Download Tunable nonlinear fiber optics using dense noble gases

Tunable nonlinear fiber optics
using dense noble gases
Abstimmbare nichtlineare Faseroptik
mit dichten Edelgasen
MOHIUDEEN AZHAR
Dissertation im Fachbereich Physik
der Friedrich-Alexander-Universität
Erlangen-Nürnberg
Tunable nonlinear fiber optics
using dense noble gases
Abstimmbare nichtlineare Faseroptik mit dichten Edelgasen
July 2013
Der Naturwissenschaftlichen Fakultät
der Friedrich-Alexander-Universität Erlangen-Nürnberg
zur Erlangung des Doktorgrades Dr. rer. nat.
vorgelegt von
MOHIUDEEN AZHAR
aus Bangalore, Indien
Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der FriedrichAlexander-Universität Erlangen-Nürnberg
Tag der Mündlichen Prüfung
Vorsitzender des Promotionsorgans:
Gutachter:
24-09-2013
Prof. Dr. Johannes Barth
Prof. Philip St.J. Russell, D. Phil, FRS
Prof. Dr. Peter Hommelhoff
Abstract
The nonlinearity and dispersion of a material media, are the pivotal parameters
influencing nonlinear light-matter interactions. By regulating the pressure in a
gas-filled hollow-core photonic crystal fiber (PCF), these two parameters can easily be controlled. The choice of monoatomic noble gases as a filling media, avoids
perturbations in the form of molecular interactions like Raman-scattering. However, the inherent material nonlinearity of these fiber systems is much lower than
that of fused silica glass core fibers. The work presented in this thesis seeks out
to address this issue by increasing the density of the filling medium in the hollowcore fiber – thereby offering an unrivaled, tunable nonlinear optical system.
Liquid Ar was used to fill the hollow-core of a kagomé-lattice PCF to scale
up the core material nonlinearity comparable to that of fused silica. For its experimental realization, a novel cryogenic fiber system was conceived. Self-phase
modulation was observed after optical pulse propagation through a column of liquid Ar,
By filling a hollow-core PCF with Ar pressures of up to 110 bar, the nonlinearity and dispersion of the fiber could be widely tuned. The zero dispersion
wavelength could be varied from the UV to IR, resulting in nonlinear studies over
various dispersion regimes which is in excellent agreement with numerical simulations. Dispersive wave emission in the modulational instability regime was
studied by positioning the zero dispersion wavelength close to that of the pump
laser. Gas pressure controllable modulational instability sidebands in the absence
of Raman-gain (inherent to fused silica) make this system a prospective, tunable
source of correlated photon pairs.
The goal to achieve fused silica nonlinearity was accomplished by a unique
blend of photonics and thermodynamics. By considering the nonlinear density
variation of room temperature Xe with pressure, supercritical Xe at 80 bar was
filled in a fiber to attain fused silica nonlinearity. Intermodal four-wave-mixing
and self focusing effects of fiber in-coupling were observed in subcritical (gaseous)
Xe. The work presented here paves the path for several novel optical systems owing to the unique properties of supercritical fluids.
Zusammenfassung
Nichtlinearität und Dispersion eines Materials sind die Schlüsselparameter, die die
nichtlineare Wechselwirkung von Licht und Materie beeinflussen. Durch die Regulierung des Drucks in einer gasgefüllten, photonischen Kristallfaser mit Hohlkern (engl. photonic crystal fiber, PCF) sind diese zwei Parameter leicht kontrollierbar. Werden monoatomare Edelgase als Füllmedium gewählt, lassen sich
Störungen in der Form molekularer Wechselwirkungen wie der Raman-Streuung
vermeiden. Allerdings ist die inhärente Nichtlinearität des Materials dieser Fasersysteme viel geringer als die einer Faser mit Quarzglaskern. Die im Rahmen
dieser Abhandlung präsentierten Arbeiten gehen diese Ausgangslage heran, indem die Dichte des Füllmediums erhöht wird; und somit ein unübertroffenes,
durchstimmbares, nichtlinear-optisches System ermöglicht wird.
Um die Materialnichtlinearität des Faserkerns auf Werte vergleichbar denen
von Quarzglas hochzuskalieren, wurde Flüssig-Argon zur Füllung des Hohlkerns
einer PCF verwendet. Zur experimentellen Realisierung wurde ein neuartiges,
kryogenisches Fasersystem konzipiert. Als Resultat der Pulspropagation durch
eine Säule aus Flüssig-Argon wurde Selbstphasenmodulation beobachtet.
Durch das Füllen einer Hohlkern-PCF mit Argon-Drücken von bis zu 110 bar
konnten die Nichtlinearität und Dispersion der Faser weitreichend abgestimmt
werden. Die Wellenlänge der Nulldispersion konnte über den gesamten Bereich
vom Ultravioletten zum Infraroten variiert werden, woraus nichtlineare Studien
über verschiedene Dispersionsregime resultierten, die in exzellenter Übereinstimmung mit numerischen Simulationen sind. Die Abstrahlung dispersiver Wellen
im Regime der Modulationsinstabilitäten wurde untersucht, indem die Wellenlänge der Nulldispersion nahe bei der Pumpwellenlänge positioniert wurde. Durch
den Gasdruck kontrollierbare Seitenbänder hervorgerufen durch die Modulationsinstabilität sowie das gleichzeitige Ausbleiben der Raman-Verstärkung (inhärent
in Quarzglas) machen dieses System zu einer aussichtsreichen, durchstimmbaren
Quelle korrelierter Photonenpaare.
Das Ziel, Nichtlinearitäten in der Größenordnung von Quarzglas zu erreichen,
wurde durch eine einzigartige Verbindung photonischer sowie thermodynamis-
cher Aspekte bewerkstelligt. Unter Berücksichtigung der nichtlinearen Dichteänderung von Xenon bei Raumtemperatur in Abhängigkeit des Drucks wurde superkritisches Xenon bei 80 bar in die Faser gefüllt, um Nichtlinearitäten vergleichbar denen von Quarzglas zu erreichen. Intermodale Vier-Wellen-Mischung sowie
Effekte der Selbstfokussierung bei der Fasereinkopplung wurden in subkritischem
(gasförmigem) Xenon beobachtet. Aufgrund der einzigartigen Eigenschaften superkritischer Fluide ebnet die hier präsentierte Arbeit den Weg für zahlreiche,
neuartige optische Systeme.
To Mummy, Abba and Mothi
Contents
I
Introduction
I.1
9
Fundamentals of nonlinear optics . . . . . . . . . . . . . . . . . . 11
I.1.1
Maxwell equations . . . . . . . . . . . . . . . . . . . . . 11
I.1.2
Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . 12
I.1.3
Fiber Dispersion . . . . . . . . . . . . . . . . . . . . . . 14
I.1.4
Nonlinear Schrödinger equation . . . . . . . . . . . . . . 15
I.1.5
Self-phase modulation . . . . . . . . . . . . . . . . . . . 18
I.1.6
Self-steepening . . . . . . . . . . . . . . . . . . . . . . . 19
I.1.7
Temporal solitons
I.1.8
Dispersive wave . . . . . . . . . . . . . . . . . . . . . . 22
I.1.9
Influence of Raman-scattering on nonlinear optics . . . . 25
. . . . . . . . . . . . . . . . . . . . . 20
I.1.10 Four-Wave-Mixing (FWM) & modulational instability (MI) 26
I.1.11 Self-focusing . . . . . . . . . . . . . . . . . . . . . . . . 28
I.1.12 Supercontinuum generation . . . . . . . . . . . . . . . . 30
I.2
Photonic Crystal Fiber (PCF) . . . . . . . . . . . . . . . . . . . . 31
I.2.1
Types of PCF . . . . . . . . . . . . . . . . . . . . . . . . 31
I.2.2
Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . 32
I.2.3
Hollow-core PCF
I.2.4
Solid core vs kagomé HC-PCF . . . . . . . . . . . . . . . 38
I.2.5
Preceding gas-filled HC-PCF research . . . . . . . . . . . 41
. . . . . . . . . . . . . . . . . . . . . 33
II Liquid Ar filled kagomé PCF
45
II.1 Experimental Implementation . . . . . . . . . . . . . . . . . . . 45
7
8
CONTENTS
II.1.1
Kagomé fiber spliced with a multimode fiber . . . . . . . 46
II.1.2
Cryogenic trap . . . . . . . . . . . . . . . . . . . . . . . 47
II.1.3
Experimental results . . . . . . . . . . . . . . . . . . . . 53
II.1.4
All liquid Cryotrap . . . . . . . . . . . . . . . . . . . . . 58
II.2 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 62
II.2.1
Calculation of the nonlinear index (n2 ) of liquid Ar . . . . 63
II.2.2
Calculation of the Sellmeier relation for liquid Ar
II.2.3
Dispersion of liquid Ar in hollow-core PCF . . . . . . . . 65
. . . . 63
II.3 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . 65
III High pressure Ar gas filled PCF
69
III.1 Supercritical fluids . . . . . . . . . . . . . . . . . . . . . . . . . 70
III.2 High pressure capable experimental setup . . . . . . . . . . . . . 73
III.3 Gaseous and supercritical Ar (25-110 bar) . . . . . . . . . . . . . 75
III.4 Correlated photon pairs . . . . . . . . . . . . . . . . . . . . . . . 81
III.5 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . 82
IV Supercritical Xe filled PCF
87
IV.1 Experimental setup modification . . . . . . . . . . . . . . . . . . 87
IV.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . 91
IV.2.1 Supercritical Xe 80 bar
. . . . . . . . . . . . . . . . . . 91
IV.2.2 Sub-critical Xe 25-35 bar
. . . . . . . . . . . . . . . . . 95
IV.3 Advantages over cryotrap . . . . . . . . . . . . . . . . . . . . . . 97
IV.4 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . 98
V Conclusions
101
A Appendix
103
A.1 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.2 Instructions to operate the cryotrap
. . . . . . . . . . . . . . . . 105
A.3 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 107
Bibliography
111
Chapter I
Introduction
The advent of lasers in the 1960s led to widespread excitement and progress in the
study of light-matter interactions leading to the diverse field of nonlinear optics
[1, 2, 3]. Exciting phenomenon like supercontinuum generation [4, 5], high power
laser development, etc led to applications and research advancements in virtually
all fields of science spanning from high energy physics to biomedical applications.
In the quest to find ways to increase interaction lengths for light-matter interactions, research groups around the world used optical cavities [6] , capillaries [7] ,
doped optical fibers [8] , among the many attempted solutions. These techniques
were useful for their time, but the high losses and low material damage thresholds
limited the progress made.
The 1990s heralded the development of photonic crystal fibers (PCF) (invented
by Philip Russell) [9], that offered to revolutionize nonlinear optics. The periodical arrangement of holes in the cladding offered a lattice structure that allowed
for customizable dispersion and enhanced nonlinearity owing to a smaller core
size. Solid core PCFs lead to the development of endlessly single-mode fibers
[10], compact and inexpensive supercontinuum sources [11], frequency combs
[12, 13, 14], etc. However the photonic bandgaps offered by means of an appropriately designed cladding structure, allowed photonic bandgaps fibers to guide
light in the core-even in a hollow-core. The prospect of hollow-core photonic
crystal fibers (HC-PCF) offered long interaction lengths and excellent light-matter
9
10
CHAPTER I. INTRODUCTION
spatial overlap. Experiments like [15, 16] made use of these fibers for low power
Raman-scattering interactions. A HC-PCF with a kagomé-style cladding lattice
was found to support a much broader transmission window (from UV-IR). Such a
broad transmission range lead kagomé HC-PCF to be used in novel nonlinear experiments such as dispersive wave UV generation [17], in-fiber ionization related
effects [18, 19], spectral broadening for pulse compression [20], frequency comb
generation [13, 14], etc. In all these cases the material nonlinearity of the core
medium was much lower than that of glasses like (fused silica [21]). Hence most
of the above experiments required amplified pulsed lasers for initiation the various nonlinear processes. This experimental work in this thesis seeks out to address
this very issue – to raise the material nonlinearity of a hollow-core to that of fused
silica using dense gases or liquids. By using monoatomic noble gases, molecular interactions like Raman-scattering (inherent to fused silica [22, 21]) could be
avoided, enabling study of nonlinear optics without Raman-related perturbations.
Pivotal parameters for nonlinear optics such as dispersion and nonlinearity could
be tuned merely by adjusting the gas pressure in the fiber – a level of flexibility not
seen before in nonlinear fiber optics. The hollow-core also means that the damage
threshold is significantly higher than fused silica fibers [23].
Chapter I begins with an introduction to nonlinear fiber optics and nonlinear
phenomena relevant to this thesis. The paragraphs in italic font point to the part
of the thesis where the corresponding phenomenon has been observed.
The experimental work presented in this thesis follows a chronological order
where the efforts to raise the material nonlinearity began with filling the core of
a HC-PCF with liquid Ar. Chapter II deals with the experimental implementation
and results of liquid Ar filled HC-PCF [24].
Chapter III details the high-pressure gas-filled HC-PCF experiments. Almost
all of the technical problems encountered in Chapter II were solved using this
system. The results presented here include the ability to tune the zero dispersion
wavelength from the UV to the IR by varying the gas pressure. This remarkable
level of flexibility comes with the added advantage of the core medium being free
of Raman-scattering [25].
I.1. FUNDAMENTALS OF NONLINEAR OPTICS
11
Chapter IV is about the supercritical Xe filled HC-PCF where the goal to
achieve fused silica nonlinearity in the core of a HC-PCF was achieved. Interesting nonlinear optical phenomenon such as intermodal four-wave-mixing and selffocusing related effects were observed. Concluding remarks and future prospects
have been outlined for each of the experimental chapters [26].
I.1
I.1.1
Fundamentals of nonlinear optics
Maxwell equations
Maxwell’s equations can be used to describe all electromagnetic phenomena including optical field propagation in optical fibers [21]. They are given by
∇×E = −
∂B
∂t
∇×H = J+
∂D
∂t
(I.1a)
(I.1b)
∇·D = ρ
(I.1c)
∇·B = 0
(I.1d)
where E and H are the electric and magnetic vectors and D and B are their electric and magnetic flux densities. Since free charges are mostly absent in optical
fibers, the current density J =0 and charge densityρ =0. The following relations
relate the flux densities
D = ε0 E + P
(I.2a)
B = µ0 H + M
(I.2b)
where ε0 and µ0 are the vacuum permittivity and permeability respectively. P and
12
CHAPTER I. INTRODUCTION
M are the induced electric and magnetic polarizations. However, since optical
fibers are nonmagnetic, M = 0.
To derive the wave equation describing optical field propagation in optical
fibers, Eq. I.2a and I.2b are inserted into the Maxwell equations and the curl of
Eq. I.1a is taken
1 ∂ 2E
∂ 2P
∇ × ∇ × E = 2 2 − µ0 2
c ∂t
∂t
(I.3)
From vector algebra it is well know that
∇ × ∇ × E = ∇ (∇ · E) − ∇2 E = −∇2 E
(I.4)
because ∇ · D = ε∇ · E = 0, therefore Eq. I.3 takes the form of a Helmholtz
equation
∇2 Ẽ + n2 (ω)
ω2
Ẽ = 0
c2
(I.5)
Eq. I.5 is the wave equation for a propagating optical field Ẽ.
I.1.2
Nonlinearity
The origin of optical nonlinearity of any dielectric medium is fundamentally the
response of the motion of the bound electrons in the media to the applied electric
field of the optical field. This motion is nonlinear. The electronic response to the
optical field can be understood from the polarization induced by electric dipoles
in the medium, and is given by
P = ε 0 (χ (1) · E + χ (2) · EE + χ (3) · EEE · · · )
(I.6)
ε0 is the vacuum permittivity and χ ( j) , ( j = 1, 2, 3, · · · ) stands for the jth order
susceptibility. χ (2) vanishes for fused silica as SiO2 is a symmetric molecule.
χ (2) is nonzero only for media which lack inversion symmetry at a molecular level
[21]. χ (2) is associated with effects such as second harmonic generation or sumfrequency generation. Therefore, in the absence of χ (2) , χ (3) plays a dominant
I.1. FUNDAMENTALS OF NONLINEAR OPTICS
13
role in nonlinear fiber optics. The refractive index of a nonlinear medium is given
by
ñ(ω, I) = n(ω) + n2 I
(I.7)
where I represents the optical intensity and is propotional to the square of the
electric field. The first term n(ω) is the linear refractive index and the second
term n2 represents the nonlinear refractive index which is related to the third order
susceptibility χ (3) by [27]
n2 =
3
χ (3)
4nε0 c
(I.8)
The nonlinear refractive index n2 is representative of the material nonlinearity
of a nonlinear medium and is tabulated in I.1 for a few material relevant to this
thesis.
material
~n2 ( x10−20 m2 /W)
fused silica glass
2.74
air (1 bar)
0.0029
Ar (1 bar)
0.00084
liquid Ar
1.7
Xe (1 bar)
0.0093
supercritical Xe (80 bar)
2.8
Table I.1: Nonlinear refractive index n2 of material media relevant to this thesis.
[21, 28, 29, 23]
To represent the nonlinearity of an optical fiber, the nonlinear parameter γ is
useful. It takes into account the spatial mode area of the light in the core of the
fiber and is defined by
14
CHAPTER I. INTRODUCTION
γ=
n2 ω0
cAeff
(I.9)
where n2 is the material nonlinear refractive index of the core media. Aeff is the
effective modal area of of the propagating mode in the fiber core. c is the speed of
light.
Matching the n2 of a gas/liquid filled hollow-core fiber to the n2 of fused silica
∼ 2.74 × 10−20 m2 /W is one of the main motivations of the work reported in this
thesis. Liquid Ar and supercritical Xe at 80 bar were two dense noble gas media
which was used in the hollow-core fibers in the work reported in this thesis.
I.1.3
Fiber Dispersion
Fiber dispersion is important for short optical pulses as it defines speed for various
spectral components given by c/n(ω). This in turn defines the phase-matching
conditions for various nonlinear processes like four-wave mixing [30] or dispersive wave generation [31]. The chromatic dispersion is the change of linear refractive index with the optical frequency (Fig. I.1) and is given by n(ω). The
mode-propagation constant (the wave-vector component in the direction of propagation) β (ω) is expanded as a Taylor series as its exact functional form is not
always known.
β (ω) = β0 +
(ω − ω0 )
(ω − ω0 )2
(ω − ω0 )3
β1 +
β2 +
β3 + · · ·
1!
2!
3!
(I.10)
where β0 = β (ω0 ) and
βj =
�
∂ jβ
∂ω j
�
ω=ω0 , ( j
= 1, 2, 3 · · · )
The group velocity is given by vg = 1/β1 , β1 = c/ng and the group velocity
dispersion (GVD) is given by β2 . β2 represents the dispersion of the group velocity of the pulse and influences pulse broadening. The sign of GVD or sgn(β2 ) is
I.1. FUNDAMENTALS OF NONLINEAR OPTICS
15
Figure I.1: Variation with wavelength of fused silica (left) refractive index n and
group index ng (right) β2 and D. β2 and D both reach 0 around the zero dispersion
wavelength (ZDW) which is around 1.27 µm [21].
important as it signifies whether the dispersion is normal (β2 > 0) or anomalous
(β2 < 0). If the dispersion is anomalous then interesting soliton related dynamics would occur. When β2 = 0, the corresponding wavelength is called the zero
dispersion wavelength (ZDW) and is shown in Fig. I.1. Obviously, in the vicinity
of the ZDW, the higher-order dispersion (βi , i ≥ 3) becomes more significant. In
the context of soliton, the propagation of such a pulse close to the ZDW can be
perturbed by higher-order dispersion, leading to the possibility of the emission of
a dispersive wave [32, 33].
One of the most significant technical advancements of the work reported in
this thesis is the large range of tunability (UV-IR) of ZDW in the high pressure
gas-filled fiber systems. This degree of flexibility for ZDW tunability is unrivaled
in nonlinear fiber optics.
I.1.4
Nonlinear Schrödinger equation
The generalized nonlinear Schrödinger equation (GNLSE) is derived by solving
the wave equation (Eq. I.5) with the method of separation of variables [21] and
the eigenvalue wavenumber. The eigenvalue wavenumber is given by
β̃ (ω) = β (ω) + �β (ω)
(I.11)
16
CHAPTER I. INTRODUCTION
Generalized Nonlinear Schrödinger equation (GNLSE)
Dispersion contribution
i
∂ 2A
2 β2 ∂ T 2
group velocity dispersion (GVD)
1
∂ 3A
6 β3 ∂ T 3
third order dispersion (TOD)
Nonlinear contribution
γ|A|2 A
nonlinear phenomena such as self-phase modulation (SPM)
2i ∂
2
ω0 ∂ T (|A| A)
nonlinear phenomena like self-steepening
2
TR A ∂∂|A|
T
Raman-gain related nonlinearities like soliton self-frequency shift.
Table I.2: Dispersion and nonlinear contribution in various terms of the GNLSE
where �β (ω) represents the fiber loss and nonlinearity. �β (ω) is also expanded,
but its higher order components are neglected as the spectral width of the pulse
satisfies �ω � ω0 , resulting in the approximation [21] where
�β ≈
iα
+ γI
2
(I.12)
iα/2 represents the fiber loss and I is the intensity. These approximations then
lead to the GNLSE. The generalized nonlinear Schrödinger equation (GNLSE) is
given by
�
�
2i ∂
∂A
αA i ∂ 2 A 1 ∂ 3 A
∂ |A|2
2
2
=−
− β2 2 + β3 3 + i γ|A| A −
(|A| A) − TR A
∂z
2
2 ∂T
6 ∂T
ω0 ∂ T
∂T
(I.13)
where the envelope of the field A(z, T ) is taken in a reference frame moving at the
group velocity vg such that the real time t is related to the time T by T = t − z/vg =
t − β1 z.
Interestingly each term of the above equation describes an important physical
I.1. FUNDAMENTALS OF NONLINEAR OPTICS
17
phenomena that the envelope can experience as the pulse is propagating along
z and is tabulated in Table I.2. TR is related to the slope of the Raman-gain
of the media. This last term in the GNLSE is important, as it represents the
Raman-scattering contribution that is avoided in the experiments in III, by using monoatomic noble gases which have negligible Raman-scattering. In the numerical simulations, this Raman-contribution is “turned off” giving an excellent
agreement with experimental observations. The term αA/2 represents the fiber
loss.
Considering a very simplified case, where only the nonlinearity plays a role in
the propagation of the pulse, the GNLSE becomes:
∂A
= iγ|A|2 A
∂z
(I.14)
Both sides of the equation must have the same dimensions, which means that we
can define a scaling length LNL as
LNL =
1
(γP0 )
(I.15)
Similarly, in the case of a pulse propagating in a purely dispersive medium, we
would have
∂A
∂ 2A
= iβ2 2
∂z
∂T
(I.16)
and then we could derive a scaling length for the dispersion effect as:
LD =
T02
|β2 |
(I.17)
where dispersion effect must be taken into account for pulse traveling over a distance L > LD . T0 is the pulse width and P0 is the peak power of the pulse. Both LD
and LNL decrease for intense and short optical pulses. These two lengths are useful
for characterizing the nonlinear and dispersive contribution of light-matter interactions. For a length of fiber L, if L � LD , and L � LNL then neither the dispersive
nor nonlinear effects can affect a propagating pulse. If L ∼ LD and L � LNL , then
18
CHAPTER I. INTRODUCTION
dispersive effects would dominate and the nonlinear effects would not play a role
and vice versa. The possibility of solitons in the anomalous dispersion regime,
arise when the dispersive effects and the nonlinear effects match.
I.1.5
Self-phase modulation
When a pulse propagates over a distance that is longer that this characteristic
length LNL , then nonlinear effects become relevant. Since phase of an optical
field with intensity I(t) is given by
φ = ñk0 z = (n + n2 I(t))k0 z
(I.18)
it is clear the phase experiences a modulation which depends on the pulse itself
– hence the name: self-phase modulation or SPM. This effect leads to nonlinear
spectral broadening of the input pulse.
�φ (t) = −n2 I(t)k0 z
(I.19)
where n2 is the nonlinear refractive index, I(t) is the time-dependent optical intensity of the pulse, k0 is the wavenumber and z is the distance travelled by the
pulse in the Kerr medium. The spectral broadening arises from the new frequency
components generated by time dependence of the phase shift. The change in instantaneous frequency is given by
�ωi =
d(∆φ )
dI
= −n2 k0 z
dt
dt
(I.20)
For a simple pulse shape, such as a Gaussian, one half of the pulse trailing
in time would have a frequency shift toward higher frequencies (blue shifted) as
dI/dt < 0. Similarly the leading half of the pulse in time, would experience a frequency downshift (red shift) as dI/dt > 0. Hence a time dependent phase change
gives rise to new frequency shifts and spectral broadening (Fig. I.2). SPM in
the above scenario also renders the pulse up-chirped. This introduction of chirp
enables SPM to be used for pulse shaping [34] appropriately tailoring the disper-
I.1. FUNDAMENTALS OF NONLINEAR OPTICS
19
Figure I.2: Intensity, Phase and spectral time-dependence during self-phase modulation process.
sion of the spectrally broadened pulse by using for example diffraction gratings
or chirped mirrors. SPM was initially studied in optical fibers in [35].
SPM was observed in all experimental systems in this thesis including liquid
Ar (Chapter II), high pressure Ar (Chapter III) and supercritical Xe (Chapter IV)
filled fibers.
I.1.6
Self-steepening
Self-steepening is basically an extension of SPM. The intensity dependence of
group velocity of a pulse propagating in a Kerr medium results in a pulse distortion. This pulse distortion is especially relevant for femtosecond pulses. The peak
of the pulse moves relatively slowly compared to its trailing edge causing the pulse
to distort as the trailing edge keeps getting steeper (as shown in the Fig. I.3(a)).
Unlike the spectral broadening due to SPM, the broadening due to self-steepening
is asymmetric due to the pulse distortion (Fig. I.3(b)).
As discussed above, SPM results in the trailing edge of the pulse to be blue-
20
CHAPTER I. INTRODUCTION
Figure I.3: Self-steepening of a Gaussian pulse (left) in time where the input
pulse at z=0 shown with dotted lines (right) spectrum when z/LNL = 20 in the
dispersionless case [21].
shifted in frequency and the leading edge of the pulse to be red-shifted. Since
self-steepening causes the trailing edge of the pulse to steepen, the overall spectral
broadening is more blue-shifted compared to that caused by SPM alone. Although
the spectral broadening is more blue-shifted, the red-shifted peaks are more intense than the blue-shifted ones as the same energy is distributed over a narrower
spectral range. More recently, this effect was advantageously used to extend the
spectral broadening of pulse propagating in a Ar-filled hollow-core fiber, in the
blue region. This also enhanced the generation of dispersive wave in the UV region [17].
I.1.7
Temporal solitons
In the simple case, where only second-order dispersion and nonlinearity play a
role, the GNLSE can be simplified to:
∂A
∂ 2A
= −iβ2 2 + iγ|A|2 A
∂z
∂T
(I.21)
This equation is the nonlinear Schrödinger equation (NLSE).
We have already introduced a way to quantify the dispersive effect and the
nonlinear effect through the dispersive length LD (I.17) and the nonlinear length
I.1. FUNDAMENTALS OF NONLINEAR OPTICS
21
LNL (Eq. I.15). When these effects balance out each other, the pulse travels without being modified in the time or spectral domain. This is the fundamental soliton
(Fig. I.4, N=1). A solution of I.21 [36] has the form
�
�
� 2 �
γA z
T
A(z, T ) = A0 sech
exp i 0
T0
2
(I.22)
where the soliton wave-vector ksol = (γA20 )/2 is independent of ω. Note that soliton can only appear for one specific sign of β2 . For β2 < 0 (anomalous dispersion),
these are so-called bright solitons and are the type of pulse that we are referring to
when mentioning a soliton in this thesis. An intuitive approach to understanding
solitons in the anomalous dispersion regime is to consider the dispersion-induced
chirp
δ ωi = −
dφ
sgn(β2 )(z/LD )
=
dt
(1 + (z/LD )2 T /T02 )
(I.23)
This dispersion-induced chirp is negative in the anomalous dispersion regime
(β2 < 0) but since SPM induces a positive chirp on the pulse (Fig. I.2). These
competing effects can compensate each other, resulting in the pulse retaining its
shape, despite nonlinear and dispersive influences from the medium [37, 38]. For
the fundamental soliton, since both nonlinear and dispersive effects exactly balance each other, the ratio of the characteristic lengths LD /LNL is equal to unity.
Actually, more generally, we can define the soliton order N by
N2 =
LD
(γP0 T0 2 )
=
LNL
|β2 |
(I.24)
Unlike for a fundamental soliton, the interplay between SPM and GVD for a
higher order soliton (N > 1) is more complicated. In the absence of all higher
order nonlinear and dispersive effects, the higher-order soliton change shape but
regains its original form periodically. These regular periods are called soliton
periods and are given by Lsol = (π/2)LD . As a higher order soliton propagates, it
compresses temporally, then splits into (N − 1) distinct pulses around Lsol /2 and
eventually joins back to revive its original shape at the soliton period (Fig. I.4,
22
CHAPTER I. INTRODUCTION
Figure I.4: Temporal evolution of a (left) fundamental soliton N=1 (right) higher
order soliton with N=3 recovering its pulse shape after a soliton length z = Lsol .
Higher order dispersion and Raman-gain are ignored.
N=3).
In the presence of higher order dispersion or Raman scattering, higher order solitons breakup into fundamental solitons or lower amplitude pulses. This
process of the soliton breaking up into sub-pulses is called soliton fission. The
solitons are ejected one after the other with the earliest ejected solitons having
the highest peak power, shortest duration and thereby propagate with faster group
velocities (Fig. I.5).
Sometimes the spectra can be broad enough to seed resonant phase-matched
processes like dispersive wave generation (Fig. I.6). If the soliton orders are 2 ≤
N ≤ 15, the propagation dynamics are dominated by soliton fission (especially in
supercontinuum generation with anomalous dispersion pumping) [39, 40]. Soliton
fission can have a pivotal role to play in supercontinuum generation owing to the
ejected solitons from soliton-fission transferring energy to blue-shifted dispersive
waves and by red-shifting owing to Raman-induced soliton self-frequency shift.
Both of these phenomena are discussed below.
I.1.8
Dispersive wave
Higher-order dispersion can influence soliton-fission dynamics by leading to energy transfer from the soliton to a resonance in the normal GVD regime. The
position of this resonance in the normal dispersion regime can be determined by
phase-matching conditions (Fig. I.6). To introduce the notion of dispersive wave
I.1. FUNDAMENTALS OF NONLINEAR OPTICS
23
Figure I.5: Soliton fission for N=5 soliton (a) over a soliton period z/Lsol = 1
(b) over z/Lsol = 0.35 without higher-order dispersion or Raman scattering (c)
soliton fission over z/Lsol = 0.35 in the presence of higher-order dispersion or
Raman scattering.
emission, we seek a linear solution of Eq. I.21 with the Ansatz:
At = A0 exp(i(ωt − klin z))
(I.25)
This leads to the dispersion relation given by
klin = −
(β2 ω 2 )
2
(I.26)
As seen from Fig. I.61 , phase-matching condition klin = ksol between the soliton and the dispersion does not occur when only second order dispersion is considered. However, when third order dispersion is taken into account, it is seen that
the linear waves are phase-matched to the soliton when
klin = −
1 Figs.
β2 ω 2 β3 ω 3
+
2
6
I.4, I.5 & I.6 are adapted versions of figures provided by Nicolas Joly
(I.27)
24
CHAPTER I. INTRODUCTION
Figure I.6: Phase-matching and corresponding spectra for linear waves (blue) and
soliton (red dotted line) under the influence of (left) GVD (β2 ) (right) GVD+TOD
(β2 +β3 ) enabling phase-matching of solitons to linear waves or dispersive waves.
A and N represent anomalous and normal dispersion. D.W stands for dispersive
wave.
During the soliton-fission process for example, ejected fundamental solitons
having a sufficiently broad spectral bandwidth to resonate with the dispersive
wave frequencies, would ideally produce dispersive waves when the phase matching conditions are satisfied. These phase matching conditions for hollow-core
fibers are discussed in I.2.5. Energy transfer from solitons to dispersive waves
also results in a soliton recoil in order to conserve energy [31, 5]. Dispersive wave
generation was the pivotal phenomena driving the pioneering UV generation research in hollow-core fibers [17], which was performed prior to the present work.
Raman scattering can influence dispersive wave generation. Solitons can shift in
frequency due the process called soliton self-frequency shift due to intra-pulse
Raman-scattering.
Dispersive waves in the vicinity of the modulational instability regime was
studied in numerically in the case of the Ar filled HC-PCF in Chapter III [25].
I.1. FUNDAMENTALS OF NONLINEAR OPTICS
I.1.9
25
Influence of Raman-scattering on nonlinear optics
Raman scattering is a significant perturbation in several nonlinear optical phenomena such as soliton fission, dispersive waves, etc. In the current research, the main
intention of using noble gases was to isolate this Raman-perturbation from nonlinear optics. Previous research has aimed at studying collision-induced Ramanscattering in dense noble gases [41, 42]. However the ultrafast pump sources used
in the experiments reported in this thesis have a spectra which is broader than
the Raman-shift observed in [41, 42]. Therefore Raman-scattering in noble gases
is negligible in the experiments reported in this thesis. It is useful to review the
main influences of Raman scattering for a Raman-active Kerr medium that were
avoided by using monoatomic noble gases.
Soliton self-frequency shift
Solitons propagation in a Raman-active Kerr media with non-instantaneous nonlinearity, undergo frequency red-shifting and pulse reshaping due to Raman effect
[43, 44, 45]. This phenomenon is called soliton self-frequency shift (SSFS). The
solitons experience continuous shift to longer wavelengths due to their spectral
bandwidths overlapping with Raman gain. As the spectrum of the solitons is redshifted, its experiences increasing GVD, its pulse width increases and this slows
down the SSFS. SSFS is proportional to the input pulse energy and the frequency
re-shifting is higher for higher pulse energies. The rate of SSFS too is proportional with the pulse energy [5]. This shift in frequency of the soliton also reduces
the gain of the dispersive wave due to phase-mismatch. In the absence of Ramanscattering and higher order dispersive wave perturbations, the chances of higher
order solitons regaining their original shape after every soliton period is higher
(soliton breathing) as shown in I.5(a) for an N=5. This is because the absence of
SSFS prevents the higher order solitonic components from shifting in either time
or frequency.
In the experimental and numerical analysis of our high pressure Ar filled fiber
experiments (chapter III), we found that solitons do not shift in frequency due to
26
CHAPTER I. INTRODUCTION
the absence of Raman-scattering in Ar [25].
I.1.10
Four-Wave-Mixing (FWM) & modulational instability
(MI)
Four-wave-mixing (FWM) is one of the most important third order nonlinear processes for spectral broadening. It is a process by which three frequencies interact
under phase matching conditions brought about by the Kerr medium, leading to
the creation of a fourth frequency. When the initial three frequencies ω have the
same frequency, then a third harmonic is generated where ωFW M = 3ω. If all
frequencies are the same including the generated frequency, then the process is
called degenerate FWM (DFWM) [21, 34]. If the difference of two frequencies is
tuned to a Raman resonance with a mode in the nonlinear medium then the FWM
is referred to as coherent anti-Stokes Raman scattering (CARS). Phase matching of the interacting frequencies is vital to the efficiency of the FWM frequency
generated.
In the time-domain, the instability of the temporal modulation or envelope of
the pulse is referred to as modulational instability. For a CW pump, the MI can
result in a train of ultra-short pulses. For optical pulses too MI can cause pulsebreakup into several femtosecond solitons or pulses. These MI generated solitons
too are influenced by Raman induced SSFS and dispersive wave generation. The
noise in the system acts as a seed for MI. Hence the frequency components generated from MI are not coherent with the pump. Processes that are not seeded
by noise, for example, dispersive waves that are seeded by solitons, form coherent spectral components. So the coherence of spectral components more or less
depends on the soliton number of the system. If the soliton number <22 [40] ,
then soliton fission is the main driving process behind the spectral components
leading to enhanced coherence. For higher soliton orders MI would be the dominant process, and since it is noise seeded would be detrimental to the coherence
of the spectral components. MI is mainly influenced in the anomalous dispersion
regime.
I.1. FUNDAMENTALS OF NONLINEAR OPTICS
27
In the scalar approximation of MI, two pump photons are shifted in frequency
by Ω. Therefore the generated MI sidebands are given by idler frequency ωi =
ω p − Ω and the signal frequency ωs = ω p + Ω. At MI phase matched conditions;
the maximum frequency shift Ω and the spectral gain maximum are [21] given by
�
Ω=±
2γP0
|β2 (ω p )|
(I.28)
By convention the sidebands in the normal dispersion are referred to as FWM
and those with solutions in the anomalous dispersion regime (where the sidebands
are much closer to the pump) are MI sidebands. As seen from Eq. I.28, MI is
power dependent, which is not the case for FWM. This is well illustrated in Fig.
I.7.The phase-matching conditions in Fig. I.7 [22] are given by
2ω p = ωi + ωs
(I.29)
where 2 pump photons ω p can undergo FWM to generate an idler ωi and signal
ωs frequency where ωs > ω p > ωi . Their respective propagation constant phasematching conditions given by
βi + βs − 2β p + 2γP0 = 0
(I.30)
As mentioned earlier, sidebands in the anomalous dispersion regime are called
modulational instability (MI) sidebands. The appearance of these sidebands in the
normal and anomalous regime have been attributed the influence of higher order
dispersion [46]. For instance, in the absence of higher order dispersion (β j , j ≥ 3)
, β2 results in frequency-shifted sidebands by Eq. I.28 in the anomalous dispersion
regime. In [46] , the dispersion of the solid core PCF was tailored to make β2 low
and flat, in order to increase the influence of higher order dispersion terms like β4
and β6 . By doing so, the sidebands were found in the normal dispersion regime
too with larger frequency shifts. This allowed for the sidebands to be generated
outside the Raman-gain band of silica [22] as shown in Fig. I.7. In [22], FWM
was used to generate a pair of correlated photon pairs. MI sidebands being close
28
CHAPTER I. INTRODUCTION
Figure I.7: (left) Nonlinear phase matching for FWM and MI (right) The calculated MI and FWM sidebands shown along with the Raman-gain band of fused
silica glass [22].
to the pump, overlapped with the Raman-gain band-which significantly introduces
noise via a high level of background photons. By using the appropriate dispersion,
FWM sidebands were generated outside the Raman-gain band (Fig. I.7).
Wavelength tunable MI sidebands in the absence of Raman-gain band have
been achieved in Ar-filled HC-PCF, which is reported in Chapter III.
I.1.11
Self-focusing
All of the above nonlinear effects, stemmed from the time-dependent intensity
variation of the optical field. The intensity can also similarly depend on the spatial
beam profile. Self-focusing is the spatial equivalent of SPM. SPM arises from the
time-dependent intensity envelope I(t) of the optical pulse. Self-focusing on the
other hand stems form the radial-dependence of intensity I(r) across the spatial
profile of the beam with radius r [27, 34].
SPM (time-dependent intensity)
n(t) = n0 + n2 I(t)
Self-focusing (radial-dependent intensity) n(r) = n0 + n2 I(r)
I.1. FUNDAMENTALS OF NONLINEAR OPTICS
29
Figure I.8: (a) Radial dependence of intensity I(r) in the beam spatial profile. (b)
self-focusing (c) self-channeling in the Kerr medium.
The refractive index n(r) experienced by the beam varies across its spatial
profile. n0 is the linear refractive index. For example, a Gaussian spatial beam
profile, would have maximum intensity at the center of the beam where r = 0.
This would also mean that the change in refractive index would be maximum at
r = 0 which results in a variation of refractive index experienced across the beam
profile-leading to focusing of the beam (n2 > 0). Hence the phenomenon is called
self-focusing. In some cases defocusing can also occur like in the presence of free
electrons in the ionization regime [47, 48]. It is clear that the intensity needs to
be sufficiently high for self-focusing to occur. At a certain peak power, the selffocusing effects can counteract the diffraction effects leading to self-channeling
or a waveguide of light. This occurs at a critical power Pcr given by [27]
Pcr =
π(0.61)2 λ 2
8n0 n2
When the peak power of the beam is is much higher than Pcr , beam breakup
can occur due to imperfections on the laser from. The beam components carry
roughly Pcr of power. At such high powers, the possibility of ionization especially
in case of gases arises. The free electrons ejected can result in defocusing of the
beam and sometimes ring-like spatial beam profiles [47, 48].
The self-focusing regime was observed in bulk supercritical Xe at 90 bar was
explored experimentally and numerically in Chapter IV.
30
CHAPTER I. INTRODUCTION
I.1.12
Supercontinuum generation
A combination of many of the above discussed processes gives rise to supercontinuum spectra. SPM, FWM, MI, soliton fission, dispersive wave, SSFS are responsible for supercontinuum generation in PCF [5]. Supercontinuum generation was
first observed by Alfano in bulk glass [4]. The smaller core sizes, single-mode
propagation over broad wavelength ranges and customizable dispersion, made
PCF an ideal medium for supercontinuum generation [11, 9]. Ideally pumping
close to the ZDW, leads to efficient spectral broadening. The GNLSE can be used
to numerically study supercontinuum generation. Supercontinuum generation in
PCF has been studied in various dispersion regimes, with primary nonlinear processes being soliton fission [11] , FWM & MI [49] and CW pump induced MI
[40].
By filling high pressure Ar gas in HC-PCFs, we observed and analyzed the
causes of supercontinuum generation in various regimes such as soliton fission
and MI initiated spectral broadening (Chapter III)
I.2. PHOTONIC CRYSTAL FIBER (PCF)
I.2
I.2.1
31
Photonic Crystal Fiber (PCF)
Types of PCF
Photonic crystal fibers (PCFs) are fibers with a periodic transverse microstructure
consisting primarily of air holes and glass. The development of PCFs in the 1990s
lead to improved versatility in nonlinear optical research as dispersion, nonlinearity, and birefringence could be adjusted by the size of air holes and glass in the
cladding lattice structure [9]. Based on the central core, where the light is guided,
PCFs can be broadly classified into two types - solid core and hollow-core. In
solid core PCFs, these air holes reduce the effective cladding refractive index with
respect to the glass core, enabling optical guidance by total internal reflection.
Solid core PCFs have been instrumental in the development of endlessly single
mode fibers [10].
With high air-filling fractions in the cladding, the core can be more isolated,
leading to tighter light confinement and enhanced nonlinearity. Another advantage
was that the dispersion could be tailored by changing the hole size and pitch – enabling the ZDW to be positioned close to the pump wavelength. These features of
the solid-core fiber have been exploited most notably in supercontinuum sources
[11]. Solid-core fibers have contributed immensely to studies in nonlinear optics
owing to its ability to guide light in a confined core over long distances. Solid core
fibers largely depend on total internal reflection as a guidance mechanism. Their
properties have been adjusted traditionally by doping the core material chemically
[50] , tapering [51, 52] , dispersion tailoring [46] , etc. The fundamental intention
Figure I.9: Types of PCF: Solid core fibers (left) endlessly single-mode (middle)
high air filling fraction in cladding (right) kagomé hollow-core PCF.
32
CHAPTER I. INTRODUCTION
to reduce the core diameter was to decrease its effective area hence increasing the
nonlinear parameter γ (Eq. I.9).
The air-holes in the cladding lattice can also support photonic bandgaps (PBG)
which prevent light from coupling into the cladding, hence enabling guidance in
the core. Unlike the prerequisite of TIR guidance, where the core refractive index
must exceed that of the cladding, PBG guidance can support hollow fiber cores. A
hollow-core meant that fibers could now be filled with appropriate media such as
gases/liquids [53, 23], for particle guidance [54, 55] and performing photosensitive chemical reactions efficiently [56, 57]. The PBG however offered a restricted
transmission range. The kagomé lattice hollow-core PCF offered a much broader
transmission range extending from the UV to IR.
I.2.2
Fabrication
PCFs are usually fabricated using the as “stack-and-draw” method [9]. The fundamental idea in this method is to scale down the original macroscopic structure
down to the microscopic dimension of the fiber. This process starts with stacking
capillaries and/or rods (of around 1 mm outer diameter and 1 m in length) in the
required structural pattern. This stack is then placed into a jacket tube-thereby
forming a preform.
Care is taken to maintain a desired ratio between the air hole size and spacing
as this defines the properties of the PCF such as dispersion. The preform is then
clamped onto the drawing tower as shown in Fig. I.10. The Fig. I.10 shows
the drawing process being demarcated by two parts. In section (a), the preform
is drawn down to a cane. The furnace heats the preform and the glass structure
is allowed to drop. In the process, the cane is drawn down to be narrower than
the preform. The cane is then cut and screwed onto the drawing tower where
the process in repeated and the cane is drawn into a fiber. The structure can be
maintained by using pressure and vacuum appropriately in the drawing tower.
Using this process most PCFs both solid and hollow-core are drawn. A polymer
coating (cured by UV light) is coated onto the drawn fiber for protection before
the fiber is wound onto the spool.
I.2. PHOTONIC CRYSTAL FIBER (PCF)
33
Figure I.10: Schematic of the fiber fabrication process.The cane is drawn in section (a). The fiber is drawn by drawing the cane through sections (a) and (b).
I.2.3
Hollow-core PCF
Total internal reflection (TIR) is the primary guidance mechanism in solid core
PCFs as long as the air holes in the cladding effectively reduce the cladding refractive index relative to the core. Traditionally in fiber optics, the mechanism
of TIR meant that the hollow-core fiber was not feasible as the cladding had to
have a lower refractive index than the core. Bragg fibers were seen as a possibility for a hollow-core, but their fabrications techniques were complex. Capillaries
were popular for large core sizes, but for core sizes <100 micron [23] their losses
increased exponentially.
The guidance mechanism of PCFs via photonic band gaps in the cladding by
means of a periodic lattice of air holes meant that certain frequencies of light could
propagate in the core-even in a hollow-core. A hollow-core fiber guiding light
had several advantages that could not have been possible in earlier light-matter
interaction systems. Long interaction lengths, excellent modal overlap and confinement for light-matter interactions, high material damage threshold, dispersion
tunable optical systems, to name a few of the advantages offered by the hollow-
34
CHAPTER I. INTRODUCTION
Figure I.11: SEM image of a (left) Photonic bandgap HC-PCF (right) kagomé
HC-PCF.
core fibers. Hollow-core PCF (HC-PCF) is mainly categorized as being either
photonic bandgap or kagomé HC-PCF (Fig. I.11). They both have a hollow-core
but differ significantly in properties such as transmission window, loss, dispersion
and guidance mechanisms.
Photonic bandgap HC-PCF
Photonic bandgap (PBG) HC-PCFs were the first HC-PCFs to be fabricated. Their
guidance mechanism stems from the PBGs in the cladding owing to the periodic
lattice structure of air holes. Their losses are in the order of a few dB per kilometer (down to 1dB/km at 1550nm [58]). These fibers have restricted transmission
wavelength windows, and these can be determined by finite element modeling
(FEM). Filling the fiber holes (with water for example) can shift the transmission window according to the scaling law [59]. The narrow transmission windows
in PBG fibers are suitable for experiments where narrow bandwidth interactions
[60, 53] are studied or where frequency components need to be filtered out in
Raman-scattering experiments in which unwanted vibrational and higher order
rotational frequency components were suppressed. [16].
Kagomé HC-PCF
Kagomé hollow-core fibers offer significant advantages over the PBG fibers in
terms of broadband transmission. The name kagomé stems from the close resem-
I.2. PHOTONIC CRYSTAL FIBER (PCF)
35
Figure I.12: Approximation of a kagomé fiber to Bragg fiber. (a) SEM of a
kagomé fiber with a ‘Star of David’ pattern marking out the unit cell of the lattice
structure. (b) Concentric hexagonal approximations (c) Further approximations to
concentric rings of resembling a Bragg fiber. Adapted from [62].
blance of the Japanese kagomé weave to that of the fiber cladding lattice structure
[61]. The unit cell of the cladding lattice structure resembles a “star of David”
as shown in Fig. I.12(a). It is clear from its broad transmission window (extending from UV-IR) that PBG are not the guidance mechanism of the kagomé fiber
[61, 62]. A few theories have been put forth to explain the guidance mechanism
of kagomé fiber. These include two-dimensional anti-resonant reflection [63] and
approximations to Bragg fiber propagation models [62]. If the light in the core
is unable to phase-match (resonate) to states in the cladding, then it is effectively
confined to the core and is unable to leak out [9]. Two-dimensional geometry of
the multilayered fiber cladding is to be considered in the anti-resonant reflection
model, which is analogous to the one-dimensional approach that has been traditionally applied to some fields in photonics.
Another intuitive approach taken to explain the guidance mechanism of the
kagomé fiber is to approximate the cladding structure to that of a Bragg fiber
with alternating concentric rings of higher and lower refractive index layers in
the cladding around the core [62]. As shown in Fig. I.12, the concentric cladding
rings of the kagomé fiber are first considered as concentric hexagonal rings around
a hexagonal core and then approximated to concentric circular rings – analogous
to the Bragg fiber. However a comprehensive model to describe the guidance
mechanism of a kagomé fiber conclusively is a subject of ongoing research. A
36
CHAPTER I. INTRODUCTION
comparison between PBG and kagomé fibers is done in Table. I.3.
Dispersion
Interestingly a convenient and simplified capillary approximation is used to determine the waveguide dispersion of the kagomé HC-PCF. The model is based on
the dispersion formalism put forth by Marcatilli and Schmeltzer [64] for hollow
metallic and dielectric waveguides. Material dispersion of gas/liquid used to fill
the fibers could also be incorporated into the model, enabling a fairly reliable and
simple dispersion model to study the gas/liquid filled kagomé fiber.
The field distribution in the core of cylindrical waveguides is given by Bessel
functions [21]. Marcatilli and Schmeltzer arrived at a similar formulation for
hollow cylindrical metallic and dielectric waveguides. Approximations such as
waveguide radius being much larger than free-space wavelength and treatment of
only low-loss (lower order) modes were made in order to simplify the analysis.
The efficacy of applying the above model to predict the kagomé dispersion was
established when finite element modeling (FEM) of the fiber kagomé fiber structure was computed and the calculated dispersion matched the simple dispersion
model very well [23]. Phase-matching conditions for THG and dispersive waves
calculated using this model have been verified by experiments [65, 17]. The propagation constant for a hollow dielectric was given in [64] can be modified for a
gas-filled kagomé fiber [65]
β (ω) =
�
u2
n2gas (ω)k2 (ω) − nm
2
reff
�
�
δ (ω)
u2nm c2
≈ k(ω) 1 +
− 2 2
2
2reff ω
(I.31)
where unm is the mth zero of the (n − 1)th order Bessel function, ω is the optical
frequency, c is the speed of light, k is the wave-vector, δ (ω) is the Sellmeier
expansion for n2gas (ω) which is refractive index of the gas, reff is the effective core
radius given by
I.2. PHOTONIC CRYSTAL FIBER (PCF)
37
photonic bandgap HC-PCF
kagomé HC-PCF
guidance
photonic bandgap
2-dimensional
mechanism
in cladding
anti-resonant reflection
loss
dB/km
dB/m
transmission
window
narrow bandwidth <200 nm
UV-IR
unit cell
honeycomb
star of David
dispersion
Table I.3: Comparison between PBG and kagomé HC-PCF. Dispersion and loss
plots taken from [23].
38
CHAPTER I. INTRODUCTION
� √
2 3
reff =
rhex = 1.0501rhex
π
(I.32)
The radius of circle with the same area of the hexagonal core of the kagomé fiber
is represented by rhex [23, 66]. The modal refractive index of the EH11 mode for
example, in a gas-filled kagomé HC-PCF is accurately approximated by that of a
glass capillary and is given by [64, 65]
λ 2 u201
ρ
n(λ , p, T ) ≈ 1 + δ (λ )
−
2ρ0 8π 2 a2
(I.33)
where λ is the vacuum wavelength, δ (λ ) the Sellmeier expansion for the dielectric susceptibility of the filling gas [6], ρ is the density of the gas at a particular
temperature and pressure, ρ0 is the density of the gas at 293 K and atmospheric
pressure. a the core radius and u01 is the first zero of the Bessel function. Finite
element simulations and numerous experiments have confirmed the reliability of
this expression [64, 65, 67, 23].
I.2.4
Solid core vs kagomé HC-PCF
One of the pivotal aims of the work reported in this thesis is to scale up the material
nonlinearity of the core gas (filled in a HC-PCF) to that of fused silica. In many
of the nonlinear HC-PCF experiments, high power amplified pulsed lasers were
required as a pump source as the inherent material nonlinearity of gas-filled HCPCF was low in comparison with solid core fibers. The solutions presented in this
thesis rely upon the density dependence of nonlinearity. Hence liquids and high
pressure gases were used to fill the fibers as the increased number density of atoms
leads to increased nonlinearity. Liquid Ar, high pressure Ar and Xe were used.
Supercritical Xe provided the best option of matching fused silica nonlinearity at
80 bar.
There are several advantages offered by highly nonlinear gas-filled HC-PCF.
Using noble gases eliminated the possibility of Raman scattering, which is a significant perturbation in nonlinear optics – undesirable in correlated photon pair
I.2. PHOTONIC CRYSTAL FIBER (PCF)
39
sources for example. Nonlinear optics could be studies in the absence of Raman
gain. A Raman active gas can also be introduced when needed, to study Ramanrelated phenomena like in [15, 16].
The other main advantage is the remarkable range of tunability of ZDW from
UV to IR by adjusting the pressure of noble gases between 1-150 bar. This level
of ZDW tunability is not available in the solid core fiber even by adjusting the
cladding holes and pitch. Moreover the dispersion can be varied over this large
wavelength range on a given piece of HC-PCF and used repeatedly – unlike in a
solid core fiber where a new fiber needs to be drawn for a specific ZDW requirement. The low and smooth dispersion profile over the transmission window of the
kagomé fiber, encourages phase-matched processes such as dispersive wave generation and FWM. The phase-matching conditions can be adjusted easily by gas
pressure in the fiber-a feature unavailable in solid core fibers. The dispersion could
be tuned from anomalous to normal or vice versa for a fixed laser wavelength.
Photo-darkening is a common problem in solid core fibers with a relatively
low damage threshold. HC-PCFs however do not have this problem even when
filled with gases. This allows for soliton compression leading to optical powers
high enough to study ionization in gases. The HC-PCFs are also transparent in the
UV which is not the case for fused silica.
The above advantages helped us explore nonlinear fiber optics in various dispersion regimes. However it is important to take note of the fact that kagomé
HC-PCF have much higher optical loss compared to solid core fibers. In most
of the experiments reported here, relatively short lengths of kagomé fiber (<50
cm) were used-hence the high fiber loss was not an experimental hinderance. In
comparison, the average core sizes of kagomé fibers are also larger. This compromises on the nonlinear parameter γ which is a few orders of magnitude larger
than that of gas-filled HC-PCFs as [23] is inversely proportional to the effective
core modal area. Hence by scaling up the pressure, we increase the n2 to match
that of fused silica but not γ. Reducing the losses and the core size of the HCPCF fiber are fabrication challenges, which when overcome, can make HC-PCF a
major competitor for solid core fibers. These comparisons are tabulated in I.4.
40
CHAPTER I. INTRODUCTION
Fused silica solid core PCF
kagomé HC-PCF
guidance mechanism
Total internal reflection
2d anti-resonant reflection
fiber core material
Fused silica
Hollow (gas/liquid)
core n2
2.74
<2.8 (at 80 bar Xe)
controllable n2
no
yes
γ (W−1 km−1 )
~240 at 850 nm
~1.9x10−3 Xe (80 bar)
for a core diameter
1 µm
18 µm
avg. core size
~1-20 µm
~15-50 µm
damage threshold*
< 1013 − 1014
> 1014
losses
1-100 dB/km
>1 dB/m
ZDW
VIS-IR
UV-NIR
dispersion tuning
cladding hole-size & pitch
gas pressure
transmission window
350 nm-IR
UV-IR
Raman gain
yes
no (for noble gases)
(×10−20 m2 /W)
Table I.4: A comparison of solid core Vs kagomé HC-PCF. The shaded rows mark
out the advantages the noble gas-filled kagomé HC-PCF systems have over fused
silica core PCFs. The above values are for an overview and can vary with factors
such as core size, pitch, etc. * [68, 23]
I.2. PHOTONIC CRYSTAL FIBER (PCF)
I.2.5
41
Preceding gas-filled HC-PCF research
The research prior to the results presented in this thesis, included pioneering work
conducted primarily by Nicolas Joly, Philipp Hölzer and Johannes Nold. Initial
technical inroads and techniques developed into HC-PCF gas filling significantly
influenced the research presented in this thesis. Challenges such has fabrication of
low-loss kagomé, theoretical analysis, etc were overcome – leading to pioneering
experimental results supported by rigorous theoretical analysis [65, 17, 66, 67, 19,
23].
Third harmonic generation
The low and smooth dispersion profile of gas-filled kagomé fibers encourages
phase matching processes such as third harmonic generation. The intrinsic anomalous dispersion of the kagomé fiber can be compensated by appropriate gas pressure, which can be regulated to tune the fiber dispersion, and hence its ability to
take part in phase matching processes. 30 fs, 800 nm at 1 kHz repetition rate was
used as pump pulses were launched into a kagomé fiber with a core diameter of
29.6 µm filled with Ar gas. At around 5 bar pressure of Ar, the phase-matching
conditions were satisfied wherein the pump in the HE11 mode generated its third
harmonic in the HE13 mode. At higher powers, the Kerr nonlinearity increases the
phase mismatch and in doing so the peak third harmonic frequency components
are shifted to lower frequencies as seen in Fig. I.13 [65].
Using Xe filled HC-PCF at 25 bar, clear intermodal FWM was observed
and confirmed theoretically using phase-matching calculations similar to those
in [65].
Dispersive wave UV generation
As discussed earlier, dispersive waves arise when the soliton spectral bandwidth
broadens into the normal dispersion, perturbing the soliton to shed energy as resonant dispersive wave radiation. For this experiment, a 20 cm long kagomé PCF
with a core diameter of 29.6 µm was filled with Ar gas and varied in pressure up
42
CHAPTER I. INTRODUCTION
Figure I.13: Third-harmonic spectrum as a function of Ar pressure for two pump
pulse energies: (a) 0.7 µJ and (b) 1.3 µJ. Theoretical phase-matching curves for
different core radii, the solid curve corresponding to a core radius of 14.9 µm.
(c) Experimental (d) Theoretical calculated near-field mode profiles of the third
harmonic at the fiber end. Figure taken from [65].
to 10 bar. The pump used was an 800 nm, 30 fs laser. A similar experimental
setup as the above experiment was used with two gas cells on the ends of the fiber
[17].
For efficient dispersive waves to occur into the deep- UV frequencies, a few
other perturbations need to be taken into account. Optical shock and self-steepening
were instrumental in bringing about asymmetry in the SPM broadened spectrum.
This asymmetry helps to push the dispersive waves further into the UV and improve its conversion efficiency. Numerical propagation modeling elucidated the
self-steepening or optical shock significance, with the self-steepening switched on
and off. The efficiency of the dispersive wave UV considerably increased when
the self-steepening term is turned on. Importantly, the dispersive wave in emitted
in the fundamental spatial mode, making it a very suitable candidate for a tunable
UV source. Compared to the third harmonic generation in the HE13 mode discussed above, the UV dispersive waves had orders of magnitude higher efficiency
(∼7% of the transmitted power is in the 240-350 nm range) in the fundamental
mode. The UV dispersive wave wavelength is also tunable by means of the gas
pressure. The soliton phase-matching condition for dispersive wave generation
I.2. PHOTONIC CRYSTAL FIBER (PCF)
43
Figure I.14: Pressure dependence of the UV dispersive wave in the fundamental
mode (inset: the spatial mode is superimposed on the scanning electron micrograph of the fiber used) at 1.5µJ pulse energy. Numerical simulations solved with
the self-steepening term turned on. N and A are normal and anomalous dispersion.
Figures taken from [17].
was given by
βsol (ω) = β (ωsol ) + β1 (ωsol )[ω − ωsol ] + (γPc )/2
(I.34)
where ωsol is the soliton frequency and γPc /2 ∼ 2.3γP0 N [23].
Ionization regime
Soliton perturbations such as dispersive waves and higher order dispersion owed
their presence to Ar gas pressure in the fiber. For this experiment, a 34 cm long
kagomé PCF with a core diameter of 26 µm was filled with Ar gas. An 800 nm, 65
fs pump laser was used with energies up to 9 µJ. When the pressure was reduced
to <1.7 bar, the ZDW shifts to the UV region, and the soliton spectra is not broad
enough to phase-match to dispersive waves. Thus the solitons propagate without
shedding energy to dispersive waves or being affected by higher order dispersion.
This results in very efficient soliton pulse compression and the consequential peak
intensities are high enough to partially ionize the Ar gas. Free electron densities of
~1017 cm−3 were ionized due to soliton self-compression induced peak intensities
44
CHAPTER I. INTRODUCTION
of 1014W /cm2 . The self-compression of higher order solitons to duration of a few
optical cycles lead to the in the release of blue-shifting solitons due to the free
electron densities of the plasma. This unique combination of soliton dynamics
and ionization helped to observe the novel phenomenon of soliton-blue shift in
frequency. Soliton blue-shifting described here is in some ways analogous to
Raman-induced SSFS [45]. The numerical treatment was presented in [67, 19].
In Chapter I, the main nonlinear effects and the basics of photonic crystal
fibers were discussed. The experimental implementations and results of filling a
HC-PCF with dense noble gases are discussed in the following chapters.
Chapter II
Liquid Ar filled kagomé PCF
Noble gases do not exhibit any significant stimulated Raman Scattering owing
to them being monoatomic. In the liquid phase, Ar has a nonlinear refractive
index (n2 ) of the same order of magnitude as silica while maintaining a negligible
Raman contribution [41, 42]. This results in a novel nonlinear system, where
Raman scattering does not add to the noise of the system. Although some research
has been done into collision induced scattering in liquid Ar [41, 42], the effect is
negligible in our case as the pump spectra is larger than the Raman frequency
shift. The path to achieve the goal of scaling up the nonlinearity of a HC-PCF to
that of fused silica, began by filling the fibers with liquid Ar. During the course of
this work, several technical challenges such as phase transitions in the fiber core,
were faced. All these technical issues were sorted out when the high-pressure
gas-filled fiber system (discussed in III and IV) was developed. However the
liquid Ar filling techniques developed in this work are novel and have significantly
increased the understanding of low temperature operation in PCF experiments.
Nonlinear optical effects like SPM have been observed in liquid Ar filled PCF.
II.1
Experimental Implementation
Filling a HC-PCF with a cryogenic liquid has not been attempted before. Hence
a new experimental setup called cryogenic trap (cryotrap in short) had to be de45
46
CHAPTER II. LIQUID AR FILLED KAGOMÉ PCF
signed and manufactured after initial attempts and methods failed. The initial
methods used, are also discussed in this chapter along with an explanation as to
why they failed to transmit light through the filled fiber. In all these attempts liquid
N2 is used as a coolant to achieve liquefy Ar gas temperatures since the boiling
point of liquid N2 (77 K) is lower than that of liquid Ar (86 K at atmospheric
pressure).
II.1.1
Kagomé fiber spliced with a multimode fiber
A kagomé fiber with a core diameter ~30 µm of was spliced to a standard multimode solid core fiber with a similar dimensions, using a fusion splicer. Transmission of the spliced fibers varied from 20-25% depending on the efficiency of
the splice. The intention here was to provide a flat interface at the splice, for the
liquid Ar filled in the kagomé fiber core. This flat splice interface was to effectively collect light that was in-coupled into the kagomé fiber. The spliced fiber
was installed vertically with the kagomé fiber serving as the in coupling end of
the fiber (Fig. II.1). When the kagomé-multimode spliced fiber was lowered into
a liquid N2 bath, the Ar gas in the kagomé would liquefy just above the level of
liquid N2 , as the boiling point of Ar (86 K) is higher than that of N2 (77 K). Liquid Ar was formed in a liquid column in the kagomé fiber. Formation of Liquid
Ar can be verified from the scattering point caused by the vapour-liquid interface.
Meniscus formation due to liquids in a capillary makes this gas-liquid interface
curved, similar to a water droplet meniscus in a capillary.
Despite several attempts, effective coupling of light into the core was not established using this method. There were a few disadvantages of this system. Sharp
gradient in the core and cladding holes of the kagomé due to the corresponding
gradient in density of Ar was observed. So the light propagates from a region of
refractive index of around 1.0 (refractive index of gaseous Ar) to about 1.2 (refractive index of liquid Ar) through a gas-liquid interface. The kagomé could not
be lowered further into the liquid nitrogen Dewar, as solid Ar might form, lead to
increased scattering (due to crystallization) [69] and lesser optical guidance in the
core. Hence the length of liquid Ar created in the fiber was limited to a few (2-4
II.1. EXPERIMENTAL IMPLEMENTATION
47
Figure II.1: (a) Kagomé-multimode spliced fiber dipped into liquid nitrogen to
liquefy some of the Ar gas-filled in the kagomé fiber. (b) the two distinct scattering
points arising due to the liquid-vapour meniscus and the kagomé-multimode fiber
splice seen through a colored filter.
cm) centimeters. These various changes in density and thereby refractive index of
Ar lead to the loss of the core mode.
The kagomé-multimode splice also leads to optical losses. If the light collected by the multimode fiber were sufficiently intense, then there would show
nonlinear effects in the multimode fiber too thereby possibly masking nonlinear
effects from liquid Ar. A reliable studying of intermodal effects might not be
possible in such a system.
II.1.2
Cryogenic trap
Since the above-described first attempts proved to be futile, a novel design for an
experimental setup to cool the fiber was conceived. For Ar at 1 bar, the melting
point of is ~83 K, the boiling point is ~87 K [29]. Since the temperature of liquid
N2 is around ~77 K, it serves a good coolant to liquefy Ar gas. However, this
48
CHAPTER II. LIQUID AR FILLED KAGOMÉ PCF
also meant that cooling the fiber using the open Dewar filled with liquid N2 could
result in formation of solid Ar which was undesirable. Solid Ar has a tendency
to crystallize [69] and thereby would create scattering defects in the core of the
fiber. Hence the idea to fabricate a novel system to achieve this was conceived.
The fabrication of the system had to be carefully thought of and a few engineering
hurdles had to be cleared. The system would still rely on liquefying Ar by cooling
down Ar gas. The following fabrication issues were resolved in the final version
of the cryotrap (Fig. II.4).
• A removable ferrule was utilized to mount the fiber into the cryotrap. The
removable ferrule also ensured that the fiber ends could be cleaved well and
clamped if required.
• The ferrule had to be uniformly cooled so as to have a homogeneous liquid
Ar medium in the middle of the fiber.
• Accurate temperature sensors were placed near the fiber to make sure that
the required temperatures were actually achieved.
• Condensation on the windows of atmospheric water vapour was to be prevented. This was done in two ways. One was to heat the windows with
10 Ω resistors. The other alternative method is purging the windows with
nitrogen gas, so as to prevent water vapour from condensing.
• An appropriate insulation was implemented to conserve the liquid N2 used
as a coolant.
• The system had to be rigid enough to handle a pressure of 10 bar and low
temperatures as low as 90 K simultaneously.
Description of Cryotrap
A cryotrap system was designed with the help of IMT Gmbh, Moosbach, Germany. It consisted of a primarily 3 parts (Fig. II.2).
• Temperature Controller
II.1. EXPERIMENTAL IMPLEMENTATION
49
Figure II.2: Schematic of the cryotrap. The insulation layers are not shown. Inset:
removable fiber ferrule.
• Cryotrap
• Cryo-Valve
A platinum resistance temperature sensor measured the temperature of the
ferrule inside the cryotrap. This sensor is connected to the temperature controller,
which in turn controlled the cryo-valve. The cryo-valve regulated the amount of
liquid N2 being pumped into the cryotrap, which cooled the cryotrap. A 100liter capacity Apollo cryogenic cylinder was used as a reservoir for liquid N2 .
The temperature controller sensed the temperature of the cryotrap and regulated
its temperature by controlling the amount of liquid N2 via the cryo-valve (Fig.
II.2). A standalone sensor and measuring device (to check if the ferrule had a
uniform temperature) also simultaneously measured the temperature of the fiber
ferrule. The two temperature sensors are placed close the two ends of the ferrule.
When the two sensors read out almost the same temperatures, the cryotrap was
confirmed to be cooling the fiber length uniformly.
50
CHAPTER II. LIQUID AR FILLED KAGOMÉ PCF
Figure II.3: The three main components comprising the cryotrap system.
The cryotrap consists of a central region housing the ferrule and this region
is welded to the two gas cells on either end (Fig. II.2). The ferrule holds a 5 cm
fiber, which the central region of the cryotrap cools down, uniformly to liquid
Ar temperatures (Fig. II.2). The ferrule is removable with the help of tweezers
through the input window. After the 5 cm of kagomé fiber is cleaved well and
loaded into the ferrule, it is secured to the body of the cryotrap by means of a
couple of screws. A ferrule had the provision for a graphite ring that could be
used to secure the ferrule to the fiber. Graphite was chosen, as it is a relatively soft
material and does not clamp the fiber too hard and avoid any damage to the fiber.
The window was designed to accommodate a glass window or an aluminum plate
with a hole for the fiber. This choice depended on the requirement to cool the fiber.
If whole length of fiber was cooled in the ferrule, then glass windows (sapphire or
fused silica) were used. When only the central part of the fiber needed to be cooled
(for fibers longer than the ferrule length), the aluminum plates were substituted in
the place of the glass windows. These aluminum plates have a hole in their center
to allow the fiber to pass through. The hole was then sealed off with glue. In
subsequent improvements the glue was replaced by Swagelok adaptations. The
pressure and the temperature of the Ar gas were optimized [29] , to achieve the
filling of liquid Ar in the kagomé fiber. At 10 bar Ar gas pressure and around 110
II.1. EXPERIMENTAL IMPLEMENTATION
51
K (approx. -160o C) liquid Ar formation was expected to begin.
Since the cold part of the cryotrap was welded to the two gas cells, the glass
windows get cold too, resulting in condensation of atmospheric water vapour.
One of the methods to avoid this problem was to heat the windows using resistors attached to the two windows as the system is cooled. This resulted in more
liquid N2 being pumped into the cryotrap to compensate for the heating of the
windows. This extra pumping of liquid N2 resulted in increased vibration of the
system, thereby affecting the in coupling of light into the fiber (placed inside the
cryotrap). Warm windows and the cold interiors of the cryotrap introduced a undesirable convection of Ar gas. The extra vibration due to the heating of windows
were avoided by introducing a N2 gas purging system to the windows to prevent
condensation of atmospheric water vapour. By avoiding heating the system, the
thermal stability of the system was be achieved.
Effective filling of the kagomé fiber with liquid Ar
To fill the kagomé fiber with liquid Ar at the appropriate pressure and temperature,
it is important that any impurities (like water), which condense or solidify within
the same thermodynamic regime, were eliminated. Water is eliminated at room
temperature by securely closing all openings of the cryotrap, and subsequently
purging the system with Ar gas. Alternatively purging and pumping out Ar gas
repeatedly eliminates the water to a great extent. Heating the cryotrap with the
resistors on the windows can also help to remove any adsorbed water on the inner
walls of the system. Using fiber was mounted in the cryotrap in 2 ways:
1. Fiber clamped in the ferrule (cooled along its whole length) as shown in
II.4.
2. Fiber is fixed on both ends to gas cells outside the cryotrap (at room temperature) as shown in II.5.
The fiber length is much longer than the cryotrap length (approx. 30 cm), with its
middle length passing through the ferrule in the cryotrap. This method prevents
52
CHAPTER II. LIQUID AR FILLED KAGOMÉ PCF
Figure II.4: Fiber installed inside the cryotrap.
the fiber from being uniformly cooled along its length, but is better for coupling
light into the core mode of the fiber.
The first method with the fiber clamped in the ferrule was not efficient in
coupling light into the core mode. Convection of Ar gas arose in the cryotrap due
to differential temperatures of the cold ferrule and the windows of the cryotrap
at room temperature. Since the fiber length is almost the same as the ferrule, the
thermodynamic phase transitions from gas to liquid were occurring near the in
coupling end of the fiber. All these factors prevented efficient optical coupling
into the fiber.
The second method on the other hand, used a longer fiber with its ends fixed
in gas cells at room temperature, with the middle of the fiber passing through
the ferrule. Although this prevented the entire fiber length from being uniformly
cooled, the transmission efficiency in the core mode through the liquid column
increased significantly to about 5-10% of the incident power. The transmission
through the kagomé fiber filled with only gas, varied between 30-50% in various attempts and fibers used. Aluminum plates with a hole in their center for
the fiber to pass through, replaced the glass windows on the cryotrap. Holes in
the aluminum plates and gas cells (made to accommodate the fiber) were sealed
with epoxy glue. Most of the experiments were performed, employed the above
II.1. EXPERIMENTAL IMPLEMENTATION
53
leakage prevention technique. However it was found that by connecting the gas
cells to the cryotrap by means of Swagelok adaption and plastic tubing/pipes (Fig.
II.5), significantly increased the transmission through the liquid filled column.
The reason being that the epoxy glue secured the fiber firmly, but the contraction
of the fiber and cryotrap at cold temperatures caused stresses on the fiber. This
was evident when hollow-core SF6 glass was briefly used in system with epoxy
glue. At low temperatures, the fiber was found snap and break due to contraction
of various parts of the system at low temperatures. Using the transparent pipes
and Swagelok adaptions, this problem was avoided as the fiber was secured only
by magnets to the V-grooves in the gas cells and was not stressed by contractions
in the cryotrap. However, this method led to formation of liquid gas interfaces
or capillary menisci, as the fiber was not uniformly cooled. Between the two
methods discussed above, the second method was most instrumental in getting
the experimental results from a PCF filled with a liquid column.
II.1.3
Experimental results
A schematic diagram of the experimental setup is shown in Fig. II.6(b). It consists
of a 30 cm of kagomé HC-PCF (Fig. II.6(a)) connected to two gas-cells at either
end. The fiber core is 28 µm in diameter and is first filled with Ar gas to a pressure
of 6 bar. The pressure of Ar gas was kept at 6 bar in order to increase the boiling
point of Ar to 108 K (from 87 K at 1 bar) [29]. This also ensured that liquid Ar
could be condensed in the fiber.
To liquefy the gas, a section of the fiber passes through the cryotrap using
liquid N2 as coolant. Precise temperature control from ambient down to 85 K
was possible. This unique setup is interesting for studying nonlinear effects, a
central liquid Ar section being sandwiched between two gaseous sections. The
pulse experiences a change in nonlinearity, dispersion and refractive index as it
propagates along the length of the fiber. The system was designed and optimized
so as to achieve optical guidance in the fundamental mode through the liquid
Ar. Diagnostics included a CCD camera, which images the near-field at the fiber
output, and an optical spectrum analyzer.
54
CHAPTER II. LIQUID AR FILLED KAGOMÉ PCF
Figure II.5: (top) Cryotrap cooling the middle section of a 30 cm long fiber with
Ar gas cells at room temperature on both ends. (bottom) Transmission and loss
due to the liquid Ar column in the fiber.
For this experiment, ~160 fs long pulses coming from an amplified Ti:sapphire
system (~800 nm) was used. Previous work has shown that the dispersion of
kagomé-lattice HC-PCF filled with gas can be accurately approximated by a filled
capillary of similar core diameter [65]. The same approach to calculate the dispersion properties of the liquid-Ar-filled fiber [64, 65]. As the pulse propagates
through the system, it experienced anomalous dispersion and low nonlinearity in
the gaseous Ar section (zero-dispersion wavelength (ZDW) ~350 nm) and then a
much higher nonlinearity but normal dispersion when it enters the liquid Ar region
(ZDW ~1.6 µm). Although transmission through the complete system was only
~10% due to the presence of menisci between the gas/liquid phases, the spatial
mode remains fundamental. The main contribution to the loss is scattering at the
first meniscus and the residual pump power in the second gas-filled section is too
weak to cause any significant nonlinear effects.
II.1. EXPERIMENTAL IMPLEMENTATION
55
Figure II.6: (a) SEM of the kagomé-lattice HC-PCF (b) Schematic diagram of the
experimental setup.
Experimental output spectra measured with and without the cryogenic system
are shown in Fig. II.7. When the central part of the HC-PCF is filled with liquid
Ar, the output spectrum (blue trace) broadens distinctly due to SPM. This is confirmed by the spectral peaks (blue trace), which are equidistant in frequency as
shown in Fig. II.7. The pulse propagates from a region of anomalous dispersion
to a region of normal dispersion as it propagates from gas to liquid. The pulse
energy launched into the first gas-filled section is estimated to be 1.7 µJ (12 MW
peak power).
We believe this to be the first demonstration of nonlinear optical effects in
liquid-Ar filled HC-PCF. We have successfully built a controllable system, which
fills a HC-PCF with a cryogenic liquid and allows measurement of optical transmission through it.
Emission spectra liquid Ar?
In another experiment, 60 fs, 800 nm pulses were used to pump the liquid-Ar filled
HC-PCF. When the fiber was filled with only Ar gas, the blue shifting soliton
was observed, like in [18] at around 1.5 mW of launched average power (Fig.
II.8(a)). Interestingly, when the fiber was filled with liquid Ar, distinctly new
spectral features appear at around the same incident energy for which the blueshifting soliton was observed in gas (Fig. II.8(a)). The spectral peaks of these new
features (green trace in Fig. II.8(b)) have an striking resemblance to the emission
spectra of liquid Ar (blue trace in Fig. II.8(b)) obtained in [70]. We believe that
56
CHAPTER II. LIQUID AR FILLED KAGOMÉ PCF
Figure II.7: Experimental output spectra shown more spectral broadening due to
SPM when the fiber is filled with gas and liquid when compared to just gas. The
SPM peaks are seen to be symmetric in frequency as expected.
II.1. EXPERIMENTAL IMPLEMENTATION
57
Figure II.8: Experimental spectra of (a) a fiber filled with gas only and with gas
and liquid in the ionization regime. (b) Experimental spectrum at 2 mW incident
energy on a fiber filled with liquid Ar (green trace). The spectral peaks obtained
resemble emission spectra of liquid Ar from Ref. [70] (blue trace).
liquid Ar might have been partially ionized and the recombination of the free
electrons resulted in the emission spectra. Its possible that the ionization of Ar is
followed by recombination when the liquid is present, due to the increased number
density and hence more free electrons [71]. This is not the case when the fiber is
only filled with gas as the number density is not high enough for recombination.
The spectral and temporal light emission properties of liquid argon are of interest
for developing large liquid rare-gas particle detectors in high energy physics.
However, a conclusive explanation has was not been obtained due to experimental uncertainties such as the pulse dynamics in density gradients due the ends
of the fiber being at room temperature and the middle section of the fiber at cryogenic temperatures. The exact role of the gas-liquid interface or meniscus and
the scattering losses was unknown. The output spectra was also noisy and unstable in the presence of liquid Ar as seen in (Fig. II.8(a)). A numerical analysis
of the above hypothesis could not be efficiently performed as the gas-liquid-gas
system in inherently complicated to model especially at the boundary conditions
58
CHAPTER II. LIQUID AR FILLED KAGOMÉ PCF
of these thermodynamical phases. A further study of these experimental spectra
would be interesting for further studying ionization regimes at thermodynamic
phase-transitions in the fiber.
II.1.4
All liquid Cryotrap
Components of the system
In order to solve the problem of the gas-liquid meniscus scattering loss in the fiber
using the cryotrap, a new method had to be devised to fill the fiber. Attaching
windows to the fiber end was ruled out as the glue holding them in place would
not be suitable at cryogenic temperatures. Filling up a HC-PCF by immersing the
fiber in the cryogenic liquid was a possible solution. For this, the fiber would need
to be evacuated (to prevent any gas bubbles) and then filled directly with the given
cryogenic liquid by immersing the fiber in it. A special cryogenic would be needed
to be designed. With the help of the company IMT, a new cryogenic system was
developed. This system offered several advantages over the cryotrap. The new
system had several transparent glass parts, thereby enabling a visual on the fiber.
The fiber would be filled from end to end, thereby filing the fiber uniformly with
liquid and preventing the formation of any gas-liquid interface. Capillary action
would also ensure that the fluid fills the fiber uniformly, just like a liquid rises
in a capillary when it is immersed into the liquid. Liquid would be between the
window and the fiber in coupling end, thereby creating no hindrances in the beam
focusing into the fiber. The system could successfully maintain the fiber immersed
in the cryogenic liquid for at least half an hour in the tests. The system however
was not used for optical experiments due to the simultaneous development of the
high pressure gas systems (discussed in), which enabled primary goals such as
fused silica nonlinearity to be easily achieved. The cryogenic system developed
is a very useful technical advancement for future low temperature experiments.
The cryogenic chamber consists of an inner fused silica cylinder (outer diameter 33 mm, wall thickness 4 mm) and an outer cylinder made of Duran glass
(outer diameter 80 mm, wall thickness 5 mm) as in Fig. II.9. Vacuum (approx.
II.1. EXPERIMENTAL IMPLEMENTATION
59
Figure II.9: Schematic of the all –liquid cryogenic system.
500 mbar) between these two glass cylinders provided thermal insulation to the
inner fused silica chamber containing the cryogenic liquid (and the fiber immersed
in it). In comparison to the previous cryotrap, the vacuum insulation between the
glass chambers provides for a visual on the fiber during the filling of a cryogenic
liquid. The vacuum between the two glass chambers not only provides thermal insulation but also avoids condensing of the atmospheric vapour on the inner fused
silica chamber, and thereby providing a clear visual on the fiber.
The inner cylinder is filled with liquid N2 and in the axis of the inner tube an
optical fiber is placed and is held in place by a simple stainless steel holder. This
cylinder has two sapphire windows (23.75 mm in diameter and 1 mm in thickness)
to allow the coupling of light into the fiber. Sapphire was used as it is a stronger
glass with a large spectral transmission and is appropriate for low temperature
applications. The outer glass chamber has also two sapphire windows. The construction design helped realize an easy opening and changing of the fiber. To avoid
moisture directly at the outer sapphire windows the windows can be flushed with
dry nitrogen to enable clear windows during the optical experiment. All screws
holding the sapphire window are fixed with a torque wrench with 10 Ncm. This
is necessary to avoid breaking of the 1 mm thick sapphire windows. 4 nuts for
60
CHAPTER II. LIQUID AR FILLED KAGOMÉ PCF
Figure II.10: (a) All –liquid cryogenic system opened up to show the constituent
chambers. (b) the fiber holder which fits into the inner chamber.
fixing the stainless steel plate are fixed with a torque wrench with 40 Ncm. All
other screws are fixed with a torque wrench of 20 Ncm. At both ends of the Duran
glass cylinder, silicone o-rings are glued on.
Before the chamber is cooled down the inner and outer chamber are flushed
with N2 . This will avoid condensing / freezing of water in the chambers. The outer
chamber should be evacuated with a rotary vane pump. There is a filling funnel
with Armaflex isolated tubing for the liquid N2 inlet. During the filling procedure,
the funnel should be always filled with liquid N2 , in order to prevent any ice from
forming in the inner tube. A plexi glass protective box is placed over the glass
parts of the setup for safety. The cryogenic system was designed to be installed
on a standard Elliot XYZ stage. The vacuum was an efficient thermal insulator.
This cryogenic system could be used to fill a hollow-core fiber or capillary from
end to end with cryogenic liquids like liquid N2 , Ar, etc.
II.1. EXPERIMENTAL IMPLEMENTATION
61
Figure II.11: a) The all-liquid cryogenic system during testing at IMT. The funnel
allows cryogenic liquid to be poured down. (b) N2 gas purging systems to prevent
atmospheric vapour condensation on the windows. (c) Side view of the inner
chamber of the cryotrap with liquid N2 filled inside. The level of liquid N2 in the
inner chamber can be seen and adjusted so as to immerse the fiber in the liquid.
62
CHAPTER II. LIQUID AR FILLED KAGOMÉ PCF
cryotrap
all-liquid cryo-system
Coolant (liquid N2 )
required
not required
Incremental cooling
yes
no
fiber filling method
liquifying gas
liquid immersion
removable ferrule
yes
yes
insulation
insulating material
vacuum
visual on fiber
no
yes
Table II.1: Some key comparisons between the earlier cryotrap and the subsequent
all-liquid cryogenic system.
II.2
Theoretical Analysis
Prior research into liquid Ar mainly depended on liquid Ar being considered a
classical simple fluid and can be analyzed as a Lennard-Jones fluid [72]. Inelastic
Raman [41] and Brillouin [73] scattering has been predicted theoretically and observed in experiments. The Raman shift however is negligible in our experiments
as the pump spectra is broader than the Raman frequency shift in [41]. The previous research that helped in the theoretical analysis of a liquid filled hollow-core
fiber :
• Self-phase modulation in liquid Ar [74]
• Refractive indices of noble gases and liquids [75, 76, 77]
• Thermodynamic studies of liquid Ar [72, 78]
The nonlinear refractive index of liquid Ar was calculated and verified. The Sellmeier relation (chromatic dispersion) of liquid Ar was derived. This relation was
also verified with experimental data points at various wavelengths [75]. The dispersion of Ar gas and liquid Ar in hollow-core fiber was studied.
II.2. THEORETICAL ANALYSIS
II.2.1
63
Calculation of the nonlinear index (n2 ) of liquid Ar
The effective refractive index of a nonlinear medium is given by
n (ω, I) = n0 (ω) + n2 I
(II.1)
where is the linear part of the refractive index and is the nonlinear-index coefficient which depends on the third order susceptibility of the nonlinear medium
(discussed in Chapter I). The nonlinear-coefficient is given by Eq. II.2. γe denotes
the electronic distortion due to intense electric fields and is called hyperpolarizability [74]. For liquid Ar γe = 5.9 × 10−37 esu
� 2
�4
n0 + 2
n2 =
πNγe
81n0
(II.2)
Where N is the number of atoms per unit volume. This can be calculated by
dividing density [72] by the atomic weight of Ar (ZAr ).
N=
ρ
1.23 kg/m3
=
= 1.8608 × 1028
ZAr 39.948 × 1.66 × 10−27 kg
(II.3)
Substituting in Eq. 2, the n2 is obtained as ∼ 5.2 × 10−14 esu/cm3 which is
in good agreement with Alphano’s predicted value of ∼ 6 × 10−14 esu/cm3 [79].
Converting to S.I units we get [80, 81] or
n2 ≈ 1.7 × 10−20 m2 /W
The interesting aspect of this calculation is that the nonlinearity of silica glass,
out of which the fibers are made, is of the same order of magnitude (n2 ≈ 2.7 ×
10−20 m2 /W for fused silica [21]).
II.2.2
Calculation of the Sellmeier relation for liquid Ar
Recently the following method was used to calculate the dispersion relation for
liquid Xe [82]. The same approach was applied for liquid Ar.
The refractive index n of an isotropic dielectric medium, satisfies the disper-
64
CHAPTER II. LIQUID AR FILLED KAGOMÉ PCF
sion relation given by
n2 − 1 4πNe2
fi
=
∑
2
2
n +2
3m i ωi − ω 2 − iΓi ω
(II.4)
N is the number density of atoms or molecules in the medium, e and m are
the charge and mass of an electron respectively, fi is the oscillator strength for the
transition frequency of ωi and Γi is the width of the line corresponding to ωi . The
term ωi2 − ω 2 is large when compared to Γi ω in a wavelength region far away
from the resonance lines. Hence the imaginary contribution of Eq. II.4 vanishes
and becomes real. For low density gases with refractive index close to 1, the above
equation can be simplified to a form which is referred to as the Sellmeier relation
n−1 ≈
2πNe2
fi
∑
2
3m i ωi − ω 2
(II.5)
The values for the Sellmeier coefficients for Ar gas are obtained from [77].
To obtain the Sellmeier relation for liquid Ar, the approximation that the ratio of
liquid Ar density to that of gaseous Ar density at STP. This ratio, given by Nl /Ng ,
takes into account the primary difference between gaseous and liquid state, which
is the Van der Walls interaction between atoms. Nonlinear contributions arising
from density should in principle be taken care of in this ratio. The following
equation which is a good estimate of the Sellmeier of liquid Ar [82].
n2 − 1
2 Nl
= 1.2055 × 10−2
2
n +2
3 Ng
�
�
0.2075
0.0415
4.3330
+
+
91.012 − λ −2 87.892 − λ −2 214.02 − λ −2
(II.6)
When the refractive index n was plotted versus wavelength λ (Fig. II.12).
This was verified with experimental data points of refractive index of liquid Ar at
various wavelengths [75, 83]. The approach used to derive the Sellmeier relation
for liquid Ar was verified by the excellent fit between the data points in [75, 83]
and the plot of n versus λ from Eq. II.6.
II.3. CONCLUSION AND OUTLOOK
65
Figure II.12: Calculated Sellmeier relation for liquid Ar is in excellent agreement
with existing discreet data points [75, 83] .
II.2.3
Dispersion of liquid Ar in hollow-core PCF
Since the Sellmeier dispersion relation of liquid Ar was derived, it could be used
to approximately model a hollow-core fiber filled with liquid Ar using the Marcatili model [64] or Eq. I.31 in Chapter I.2.3. Liquid Ar is also of interest, because
being denser; it has a higher Kerr nonlinearity. In contrast to previous experiments
where the zero-dispersion wavelength (ZDW) was shifted by changing the pressure [65], in the liquid-phase the ZDW can only be tuned by varying the core
diameter. We show that the ZDW can be tuned over more than 600 nm for core
diameters between 20 and 40 µm.
II.3
Conclusion and outlook
In this chapter, a novel method to raise the material nonlinearity of a fiber core by
filling a hollow-core fiber with liquid Ar was discussed. A large number of tech-
66
CHAPTER II. LIQUID AR FILLED KAGOMÉ PCF
Figure II.13: Variation of ZDW with core diameter of a liquid filled hollow-core
fiber.
nical challenges were overcome to perform this complex experiment significantly
enhancing our low temperature capabilities for fiber optical experiments.
Liquid Xe, with a refractive index of around 1.4 (visible wavelengths) [82]
making it close to the fused silica refractive index of 1.45. By sealing of the
cladding holes at the fiber ends, it might be possible to fill only the core with
liquid Xe. If the effective index of the cladding can be reduced to less than the refractive index of Xe, then total internal reflection could act as a possible guidance
mechanism, thereby reducing the losses of the kagomé fiber. Liquid Xe can be
created in the fibers by reducing the temperature of Xe gas to 283 K at a pressure
of about 50-80 bar. The possibility to create liquid Xe at temperatures as high as
~289 K (the critical temperature higher than which Xe is supercritical and cannot
be liquefied) makes the experimental realization much easier than filling the fiber
with liquid Ar. PBG fibers could also be used in the cryotrap after taking account
of the inevitable shift of photonic bandgaps given by the scaling law [59]. Liquid Ar experiments performed here could in principle be performed at with pump
wavelengths around 1500 nm, enabling the possibility of pumping close to the
ZDW of the filled fibers (with a core diameter of around 20 µm). However filling
II.3. CONCLUSION AND OUTLOOK
67
the fiber from end to end uniformly with a cryogenic fluid is the main technical
challenge of the low temperature studies performed here. The all-liquid cryotrap
is a step in the direction to solving this issue. Interestingly, most problems such
as scattering loss due to meniscus formation were avoided once the high pressure
gas-filled fiber was devised which significantly reduced the technical complexity
of filling the fibers with dense noble gases.
68
CHAPTER II. LIQUID AR FILLED KAGOMÉ PCF
Chapter III
High pressure Ar gas filled PCF
Hollow-core photonic crystal fiber (HC-PCF) offer long interaction lengths while
avoiding beam diffraction, thus providing an effective environment for nonlinear
optics in gases [9, 23]. Kagomé-style HC-PCF [9] offers in addition broadband
transmission with moderately low loss, features that are useful for exploring nonlinear effects (such as self-phase modulation), which generate broadband optical
spectra. It also uniquely offers a smooth pressure-tunable variation of dispersion
over a very wide wavelength range, providing a perfect environment for demonstrating many different nonlinear effects, such as efficient generation of tunable
deep UV light via dispersive wave generation [17] and the first observation of a
plasma-related soliton blue-shift [18, 19]. In the first of these cases the zero dispersion wavelength λ0 , was placed closer to the pump wavelength so as to allow
dispersive wave generation in the UV. In the second case it was pushed far into the
UV so as to prevent dispersive-wave perturbations to soliton compression at 800
nm [18] ; the ensuing self-compression created peak intensities as high as 1014
W/cm2 , sufficient to ionize the gas.
An inherent limitation in these systems is the rather low nonlinearity provided by the gas at the pressures used (<10 bar). Since the efficiency of selfcompression decreases significantly with soliton order N [23], it is necessary to
use short (~50 fs) and high energy (a few µJ) pump pulses. When the pressure of
the gas used to fill the kagomé HC-PCF is increased to above 100 bar (the fibers
69
70
CHAPTER III. HIGH PRESSURE AR GAS FILLED PCF
are capable of withstanding inner pressures of at least 1000 bar), the Kerr nonlinearity that approaches that of silica glass [25]. Uniquely, the dispersion remains
low and flat from the UV to the near-IR. These features allow exploration of a
range of different nonlinear regimes at pulse energies of a few 100 nJ, without
changing the fiber or the pump laser. Using noble gases adds an extra twist to
the system by eliminating Raman effects, allowing us to study Kerr-related phenomena in the absence of perturbations such as the soliton self-frequency shift or
Raman-induced noise.
III.1
Supercritical fluids
Supercritical fluids are in the thermodynamic state where phase transition between
liquid and gaseous state cannot be distinctly made. A supercritical fluid is obtained when its pressure and temperature are above the critical point, which is
defined by critical temperature and pressure (Fig. III.1(a)). The critical points are
(48 bar, ~150 K) for Ar, (55 bar, ~209 K) for Kr and (58 bar, ~289 K) for Xe.
Among the noble gases, Xe has the most convenient critical pressure and temperature for experimental realization. Thereby, supercritical Xe could be studied at
room temperature, without requiring a cryogenic system as in Chapter II,[24]. Supercritical fluids have been previously used for molecular scattering studies like
Brillouin [84] and Raman [85] scattering. Numerical predictions of the high nonlinearity of Xe at the critical point have been attributed to increased light scattering
due to critical opalescence [86]. However the experimental study of Kerr-related
optical phenomena in supercritical fluids remains a largely unexplored area.
Fig. III.1(b) shows the density variation of Ar with pressure at various temperatures (shown by isotherms). Below the critical pressure and temperature (critical
point), Ar is in gaseous phase (shown by green isotherms). The distinction of
liquid and gaseous states can be clearly made by the phase transition line. The
nonlinearity of a medium depends on its density. Therefore the variation of gas
density with pressure is important in the reported work.
Fig. III.2(a) shows the room temperature variation of density with pressure
III.1. SUPERCRITICAL FLUIDS
71
Figure III.1: (a) Pressure-temperature phase diagram showing the existence of the
supercritical region above the critical temperature and pressure. (b) The variation
of Ar density with pressure at different temperatures (shown by isotherms). Closer
to the critical temperature, the density varies nonlinearly with pressure, whereas
for isotherms of temperatures much higher than the critical temperature are more
linear. Data obtained from [29] .
for Ar, Kr and Xe. Tabulated density data at ambient temperature (293 K) from
[29] was used. The n2 values for Ar, Kr and Xe are plotted against pressure in
Fig. III.2(b), assuming that n2 is proportional to density. The curves are isotherms
at 293 K [29]. The critical pressures for Ar, Kr and Xe are 48, 55 and 58 bar
respectively. The shape of the isotherms significantly depends on the critical temperatures that are about 150, 209 and 289 for Ar, Kr and Xe respectively. At room
temperature the influence of the critical point is weak for Kr and Ar and therefore
the gas density, and hence n2 , varies more or less linearly with pressure (even in
the supercritical region for Ar), reaching respectively ~5% and ~23% of the value
in fused silica glass at 150 bar [25].
This was confirmed for Ar in the following work [25] where there was excellent agreement between theory and experiment. Under experimental room temperature conditions of 293 K, the density of Xe varies sharply above its critical
pressure of 58 bar as the influence of the critical point is much stronger, leading
to a sharp increase in n2 when the pressure reaches ~60 bar [29]. Experiments
72
CHAPTER III. HIGH PRESSURE AR GAS FILLED PCF
Figure III.2: The variation of density, nonlinearity and dispersion of Xe, Kr and
Ar with pressure at experimental temperature (293 K). Pressure dependence of
(a) density (b) nonlinearity compared to fused silica n2 . The green circles mark
the pressures at which the experiments in chapter IV were performed and the
orange circles mark the experimental pressures discussed in this chapter. (b) zero
dispersion wavelength (ZDW) in a gas-filled kagomé-PCF with a core diameter of
18 µm.
III.2. HIGH PRESSURE CAPABLE EXPERIMENTAL SETUP
73
related to supercritical Xe filled fibers are reported in detail in Chapter IV.
As described in previous papers, the dispersion of the guided mode in kagoméPCF can be described using a capillary model [64, 65]. As the pressure increases,
the normal dispersion of the gas counteracts the weak anomalous waveguide dispersion of the empty kagomé PCF, creating a pressure-tunable zero dispersion
wavelength (ZDW, λ0 ). Fig. III.2(c) shows the variation of λ0 with pressure for
Ar, Kr and Xe. Although Xe clearly extends the range of tunability of λ0 compared to Ar and Kr, critical opalescence makes it unusable in the vicinity of the
supercritical transition. Note that the dispersion can also be tuned by changing the
fiber core diameter, for example, λ0 could be shifted further into the infrared with
larger core diameters [23, 65].
III.2
High pressure capable experimental setup
The experimental setup consisted of a 28 cm length of the kagomé HC-PCF having
a core diameter of 18 µm with gas cells at both ends. The gas cells were made out
of steel and tested with water pressure up to 1000 bar and with gas pressure 300
bar. The cells were about 10 cm long and 7 cm broad. The lids on the gas cells
were initially made of steel, but were then later upgraded to acrylic in order to
get a visual of most of the input and output lengths of the fiber. M10 Allen bolts
were used to secure the lid to the rest of the gas cell. Removable lids were used to
install the fiber in the system. The manufacturing and safety test were conducted
at Gastechnik Geburzi GmbH, Nürnberg. Simple methods were used to secure the
fiber in the v-groove such as scotch tape, magnets or plasticine. The cells were
mounted on XYZ stages to facilitate efficient in-coupling of light into the installed
fiber in the cell.
Care was taken to keep the weight of the gas cells well within the 4 kg load
limit of the XYZ stages used. The gas cells used in the experiment could be used
only up to 150 bar. This limitation was brought about by the thickness of the input
and output glass windows in the gas cells. The glass thickness was limited to 2.3
mm and a gap of a few centimeters was maintained between the in-coupling end
74
CHAPTER III. HIGH PRESSURE AR GAS FILLED PCF
Figure III.3: Experimental setup with the 2 gas cells used for the high pressure
experiments. The fiber is secured in the v-grooves of both gas-cells with scotch
tape and passes through the steel tubing in between the cells.
of the fiber and the window, which kept the focal spot of the beam away from the
window. This minimized the possibility of any nonlinear optical effects from the
window influencing the experimental results from the fiber. UV fused silica windows have a broad wavelength range of uniform optical transmission (from ~150
nm to 2 µm) and strength that make it ideal for use in the high pressure system.
The fiber was enclosed Pin a steel tube joining the two gas cells, which has the
advantage that the pressure is equalized inside and outside the PCF, thus avoiding any possible structural distortion. In case the length of the fiber needed to be
changed, an appropriate length of steel tubing was used between the cells. Needle
valves were used for easily fine-tuning the gas pressure at the inlet and outlet of
the cells. Multiple pressure gauges ensured that any leak could be quickly identified. The system is very robust, and is capable of maintaining a fixed pressure
even for several weeks.
The pump laser used, was an amplified Ti:sapphire laser system oscillating at
a center wavelength of 800 nm, delivering pulses of duration 140 fs at a repetition
rate of 250 kHz. The maximum pulse energy launched into the fiber was 450 nJ.
The diagnostics included a CCD camera and an optical spectrum analyzer, and the
input pulse was characterized using a frequency resolved optical gating (FROG)
system. A 50 litre Ar cylinder filled at 300 bar was used as a gas source.
III.3. GASEOUS AND SUPERCRITICAL AR (25-110 BAR)
III.3
75
Gaseous and supercritical Ar (25-110 bar)
Ar gas was the first gas to be used in our high pressure experiments. The absence
of Raman scattering, linear variation of gas density with pressure, experience with
Ar in previous experiments and its inexpensive cost in comparison with Xe influenced the decision to use Ar gas in the first experiments. The critical temperature
(150 K) of Ar is much lower than the experimental operating conditions at room
temperature. Hence the density of Ar varies linearly with pressure even in the
supercritical regime (pressures above 48 bar, which is the critical pressure for Ar).
The low critical temperature also enabled very smooth tuning of fiber dispersion
and nonlinearity without the influence of critical opalescence (as observed in case
of Xe in Chapter IV). This ensured that previous numerical modeling in [67, 23]
could be used without considering extra scattering perturbations. Excellent agreement between numerical modeling
1
and experimental data was observed. The
dispersion of Ar filled PCF could also be smoothly tuned. In comparison, Xe has
a nonlinear density change with pressure owing to its critical temperature being
close to room temperature. Hence the dispersion tuning was not as smooth as in
Ar. At an Ar gas pressure of 90 bar, λ0 coincides with the pump laser wavelength.
At this pressure the nonlinear refractive index n2 is only one order of magnitude
lower than that of pure silica. By varying the gas pressure we are able to observe soliton fission, supercontinuum generation, dispersive wave emission and
modulational instability (MI) at relatively low pulse energies (~250 nJ).
The modal refractive index of the EH11 mode (azimuthal order 0, radial order
1) in kagomé HC-PCF is accurately approximated by that of a glass capillary and
is given by [64, 65]
n(λ , p, T ) ≈ 1 + δ (λ )
λ 2 u2
ρ
− 2 012
2ρ0 8π a
(III.1)
where λ is the vacuum wavelength, δ (λ ) the Sellmeier expansion for the dielectric susceptibility of the filling gas [6], ρ is the density of the gas at a particular
pressure and temperature, ρ0 is the density of the gas at 293 K and atmospheric
1 The
numerical simulations were done by Wonkeun Chang.
76
CHAPTER III. HIGH PRESSURE AR GAS FILLED PCF
pressure, a the core radius and u01 is the first zero of the Bessel function as discussed in Chapter I. Finite element simulations and numerous experiments have
confirmed the reliability of this expression[64, 65, 67, 23].
25 bar
Fig. 4 shows the experimental and theoretical output spectra with increasing input
power at five different gas pressures. The expected zero dispersion wavelengths
are calculated using Eq. III.1 and Eq. I.31 and are represented by the vertical black
and white dashed lines. Remarkable agreement is found between experiment and
numerical simulations using the unidirectional field propagation equation [67]. At
25 bar (Fig. III.4 (a)), when the pump wavelength is far from λ0 .
The soliton order is ~27 for 450 nJ pulse energy at 25 bar ( γ ≈1.34×10-5W -1 m-1 ).
Under these circumstances the solitons break up and phase-match to higher fre-
quency dispersive waves in the normal dispersion regime. The emission of dispersive waves causes soliton recoil to lower frequencies, thus conserving energy [31].
Although the spectrum analyzer was unable to detect the dispersive waves directly
(the simulations show that they should appear at ~300 nm), the experimental measurements (Fig. III.4 (a) denoted by arrows) show the accompanying soliton recoil
at ~940 nm (~0.32 PHz). The asymmetric extension toward shorter wavelengths
can be explained by the frequency-dependence of γ or self-steepening [5, 40].
50 bar
As the pressure is increased to 50 bar, the nonlinearity increases and λ0 . moves
closer to the pump wavelength. Consequently spectral broadening appears at a
much lower pump power. Numerical simulations show the appearance of multiple solitons at ~1000 nm. In the experiments there is no evidence of any Ramaninduced self-frequency shift with increasing pump power. This is expected as
noble gases do not exhibit Raman-scattering.The generated soliton remains fixed
in a narrow wavelength band (indicated by the arrow on Fig. III.4(b)). The spectral broadening is greater than an octave, because the low and flat dispersion pro-
III.3. GASEOUS AND SUPERCRITICAL AR (25-110 BAR)
77
Figure III.4: Experimental and numerical evolution of the output spectra with
launched pulse energy for five different gas pressures. The black and white dashed
vertical lines indicate the location of λ0 . A and N denote anomalous and normal
dispersion regimes. The pump frequency is kept constant at 0.375 PHz (800 nm).
The arrows show soliton recoil.
78
CHAPTER III. HIGH PRESSURE AR GAS FILLED PCF
Figure III.5: Numerical X-FROG traces at 75 bar. (a) MI bands at 170 nJ launched
pulse energy and (b) asymmetric spectrum at 250 nJ pulse energy when the dispersive wave band overlaps with the high frequency sideband of the MI. The black
and white dashed lines indicate the position of the zero dispersion frequency c/λ0 .
The red curve indicates the mismatch in propagation constant β between the solitons and dispersive waves at a given frequency. The β mismatch is zero at the
white circle for a soliton at the pump frequency, resulting in generation of a dispersive wave in the normal dispersion regime. The red arrow marks the pump
frequency.
file of the gas-filled kagomé HC-PCF (compared to solid-core systems) significantly reduces group-velocity walk-off between different frequency components
and ensures long interaction lengths. This enhances the effectiveness of fourwave mixing as a broadening mechanism, resulting in the generation of a cascade
of sidebands, allowing it to dominate the spectral broadening process. By varying
the gas pressure, a great variety of spectral broadening regimes, governed by processes such as soliton fission and MI, can be accessed without changing the pump
laser or the fiber. Such flexibility is unique to the PCF-based system.
75 bar
On increasing the pressure to 75 bar (Fig. III.4(c)), λ0 moves even closer to the
pump wavelength and two distinct MI sidebands can be seen at pulse energy of
III.3. GASEOUS AND SUPERCRITICAL AR (25-110 BAR)
79
~200 nJ. It is curious that these sidebands appear to be asymmetrically distant
from the pump frequency. This is caused by phase-matching of the solitons to
dispersive waves [5, 87] , which appear only in the normal dispersion regime, creating asymmetry in the observed spectrum. Hence there is an overlap between
MI sideband and dispersive wave in the normal dispersion regime, whilst only MI
sideband is present in the anomalous dispersion regime. A simple phase-matching
analysis between the propagation constants of solitons and linear waves confirms
that the dispersive wave band appears at a higher frequency than the high frequency MI side-band [23]. This is also seen in Figs. III.5(a) & (b), which show
the results of a numerical X-FROG analysis of the signal after 28 cm of propagation for launched energies of 170 and 250 nJ at 75 bar. At the lower pulse energy
(Fig. III.5(a)), where the dispersive wave contribution is small, the MI sidebands
are relatively symmetric in spacing and intensity. As the energy is increased,
however, the broad dispersive wave band overlaps with the high frequency MI
side-band, overwhelming it (Fig. III.5(b)) and producing strong asymmetry between the side-bands. Recently Droques et al. studied the interaction between a
MI side-band and a dispersive wave in a solid core fiber [87]. To avoid Raman
perturbations, however, they were forced to work at low CW power levels – a
limitation that is entirely absent in our PCF-based system. Of course, if required,
a Raman-active gas such as hydrogen can be used if Raman effects are needed,
offering yet another degree of freedom compared to existing systems.
90 bar
At 90 bar and 250 nJ pulse energy, λ0 coincides with the pump wavelength and
the sidebands are symmetric in frequency (Fig. III.4(d)). The X-FROG trace in
Fig. III.6(a) also shows a pair of distinct symmetric MI sidebands at zero delay.
The self-phase modulation (SPM) trace in the X-FROG is distorted at a delay
of ~110 fs, and we think this is due to the effects of higher order dispersion,
which become important close to λ0 Using numerical simulations, we followed
the propagation of the pulse over a longer length (58.5 cm). Both the dispersive
wave band (overlapping with the high frequency MI band) and the soliton regime
80
CHAPTER III. HIGH PRESSURE AR GAS FILLED PCF
Figure III.6: Numerical X-FROG traces for 250 nJ launched pulses at 90 bar. (a)
Symmetric MI spectrum after 28 cm of propagation. Arrow denotes the asymmetric SPM due to higher order dispersion. (b) after 58.5 cm. Dotted lines have same
meaning than for Fig. III.5.
(anomalous dispersion) are clearly seen in Fig. III.6(b).
110 bar
At 110 bar (γ ≈ 5.65 × 10−5W −1 m−1 , Fig. III.4(e)), the dispersion is normal and
spectral broadening due to SPM is observed. Soliton effects would be minimal
as the pump is predominantly in the normal dispersion regime. Being predominantly SPM induced broadening, this regime could potentially be used for pulse
compression with the aid of dispersion compensation.
For all the pressures discussed above, it can be seen that SPM is the dominant process at lower launched energies (< 0.1 µJ). The dispersion tunability (Fig.
III.7(a)) and increase in nonlinearity with pressure can be well illustrated by comparing the SPM induced spectral broadening at these low pulse energies as show
in Fig. III.7(b).
III.4. CORRELATED PHOTON PAIRS
81
Figure III.7: Pressure controlled dispersion and nonlinearity (a) dispersion landscape of the Ar-filled kagomé fiber of 18 µm core diameter for experimental pressures and that of a solid-core silica fibre used for supercontinuum generation with
a ~1 µm core [5]. (b) Sections of experimental results from Fig. III.4 illustrating the increase in nonlinearity by SPM spectral broadening at lower powers for
increasing pressures.
III.4
Correlated photon pairs
Fields such as quantum cryptography would find bright, single-mode sources for
correlated photon pairs. Three-wave-mixing in birefringent crystals lead to low
power wide bandwidth pair sources despite having high nonlinearity [22].
A significant motivation of Raman-free nonlinear optics is to generate correlated photon pairs close to the pump without Raman-related perturbations. By
pumping close to the ZDW in the anomalous dispersion regime, MI sidebands
can be generated. Long pulses are favored in this process. Hence we broaden the
pulse temporally by using a 5 nm bandpass filter centered at 800 nm is used. This
broadens the originally 140 fs pulse to about 210 fs. The pressure is varied from
65 bar to close to 85 bar in the anomalous dispersion regime. At 90 bar the ZDW
would coincide with the pump wavelength at 800 nm. Excellently controllable
sidebands were obtained as shown in Fig. III.8. Changing the pressure meant
changing the dispersion, hence the phase-matching conditions too. The sidebands
82
CHAPTER III. HIGH PRESSURE AR GAS FILLED PCF
shift apart in frequency with increasing pressure. For higher launched average
powers, the appearance of secondary side lobes are apparent which arises due to
cascaded MI.
From the phase-matching analysis like the one shown in Fig. I.7, it is known
that with increasing power, the sidebands shift as seen in Fig. III.9. Here for a
fixed pressure of 65 bar, the power launched into the fiber was increased and the
sideband shift in frequency was seen to be small.
These tunable MI sidebands are expected to be correlated. Ongoing experimental analysis by balanced homodyne detection is in progress to establish correlation between the sidebands [88]. Since the sidebands are close to the pump,
silicon photodiodes are being used in the two photo-detectors with quantum efficiencies suitable for the respective wavelengths of the sidebands. This is convenient compared to photon pair sources which are far separated in frequency from
the pump-thereby requiring silicon based photodetectors and GaAs based detectors for each sideband respectively. Most importantly the absence of Raman gain
coupled with the pressure based wavelength tunability of the sidebands, make the
system presented here a potentially revolutionary photon pair source.
III.5
Conclusion and Outlook
In conclusion, noble gases at high pressure can be introduced into kagomé-style
HC-PCF, providing a unique and highly versatile single-mode fiber system for
exploring gas-based nonlinear optics in the absence of Raman scattering. The
pressure-tunable system allows studies of nonlinear dynamics in different dispersion landscapes and over a wide range of different nonlinearity levels and soliton
orders. A fixed-frequency laser can access regimes of normal and anomalous dispersion merely by tuning the gas pressure. The gas-filled hollow-core allows very
high energies to be launched without optical damage or photo-darkening – serious
problems in fibers with solid glass cores. The system is simple and remarkable
agreement can be reached between experiment and numerical simulations based
on the GNLSE. High nonlinearity and normal dispersion at pressures above 90
III.5. CONCLUSION AND OUTLOOK
83
Figure III.8: Pressure tunable (65 to 85 bar) MI sidebands at the fixed launched
average powers of 50, 60 and 70 mW.
84
CHAPTER III. HIGH PRESSURE AR GAS FILLED PCF
Figure III.9: Power-dependence of the MI sidebands at a fixed pressure of 65 bar.
The shift in sideband frequency is small as expected with increasing power.
bar, together with the absence of Raman scattering, make this system a promising candidate for novel studies of nonlinear optics. It could also be important for
the generation of correlated photon pairs by eliminating the problem of Ramangenerated noise [22]. Other noble gases may also be used. The results pave the
way for a new series of experiments on ultrafast nonlinear dynamics in highly
nonlinear Raman-free systems. Hollow-core kagomé-style PCF, filled with gases
at very high pressure, allows us for the first time to transform gases into "honorary solid state materials", with the added advantages of tunable dispersion and
extremely high optical damage resistance.
The remarkable flexibility offered by a gas-filled HC-PCF system as a vehicle
for studies in nonlinear optics is unrivaled. Merely regulating the gas pressure in
the fiber can control dispersion. This means that this system can switch seamlessly
between a normally dispersive to an anomalously dispersive regime – on a given
piece or length of kagomé HC-PCF. A desired dispersion regime can be chosen
III.5. CONCLUSION AND OUTLOOK
85
irrespective of the laser wavelength range in the UV-IR. The system is robust as
the same length/piece of fiber has been used for experiments over several weeks.
This large range of dispersion tunability is accompanied by scalable nonlinearity
by virtue of gas density.
Raman scattering introduces significant perturbations to nonlinear optics. This
system can either “turn on” or “turn off” the Raman effect by the choice of gas
used to fill the fiber. In the work reported here noble gases were used in order to avoid Raman scattering. However the use of Raman active gases at high
pressures like H2 for example can lead to interesting studies since the Raman
gain would proportionally increase with density and might result in enhanced frequency combs [13, 14]. An experimental setup for this experiment is being setup
at the time of writing this thesis. High pressure fiber filling systems might also
lead to tunable transmission windows in PBG fibers, where the shift is given by
the scaling law [59].
Another possible post-processing application of the high pressure system can
be the adjusting the thin glass structures of a kagomé fiber lattice by selectively
filling holes. This could help to move the glass struts by a few nanometers and
might help in better understanding of kagomé fiber guidance mechanisms or reducing losses. Steep pressure gradient (up to 150 bar of pressure difference) experiments can also be performed as this would lead to varying nonlinearity and
dispersion along the length of the fiber. Pressure gradients are expected to enhance UV generation via dispersive wave generation [89]. The needle valves used
in these setups enable gentle flows or gradients.
The work reported here has overcome most of the technical hurdles for a high
pressure system. Efforts in improving and enhancing the pressure capability of
these systems are continuing. The simplicity, stability and remarkable flexibility
offered by the high pressure fiber system reported here make it an unparalleled
nonlinear optical system.
86
CHAPTER III. HIGH PRESSURE AR GAS FILLED PCF
Chapter IV
Supercritical Xe filled PCF
The critical points are (48 bar, ~150 K) for Ar, (55 bar, ~209 K) for Kr and
(58 bar, ~289 K) for Xe [29, (NIST)]. At room temperature the influence of the
critical point is weak for Kr and Ar and therefore the gas density, and hence n2 ,
varies more or less linearly with pressure, reaching respectively ~5% and ~23%
of the value in fused silica glass at 150 bar [25]. For Xe, the influence of the
critical point is much stronger, leading to a sharp increase in n2 when the pressure reaches ~60 bar [29, (NIST)] ; recent linear measurements have confirmed
this [90]. When the pressure and temperature lie above the critical point – easily achievable in experiment without the need for a cryogenic system [24] – Xe
becomes a supercritical fluid, (Table. IV.1) with a nonlinearity that can exceed
that of fused silica (Fig. IV.1). Theory predicts that Xe will exhibit a temporally
non-local (response times in the µsec range) nonlinearity close to the critical point
[86] , due to intense scattering arising from critical opalescence (see Fig. IV.9).
This regime is avoided by operating sufficiently above or below the critical point,
where Xe remains transparent.
IV.1
Experimental setup modification
The experimental setup for the supercritical Xe was similar to the one used
in Chapter III but with some important modifications. Commercially available
Xe cylinders are filled usually up to around 40 bar. Supercritical Xe however is
87
88
CHAPTER IV. SUPERCRITICAL XE FILLED PCF
Figure IV.1: (a) Density and (b) nonlinear refractive index n2 variation of Ar, Kr
and Xe with pressure.
formed above a pressure of 58 bar at 293K. Hence using just a Xe cylinder as a gas
source (like in the Ar experiments), was not an option for Xe experiments above
40 bar (below the supercritical pressure of 58 bar). This problem was overcome
by the following simple technique.
A length of steel piping was made into a spiral and placed in a Styrofoam box
containing solid CO2 (dry ice). The spiral piping system had gas regulators at
both ends. The input regulator (shown in green in Fig. IV.2) was connected to the
source cylinder. Xe at 40 bar is introduced into the pipe and the input regulator
closed. Dry ice around the spiral pipe helps liquefy the Xe, thereby reducing its
volume. The input regulator is again opened to introduce more Xe into the pipe
and closed.
This process is repeated about 3-4 times until a sufficient amount of liquid Xe
Xe
Kr
Ar
Critical T (K)
289
209
150
Critical P (bar)
58
55
48
Table IV.1: Critical pressures and temperatures for Xe, Kr and Ar. Xe has to
closest critical temperature to 293 K (ambient/experimental temperature).
IV.1. EXPERIMENTAL SETUP MODIFICATION
89
Figure IV.2: (a) Schematic of the experimental setup (b) Liquefying Xe in the
steel pipes at 40 bar and then warming it up, sequentially raises the pressure to
well above 58 bar, thereby supercritical Xe is formed. The red dotted circle marks
the input valve (discussed below).
is collected in the pipes. The pipe is then removed from the box of dry ice and
allowed to warm up. The warming up process can also he quickened by using a
heat gun. Once the liquid Xe is warmed up, supercritical Xe is formed in the pipes
as the pressure exceeds 58 bar. Pressures of up to 200 bar have been achieved by
this method. When the intended pressure is achieved, then the input valve for
the cells (marked by a red dotted circle in Fig. IV.2) is opened to introduce the
supercritical Xe into the gas cells and fiber. This simple technique avoided the
need for costly compressors and thereby maintaining the high purity of Xe in the
experiments.
Residue from valves
The input valve as shown in Fig. IV.3 and marked by a red dotted circle in Fig.
IV.2, was used in the supercritical Xe experiments. The valves were lubricated
during manufacturing (Fitok Inc.) so as to make the valve operation smooth. The
manufacturer catalog mentions nickel anti-seize with hydrocarbon carrier as the
lubricant used in its MH series valves like the one above. However supercritical
Xe, like many other supercritical fluids is an excellent solvent owing to the absence of surface tension [91]. This created the problem of the supercritical fluid,
displacing the lubricant from the inside of the valves (especially parts 2 and 3 in
90
CHAPTER IV. SUPERCRITICAL XE FILLED PCF
Figure IV.3: Schematic of the valves used in the experiment, provided by their
manufacturer Fitok Inc. Parts on the schematic and table marked in red are in
contact with a lubricant.
Fig. IV.3), thereby contaminating the pure Xe. When the pressure of the system
was decreased to subcritical (<58 bar) pressures, this lubricant was deposited as
residue on the inner walls of the cell and on the fiber. The residue on the input
fiber end made the fiber unusable. Interestingly there was much lesser residue in
the output gas cell, which had the outlet valve.
The difference in the amount of deposit on the input and output fiber ends is
shown in Fig. IV.4 (a) & (b) respectively. This difference is due to the outlet
valve being on the gas cell at the output of the fiber. Hence when the pressure
was reduced, the gas in the input gas cell was sucked in through the fiber and the
enclosing pipe. This resulted in more deposition on the input fiber end, thereby
rendering the fiber unusable for optical experiments. The deposits formed in layers according to their weight and size (Fig. IV.4 (d)). By disassembling the valve
into its components and then cleaning them thoroughly in an ultrasonic cleaner,
the problem of the deposit was solved. This simple solution prevented the cells
from getting contaminated thereby allowing the experiment to be performed.
IV.2. EXPERIMENTAL RESULTS
91
Figure IV.4: Microscopic images showing (a) Deposit on input fiber end (b) output
fiber end (c) side view of input fiber end (d) deposit on the input gas cell window.
IV.2
Experimental results
The experimental set-up consisted of a 28 cm length of kagomé-PCF (core
diameter 18 µm) with a high pressure gas cell at each end. The pump laser was an
amplified Ti:sapphire system (wavelength 800 nm) delivering pulses of duration
150 fs and energy ~1.8 µJ at a repetition rate of 250 kHz. Diagnostics included a
UV-sensitive camera for modal imaging and a spectrometer sensitive from 200 to
1100 nm. To prevent the spectrometer from saturating, the signal was attenuated
by reflection at two wedged glass plates. A parabolic mirror was then used to
focus light into the spectrometer. Supercritical Xe was collected by liquefying Xe
in steel pipes cooled by dry ice. After a sufficient amount of Xe had collected, the
pipes were warmed up to room temperature. This simple procedure allowed us to
reach Xe pressures of 200 bar from a 40 bar gas cylinder, while maintaining high
Xe purity.
IV.2.1
Supercritical Xe 80 bar
We filled the fiber with 80 bar Xe, well inside the supercritical regime at 293
92
CHAPTER IV. SUPERCRITICAL XE FILLED PCF
Figure IV.5: Experimental output spectral broadening due to SPM in supercritical
Xe at 80 bar for launched pulse energies 15, 30 and 60 nJ.
K. Operating the experiment between ~60 to ~70 bar was avoided due to sharp
density changes across this pressure range. 80 bar was chosen as a convenient
working pressure. At 80 bar the nonlinear refractive index is ~2.8×10-20 m2 /W
(calculated by multiplying n2 at 1 bar with the ratio of Xe densities at 80 bar to 1
bar) [28, 92] , which matches the value for fused silica [21].
Clear SPM was observed experimentally (Fig. IV.5), with distinct spectral
broadening on increasing the launched pulse energy in the fiber from about 15 nJ
to 60 nJ (~15% transmission of the fiber filled with supercritical Xe was taken into
account). This continued up to ~80 nJ, when the broadening abruptly collapsed, a
dramatic effect that we attribute to disruption of the in-coupling by self-focusing
effects in the input gas-cell. To verify this, we performed experiments in a simple
gas cell, obtaining reasonable agreement with full spatio-temporal numerical simulations1 using the methods described in [93] , and simple numerical estimates of
nonlinear focusing described in [94].
Among the noble gases (with the exception of radon), Xe has the highest nonlinearity for a given pressure but also the lowest ionization threshold. Hence for
higher powers free electron densities can be expected (>1015 for 1 µJ) [71]. To
1 The
numerical simulations were done by John Travers.
IV.2. EXPERIMENTAL RESULTS
93
Figure IV.6: (a) Variation of output beam radius with incident beam energy propagating in a bulk cell (without a fiber). Inset: experimental setup. Beam profiles
(i) at low energy (ii) Kerr focusing dominating the plasma defocusing (iii) ring
formation when the plasma defocusing dominates the Kerr focusing effects.
analyze the possible interplay of Kerr and plasma effects at higher pressures, we
devised a simple experiment in the absence of the fiber, in a cell filled with bulk
supercritical Xe at 90 bar. The intention of this experiment was to image the spatial profile near the beam waist. The cell was kept between two lenses of the same
focal length separated by the twice the distance of their focal length. A CCD was
used to image the beam profile. Fig. IV.6 shows the output beam radius varying
with increasing incident energy.
The plasma and Kerr influences cause defocusing and focusing of the beam respectively [47, 48]. At low powers, the plasma has negligible influence compared
Kerr effect. At 1.2 µJ and higher energies, the Kerr dominates the plasma effects,
resulting in effective focusing of the beam at its center also seen in the experiment
94
CHAPTER IV. SUPERCRITICAL XE FILLED PCF
in Fig. IV.6(ii)). For powers higher than 2 µJ, there are sufficient free electrons
for the plasma to dominate the Kerr effect resulting in effective beam defocusing in the beam center, thereby forming a ring profile. Similar ring formations
in the spatial beam profile have been investigated in [48]. A sharp threshold for
supercontinuum generated was also observed, which is consistent with a similar
experiment done at lower pressures in [95]. The analysis in [95] however did not
consider the effect of plasma formation. The numerical analysis was done using
full spatio-temporal numerical simulations using the methods described in [93]
, and simple numerical estimates of nonlinear focusing described in [94]. Other
methods were also used to analyze the self-focusing effects on fiber coupling.
Kerr lens formulation over several consecutive “slices” of bulk Xe gas [96, 34] (in
the length from the window up to the fiber input end) at 30 bar was carried out2 .
Another possible explanation could be the change of focal plane of a pre-focused
beam in a Kerr medium as discussed by [97]. All these approaches lead us to
believe that the optical powers used in the experiment were near the self-focusing
regime. Hence we can conclude that the self-focusing effects and/or competing
Kerr and plasma influences would have an effect of the beam profile at higher
pressures. This could be a possible hindrance to study high pulse energy effect in
the fiber, as the fiber in coupling would get affected by these spatial beam modifications. By reducing the Xe density between the fiber and cell window (possibly
fixing the window to the fiber in coupling end), this shortcoming might be overcome in the future. Another solution might be to use a pressure gradient [20] with
lower pressures at the input end. Noble gas-filled capillaries [98, 99, 100] and
photonic bandgap HC-PCFs [60] have been used for pulse compression where the
nonlinearity of the gas was used to broaden the spectrum of the launched pulse.
Xe-filled kagomé-lattice [20] has also been used for pulse compression at low
pressures. Corkum et al. observed self-focusing induced supercontinuum generation in bulk gaseous Xe [95].
It is important to note that the n2 of supercritical Xe at 80 bar (∼ 2.8 ×
10−20 m2 /W ) also exceeds the nonlinearity of liquid Ar (∼ 1.7 × 10−20 m2 /W ),
2 The
calculations were done by Nicolas Joly
IV.2. EXPERIMENTAL RESULTS
95
which we used in a hollow-core previously [24] and Chapter II. Almost all drawbacks of the liquid Ar system reported earlier in Chapter II such as thermodynamic phase transitions, lack of tunability of dispersion and technical complexity
of working at cryogenic temperatures were overcome in the system reported here.
IV.2.2
Sub-critical Xe 25-35 bar
The disruptive effects of self-focusing are also apparent in the sub-critical
regime (Fig. IV.7(a)), where spectral broadening is abruptly attenuated above a
certain critical launched energy (marked by the red arrows in Fig. IV.7 (a)) that
depends inversely on the pressure. A 70% drop in transmitted power accompanies this spectral collapse and its threshold energy clearly drops with increasing
pressure and nonlinearity (Fig. IV.7(a)). At 25 bar, λ0 ~ 890 nm and the pump
wavelength lies in the normal dispersion regime. In addition to SPM-induced
spectral broadening, an unexpected band of UV light appears at ~330 nm. Using a
narrow-band filter to isolate the near-field pattern at this wavelength (Fig. IV.7(c)),
we were able to identify this signal as being in the HE12 mode. We attribute its appearance to intermodal four-wave mixing (iFWM). Fig. IV.7(b) shows the results
of a phase-matching analysis, based on the Marcatili model [64, 65], assuming
that pump and idler are in the HE11 mode and signal in the HE12 mode. As the
spectrum broadens, it reaches beyond 1 µm wavelength and is then able to act as
an HE11 idler seed for iFWM, pumped by the green spectral edge at ~550 nm.
These two signals result in the generation, via iFWM, of signal photons in the
HE12 mode at ~375 nm. The analytical theory predicts wavelengths that are in
good agreement with the observations; the slight disagreement can be attributed
to deviations of the actual fiber dispersion curve from that predicted by the Marcatili model. The weak iFWM signal at ~375 nm and 30 bar, visible in the middle
panel of Fig. IV.7(a), was also experimentally confirmed to be in the HE12 mode.
Pulse propagation numerical simulations were also performed taking into account higher order modes. The simulations were run for HE11 and HE12 modes
separately. It was clearly seen that a distinct UV band was visible in the HE12
mode. The spectral evolution of the two modes were then overlapped to give
the below image. The slight disagreement between experiment and simulations
96
CHAPTER IV. SUPERCRITICAL XE FILLED PCF
Figure IV.7: (a) Experimental spectral broadening with launched pulse energy at
25 bar, 30 bar and 35 bar; the dashed vertical lines indicate the position of λ0 (N =
normal, A = anomalous); the red arrows indicate the onset of self-focusing in the
input cell. (b) Theoretical phase-matching wavelengths (2/λ p = 1/λ s + 1/λ I ) for
iFWM at 25 bar; for a ~550 nm pump, the signal and idler wavelengths are ~325
and ~1037 nm. (c) Experimental near-field image of the light emitted in the HE12
mode at 325 nm (indicated by the black arrow in (a)).
IV.3. ADVANTAGES OVER CRYOTRAP
97
Figure IV.8: Higher order modal pulse propagation numeric confirms the appearance of the UV in the HE12 mode (dark purple arrow). The light purple arrow
marks the experimentally observed UV band.
(shown in Fig. IV.8 by the light and dark purple arrows respectively) can be attributed to deviations of the actual fiber dispersion curve from that predicted by
the Marcatili model3 .
IV.3
Advantages over cryotrap
It is also important to note that the n2 of supercritical Xe also matches nonlinearity of liquid Ar, which we used in a hollow-core previously [24]. Almost all
drawbacks of the liquid Ar system (cryotrap) reported earlier in Chapter II such
as thermodynamic phase transitions, lack of tunability of dispersion and technical
complexity of working at cryogenic temperatures were overcome in the system reported here. Moreover this system is much easier to operate and more stable than
the cryogenic system in Chapter II. The length of fiber is also variable unlike the
cryotrap and the conditions experienced by the fiber are constant throughout the
fiber length. Supercritical fluids do not exhibit surface tension [101] , hence the
occurrence of a meniscus in the fiber holes can be ruled out. Care must be taken to
avoid operation near the critical point of Xe, as supercritical fluids are opaque owing to density fluctuations in this thermodynamic regime [86]. The phenomenon
3 The
numerical simulations were developed by Francesco Tani and John Travers.
98
CHAPTER IV. SUPERCRITICAL XE FILLED PCF
Figure IV.9: A sequence of images of the gas cell taken as the pressure was being
reduced from ~72 bar (supercritical Xe) to ~48 bar (gaseous Xe). The middle
panel shows critical opalescence at the critical pressure (~58 bar). Xe is transparent when the temperature and pressure are not close to the critical pressure at 293
K.
is called critical opalescence and it was observed around the critical pressure of
Xe (58 bar) at ambient experimental temperature. Hence, all factors considered,
the supercritical Xe system is an excellent system to scale up the material nonlinearity of a gas-filled HC-PCF in comparison to previous attempts with liquid Ar
in the cryotrap.
IV.4
Conclusion and Outlook
In conclusion, clear SPM broadening was observed in a kagomé PCF filled
with supercritical Xe, which at 80 bar has the same Kerr nonlinearity as silica.
As a result of self-focusing effects in the launching cell, the spectral broadening
was observed to collapse abruptly at a critical energy level that scaled inversely
with the gas pressure; this effect could be eliminated by placing the glass window
closer to the fiber end-face. Intermodal four-wave mixing was observed to occur
at the point of spectral collapse, resulting in generation of UV light in the HE12
mode. Compared to all-silica fiber systems, noble-gas-filled HC-PCF offers a
much higher damage threshold, excellent transparency at ultraviolet wavelengths,
pressure-tunable dispersion and Raman-free operation.
Filling supercritical fluids in HC-PCFs has a potential to significantly aid pho-
IV.4. CONCLUSION AND OUTLOOK
99
tonics research. Supercritical fluids can have densities almost as high as liquids,
yet do not exhibit surface tension – hence meniscus formation in fibers cannot
occur. These fluids can be controlled with standard gas valves (without lubricants) and their density can be controlled over a much larger range than liquids –
thereby enabling a larger range of control of dispersion and material nonlinearity
in the fiber. The lack of surface tension gives supercritical fluids the ability to
occupy inter-molecular spaces, which makes it an excellent solvent. Supercritical
CO2 for example is widely used in industry as a solvent. The solvent ability of
supercritical Xe was seen earlier, where the lubricant of the needle valves in the
setup was dissolved and deposited in the gas cell and on the fiber. This solvent
ability can be put to interesting applications like in [102] where the Rhodamine
was dissolved in supercritical CO2 with the help of a cosolvent like methanol. In
principle this can be used to coat the inner walls of a fiber with the appropriate
solute. For this the supercritical fluid would need to dissolve the required solute
using a cosolvent if required as CO2 and Xe can only dissolve non-polar solutes,
as they are non-polar molecules. The solution can then be filled into the fiber
and the pressure reduced to below the critical pressure. The solute could then be
deposited on the inner walls.
Towards the final stages of experimental work, supercritical CO2 was also obtained paving the way for further interesting experiments such as Raman-scattering
studies [85] or solvent related experiments. Supercritical CO2 was created merely
by placing pieces of dry ice in a specially designed gas cell and then sealed. After a few minutes at room temperature, supercritical CO2 would be formed from
the dry ice. It was also found that if left for a few weeks, the supercritical CO2
would swell the acrylic windows of the gas cell – as it is a well-known swelling
agent [91]. This has been used in [102] to enable Rhodamine to seep into polymer
fibers.
Using appropriate dispersion compensation techniques, the large spectral broadening due to SPM can be used for pulse compression [20].
100
CHAPTER IV. SUPERCRITICAL XE FILLED PCF
Chapter V
Conclusions
The technical know-how developed in this thesis has extended the capabilities of
a gas-filled HC-PCF significantly. The use of dense noble gases avoids Ramanscattering related perturbations and simultaneously scales up the material nonlinearity of the core of gas-filled HC-PCF to a value comparable to that of fused
silica. Interestingly a blend of seemingly disparate fields of photonics and thermodynamics has led to the interesting results reported here.
In Chapter II, novel cryogenic systems were used to condense liquid Ar in
sections of kagomé HC-PCF. The system presented a unique opportunity to study
nonlinear fiber optics in a liquid media sandwiched between lengths of gas in
the fiber core. SPM was observed in a kagomé HC-PCF filled partly with liquid
Ar. Systems to fill a cryogenic liquid throughout the length of the fiber were
also developed. Future directions in this study can make use of the possibility of
obtaining liquid Xe at ~280 K by condensing using high pressure Xe gas.
Almost all technical challenges faced in Chapter II, such as scattering losses
at gas-liquid interfaces and overall complexity of the system, were solved with
the high-pressure gas fiber systems (Chapter III). Nonlinear effects such as soliton fission, modulational instability and dispersive wave generation in the absence
of Raman-scattering have been observed experimentally. Excellent agreement between experimental observations and numerical simulations were obtained when
the Raman-related perturbations were ignored. Moreover a wide range of tunabil101
102
CHAPTER V. CONCLUSIONS
ity of ZDW from the UV to the IR was achieved – allowing the access of various
dispersion regimes, using the same length of fiber and for a given laser frequency.
The nonlinearity also was proportionally scalable with gas pressure.
Chapter IV relied on a modified high pressure gas fiber system. By accessing
the supercritical properties of Xe at room temperature, a sharp density change
around the critical pressure of 58 bar helped to raise the the material nonlinearity
to match (and exceed) that of fused silica. Self-focusing effects where found to
affect fiber in-coupling. More studies need to be carried out to understand this
phenomenon rigorously. Intermodal FWM was observed in subcritical Xe and
explained using multimodal phase-matching analysis.
The successful impact of the versatile high pressure gas fiber system is already
generating new ideas. In just a few months after the first prototype was developed,
other similar experimental setups have already been installed. For example, to
explore the possibility of enhanced Raman-gain at higher pressures of Ramanactive gases [14]. Plans to implement the high pressure gas fiber systems for fiber
ring cavities [103] are underway.
Supercritical fluids can be exploited for the solvent abilities and used to coat
the inside of fibers with an appropriate solute when the pressure is decreased to
subcritical pressures. A potential for this functionality was seen in IV.1. Polymer
fibers could use the supercritical fluid as a swelling agent for post processing
[102].
The potential for further experiments with high pressure gas-filled fibers is
indeed huge. By raising the nonlinearity, compact fiber-lasers could be used as
pump sources rather than expensive amplified-oscillator lasers. Low loss, smaller
core diameter hollow-core fibers filled with high pressure gases may prove to be
competitors to solid core fibers, with considerably enhanced versatility.
Appendix A
Appendix
A.1
List of Publications
1. M. Azhar, N. Y. Joly, J. C. Travers, and P. St. J. Russell. Nonlinear optics
in xenon-filled hollow-core PCF in high pressure and supercritical regimes.
Applied Physics B, DOI:10.1007/s00340-013-5526-y:1–4, 2013
2. M. Azhar, G. K. L. Wong, W. Chang, N. Y. Joly, and P. St. J. Russell.
Raman-free nonlinear optical effects in high pressure gas-filled hollow-core
PCF. Optics Express, 21(4):4405–4410, February 2013.
3. M. Azhar, G. Wong, W. Chang, N. Joly, and P. St. J. Russell. Nonlinear optics in hollow-core photonic crystal fiber filled with liquid argon. In CLEO:
Science and Innovations, OSA Technical Digest (online), page CTh4B.4.
Optical Society of America, May 2012.
Conferences (oral presentation)
1. Conference on Lasers and Electro-Optics (CLEO)-Europe, Munich, Germany (2013).
2. Conference on Lasers and Electro-Optics (CLEO), San Jose, USA (2012).
3. Photonics 2010, Guwahati, India (2010).
103
104
APPENDIX A. APPENDIX
Workshops (poster presentation)
1. International workshop on light-matter interaction, Porquerolles, France (2012).
2. Nonlinear optics and complexity in PCF and nanostructures, Erice, Italy
(2011).
Awards
Optical Society of America Best Student paper award at Photonics 2010, Guwahati, India.
A.2. INSTRUCTIONS TO OPERATE THE CRYOTRAP
A.2
105
Instructions to operate the cryotrap
Starting the cryotrap
1. Check if the liquid N2 cylinder is filled
2. Install the cryo-valve with the insulated tube to the liquid N2 cylinder liquid
outlet.
3. The insulated tube is then connected to the liquid N2 inlet of the cryotrap.
The tube must be clean and dry-otherwise frozen particles might affect the
temperature controlling system. Preferably purge with gaseous N2 .
4. Attach the silicone tube to the output of the cryotrap.
5. Connect the cryo-valve plug and the PT100 temperature sensor plug to the
temperature controller.
6. Connect the temperature controller unit to 230 V ac power supply and turn
on.
7. Open the valve on the liquid N2 cylinder to let the liquid N2 flow into the
cryotrap when the cryo-valve is active (the red LED on the cryo-valve wire
would turn and is accompanied by a characteristic sound of the valve opening).
8. The temperature controller stores the last parameter setting and also the last
set temperature. This last set temperature is shown on the display (green),
the actual temperature is shown on the display in red numbers. The temperature can be changed in the following way: press key “P”, change the
temperature with the arrow up, arrow down key and confirm the temperature by pressing the key “P” again. The newly set temperature is displayed
in green numbers.
106
APPENDIX A. APPENDIX
Caution
1. Do not disturb the tube between the liquid N2 cylinder and the cryotrap
when the system is cooled down. The tube becomes brittle when cold, and
might break if disturbed.
2. Check that the PT100 sensor is well fitted in the cryotrap. This is vital for
accurate functioning of the temperature controller.
3. The output pipe for the gaseous N2 must not have a blockage.
4. The exhaust N2 must be safely released to the outside environment to prevent oxygen depletion within the confines of the lab.
Switching off the system
1. Close the valve on the liquid N2 cylinder to shut off the supply of liquid N2 .
2. Switch off the temperature controller.
3. Wait until the entire system is warmed up to disassemble the setup if needed.
A.3. ACKNOWLEDGMENTS
A.3
107
Acknowledgments
A few years back Prof. Russell gave me the opportunity to work for one of the
best fiber optics research groups in the world. It is an opportunity for which I shall
forever be indebted to him. I admire the sheer “audacity” of some of his ideas and
his guidance in helping to achieve them. I am most grateful to him for a wonderful
experience, from giving me a challenging project to standing by me during some
tough times – eventually helping me achieve and even exceed the goals we set off
to accomplish. Thank you Philip.
Prof. Nicolas Joly, merci beaucoup. His suggestions and ideas have had a
major impact on this thesis. His patience and support during the past few years
meant a lot to me – thanks Nick. Many years from now, I can imagine him cheerfully narrating the story of how his new Indian student saw his first snowfall while
in his office. And don’t challenge him to a badminton match – he will beat you!
Gordon Wong too helped me greatly during the course of my PhD. It was a pleasure having the many scientific discussion we had. I am also grateful to Wonkeun
Chang and John Travers for their ideas and numerical simulations. John and me
tried our best to educate the Germans about the nuances of cricket (und dessen
Unterschiede zu Crockett).
Philipp Hölzer, Amir Abdolvand, KaFai Mak, Michael Schmidberger, Francesco
Tani, Barbara Trabold, Martin Finger and Federico Belli provided an excellent
environment in the lab, though their ideas, encouragement, lab music and the occasional stray UV beam. Thank you Martin Butryn for helping me out with the
lasers. My office colleagues, Sarah, Thang, Nicolai and Patrick made great company. It is a joy to see the whole office graduating in the same year. I would
like to thank Xiaoming Xi, Stan, Ana Cubillias, Anna Butsch, Tijmen Euser,
Oliver Schmidt, Ralf Keding, Basti, Stan, Fehim, Jimmy, Alexey, Johannes, David
Novoa, Alessio Stefani and Michael Frosz. Among the former members I would
like to thank Martin Garbos, Marta Ziemienczuk, Andreas Walser, Silke Ramler,
Howard Lee, Johannes Nold and Sebastian Stark. Colleagues at the Russell Division came from all parts of the world and it was an honour to work with such a
fine group of scientists.
108
APPENDIX A. APPENDIX
I have to thank Heike Schwender for her immense help in my first few weeks in
Germany and for being the first of my many friends in Germany. Bettina Schwender is a great help as she is so efficient with all the administrative aspects. A
big dhanyavaad to my compatriots at the MPL Hemant Tyagi, Nitin Jain, Bharat
Navalpakkam and Samudra Roy for the great times.
I am also most grateful to Erwin Strigl from IMT Moosbach and Klaus Geburtzi
from Geburtzi Gastechnik Nürnberg for their invaluable help in manufacturing the
cryo-trap and high pressure systems. I have met wonderful people from several
countries during my stay in Germany. It has been an unforgettable last few years,
rich with fascinating experiences. Thank you to all the friends who made my stay
in Germany memorable and absolutely wunderbar!
Several important people have helped me throughout my career. Prof. Srinivasan, Hema Ramachandran and Andal Narayan gave me an excellent start in the
field of experimental optics at Raman Research Institute, Bangalore, India. My
FIMSc classmates who were an excellent peer group and are now working in labs
around the world.
It is not easy when your family is spread over three continents. I am grateful
to Mummy, Abba and Mothi for being there for me all these years. This thesis is
dedicated to them.
List of Tables
I.1
Nonlinear refractive index n2 of material media relevant to this
thesis. [21, 28, 29, 23] . . . . . . . . . . . . . . . . . . . . . . . 13
I.2
Dispersion and nonlinear contribution in various terms of the GNLSE 16
I.3
Comparison between PBG and kagomé HC-PCF. Dispersion and
loss plots taken from [23]. . . . . . . . . . . . . . . . . . . . . . 37
I.4
A comparison of solid core Vs kagomé HC-PCF. The shaded rows
mark out the advantages the noble gas-filled kagomé HC-PCF systems have over fused silica core PCFs. The above values are for
an overview and can vary with factors such as core size, pitch, etc.
* [68, 23] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
II.1 Some key comparisons between the earlier cryotrap and the subsequent all-liquid cryogenic system. . . . . . . . . . . . . . . . . 62
IV.1 Critical pressures and temperatures for Xe, Kr and Ar. Xe has to
closest critical temperature to 293 K (ambient/experimental temperature). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
109
110
LIST OF TABLES
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