns, fluctuating and asymmetric with a long trailing edge. The total
BSRS process can be divided into two stages [68].
In the first stage, linear stage, pump energy is increased from 0µJ to below 10µJ, the seed
pulse is uniformly amplified. Its sharpening and shortening are negligible, the Stokes
pulse shape is similar to the original one (seed Stokes). Consequently, its peak’s position
in time is nearly constant or its maximum moves with light velocity in vacuum.
68
Figure 4.9 Evolution of a pulse train in transient BSRS via the increasing of pump
energy. Stokes intensity, time are described by vertical and horizontal axis respectively.
The third axis shows the direction of pump energy: black and red arrows show linear and
nonlinear stages.
In the second stage, the nonlinear stage, pump energy is increased from 10 µJ to 18µJ. At
the pump energy 12µJ (nonlinear threshold), which is approximately equal to the FSRS
threshold, additional asymmetric peaks whose shapes look similar to the initial seed pulse
begin to emerge from the trailing edge of the first pulse. As the pump energy is kept
rising, the first peak amplification seems to be saturated in amplification. Interestingly,
these additional asymmetric peaks are amplified, shortened and reshaped to relatively
symmetric shapes, similar to the first Stokes. At the pump energy 18µJ, the signal
consists of three consecutive solitary-like pulses. The duration of the first, second and
third peaks are 6.5ns, 5.5ns and 4.4ns, respectively.
Experimental observations can be explained as following: for low pump intensity, the
rate of the coherence wave (nonlinear polarization) creation cannot overcome the effect
of dephasing (or relaxation), so that the Stokes pulse will be amplified uniformly. When
pump intensity is sufficiently strong, it can overcome the effect of dephasing [69] and the
pump duration required for transient SRS regime is larger (see section 3.3.6.4). This
results in the generation of a strong coherence wave that lasts for the duration of the first
69
Stokes peak, creating a channel for back-flow of energy at its trailing edge to the pump
[31]. This part of the pump in turn can use the already existing coherence to amplify the
long trailing edge of the first Stokes peak, resulting in the generation of a multi-peak
structure. Peak number will be increased with the increasing of the interaction length,
pump duration and intensity.
One difficulty here is the presence of forward Stokes that results in the saturation of the
first peak intensity (seed). Moreover, the length of the active medium 1.4m is short
compared with the physical length of pump pulse-12m (40ns). This means that after the
passage of the first peak and its satellite through the fibre one could expect the
appearance of a similar dynamical behavior.
4.6 Conclusion
To summarize, by using the advanced characteristics of PBG-PCF we have generated
efficiently pulses 20 times shorter than that of original pump by using only a laser source
with low peak power (maximum 2.5kW) in hydrogen medium. Of course, this
experimental configuration is also applied to other gases and laser sources for optical
pulse compression. Other heavier gases such as CH4, SF6, CF4, SiH4, etc., providing the
higher backward/forward Raman gain ratio should also be interesting in this regard.
Hence, the Stokes amplification and shortening effects may be higher than in H2 gas [8].
The opto-electrical modulator may also be used to cut the leading edge of seed Stokes for
a shorter rising time. This may improve significantly the compression efficiency because
the shortening in transient stimulated Raman scattering depends significantly on the
rising time and the timing advancement of Stokes seed [66,67]. In the transient BSRS
regime, a new dynamical process generating a train of solitary-like Raman pulses with
flexibly controllable peak intensities have been observed.
70
Chapter 5 Phase-coherent Raman frequency comb in
gas filled HC-PCFs
For thus experiment I participated partly in performing the experiments for the generation
part of Raman frequency comb. This work was presented in [79].
5.1 Introduction
Optical frequency comb has a wide range of applications such as highly precise optical
atomic clocks and generation of ultrashort pulses which has led to measure and control of
previously inaccessible physical and chemical processes [70,71,72]. These applications
require mutual coherence, phase-coherent (stable relative phase) within precisely
equidistant comb lines [73,74,75,76]. Stimulated Raman scattering in gas has recently
been an attractive approach for creating a broadband frequency comb. It has been shown
that the spectra of frequency comb as much as four octaves can be adiabatically generated
in hydrogen gas medium by using two-pump lasers whose frequency difference nearly
equal to a resonant Raman transition [49]. Following efforts in controlling the carrier-
71
envelop phase (CEP) for the synthesis of the single-cycle or sub-cycle optical pulses have
been carried out. Normally, these approaches have to use complex setups, for example, a
cryogenic temperature system with the support of the phase modulator [77], the different
mixing of a pulsed dye laser and a pulsed Ti:Sapphire lasers [76], dual-wavelength laser
radiation locked on a single cavity [78].
In parallel developments, HC-PCF with Kagomé lattice possesses unique characteristics
such as low loss and dispersion, ultrabroad transmission bandwidth makes it an excellent
candidate for muti-octaves frequency comb. Kagomé-PCF for frequency comb generation
allows us to reduce six orders of magnitude in the required laser powers over previous
equivalent techniques [29]. Currently, frequency comb generation via H2 gas-filled HC-
PCF is mainly initiated from quantum fluctuations resulting in a broad frequency comb
consisting of both rotational and vibrational Raman lines [33] which is not convenient for
synthesizing ultrashort optical pulses [73,74]. In this work, we generated a broad,
mutually coherent, phase-coherent, purely rotational-Raman frequency comb using a
relatively simply setup consisting of a microchip pump laser source and two hydrogen-
filled HC-PCF [79].
5.2 Purely rotational Raman frequency comb generation
Previous schemes of Raman comb generation using gas filled HC-PCF mainly depend on
the spontaneous generation of Stokes field [29], where the molecular transition is driven
only one pump laser field. In the present work, the molecular transition is resonantly
driven from the frequency difference of two monochromatic pump and seed Stokes fields
(resonant excitation scheme). This approach provides some advantages: the well defined
input elements such as frequency, delay time and high selection of the molecular
excitation.
72
Figure 5.1 Frequency-propagation constant diagram for Raman comb generation based
on a resonant excitation scheme. Red and green arrows describe Stokes and anti-Stokes
processes respectively. The Raman transition is equal the difference of two-driving
frequencies of . 10 ω-ω=Ω
Consider the dispersion curve of a HC-PCF to be as presented in fig.5.1, the process of
comb generation is described as following. The molecular excitation was strongly
modulated by driving frequencies (pump and Stokes seed) which create the strong
coherence wave with the rotational Raman frequency of 10 ω-ω=Ω . The first Stokes
and pump wave combine this coherence wave to generate a broadband comb
including many sidebands of Stokes and anti-Stokes lines shifted from the pump
frequency by multiples of the rotational Raman frequency is generated. These sidebands
and their propagation constants are expressed on vertical and horizontal axis respectively
in figure 5.1. In which red arrows are described for phonons generated in Stokes process
and green arrows are given for phonons created in anti-Stokes process.
1ω
0ω
A schematic of the experimental setup is shown in fig.5.2a. It includes two stages: Seed
preparation in a narrowband guiding hollow-core photonic crystal fiber (PBG-PCF) and
comb generation in a broadband guiding Kagomé-PCF. The output of a 1064 nm
73
microchip pump laser, delivering pulses of 100 μJ energy and 2 ns duration at 1064 nm,
is split into two parts. The first part (~10µJ) is coupled into a 2 m long PBG-PCF with
loss of 0.13 dB/m and a transmission window of 150 nm wide centered at 1100 nm (its
loss spectrum was shown in fig.2.5).
Figure 5.2 a) Experimental diagram for frequency comb generation. Initially, seed pulse
(the first Stokes) was created by 2m H2 gas-filled PBG-PCF (left-hand side). Then,
commensurate sidebands were generated by driving resonantly with Raman transition by
the coupling of pump field and seed field in a 60cm Hpω sω 2 gas filled Kagomé-PCF
(right hand side). A right-above inset compares the narrow loss spectrum of PBG-PCF
74
(grey shaded region) and the broadband loss spectrum of Kagomé-PCF. b) A purely
Raman rotational comb (purple lines) was generated by Kagomé-PCF. The solid green
curve indicates the total wavevector mismatch (waveguide + gas) across the
frequency comb [
Δβ
79].
The fiber is filled with 6bar hydrogen gas. The limited frequency bandwidth of the PBG-
PCF (30 THz) only supports for the pump wavelength 1064nm and the first rotational
Stokes generation at wavelength 1134 nm (Raman transition 31 =→= JJ or frequency
shift of ). After filtering out the residual pump from the output of the PBG-
PCF, the rotational Stokes was used as a seed for the second stage of comb generation.
The seed Stokes pulse generated in PBG-PCF is timed with the arrival of the pump pulse
of the second part (~60µJ) and coupled into a 60 cm long Kagomé-PCF. Kagomé-PCF
has a broad transmission window extending from 800 nm to more than 1750 nm with an
average loss of 2dB/m. Its transmission loss is shown by a top-right inset in fig.5.2a
showing much broader transmission spectrum than that of PBG-PCF. The wide
transmission window of Kagomé-PCF supports multiple rotational Raman lines of ortho-
hydrogen. Kagomé-PCF was also filled to the same pressure as the PBG-PCF, two gas-
filling systems being physically connected, and the experiments were carried out at room
temperature. This scheme is different from the two-color excitation scheme in which the
Raman coherence is prepared adiabatically in a cryogenic environment [
18THzΩ =
49].
As a result rotational Raman comb consisting of four anti-Stokes frequency components
and five Stokes lines spanned almost over Kagomé-PCF’s transmission window was
generated (fig.5.2b). Generated frequency comb does not contain any vibrational Raman
lines irrespective of the input polarization of the input pump pulse. This shows the highly
selective excitation of molecular motion in this technique.
In order to examine the coherent characteristic of the Raman frequency comb-lines we
first carried out a frequency-doubling process. Output spectrum of the Kagomé-PCF was
collimated and focused into a 5 mm thick BBO nonlinear crystal (fig.5.3a). The phase-
matching conditions of the frequency-doubling crystal can be controlled by changing the
angle of BBO crystal. Figure 5.3b shows a typical frequency-doubled spectrum recorded
by a spectrometer. We also recognized that that signal spectrum contains both second
75
harmonic (SH) and the sum-frequency (SF) components. Figure 5.3c shows photographs
of the doubled signal (cast on a screen) for three different angles of the BBO crystal. Any
uncorrelated temporal phase variations in the Raman comb lines would result in heavily
decreased levels of sum-frequency signal strength. The efficient generation of sum-
frequency signals hints toward the existence of a mutual- and self-coherence between
individual comb components.
Figure 5.3 a) Setup schematic of the frequency-doubling used a BBO nonlinear crystal. b)
Frequency-doubling spectrum of was recorded by a spectrometer. c) Signal photographs
were casted on a screen at three different angles of BBO crystal [79].
Experimental observations can be explained by defining the frequency of nth-order
Raman line as (n is integer), where (or ) and are the pump and
Raman transition frequency respectively. The m
nΩωω pn += pω 0ω Ω
th-order frequency-doubled sideband is
mΩω2ωωω~ 0n-mnm +=+= (m is integer). Hence, the magnitude of the mth-order signal
76
frequency (wavelength) will
he signal strength at the mth -order frequency is given by
(5.1)
Where is the complex amplitude of the nth-order Raman frequency sideband.
be the sum over all possible second harmonic (SH) and sum
frequency (SF) contributing to this frequency.
T
( )nmn ΦΦi
nnmnm eAAS −+
−∑∝
niΦneA
Eq.(5.1) indicates that any uncorrelated phase variations of nΦ will result in the
significant reduction of signal strength mS when averaged over m y laser shots. If the
signal strength is steady and efficient, it demonstrates that comb-lines are mutually
coherent. In the next section, the phase characteristic of individual comb lines was
considered in detail using a doubled-frequency interferometry [
an
73].
5.3 Stable phase-locking characteristic between comb lines
he technique of frequency-doubled interferometry allows us to measure directly the
e assume that all spectral lines are monochromatic and well separated in frequency.
T
relative phase difference of the comb components. For this purpose, the generated Raman
comb was separated into two equal parts by a non-polarizing splitter (B). The delay time
of the first part was controlled by a micro-stepper and filtered so that only pump and the
1st Stokes remained. Then, it was mixed with the second one (comb spectra) in the BBO
crystal using a concave mirror (M). The output of interfering doubled frequency spectra
were dispersed by a diffraction grating and detected by a detector. The set-up schematic
of interfering sum-frequency process is illustrated in figure 5.4a.
W
The signal strength at the sum frequency mω~ is the contribution of two sum-frequency
lines m0 ωω + and 1m1- ωω ++ , where R0m mΩωω += is the frequency of mth-order
Rama ty as a fun lay time τ at this frequency is n line. The signal intensi ction of the de
given by
77
( ) ( ) ( ) 2ΦΦτωi1m1-
ΦΦτωim0
mSF
1m1-1-m00 eAAeAAI ++++
++ +∝τ
( ) ( ) ( )m1mm1-02
1m1-2
m0 ΔΦΩτcosAAAA2AAAA +++= ++ (5.2)
Where is the phase difference of the consecutive n
sidebands. The bracket
Rama1mm1-0m ΦΦΦΦΔΦ +−+−=
... is an ensemble average over many laser shots. If the phase
difference of is stable, the shot-to-shot averaged signal intensity m ( )ΔΦ τmSFI should
show a clear sinusoidal variation as a function of the delay time with a fixed period τ
Ωπ2 . Then, w tain the value of by measuring the phase of the periodic change e ob
of
mΔΦ
( )τmSF
I . In contrast, if the fluctuation of phase difference is strong, the modulation
term in Eq.(5.2) will be vanished when averaged shot-to-shot.
78
Figure 5.4 a) Experimental schematic for extracting the phase information between
Raman comb lines via the sum-frequency spectral interferometry. b) Intensity variations
of ( )τmSFI respect to the delay time (black-dot) for sum-frequency lines and their
fitting curves (red-solid), where each intensity of m
τ
th-order doubled-frequency line is the
mixing of two SF and SH pairs expressed respectively with nΩωω 0n += [79].
The intensity variations for four sum-frequency lines against delay time when averaged
over many laser shots are shown in fig.5.4b. They displayed a clean sinusoidal
modulation respect to the delay time at the same period of 57 fs
τ
( )Ω2π= without any
active stabilization of setup, in good agreement with the theoretical formula Eq.(5.2),
indicating the existence of phase-coherence among the comb-lines.
The relative temporal shift of the peaks provides directly the phase difference between
the comb lines. As shown in fig.5.4b, the magnitudes of sum-frequency signals at the
starting positions are different. This indicates that they are not in phase at the input face
of the BBO crystal. This mismatch can be caused by fibre dispersion and propagation of
laser beam through optical elements. To bring all the spectral components into phase, it
needs to adjust appropriately the optical path lengths of Raman lines, for example using a
79
pair of prisms [77] or a liquid crystal modulator [10]. This would result in the generation
a train of ultrashort pulses at a repetition rate of 18 THz.
The level of the phase-coherence will be affected by fiber dispersion because the efficient
coupling between the comb-lines relies on the same coherence wave being able to couple
efficiently between all the comb lines. If this is not the case, for example for Stokes and
anti-Stokes bands far away from the pump and first Stokes frequencies, uncorrelated
coherence waves will grow from noise and disturb the overall coherence of the system.
The rate of linear dephasing for the mth -order coherence wave (beat note signal) relative
to the 0th-order one is considered by the wavevector mismatch of the scattering system
(gas + fibre), plotted in fig.5.2b (solid-green
curve). Moreover, the nonlinear phase-locking can play an important role in maintaining
coherence, for example it can cause efficient anti-Stokes generation in gas-filled HC-
PCF, even in the presence of large linear phase-mismatch [
( ) ( ) coh1,0-
cohm1,-m01m1mm ββββββΔβ −=−−−= −−
80].
5.4 Summary
By using of the novel properties of Kagomé-PCF, we have generated efficiently a broad,
purely rotational Raman frequency comb in hydrogen gas. Sum-frequency spectral
interferometry confirms that these comb-lines are stable phase-coherent. From a practical
point of view, this makes the frequency comb attractive for Fourier synthesis of ultrashort
pulses. This scheme is much simpler than other methods requiring sophisticated setup
such as a two synchronized high-energy laser sources, a cryogenic environment and a
dual-wavelength injection-locked pulsed laser.
80
Chapter 6 Raman linewidth broadening in gas filled HC-PCFs
For the work presented here I carried out preliminary measurements of the Raman
linewidth using a different technique based on a PCF-based supercontinuum and a Fabry-
Perot. Setup was further improved using a narrowband laser source at Stokes frequency
and a pulsed nanosecond laser at pump frequency, as described in the text.
6.1 Introduction
It is known that the Raman spectral linewidth provides the important information on the
properties of the scattered medium, like the inter-molecular forces, the dephasing rate of
the molecular coherence and population. When Raman-active gases are tightly confined
in the micron-size cell like as in HC-PCF, it will result in further broadening of the
Doppler- and pressure-broadened Raman-gain linewidth by the collision between gas
molecules and the fibre core wall. This effect is particularly appreciable at low gas
81
pressures (below 1bar) where the molecular mean-free path is comparable with the fibre’s
core radius, on the order of a few microns [30]. However, this effect has not been studied
completely until now [81,82], probably because of the requirement of micro-size core
radius of gas cell. The invention of HC-PCF allows us realize this by experimental
measurement. In the present chapter, by using hollow-core bandgap fibre (PBG-PCF)
with a radius of 5.5µm, the effect of molecular core-wall collision on the rotational
spectral linewidth of S-branch of hydrogen in forward stimulated Raman scattering will
be considered.
6.2 Analysis of Raman linewidth change in gas medium
The dependence of Raman spectral shape and linewidth on the gas density is rather
complex. A simplified description of this is given in the figure 6.1.
For the low gas pressure region (<10mbar) where no significant molecular collision is
expected, the linewidth broadening is caused by the Doppler. This effect arises from the
frequency shift caused by the translational motion of gas molecules relative to a
spectrograph. Assume these motions to be in thermal equilibrium, their velocity
distribution will obey the Maxwell-Boltzmann distribution. This distribution contributes a
usual Gaussian spectral profile with its width determined by Eq.(4.2), Chapter 4. DΓ
( )m
T2ln2kωω2Γ BSPFD c
−= (6.1)
Where T is the temperature in Kelvin (here we assume T= 298K) and m is the H2 mass,
for rotational Raman transition, k(MHz1017.4ωω 6SP ×=− ) B=1.38.10-23 (JK-1)
Boltzmann constant, c=3.108m/s. The Doppler linewidth is shown with the black line in
figure 6.1.
82
Figure 6.1 Dependence of the rotational forward Raman linewidth on the Doppler
broadening and pressure-broadened linewidth and core-wall collision in H2 gas filled
PBG-PCF (core radius-5.5µm). The vertical axis describes the linewidth in MHz units,
the horizontal axis presents in the log scale of the gas pressure in bar units. The black,
pink, blue and purple-dash curves show the contribution of the Doppler shift, collisional
narrowing, pressure broadening and combination respectively. The red curve presents the
effect of the collision between gas molecules and fibre core’s wall on the molecular
coherence [57,81,82].
When the gas density increases or collision frequency increases, the molecular collision
begins to contribute to the spectral lineshape. If this collision is an elastic velocity-change
process (do not affect the internal state of molecules), the collisional narrowing will
occur. The collisional narrowing was explained as a result of the velocity-changing
collision and the uncertainty principle [83]. From the uncertainty relation, a photon of
momentum λh only gives information on the displacements of the molecule larger than
the value of πλ
2. The mean velocity of the molecule in the direction of observation for
83
displacements of πλ
2 is considered by the Doppler shift on this photon. If collisions are
rare during the time that it takes a molecule to travel that displacement, the mean velocity
is the thermal velocity of the molecule. The Doppler shift will be proportional to this
velocity and the radiating molecules will contribute a Gaussian spectral profile whose
linewidth is expressed above, Eq.(6.1).
In contrast, many collisions of a molecule by increasing gas density, the mean velocity of
a molecule will reach zero as averaged over many collisions because the collisions take
the molecule through all possible velocity states. Hence, the mean velocity of a molecule
averaged over πλ
2 will be decreased, resulting in linewidth narrower than the usual
Doppler width. In the limit of high collision frequency between molecules, the velocity-
changing collision contributes a Lorentzian line profile and the linewidth is given
approximately by a diffusion model as
NΓ
ρAΓN = (6.2)
Where A=6.16 (MHz.amagat) is a coefficient for rotational Raman scattering,
proportional to the self- diffusion coefficient Do (cm2.amagat.s-1); ρ (amagat=1.1bar) is a
gas density. This model is divergent at zero gas density. The gas pressure dependence of
the linewidth is presented by the pink curve in figure 6.1.
In reality, as the gas density increases significantly (>1bar), it will cause the collisional
broadening by the internal state-changing in the gas molecules, leading to the disruption
of scattering. Collisional broadening contributes a Lorentzian line profile, the linewidth is
linearly proportional to the gas density, expressed by a blue curve in fig.6.1.
ρB=RΓ (6.3)
84
where B (MHz/amagat) is a broadening constant of rotational Raman linewidth B=110
(MHz/amagat) [57].
The further considerations of the gas density dependence of the linewidth giving the more
complete physical pictures have also been considered. For example, a “soft-collision”
model in which the velocity change in a single collision is much less than the mean-
thermal velocity [84], or a “hard-collision” model in which the velocity change in a
single collision is comparable to the mean-thermal velocity [85] that allow to cover the
gas pressure range between 0 and 1bar and eliminate the divergence at zero density in
diffusion model Eq.(6.2). When compared with the “hard-collision” model, the validity
of the diffusion model in Eq.(6.2) is restricted to densities higher than a cut-off density
137mbarΓ3.33Aρ
Dc == [57].
The dominant contribution to the linewidth broadening at the low gas pressure below
1bar is the collision between molecules and fibre’s core wall and is shown by the red
curve in figure 6.1. In this case, we assume that the molecules collide with the wall and
come back “fresh”, so that they lose coherence. The linewidth caused by the collision of
the core-wall collision is considered by [86].
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
c
2c
o2W
rl6.81ρr
D2.405Γ (6.4)
Where ρπσT4kl 2
B= is the mean free path, the average distance of a molecule travels
between successive collisions; kB(J.K-1) is a Boltzmann constant; Do =1.32 cm2.amagat.s-
1 is the molecular self-diffusion coefficient at the room temperature (298 K); ( )0A2.8σ =
is the collision diameter of the H2 molecule; rc =5.5µm, the radius of HC-PBG; ρ is the
gas density (amagat).
85
From above analysis, we predict the evolution of forward Raman linewidth in gas density
as following: if the collisional narrowing, collisional broadening and wall collision
processes are considered independent at gas density (the valid density of the
diffusion model), the convolution of several Lorentzian terms gives a Lorentzian with the
linewidth to be the sum of individual widths (moderate pressure region). When the
pressure (low pressure region), the dominant contribution of the molecular-wall
collision will govern the resulting line profile, namely a Lorentzian, with the linewidth is
given by in expression (6.4).
cρρ >
cρρ <
WΓ
6.3 Experimental setup and results
A schematic for measuring the Raman linewidth is illustrated in figure 6.2 consisting of
pump and probe lasers. The pump laser is a single-frequency laser producing the
wavelength 1064nm, pulse duration 17ns, 30MHz spectral linewidth. The probe one
is a CW tunable diode probe laser generating 80mW power, linewidth <1MHz.
They are combined on a dichroic mirror (DM) and focused into a 1.2m PBG-PCF-core
radius of 5.5µm filled with H
pumpω
tunableprobeω
2 gas. The transmission loss of PBG-PCF is shown in figure
2.5 (Chapter 2). The change of gas pressure is measured by a low pressure gauge (G).
The signal power was detected by the photodiode (PD) after filtering the output beam by
a bandpass filter-1134nm (BP). The small detuning range of the probe laser was
calibrated by a scanning confocal Fabry-Perot interferometer (FPI, free spectrum range,
4GHz; finesse, 500) and displayed by the oscilloscope.
86
Figure 6.2 Block schematic of Raman linewidth measurement: dichroic mirror (DM),
mirror (M); a sensitive gas gauge (G); bandpass filter 1134nm (BP); neutral beam splitter
(BS); photodiode (PD); a scanning confocal Fabry-Perot interferometer (FPI).
The Raman line profiles were measured by shifting the frequency of probe laser around
the Raman resonance while the frequency of pump laser was fixed. The pump power was
also fixed and kept below its Raman threshold power. Figure 6.3a, 6.3b, 6.3c & 6.3d
shows Raman gain profiles at 10mbar, 360mbar, 1.928bar and 4.04bar respectively. In
which, the spherical points shows the signal powers measured at different tuning
frequencies of probe wave and fitted by a solid curve with a Lorentzian profile. The good
agreement between fitted line Lorentzian profiles and experimental measurements shows
the suitability of the theoretical model used.
87
Figure 6.3 Raman spectral profile at gas pressures: a) 10mbar, b) 360mbar,
c) 2bar, d) 4bar
The Raman linewidth (FWHM) is calculated using the fit and the pump linewidth
(30MHz) is subtracted. As a result we obtained 275MHz, 110MHz, 244MHz and
498MHz at gas pressures 10 mbar, 360 mbar, 2 bar and 4 bar, respectively.
The linewidth in this region is the result of three contributions: wall molecular collision,
collisional narrowing and collisional broadening. The linewidth is the sum of three
individual linewidths. The red-solid curve presents Raman linewidth only caused by the
88
Figure 6.4 Dependence of rotational Raman linewidth on the H2 gas pressure filled in a
radius-5.5µm PBG-PCF.
collision of gas molecules with the fibre core wall in the low pressure region smaller than
the cut-off density ( ). Experimental measurement nicely agrees with our theoretical
prediction. This result confirms the significant effect of wall molecular collisions on the
coherence of Raman radiation as carried out into areas with radius of few microns.
cρ<
6.4 Conclusion
The effect of the collision between gas molecules and the fibre’s core wall on the
rotational Raman linewidth in small-core hollow-core photonic crystal fibre is studied.
This effect is especially strong in a range of low pressure and decreases gradually as the
pressure is increased. Experimental measurements are in agreement with the theoretical
calculation. The results provide the reliable data for the investigation of the optical
coherence of the forward Raman scattering in a small core HC-PCF.
89
Chapter 7 Summary and outlook
By exploiting the novel characteristics of HC-PCF, I have studied new interaction
regimes of the stimulated Raman scattering in hydrogen gas. In this chapter I summarize
the results of our studies and briefly mention possible further research directions of work
done here.
In chapter 2 I introduced the advanced properties and the potential applications of HC-
PCF. HC-PCF has novel characteristics such as low loss power, controllable transmission
window, diffraction-free interaction length and effective cross-section of only a few µm2.
As a result, the performance of HC-PCF in nonlinear interaction of light with low density
media is several orders of the magnitude higher than those of previous configurations.
These make HC-PCF become a desired candidate in maximizing the light-matter
interaction, particularly in the stimulated Raman scattering in gas.
The physical origin of Raman scattering and Maxwell-Bloch equations governing the
spatiotemporal evolution of the interaction between the laser fields and molecules in
Raman medium are described in the chapter 3. These equations are used to explain the
experimental observations in next chapters in both backward and forward stimulated
Raman scattering. SRS is divided into three regimes: spontaneous, transient and steady
state regimes. The transient regime providing the high coherence is important in
ultrashort pulse generation and easily achieved in HC-PCF by long pump durations.
Chapter 4 presents the stimulated backward Raman scattering in H2 gas filled PBG-PCF.
It includes two main results: optical pulse compression and multi-peak process in BSRS:
Optical pulse compression in a two stage pulse compression scheme: A low average
power laser source was split into three components for Stokes seed generation and its
further amplification in the first and second stages. The PBG-PCF used was filled with
H2 gas and had a narrow transmission bandwidth only supporting pump and the first
rotational Stokes wavelength. As a result the signal pulse 20 times shorter than that of
90
original pump was generated. This method is efficient, simple and useful for the
applications of a poor-quality source into a high-quality laser source. The shortening
efficiency could be significantly improved by cutting the leading edge of the Stokes seed,
for example by using an electro-optical modulator.
• An amplification scheme, similar to backward stimulated Raman scattering
discussed but using plasma as the gain medium is recently proposed. Recent
reports on generation of plasma medium in fibre using a femtosecond pulse might
open the possibility of studying backward Raman pulsed compression in plasma
contained HC-PCF. In plasma medium, which is different from conventional
Raman media, backward Raman scattering can be achieved in ultrashort pulse
regime.
• Unconventional amplification of higher order Stokes and anti-Stokes in BSRS
may be interesting in media having high backward Raman gain. Theoretically,
these regimes are possible and described in frequency-propagation diagram βω−
as following.
Figure 7.1 Phase-matching schematic for backward higher order Stokes and anti-Stokes
includes backward anti-Stokes phonon (green arrow), backward 1st Stokes phonon (pink
arrow) and backward 2nd Stokes phonon (blue arrow).
91
Multi-peak process: In this case, a Stokes pulse duration several times shorter than that of
the pump one is generated in H2 gas filled PBG-PCF by transient BSRS regime and a
new dynamical process generating a train of solitary-like Raman pulses with flexibly
controllable peak intensities is observed. The dynamics of backward amplification is
divided into two regimes: The linear stage in which the pump energy is low and the seed
pulse is uniformly amplified. Its sharpening and shortening are negligible and the Stokes
pulse shape is similar to the seed pulse. The second regime, nonlinear amplification stage,
in which pump energy is significantly increased (~ the first forward Stokes threshold),
leading to the rise of additional asymmetric peaks at the trailing edge of the Stokes pulse.
As the pump energy is increased, the amplification of first peak seems to be saturated.
Notably, these extra asymmetric peaks are spectacularly shortened and resharped to
relatively symmetric shapes similar to the first one but faster and stronger. This result
demonstrates further the unprecedented possibility for the observation of new dynamics
in complex nonlinear optic phenomena based on HC-PCF filled with gases.
• The possibility of larger number of solitary-like Raman peaks may be observed if
a seed Stokes with much shorter duration (~ ps) used.
The work presented in chapter 5 shows the efficient generation of a broad, mutually
coherent, purely rotational-Raman frequency comb by a simply setup consisting of a
microchip pump laser source and two H2 gas-filled HC-PCF. The comb is generated by
driving the Raman transition at resonance using two-color excitation scheme (pump and
seed) as described in the energy diagram of figure 7.2a. The coherence of the comb is
important for synthesizing an ultrashort pulses train and has been checked by an
interferometric technique based on the frequency doubling. Some possible directions for
further research include:
• Synthesis of ultrafast waveforms from the generated comb is possible by using the
amplitude modulator (AM) and phase modulator (PM). The careful control of
relative phase and their amplitudes of comb lines may generate waveforms in the
shape of square, sawtooth, sine pulses like in the radiofrequency regime [10].
92
• In an approach different from the approach has been carried out above, the
molecular modulation is driven adiabatically by two strong single-mode laser
fields whose frequency difference is slightly detuned from the exact Raman
resonance as described in figure 7.2b with the Stokes frequency. This can result in
an increase in the Raman coherence and further increase in the comb extension.
Figure 7.2 Two excited mechanisms for Raman comb generation. Where are the
frequency of pump and Stokes laser fields;
SP ω;ω
Ω is the molecular Raman transition
frequency: a) Resonantly molecular excitation regime Ω=− SP ωω ; b) Adiabatically
molecular excitation regime . Ωωω tunableSP ≠−
In chapter 6 we studied the effect of the collision between hydrogen molecules and PBG-
PCF’s core wall on the rotational Raman linewidth at low gas densities at room
temperature. The tight confinement of gaseous molecules in a small area (a few µm2) of
HC-PCF allows stimulated Raman scattering to be achieved in a low pump power
regime. However, this also makes the molecular wall collision effect become significant
when the molecular mean-free path is comparable with to the fibre core radius.
Experiment was performed using 1.2m BPG-PCF (radius 5.5µm) filled variable H2
pressure using two laser sources: the pump pulse 17ns with the linewidth 30MHz and
CW tunable frequency signal source. Experiments were carried out in the range pressure
from 10mbar to 4bar. The change of Raman linewidth is described in two low pressure
and moderate pressure regions. The linewidth in the moderate pressure region is the
93
mixing of three contributions including of collisional narrowing, wall collision and
pressure broadening. The spectral broadening in the low pressure region is mainly
dominated by the collisions of gas molecules with fibre core wall. The good agreement
between the theoretical calculation and experimental data has shown the suitability of the
calculation model used. This result provides directly the information about the molecular
coherence and the Stokes build-up during Raman process. Further research plans could
include:
• Because the Raman linewidth caused by molecular wall collision depends
strongly on the fibre core radius, measurement of forward Raman scattering
linewidth for HC-PCF with different core radii will provide an interesting picture
about the effect of fibre core radius. Figure 7.3 shows the change in the Raman
linewidth as the fibre core radius is changed. Here blue, red and pink curves
describes the core radius r=3µm, r=5.5µm and 10µm respectively.
Figure 7.3 The change of Raman linewidth on the fibre core radii. The blue, red and
pink curves describes the Raman linewidth corresponding to the core radius r=3µm,
5.5µm and 10µm respectively.
94
• Investigation of the Raman linewidth in backward Raman scattering may be
interesting. In backward fashion, the contribution of Doppler shift effect to the
spectral linewidth is very different from the forward case. It is huge (about
4.7GHz) compared with the effect of fibre core wall collision at low pressures.
The experimental scheme is similar to the one in figure 6.2, but pump and signal
sources are rearranged in opposite directions. The dependence of backward
Raman linewidth on the gas pressures is predicted in figure 7.4. In which, the pink
curve shows the dominant effect of the Doppler broadening compared with the
red curve from molecular wall collision in HC-BPG with the radius 5.5µm.
Figure 7.4 Backward Raman linewidth on the gas pressure filled in a HC-PCF with the
radius 5.5µm.
95
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