Institute of Radio Electronics and Telecommunications
Department of Radiophotonics and Microwave Technologies
Photonics and Optoinformatics
MASTER THESIS:
Obtain the fourwave interaction in a wide spectral range
Kebour Mohamed
Introduction
Chapter 1
1 Basics of light propagation in optical fibers
11 Conventional optical fibers
12 Manufacture of Optical Fibres
13 Light Guidance Mechanism in
Conventional Fibre
14 Birefringence
1.5 Loss
Chapter 2
2 Microstructure fibers and nonlinear optics
21 Definitions and basic concepts
22 Microstructured fibers
2.3 Sources of photons
2.4 Properties of photonic crystal
fibers
25 Nonlinear optics
26
Photon detectors
2.7 Quality of a photon source
2.8 Optical fibers
2.9 Nonlinear Susceptibility and
Nonlinear Phase Shift
210 Nonlinear propagation equation
Chapter 3
3 Fourwave mixing
32 PhaseMatching Conditions for FWM
33 Gain and Bandwidth of FWM
34 CrossPolarised FWM
35 Calculation of phase agreements of
the degenerate fourwave mixting
36 The Hamiltonian
37 Raman scattering
Chapter 4
41 Quantummechanical description of FWM
42 Pure Single Photon States
43 HongOuMandel
44 Generation of Single Photon States
45 Attenuated Coherent Light
46 Photon Pair Generation by FWM
47 Spectral Filtering
48 Photon Statistics for FWM
Chapter 5
51 Experimental
setup
51 Spectrometer Measurement
52 Monochromator Measurement
Conclusion General
Symbol
list
Abbrviations
List
Figure References
References
Introduction
Quantum optics is an expanding area of
research, it’s interested in the application of fundamental concepts from the
formalism of quantum mechanics. It basically involves the production and
detection of quantum states implemented on one or more degrees of photon
freedom. Quantum optics is found in metrology, cryptography and quantum
computing. For a laboratory wishing to orient itself towards research on
quantum optics, it is of primary importance to develop knowhow on the creation
and manipulation of nonconventional photon sources.
We will present here the approach undertaken
in order to assemble, from a preliminary plan, a source of announced photons
generated by a microstructured fiber. The project consists not only in
designing and manufacturing the source but also in carrying out tests to
validate the quantum nature of this source. Before presenting the approach
undertaken and the results obtained, it is advisable to clearly define the
nature of the project and to identify the problem.
The study and control of the properties of
light has long been a major goal in science. Early experiments into classical
interferometry, as well as studies of the polarisation properties of light,
provided seemingly incontrovertible evidence that light is a type of wave.
However, the subsequent discovery of the photoelectric effect and the proposed
quantization of the electromagnetic field in order to explain the observed the spectral profile of blackbody radiation, both implied that at a fundamental
level a light field is composed of discrete, particlelike packets of energy,
known as photons. Along with further developments in the theory of quantum
mechanics, the concept of photons proved to be of great use in understanding
the way in which light and matter interact.
However, due to the probabilistic nature by
which photons are emitted from ensembles of atoms it was, at that time, not
experimentally possible to produce a single isolated photon, as measuring such
a state to determine its presence would necessarily result in its destruction.
While at first seemingly unrelated to the topic of singlephoton generation
(and considered a solution looking for a problem at the time of its
development), the initial demonstration of lasing in 1960 was a significant moment
in the story of the control of light. For the first time, light could be
prepared in high power, low divergence beam with high coherence, and narrow
spectral bandwidth.
Among other applications, this opened up the
nascent field of nonlinear optics to experimentation for the first time.
Nonlinear effects can occur when the light propagating in a material is so
intense that the motion of the bound electrons in response to the driving
electric field is no longer linear due to the shape of the atomic binding
potential. This electron motion sets up a timevarying polarization response in
the material, leading to the generation of light at new frequencies, first
observed in 1961 for the case of secondharmonic generation.
Another major innovation that impacted on
nonlinear optics was the development of modern, clad optical fibers in the
1950s, capable of guiding light along with a narrow core by total internal
reflection. Today optical fibers are most other can be inferred, providing a
singlephoton state of known properties, localized in time. The use of
waveguides, such as optical fiber, as a generation medium, gives the
significant benefit that such a state is also guaranteed to be spatially
localized in a propagating mode of the structure. In recent years,
singlephoton sources have proven to be a valuable resource in the rapidly the developing field of quantum information, both for fundamental tests of quantum
mechanics and as nonclassical light sources to enable investigation of a range
of applications, such as quantum cryptography, metrology, and quantum
computation.
The nonlinear effect that is central to this
thesis work is the fourwave mixing, it is a nonlinear 3 order process, during
which photons from an intense laser beam (called pump beam) are annihilated in
pairs to simultaneously generate a pair of photons called signal and
complementary (or idler), of respective energies whose sum is equal to the sum
of the energies of the two pump photons (the condition of conservation of
energy is thus respected during the process).
In general, and in particular, in the case of
fourwave mixing, the efficiency of nonlinear processes increases with the
length of interaction and with the light intensity present in the medium.
Thus, despite the low nonlinear coefficients
generally exhibited by the materials that make them up, optical fibers are very
suitable for observing effective nonlinear effects: due to the low propagation
losses, the interaction lengths can be very large (from a few centimeters to
several tens  even hundreds  of kilometers), and the confinement of light in
very small hearts makes it possible to reach extremely high light intensities.
This is the reason why nonlinear optics in fibers has been a very active field
of research for several decades already, as evidenced by the numerous
applications which have emerged from this field of research since the early
1990s: generation of supercontinuum, that is to say very spectrally wide laser
beams (for spectroscopy or wavelength multiplexing for example), amplification
or conversion of wavelengths and processing ultrafast signal (exploiting the
almost instantaneous nature of the fourwave mixing process) thanks to
parametric fiber optic amplifiers, etc.
We show that this approach, characterized by
high spectral precision and short measurement times, allows one to distinguish
between almost unrelated and more closely related photon pairs.
Silicon embedded devices, such as microring
resonators, have recently been shown as effective sources of quantumbound
photon pairs. Mass production of integrated devices requires the application of
fast and reliable technologies to monitor device performance. In the case of the
association between time and energy, this is a particular challenge, because it
requires high spectral precision that cannot currently be achieved in the shell
measurements.
Here we are reconstructing the common
spectral density of the photon pairs resulting from the automatic mixing of
four waves in the silicon ring resonator by studying the corresponding
catalytic process, which is the stimulation of the fourwave mixing.
Photonic crystal fibers (PCFs) represent a
new category of optical waveguides that have unique optical characteristics.
They make it possible to strongly increase nonlinear effects with multiple
dispersion parameters. Hence the interest in these fibers, particularly in the
telecommunications field. We are interested in our work in the characterization
of the optical properties of PCF by studying the impact of geometric
deformations on chromatic dispersion and birefringence. The finite element
method was adopted for the analysis of structures with ideal and real profiles.
We have also used this study to improve the guiding properties of PCFbased
couplers in terms of coupling length, by voluntarily introducing geometric
irregularities.
After having mastered the modal analysis of
PCF, we are interested in the study of the generation of very broadband spectra
by means of the generation of supercontinuum. The study of such spectral
broadening, in particular in PCF, requires precise modeling of the propagation
of ultrashort pulses. The development of a model based on the nonlinear Schrödinger
equation (ESNL), taking into account linear attenuation and dispersion effects
and nonlinear Kerr effects, stimulated Raman scattering and selfhealing, has
been achieved. A digital tool based on the Fourier splitstep method and the
RungeKutta method was developed for the resolution of the ESNL.
An experimental device has been set up for
the investigation and analysis of supercontinua (SC), in a highly nonlinear
PCF, generated by the fundamental mode as well as the higherorder modes. Different
types of spectral broadening have been identified. The dynamics of the
construction of the SC as a function of the nonlinear effects brought into
play have been validated numerically. The influence of the cutoff wavelength
on the generation of SCs by higherorder modes has been studied.
High power fiber lasers are now the preferred
solution for industrial cutting applications. The development of lasers for
these applications is not easy due to the constraints imposed by industrial
standards. The manufacture of increasingly powerful fiber lasers is limited by
the use of a gain fiber with a small mode surface suitable for nonlinear
effects, hence the interest in developing new techniques allowing the
attenuation of these.
The experiments and simulations carried out
in this thesis show that the models describing the link between laser power and
nonlinear effects in the context of passive fiber analysis cannot be used for
the analysis of nonlinear effects in lasers. high power, more general models must
therefore be developed. The choice of laser architecture is shown to influence
nonlinear effects. Using the generalized nonlinear Schrödinger's equation, it
was also possible to show that for a copropagation architecture, Raman
scattering influences spectral broadening. Finally, experiments and simulations
show that increasing the nominal reflectivity and bandwidth of the slightly
reflective network of the cavity makes it possible to attenuate Raman
scattering, in particular by reducing the effective Raman gain.
Chapter 1
1 Basics of light propagation in optical fibers
After briefly introducing the key concepts of
nonlinear optics and identifying the outlines defining the framework of
nonlinear optics in optical fibers, the objective of this chapter is to
describe the three processes involved in the work presented in this manuscript:
phase selfmodulation, fourwave mixing, and Raman scattering.
11 Conventional optical fibers
A conventional optical fiber consists of a
cylindrical core of radius r, and of refractive index n, surrounded by a sheath
of radius r, and with a refractive index n.
The basic material constituting the optical fibers is generally silica
glass (SiO,). The light is guided by the internal total reflection mechanism.This
phenomenon requires a positive index difference between the heart and the
sheath. Consequently, the core is
composed either of pure silica or of silica doped with germanium oxide (GeO;)
or potassium (P, O,) for example.
Figure 11 : a) Schematic representation
of a conventional stepindex fibre. The core size in this case would correspond
to a multimode fibre; (b) Refractive index profile of the stepindex fibre.
[1 ]
Dopants based on
fluorine or boron can be incorporated into the sheath to induce a reduction in
its refractive index. The refractive
index profile can then be discontinuous , which corresponds to the index
hopping fiber, or be gradual, which corresponds to the index gradient fiber.
The relative
difference in the coresheath index:


The manufacture of this type of fiber is
carried out in two main stages. The
first consists of producing a preform which has a transverse structure similar
to that of the desired fiber except for a scale factor. Typically a preform has a diameter of a few
tens of millimeters. The second step is to stretch the fiber preform, heating
it until a final outside diameter of about 125 µm is obtained for standard
fibers.
The main manufacturing steps are shown
diagrammatically in Figure 12 . An enlargement of the image of the
crosssection of a PCF, recorded using an electron scanning microscope. This type of optical fiber is characterized
by two main parameters: the diameter of the air holes, noted d, and the spacing
between these holes, noted A .The control of these two parameters makes it
possible to modify the chromatic dispersion and the confinement of the optical
modes.
12
Manufacture of Optical Fibres
Both PCF and standard
singlemode fiber are produced using a tower drawing process. Firstly, a
macroscopic preform is fabricated. In the case of standard fiber, this will be
a solid silica rod with the required index profile, and for PCF it will be a
glass cane with the correct pattern of air holes to produce the desired fiber
geometry.
While
the methods of producing the preform differ, the final stage of drawing the
preform down to fiber is similar for both. The preform for a standard fiber is
fabricated from fused SiO2 silica glass, with dopants added to either the core
or cladding to produce the required index mismatch between the core and
cladding.
The
glass for the preform needs to be of extremely high purity in order to achieve
the low levels of loss associated with commercial fiber. In particular, care
must be taken to avoid contamination of the preform with water, as the presence
of OHions leads to a vibrational absorption peak that causes significant loss
near 1.4 µm, in the desired operating range close to 1550 nm, the wavelength of
minimum attenuation
Figure 12 : Schematic
of the MCVD process, where silica is deposited on the inside of the silica tube
to produce a fibre perform. [2]
Fig. 12 shows a commonly used technique of preform fabrication
based on modified chemical vapour deposition (MCVD). A multiburner torch is
moved back and forth across the outside of a silica tube, uniformly heating the
inside surface to around 1800°C. SiCl4 and O2 gases flow into the hot tube
where they react to form small silica particles that fuse on the inner surface.
Small quantities of boron or fluorine may also be added to lower the glass
index in the cladding. After the cladding layer is formed GeCl4 or POCl3
dopants are added to the gas mixture to give a layer of higher index that will
form the core. Finally, the burner temperature is raised, causing the tube to
collapse into a solid preform rod, with the required index profile.
The technique required to produce a
preform for PCF, is dependent on the desired fibre geometry and material. For
fibres with unusual or irregular cladding geometries a preform can be produced
by drilling a bulk sample of the glass, or softening the glass by heating and
then extruding the material through a metal die. These techniques are more
commonly used when working with softer, highly nonlinear materials such as
chalcogenide glasses. For preforms made from silica glass, the standard method
is the stack and draw technique.
13 Light Guidance Mechanism in
Conventional Fibre
An optical fiber is a type of
waveguide designed to confine light to a narrow core region so that it can
easily be transferred over large distances with minimal levels of loss. Early
work, showed that narrow silica strands in air were capable of guiding light
along their length through total internal reflection at the airglass
interface, a property that was already well known from the similar effect that
can be observed using jets of water. Later work showed that the properties of
fibers could be improved by taking advantage of a cladding layer. Most
conventional optical fibers still rely on this straightforward geometry
Fibres typically also include a polymer
jacketing layer to improve their mechanical strength. Due to the rapid
exponential decay in the field strength of guided light with increasing
distance from the core, the presence of this jacketing layer should not
influence the optical properties of the fiber.
Figure 1.3 : Schematic representation of ray propagation in a large
core multimode stepindex fiber.
The remaining light will be launched into the
core and propagate for some distance before encountering the corecladding
interface. If the angle between the ray and the normal to the interface
_{} 
(1.2) ^{ } 
^{ }Where

(1.3) 
And

(1.4) 
Where
The resulting equation for the electric field
distribution (expressed in the frequency domain) is given by

(1.5) 
Where

(1.6) 
Where F(ρ) is the solution of the differential
equation for Bessel functions:

(1.7) 
Where

(1.8) 
Where n_{eff} is dependent on
the angular frequency of the wave
The origin of the material contribution to
the dispersion is due to the oscillation of the bound electrons within the
silica glass in response to the applied electric field from the propagating
light. This leads to a wavelength dependent refractive index that is determined
by the resonance frequencies at which absorption occurs in the material as
described by the Kramers–Kronig relations . While silica glass has extremely
low loss in the wavelength range of interest here, from 500 – 2000 nm, strong
absorption due to electronic excitation in the ultraviolet region, and
vibrational resonances in the infrared lead to a characteristic dispersion
profile for silica glass. In the transparency region, far from these
resonances, the refractive index can be approximated by the Sellmeier equation
n^{2}(ω) = 1+ 
(1.9) 
The index of the material and
N_{g} = 
(1.10) 
In order to demonstrate the effects of
dispersion it is common to express the propagation constant as a Taylor
expansion about some central frequency

(1.11) 
Where

(1.12) 
By comparing with Eq. (1.10) with Eq. (1.12), it can be seen that the first derivative
term is related to the group velocity, and therefore the second derivative term
(the rate of change of

(1.13) 
And

(1.14) 




Generally the higher order terms in Eq. (1.12) can be disregarded for quasimonochromatic
pulses where the spectral width is significantly less than the central
wavelength such that
The dispersion profile of a fibre is commonly
described using the dispersion parameter
D = 
(1.15) 
at a wavelength of
Figure 1.4 :
Refractive index n, group index n8 and dispersion parameter D calculated for
bulk silica glass. [3]
In addition to the material contribution to
the dispersion, the confined light guided in a fibre also experiences a
waveguide contribution to the dispersion and the overall dispersion D can be
approximated by their sum.
D = D_{m} (λ) + D_{wg}
(λ) 
(1.16) 
Where
n_{CL }k_{0} 
(1.17) 
It is common to express the modes of the
waveguide instead in terms of linearly polarised modes (
An important
parameter of fibres is the refractive index contrast between the core and
cladding.
One of the reasons for this is that it defines
the numerical aperture (NA) of the fibre which gives the maximum input angle qmax over which the fibre will accept light
according to
NA = sin 
(1.18) 
For standard fibres the index contrast is
limited to a narrow range because the high levels of dopant required to achieve
a significant index mismatch also result in high levels of loss and a fibre
with poor mechanical properties. Another important parameter is the normalised
frequency
V = k_{0}α 
(1.19) 
For a wavelength of light λ, where
F (x, y) 
(1.20) 
Where
The width of the mode is also relevant for nonlinear
processes where the strength of the interaction is dependent on the intensity
of light, and therefore its crosssectional effective area during propagation A_{eff}, where
A_{eff} = πw^{2} 
(1.21) 
For fibres with a more complex core shape,
the mode field profile for the fundamental mode will differ from the Gaussian
profile seen in a circularly symmetric structure. For these types of fibres the
distribution of optical power in the core cannot be described by the simple
formula . In
this case the effective mode area can be found through integration of the of
the mode field distribution
A_{eff} = 
(1.22) 


14 Birefringence
Birefringence was first noted as a property
of certain anisotropic materials, such as calcite, whereby the crystal
structure defines an optical axis, with a difference in the refractive index of
the material depending on whether light is polarised parallel or perpendicular
to this axis.
While silica glass itself is an isotropic
material, asymmetry in a waveguide can also lead to birefringence. Even in
fibres that support only a single guided mode there are still two possible
distinct modes of polarisation. In the case of a perfectly symmetrical
waveguide the propagation constant is degenerate for these polarisation modes,
but for an asymmetrical design there will be a variation in b for the two modes, defining fast and slow
axes of the fibre.
If
light is launched into the fibre with a linear polarisation that is not aligned
to one of these axes, the state will periodically evolve to an elliptic state
and then back to a linear polarisation state as it propagates, due to the
relative phase difference that accumulates between the components of the light
on the two axes of the fibre, as illustrated in 1.4.
Figure 1.5 : Evolution
of the polarization state of an initially polarized input beam as it propagates
through a fibre with birefringence. [3]
The spatial period over which the light
undergoes this transition and returns to its initial polarisation state is
known as the beat length (
L_{B} = 
(1.23) 


Where
1.5 Loss
One of the most attractive features of light
propagation in optical fibres is the exceptionally low transmission losses that
can be achieved. Achieving low loss is of particular importance when dealing
with the delivery of single photons through optical fibre in order to realise a
high performance single photon source.
α = 
(1.24) 
Figure 1.6 :
Comparison between the theoretical minimum loss in silica fibres and a typical
measured loss. The intrinsic loss is dominated by Rayleigh scattering over most
of the range shown. The loss peak in the measured fibres is due to OH
contamination. [3]
The loss of the
fibre can be characterised by the attenuation constant α, which is
commonly given in units of dB/km. When light of power
Chapter 2
2 Microstructure fibers
and nonlinear optics
21 Definitions and basic concepts
In order to fully understand the essence of
the project presented in this document, let us first begin by briefly
describing the concepts attached to the design of a source of advertised
photons generated in a microstructured fiber.
22 Microstructured fibers
A microstructured fiber is an optical fiber
whose transverse index profile is a complex pattern of index jumps, often air
holes in pure silica. Figure 1.1 shows a specimen of photonic crystal fiber
observed using an electron microscope to scan a photon source on demand.
2.3 Sources of photons
A conventional source is a light source
operating at a power regime such that the radiation can be properly written by
wave formalism. Even if the emission lines of a laser involve the atomic
orbitals calculated by quantum mechanics, the theory of wave light can
successfully predict the behavior of its macroscopic radiation. In contrast, a
quantum source is a source whose emission characteristics involve phenomena
that cannot be described by this same wave theory.
One of these phenomena describes the tendency
of certain sources to emit photons which are more space in time than a source
whose emission follows a Poisson law. We then speak of a phenomenon of
unbundling (antibunching) where the arrival of photons follows a subpoissonnian
distribution. Let us underline the fact that the wave formalism of light cannot
explain this phenomenon. The observation of grouping is, therefore, a
sufficient criterion to consider our source as unconventional and to give it
the designation of quantum source. A photon source is said to be
"on request" if the user can cause the emission of one or more
photons using a d signal.
Trigger, while
a source of announced photons is a quantum source whose emission (unpredictable)
of each photon is accompanied by an announcement signal. This output signal can
be another photon or an electrical pulse.
A
microstructur fiber is an optical fiber whose transverse index profile is a
complex pattern of index jumps, often air holes in pure silica. Figure 1 shows
a specimen of photonic crystal fiber observed using an electron microscope to
scan a photon source on demand.
Figure 21: Image of
T431C fiber taken by a balayage electron microscope. [4]
The manufacturing method differs from
conventional indexhopping fiber manufacturing methods. To produce the preform,
a stack of capillaries is used rather than a vapor deposition process. The
stretching of this preform is also complicated if we want to avoid the collapse
of the holes or control their expansion.
The advantage
of this type of manufacturing is an increased latitude on the final geometry of
the index profile which then makes it possible to obtain interesting
propagation characteristics by means of an appropriate design. Knowing the
propagation characteristics implies knowing the different amplitude
distributions of the stationary electromagnetic fields commonly known as
propagation modes. For an index hopping fiber, the circular symmetry and the
low index contrast between the optical core and the optical cladding facilitate
analytical resolution (Bures, 2009).
The finite element method is a method of
numerical resolution using an integral formulation, the weak formulation, of a
physical problem. The trick is to reformulate a problem of a differential
equation which is difficult to solve analytically in a soluble form by an
iterative method. We then obtain an approximate solution that can fully satisfy
our needs.
2.4 Properties of photonic crystal fibers
The concept of photonic crystal fibers also
known as microstructured fibers or fibers with holes consists of a regular
arrangement or not of air channels of micron dimensions arranged parallel to
the axis of propagation. The parameters which characterize this arrangement and
adjust the optical properties of the fibers are the distance between the
centers of two adjacent holes denoted
Figure 22 :Shows
the crosssections of an ideal (right) and real (left) PCF. [5]
The arrangement of the holes can constitute a
triangular, hexagonal or random matrix. The number of rows or crowns of holes
used to form the microstructured sheath is an important criterion for reducing
guide losses. The region, at the center of the fiber, allowing the light to be
guided is considered to be the heart of the fiber. Generally, in the case of
fullcore fibers, the latter consists of pure silica.
25 Nonlinear optics
Nonlinear optics is the field of optics that
covers interactions between photons via their interaction with matter. Even in
a transparent material, a variable portion of the energy of the oscillating
electromagnetic field excites this dielectric medium. This ≪ polarizability ≫ depends on the amplitude of the electric field. For
signals of sufficient intensity, this interaction can give rise to various
frequency conversion mechanisms which are not negligible.
These various mechanisms can be grouped mainly
into three categories: threewave mixing (3ed order), fourwave mixing (4th
order) and Raman enlargement. 3ed order is a nonlinear 2 order effect that groups the phenomena causing
three waves to interact with each other: the generation of second harmonics,
the generation by the sum of frequencies and difference in frequencies.
The nonlinear
3 order effect grouping the phenomena of
phase selfmodulation, crossphase modulation, thirdharmonic generation, and
param chord. Electric.
26 Photon detectors
The analysis and processing of signals is
difficult in optical form. Information is removed from the signal by a
photodetection process. Detection of a single photon is usually done by
reading an avalanche current caused by a single photoionization. A detector
that can change state, with a certain probability, according to the presence or
the absence of radiation (of at least one photon) is called photon detector.
This term should not be confused with that of the photon counter which is
capable of giving us information on the number of incident photons.
2.7 Quality of a photon source
The criteria to take into consideration when
designing a good source of single photons are:
 The flow
rate which is defined as the quantity of single photons emitted per unit of
time;
 "Synchronizability" is the
capacity to produce the photon on demand by a signal outside a known frequency,
specific to the device.
 Failing to be able to force the
emission of a photon, an announcement signal must accompany its production and
make it possible to synchronize the assembly downstream of the source;
 Reliability is the probability that
a single photon is actually produced in response to an input signal or accompanying
the emission of an announcement signal;
 “Tunability”
describes the ease with which the user can modify the emission spectrum of the
source;
 The purity of the quantum state is
the constancy in the properties of the emitted photon which guarantees that
successive photons are identical;
 The complexity and the robustness
of the assembly: we want a minimum of adjustment and a maintenance of its
capacities of good functioning under different operating conditions. We also
want to maximize the life of the product.
These different
aspects are to be taken into consideration not only during the design but also
when the time comes to integrate this source inside a more complex experimental
setup.
2.8 Optical fibers
Optical fiber is a waveguide of choice in
many applications and for many research themes.
This is mainly due to the many advances in its manufacturing technology
which have contributed to marked improvements in their performance. The aim of
this first part is to present two main families of optical fibers: socalled
conventional fibers and microstructured optical fibers. Next, the main linear and nonlinear effects
experienced by an optical wave propagating in a fiber will be discussed. Finally, we will present the various digital
simulation tools used in this work.
In particular, it has been possible to
manufacture optical fibers with a zerodispersion wavelength in the vicinity of
the emission wavelengths of powerful lasers .The combination of a low
dispersion with a high nonlinearity made it possible to completely revisit a
whole range of nonlinear optics, as for the generation of supercontinuum and
frequency combs. In addition, in some cases, PCFS has the advantage of only
guiding a single transverse mode over the entire transparency window: this is
called infinitely single mode PCF (or "endlessly singlemode").
When light propagates in a dielectric
material its electric field component causes bound electrons within the
material to oscillate. At low field intensities the induced polarisation in the
material from this effect is linear with field intensity and the electrons
simply reradiate light at the same frequency as the applied field, albeit with
a slight time delay that leads to the lower speed of propagation for light in a
material compared to freespace. For significantly intense applied electric
fields the oscillation of the bound electrons becomes anharmonic due to the
profile of the binding potential of the electron. In this case the induced
polarisation of the material can no longer be related linearly to the electric
field and it becomes possible for the material to radiate frequency components
that were not present in the input light.
2.9 Nonlinear Susceptibility and Nonlinear
Phase Shift
When an electric field E is present in a
dielectric material the field induces an electric dipole moment, the polarisation
of the material P, such that the electric displacement field

(21) 


Where
P = 
(2.2) 


Where
n = 
(2.3) 




This linear dependence of
it can be
approximated by a Taylor expansion, giving
P = 
(2.4) 
Where x^{(i)} is the
The linear expression for the refractive
index given in can be modified to account for the addition of the

(2.5) 
Here
n_{2} = 
(2.6) 
Where
A linearly
polarised wave of frequency
E= 
(2.7) 
Where
k_{0a} =
The intensity dependent term in gives the additional
nonlinear component of the phase shift in the case of a material without loss.
For a real material with loss it is common to define an effective length of
propagation L_{eff}, that takes into account the reduction in the intensity I over the course
of propagation through a physical distance
For a fibre with attenuation constant α
L_{eff} = 
(2.8) 




After
propagation through a length of fibre

(2.9) 
This effect is known as selfphase modulation
(

(2.10) 
Where
Figure 2.2 : (a)
Temporal intensity profile of an initial pulse, and resulting time dependent
shift in the frequency after propagation dur to SPM. (b) Output spectrum after
propagation of a Gaussian high energy pulse dur to SPM. [3]
Since new frequency components will be
generated in pairs with a frequency dependent phase separation between them,
this eventually leads to a periodic structure developing in the pulse spectrum
when the maximum value of
210 Nonlinear propagation equation
The nonlinear electromagnetic wave
propagation equation describes how the different components of the
electromagnetic field present in the material medium interact, via the nonlinear
polarization of the medium, to give rise to the different nonlinear processes. Linear
that we are likely to encounter. The derivation of this propagation equation
from Maxwell's equations is carried out in detail in numerous reference works.
It generally leads 2 to the
following temporal form:

(2.11) 
Where

(2.12) 

(2.13) 
In the frequency domain, the propagation equation takes
the following form:
Δ 
(2.14) 
If we consider
plane waves, and that we note z the direction of propagation (assumed to be
common to all waves).
The complex amplitudes E (

(2.15) 

(2.16) 
Where

(2.17) 
With
and recalling that the intensity of the wave at w is expressed
I(z

(2.18) 
as
we obtain:
I(z

(2.19) 


With sin(x) = sin(x )/ (x)
When the phase mismatch
Developing and manufacturing a highpower
laser poses several scientific challenges. For industrial grade LFHP, managing
nonlinear effects is one of the most important constraints, but it is not the
only constraint that must be carefully managed when designing LFHP. For
example, if the thermal load of the LFHP is not managed properly, thermal
degradation can have catastrophic effects on longterm reliability.
LFHP are also prone to photo blackening, a
phenomenon which increases the losses of bottom which decreases the efficiency
of the laser. In addition, as is often the case in industrial applications, if
we aim to develop LFHP to make the combination of laser modules, it will also
be necessary that the output of LFHP is limited by diffraction, and therefore
single mode, which imposes additional constraints on the fiber used and the
laser architecture.
Chapter 3
3 Fourwave
mixing
The wave mixing process involves
four waves (as the name suggests) which exchange four their energy via the
nonlinear medium of 3 order which they cross (without however exchanging
energy with this medium). Schematically, the three types of energy transfer
that can occur are represented by the energy diagrams in the figure (3).
Figure 31: Energy diagrams representing the three types of
fourwave mixing. [6]
We will be interested in our part
in an energy transfer of type b), for which two waves of strong intensity at
Figure 32 :
Diagrammes show energy diagram of the
frequencydegenerate fourwave mixing. [6]
From a corpuscular point of view,
the process consists in the annihilation of a pair of pump photons at onsp,
accompanied by the simultaneous creation of a pair of signal and idler photons
which make it possible to respect the condition of conservation of l energy
Ultimately, the signal and idler
wave pairs that can be generated efficiently correspond to relatively limited
spectral ranges. To describe the process a little more quantitatively, we will
assume below that a nonlinear medium of 3 order is crossed, in the zdirection,
the total field in the middle is written:

(3.1) 
We assume that the three waves have the same state

(3.2) 
With
From
part to equation (3), we can calculate the
expression of the complex amplitudes of the nonlinear 3 polarization at frequencies
є_{0} 
(3.2.1) 
+ 6є_{0 } 
(3.2.2) 
+ 6є_{0 } 
(3.2.3) 
+ 6є_{0 } 