Institute of Radio Electronics and Telecommunications
Department of Radiophotonics and Microwave Technologies
Photonics and Optoinformatics
MASTER THESIS:
Obtain the four-wave interaction in a wide spectral range
Kebour Mohamed
Introduction
Chapter 1
1 Basics of light propagation in optical fibers
1-1 Conventional optical fibers
1-2 Manufacture of Optical Fibres
1-3 Light Guidance Mechanism in
Conventional Fibre
1-4 Birefringence
1.5 Loss
Chapter 2
2 Microstructure fibers and nonlinear optics
2-1 Definitions and basic concepts
2-2 Microstructured fibers
2.3 Sources of photons
2.4 Properties of photonic crystal
fibers
2-5 Non-linear optics
2-6
Photon detectors
2.7 Quality of a photon source
2.8 Optical fibers
2.9 Nonlinear Susceptibility and
Nonlinear Phase Shift
2-10 Nonlinear propagation equation
Chapter 3
3 Four-wave mixing
3-2 Phase-Matching Conditions for FWM
3-3 Gain and Bandwidth of FWM
3-4 Cross-Polarised FWM
3-5 Calculation of phase agreements of
the degenerate four-wave mixting
3-6 The Hamiltonian
3-7 Raman scattering
Chapter 4
4-1 Quantum-mechanical description of FWM
4-2 Pure Single Photon States
4-3 Hong-Ou-Mandel
4-4 Generation of Single Photon States
4-5 Attenuated Coherent Light
4-6 Photon Pair Generation by FWM
4-7 Spectral Filtering
4-8 Photon Statistics for FWM
Chapter 5
5-1 Experimental
setup
5-1 Spectrometer Measurement
5-2 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 know-how on the creation
and manipulation of non-conventional 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, particle-like 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 single-photon 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 time-varying polarization response in
the material, leading to the generation of light at new frequencies, first
observed in 1961 for the case of second-harmonic 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
single-photon 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,
single-photon 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 non-classical 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 four-wave mixing, it is a non-linear 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
four-wave mixing, the efficiency of non-linear processes increases with the
length of interaction and with the light intensity present in the medium.
Thus, despite the low non-linear coefficients
generally exhibited by the materials that make them up, optical fibers are very
suitable for observing effective non-linear 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 ultra-fast signal (exploiting the
almost instantaneous nature of the four-wave 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 micro-ring
resonators, have recently been shown as effective sources of quantum-bound
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 four-wave mixing.
Photonic crystal fibers (PCFs) represent a
new category of optical waveguides that have unique optical characteristics.
They make it possible to strongly increase non-linear 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 PCF-based
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 ultra-short pulses. The development of a model based on the non-linear Schrödinger
equation (ESNL), taking into account linear attenuation and dispersion effects
and non-linear Kerr effects, stimulated Raman scattering and self-healing, has
been achieved. A digital tool based on the Fourier split-step method and the
Runge-Kutta 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 non-linear
PCF, generated by the fundamental mode as well as the higher-order modes. Different
types of spectral broadening have been identified. The dynamics of the
construction of the SC as a function of the non-linear effects brought into
play have been validated numerically. The influence of the cut-off wavelength
on the generation of SCs by higher-order 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 non-linear
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
non-linear effects. Using the generalized nonlinear Schrödinger's equation, it
was also possible to show that for a co-propagation 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 self-modulation, four-wave mixing, and Raman scattering.
1-1 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 1-1 : a) Schematic representation
of a conventional step-index fibre. The core size in this case would correspond
to a multimode fibre; (b) Refractive index profile of the step-index 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 core-sheath 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 1-2 . An enlargement of the image of the
cross-section 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.
1-2
Manufacture of Optical Fibres
Both PCF and standard
single-mode 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 OH-ions 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 1-2 : Schematic
of the MCVD process, where silica is deposited on the inside of the silica tube
to produce a fibre perform. [2]
Fig. 1-2 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.
1-3 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 air-glass
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 step-index fiber.
The remaining light will be launched into the
core and propagate for some distance before encountering the core-cladding
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 neff 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
|
n2(ω) = 1+ |
(1.9) |
The index of the material and
|
Ng = |
(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 quasi-monochromatic
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 = Dm (λ) + Dwg
(λ) |
(1.16) |
Where
|
nCL k0 |
(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 = k0α |
(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 cross-sectional effective area during propagation Aeff, where
|
Aeff = πw2 |
(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
|
Aeff = |
(1.22) |
|
|
|
1-4 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 (
|
LB = |
(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
2-1 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.
2-2 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 2-1: Image of
T431C fiber taken by a balayage electron microscope. [4]
The manufacturing method differs from
conventional index-hopping 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 2-2 :Shows
the cross-sections 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
full-core fibers, the latter consists of pure silica.
2-5 Non-linear 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: three-wave mixing (3ed order), four-wave mixing (4th
order) and Raman enlargement. 3ed order is a non-linear 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 non-linear
3 order effect grouping the phenomena of
phase self-modulation, cross-phase modulation, third-harmonic generation, and
param chord. Electric.
2-6 Photon detectors
The analysis and processing of signals is
difficult in optical form. Information is removed from the signal by a
photo-detection process. Detection of a single photon is usually done by
reading an avalanche current caused by a single photo-ionization. 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: so-called
conventional fibers and micro-structured optical fibers. Next, the main linear and non-linear 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 zero-dispersion wavelength in the vicinity of
the emission wavelengths of powerful lasers .The combination of a low
dispersion with a high non-linearity made it possible to completely revisit a
whole range of non-linear 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 single-mode").
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 re-radiate 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 free-space. 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
|
|
(2-1) |
|
|
|
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
|
n2 = |
(2.6) |
Where
A linearly
polarised wave of frequency
|
E= |
(2.7) |
Where
k0a =
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 Leff, 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
|
Leff = |
(2.8) |
|
|
|
|
|
After
propagation through a length of fibre
|
|
(2.9) |
This effect is known as self-phase 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
2-10 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 high-power
laser poses several scientific challenges. For industrial grade LFHP, managing
non-linear 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 long-term 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 Four-wave
mixing
The wave mixing process involves
four waves (as the name suggests) which exchange four their energy via the
non-linear 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 3-1: Energy diagrams representing the three types of
four-wave 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 3-2 :
Diagrammes show energy diagram of the
frequency-degenerate four-wave 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 non-linear medium of 3 order is crossed, in the z-direction,
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
|
є0 |
(3.2.1) |
|
+ 6є0 |
(3.2.2) |
|
+ 6є0 |
(3.2.3) |
|
+ 6є0 |
(3.2.4) |
In this
expression, the term (3.2.1) corresponds to the phase of the pump wave
self-modulation process .The terms (3.2.2) and (3.2.3) are associated with the
cross-phase modulation process. The term (2.2.4), meanwhile, represents the
four-wave mixing process. As we assumed that the pump wave was of very strong
intensity compared to that of the signal and idler waves, the influence of the
terms (3.2.2), (3.2.3) and (3.2.4) (which do not intervening only once the
amplitude of the pump wave) is negligible compared to that of the term of :
Phase self-modulation, and we can therefore
write
|
A( |
(3.3) |
With regard to the component leads to:
|
|
(3.3.1) |
||
|
+ 6є0 |
(3.3.2) |
||
|
+6є0 |
(3.3.3) |
|
||
|
+3є0 |
(3.3.4) |
|
||
It is then the terms (3.3.1) and (2.4.2) (crossed phase modulation of the signal wave by the idler wave) that are negligible, and we keep the term for the phase modulation of the signal wave by the pump wave (3.3.3) and the term for four-wave mixing (3.3.4),which both involve twice the amplitude of the pump wave:
|
6є0 |
(3.4) |
|
+ 3є0 |
(3.4.1) |
For
|
6є0 |
(3.4.2) |
||
|
+3є0 |
(3.4.3) |
|
||
Finally, assuming that all the frequencies
involved are far from any resonant frequency of the material medium, we can
pose:
|
x(3)eff = =
= |
(3.5) |
Takes the following forms, for each of the three waves considered :
(3.6)
In recent years, the ability to produce
single photons in pure states has become highly relevant to the field of
quantum information. As a result of their many desirable characteristics,
photons are one of the most well studied systems used for the encoding and
manipulation of quantum information in the form of qubits. The continued
development of single photon sources will be of great importance, both in the
short term as a resource for the study of more complex quantum information
systems, and in the future to make commercial applications based on this work
feasible
The corresponds
to the linear contribution to the polarisation is rewritten as
|
P = ε0 (x(1)
.E) + ε0 (x(3).
EEE) = PL +PNL |
(3.7) |
In the nonlinear case, the induced nonlinear
component of the polarisation,
|
PNL-i(ω4) = ε0 |
(3.8) |
where the subscripts
|
|
(3.9) |
For propagation in a fiber, only electric
fields with a polarisation in the transverse directions x and y, are considered
meaning that the terms involving z can be ignored. Finally, the resulting
behavior should be the same regardless of the chosen coordinate system, which
leads to the remaining elements of the tensor being related to each other by
|
|
(3.10) |
Note that in the case where two of the coupled fields are polarised orthogonally with the other two, the strength of the nonlinear susceptibility is only 1/3
that of the case where all four are co-polarised. This is the
reason for the difference in the nonlinear phase shift associated with
Considering the
simpler case in which all four coupled fields are polarised in the same
direction demonstrates much of the interesting nonlinear behaviour. For four
such
|
E = |
(3.11) |
|
|
|
|
|
Where in general the frequency of each field
|
PNL = |
(3.12) |
For instance, considering the P4 amplitude of the polarisation, part of the
expression is given by substituting these
P4=
Where
Different terms correspond to various third-order
nonlinear effects, including
This term shows that a component of the
material polarisation at frequency
The presence of the relative phase factor
exp(
In
|
|
(3.15) |
Where
|
|
(3.16) |
Where
where
|
|
(3.17) |
|
|
|
|
|
While the energy matching condition seems to
suggest that coupling can occur between signal and idler pairs of any
wavelengths satisfying. Means that the process is only efficient for specific
signal and idler wavelengths, dependant in part on the dispersion properties of
the fibre.
3-2 Phase-Matching Conditions for FWM
Since the induced polarisation in a material
will be coherent with the phase of the pump pulse, the generated parametric
(signal or idler) field generated as a result of it will also have a constant
phase relationship to the pump at the point of generation. However, if the
generated waves are travelling with a different phase velocity to the pump then
in general they will tend to cancel out with the generated field produced at
earlier points in the fibre.
Only in the case where all three waves (the
signal, idler, and two degenerate fields comprising the pump) maintain a
constant phase relationship to each other is destructive interference avoided,
allowing the amplitude of the generated waves to build up to a significant
amplitude. An analysis of the coupled amplitudes of the fields, in the quasi CW
regime (assuming the pump beam is not depleted) shows that
|
|
(3.18) |
And
|
K = |
(3.19) |
Where
|
|
(3.20) |
The energy and phase-matching conditions for
|
|
(3.21) |
The condition of perfect phase-matching, when
|
k = |
(3.22) |
Note that only the even terms in the
expansion of b contribute to the phase-matching, since the odd terms
from the expansion of the propagation constant for the signal will cancel with
those from the idler. Generally the b2 term of the expansion will dominate, but
close to the
Figure 3-3 : Calculated, phase-matched signal and
wavelengths at different offset values of the pump wavelength from the ZDW
(λD), for variety of peak pump power levels.These curves were calculated for a
fibre with λD=1090 nm [3]
Due to the potential for a large wavelength separation
from the pump in the normal dispersion regime, there is less overlap between
the narrow peaks generated through
3-3 Gain and Bandwidth of FWM
In order to evaluate the signal and idler gain
for co-polarised FWM in the case of degenerate pump fields it is necessary to
consider coupled equations for the field amplitudes
|
|
(3.23) |
Where
|
Ap (z) = Ap
(0) |
(3.24) |
Where
|
Ep (r) =Fp(x,y) Ap (z) |
(3.25) |
|
|
|
|
|
Where
|
|
(3.26) |
And
|
|
(3.27) |
By making substititions Bj
= Aj
|
|
(3.28) |
And
|
|
(3.29) |
Solving these equations, the amplitude for the signal peak is then given
by
|
Bs = [C1 |
(3.30) |
|
|
|
|
|
with a similar expression for the idler
amplitude, where the parametric gain coefficient g is given by
|
g = |
(3.31) |
The coefficients
|
Gs = |
(3.32) |
and
|
Gi = |
(3.33) |
Where Pj = Bj
When the phase mismatch is small and the gain
is large (gL >> 1) the gain in the signal and idler is exponential. In
the limit of low gain and low pump power g becomes dominated by the phase
mismatch term so that g » κ/2 and sinh(gL) » sin(
The low power gain for FWM (
|
GFWM = Gs
– 1 = Gi |
(3.34) |
For perfect phase-matching in the
low power case, the generated signal and idler powers scale as the square of
both the pump power and propagation length.
3-4 Cross-Polarised FWM
In addition to the co-polarised FWM process discussed so
far, phase-matching can also be achieved for cross-polarised FWM (XFWM) with
degenerate pumping in birefringent fibres. In birefringent fibres the
difference in the propagation constant between the fast and slow axes can be
used to achieve phase-matching. For degenerate pumping, on the fast axis the
signal and idler pair will be generated on the slow axis, and with the pump on
the slow axis the generated signal and idler must both be on the fast axis in
accordance with the symmetry properties of the
|
k = |
(3.35) |
|
|
|
|
|
where
|
ns,i(λ) = np(λ) |
(3.36) |
Here
the effective index for the signal and idler is modified by an additional term δn(λ) to account
for propagation on a different axis to the pump, with the sign dependent on
which axis the pump propagates. The wavelength dependence of the refractive index
difference δn(λ) can be ignored without significantly
affecting the qualitative results of the phase-matching calculation.
3-5 Calculation of phase agreements of the degenerate four-wave mixting
We want to find the value of the
spectral difference
|
Δk = 2kp
– k8 – ki = |
(3.37) |
Where the frequency of the signal
is
3-6 The Hamiltonian
The complete Hamiltonian of the
electromagnetic field is therefore written.
|
H = h |
(3.38) |
|
|
|
Where
ϑ is the interaction Hamiltonian. This Hamiltonian in the case of an M4O takes
the form
|
|
(3.39) |
Where
The Heisenberg representation which
makes it possible to follow the evolution of operators either
|
ih |
(3.40) |
|
ih |
(3.41) |
This leads us to the coupled equations:
|
|
(3.42) |
|
|
|
(3.43) |
|
|
|
|
|
With the change of variable
|
|
(3.44) |
The situation is similar to the
case presented in section. The evolution over time of the operators is written:
|
|
(3.45) |
|
|
|
|
|
(3.46) |
In our particular case, we are in the spontaneous
transmission regime, that is to say, that no priming signal should be present.
In addition, we are interested in the number of photons after a certain
interaction length.
3-7 Raman
scattering
The modes of
vibration of the molecules lead to an inelastic scattering process. Raman scattering
occurs when a molecule in the glassy matrix absorbs a photon and then emits a
photon at a longer wavelength and a phonon corresponding to the energy
difference between the absorbed photon and the scattered photon. This
phenomenon is related to the transient component of the nonlinear response of
the material. illustrates the inelastic scattering process when a photon is
absorbed and then inelastically scattered by an atom of the glass matrix.
Figure 3-4 : Figure
shows Energy diagram representing the Raman scattering process [3].
Raman scattering is an undesirable
effect in LFHP since it converts part of the photons in the laser signal into
longer wavelength photons. New photons can be generated spontaneously in all
directions. Certain photons will be guided in the direction of propagation of
the signal and certain photons in the opposite direction .
The wavelength shift caused by
Raman scattering poses a problem for cutting processes, since the
anti-reflective coatings of the optics of the cutting heads are made according
to the wavelength of the signal, but absorbs power at length Raman wave which
generates heating. As is the case with laser amplification, there is a critical
power, that is to say, a power beyond which the Raman photons can generate
stimulated Raman emission and therefore even more damage.
In the case of passive fibers, it is possible to show that the critical
signal power (Pcr) depends mainly on the Raman gain,
|
Pcr |
(3.47) |
|
|
|
|
|
Where
|
Leff = |
(3.48) |
|
|
|
|
|
In the case of a laser, the Raman threshold
also seems to depend on other parameters, such as the pumping configuration and
the bandwidth of the fiber Bragg gratings. The Raman gain coefficient (
Figure 3-5:
Normalized Raman gain for a silica-germanium matrix (titrated data and a
sillice-phoshore matrix (data drawn as a function of wavelength for a signal at
1080 nm)
The figure compares the normalized
Raman gain as a function of the wavelength for a germanium and phosphorus-doped
silica fiber.
Chapter 4
4-1 Quantum-mechanical description of FWM
The ultimate goal of quantum
information (QI) science is to be able to understand and manipulate the quantum
states of physical systems in order to represent, communicate and process
information. Of particular interest are two-level quantum systems, known as
qubits in analogy with classical bits. These represent a fundamental resource
of many quantum information protocols. Like a regular bit that can take values
of zero or one, a qubit can exist in either of two possible eigenstates of the
quantum system. The difference arises due to the fact that a quantum system can
also exist in a superposition of these eigenstates. The general state of the
qubit can therefore be expressed as
|
| |
(4.1) |
Where |α⟩ and |β⟩ are eigenstates of system, a and B are
arbitrary complex coefficients, and the state of the system is normalized such
that
|
|α|2
+ | |
(4.2) |
In addition to the superposition states that
can be obtained with a single qubit, multiple qubits can exhibit non-local
correlations between their states which have no analogue in classical bits. For
a system consisting of two qubits, both with eigenstates |añ and | b ñ, the state of the overall system can be
described by
|
| |
(4.3) |
Where the subscripts 1 and 2 refer
to the two individual qubits and the amplitude coefficients a are again chosen such that the state of the
system is normalised.
For some values of these amplitudes it is possible to factorise the state
of the system in the form of
|
| |
(4.4) |
Such that the states of the two
qubits are independent and the measurement of one has no effect on the other.
In that case the states of the qubits are entangled. For example, the state
|
|
(4.5) |
Ates that are of significant
interest in QI applications. There are several important criteria that a
physical system must meet in order to be considered as a useful qubit. The
system must have two well defined quantum states and be able to be reliably initialised
to a desired starting state before any experiment in order to encode
information. For the purposes of quantum information processing (QIP) it is
also necessary to be able to manipulate the state of the qubits after
initialisation, including implementing interactions between the states of two
separate qubits, and be able to reliably detect the output qubit state.
Finally the qubit should be stable
against decoherence of its quantum state due to interactions with the
environment. This last requirement is necessary to ensure that the qubit state
survives for long enough for a desired QIP task to be completed. In the case of
qubits for quantum communications systems it is also necessary to be able to
faithfully transmit qubits between locations without degradation of the qubit
state.
4-2 Pure Single Photon States
Ever since the early experiments of
the 20th century into the nature of light, such as the investigation of the
photoelectric effect, it has been understood that at a fundamental level electromagnetic
waves are composed of discrete quantised excitations of the electromagnetic
field known as photons. A theory of light in which the field consists of
discrete energy levels can be developed by considering the optical modes of a
hypothetical bounded cavity. This allows the energy states of the system to be
described using the model of a quantum harmonic oscillator with energy levels
En given by
|
En = n = 0, 1, 2, … |
(4.6) |
Where
|
|
(4.7) |
Where
In order to describe nonlinear processes in which photons are created in a
mode of one frequency by the annihilation of photons in other frequency modes,
it is useful to define a creation operator
|
|
(4.8) |
And
|
|
(4.9) |
Any number state of n photons can
be therefore be expressed in terms of the vacuum state (containing zero
photons) through successive application of the creation operator
|
|
(4.10) |
The relationship between the
creation and annihilation operators for a mode and the number of photons in
that mode can be further explored by considering the effect of applying both of
these operators, showing that
|
|
(4.11) |
The product
|
|
(4.12) |
For the case where
Such a localised photon state can be
realised by considering an integral over one photon Fock states of different
frequencies
|
|
(4.13) |
Where
|
Δ |
(4.14) |
|
|
|
|
|
Where
4-3 Hong-Ou-Mandel
Interference One striking result of
the non-classical nature of single photon states is the observation of
two-photon quantum interference that can be observed when indistinguishable
single photon states are incident on the two input ports of a beam-splitter.
The effect, which is known as Hong-Ou-Mandel (HOM) interference, results in
photon bunching such that both photons exit from the same beam-splitter output.
Figure 4-1 : The input
and output modes of a beam-splitter
To understand the origin of HOM
interference it is necessary to consider the relationship between a general
input photon state to an idealised lossless beam-splitter and the resultant
output state. For a beam-splitter with input modes
|
|
(4.15) |
Where the minus sign in the second expression
is a result of the phase-shift required to satisfy conservation of energy. For
a single polarisation state these expressions are universal for all types of
beam-splitters and are also applicable to fused-fibre couplers. For a single
photon input into one of the input arms of the beam-splitter described by
|
|
(4.16) |
|
|
|
|
|
The final state after the beam-splitter will be a single
photon in a superposition of the two output modes
|
|
(4.17) |
While the photon can only be found
in one of the two output arms after measurement, the probability amplitude of
the photon wavefunction before measurement is non-zero in both arms. In the
case where two photons are input simultaneously the beam-splitter these
wavefunction contributions in the two arms can interfere. When two
indistinguishable single photon states are input to the system, one in each arm
of the beam-splitter the two-photon state is described by
|
|
(4.18) |
|
|
|
|
|
Using the input and output
beam-splitter relations given in the resultant state is given by
|
= |
(4.19) |
For the case of an ideal lossless 50:50
beam-splitter the reflectivity and transmittance must be equal, and sum to one
in order to satisfy energy conservation so that.
|
|
(4.20) |
Here r and t are assumed to be
real, although a more general analysis allowing for complex reflection and
transmission coefficients
|
|
(4.21) |
Where the difference of the factor
of
4-4 Generation of Single Photon States
The number state representation of
optical fields , allows for a simple theoretical description of a single photon
state
4-5 Attenuated Coherent Light
In classical optics the most stable type of
light source that can be realised is a perfectly coherent, single polarisation
state, monochromatic beam, which is close to what can be achieved using a
stable pulsed laser source. It may therefore appear logical to consider
attenuating such a beam until it contains only a single photon per pulse on
average. However it will be shown in this section that such a scheme gives a
relatively poor approximation of the idealised single photon source, due to the
underlying photon number statistics of a coherent light source.
The scalar electric field
|
|
(4.22) |
|
|
|
|
|
Where the peak field amplitude
|
mº |
(4.23) |
Where A is the cross-sectional area of the
beam and
|
|
(4.24) |
Then the probability of the length
L containing exactly n photons,
|
|
(4.25) |
For N ® ¥ this can be shown to be equivalent to the
Poisson distribution, where
|
|
(4.26) |
An equivalent quantum
representation of the classical coherent states can be expressed in the number
state basis as
|
|
(4.27) |
where
Since the coherent state is defined
as a superposition of individual number states, the number of photons found
upon repeated measurement will show variation as expected due to the shot
noise.
4-6 Photon Pair Generation by FWM
One standard approach to producing
a single photon source is to generate pairs of signal and idler photons via
some type of nonlinear interaction and detecting one member of the photon pair
to provide a heralding signal. The theory was based on the interaction between
classical oscillating fields. While it was posited that in the low power regime
this process could proceed in a spontaneous fashion, seeded by quantum vacuum
fluctuations, it is not clearly apparent from this treatment whether the
results for the gain of the FWM process would remain valid in this limit.
Additionally, as the pump beam was
assumed to be quasi-monochromatic, this approach does not quantitatively
describe the influence of the pump bandwidth on the widths of the generated
signal and idler, which becomes relevant for short duration pump pulses.
For describing the generation of
light by parametric processes is often simpler to work with the interaction
picture of quantum mechanics. The main difference between this approach and that
of the more commonly used Schrödinger picture is that instead of the
calculating the how the state of the system evolves over time, the time
dependence of the system is moved to the quantum mechanical operators (such as
the creation operaor
In this representation the time
evolution for any general operator
|
|
(4.28) |
|
|
|
|
|
Where
|
|
(4.29) |
Creation operators for idler or
signal photons it is possible to see how these operators vary with the time.
This then allows the expectation value for the number of photons generated in
the signal and idler modes from an initial zero-photon number state,
|
|
(4.30) |
Where
|
|
(4.31) |
And
|
|
(4.32) |
Here
The classical pump electric field in int
|
|
(4.33) |
With
|
δ= ( |
(4.34) |
The bandwidth pump pulse centred on a frequency
|
ò |
(4.35) |
In order to account for the
quantum nature of light for the generated signal and idler beams,
|
|
(4.36) |
where the subscript α denotes the
signal or idler,
These terms can be grouped together into a single function f, which is
dependent on the generated signal and idler wavelengths according to
|
|
(4.37) |
Where
4-7 Spectral Filtering
One way in which a single photon
source can be realised is by taking advantage of some form of parametric
interaction to generate signal and idler photon pairs and herald the presence
of one through the detection of the other. In order to observe high visibility
interference effects using photons produced in this manner it is necessary to
ensure that the heralded single photon is in a pure quantum state.
The effect of filtering can also be
seen in the altered form of the JSA. Narrowband filtering of either the signal
or idler photons can be incorporated into the JSA picture by multiplying the
result by an appropriate filter function. As the filtering bandwidth tends to
zero, the resultant JSA function becomes progressively more symmetrical, such
that the degree of correlation between the generated signal and idler
wavelengths is reduced. An
example of this can be seen in Fig. 3.6.
Figure 4-2 : Demonstration of the effect of
filtering on the shape od the JSA function these results were calculated by
applying energy-matched Gaussian filter profiles to the JSA shown.The filtering
FWHM bandwidths for signal and idler were (a) 3 nm and 12 nm respectively, and
(b)0.5 nm and 2 nm. [3]
Demonstration of the effect of filtering on the shape od the JSA funton these results were calculated by applying energy-matched Gaussian filter profiles to the JSA shown.The filtering FWHM bandwidths for signal and idler were (a) 3 nm and 12 nm respectively ; and (b) 0.5 nm and 2 nm
The visibility of HOM interference that can be observed
at a beam-splitter, between the heralded photons from two separate
parametric-based sources with narrowband filtering, depends on both the chosen
filtering width and the accuracy with which the filter central frequencies are
matched. For each individual source the two-photon wave function for the signal
and idler pair before the heralding photon reaches its detector and the other
photon reaches the beam-splitter.
4-8 Photon Statistics for FWM
One of the key requirements of a
single photon source is that only one photon should be output upon operation.
The concept of heralding was introduced for sources based on the generation of
time-correlated pairs of photons. Ignoring pulses of the laser in which no
pairs are generated allows sources of this type to approach the pure number
state of an idealised source by eliminating the vacuum component of the
generated state.
When both the accidentals count
rate is low and the heralding fidelity is high, the heralded state approaches
that of a true single photon. Due to the limiting effect of the higher-order
photon number terms on the source performance it is important to quantify how
the number distribution of generated pairs varies with the average pair
generation rate. When observed over a sufficiently long time period, the
distribution of the number of photon pairs generated by FWM on each pulse of a
pump laser is found to follow a Poisson distribution.However, on short time
scales the number distribution for spontaneous emission processes deviates from
the Poisson distribution and is instead found to follow a thermal distribution
, where the number of photon pairs within the measurement period is given by
|
|
(4.38) |
|
|
(4.39) |
|
|
|
|
|
|
|
(4.40) |
|
Where Χ is a function
that incorporates both the peak pump power and the appropriate nonlinear
susceptibility coefficient of the interaction. Solving for the state of the
system shows that the distribution of the number states obeys exactly only when
the photons are constrained to be generated in a single mode, and tends towards
the Poisson distribution as the number of available modes becomes large , can
describe both χ (2) and χ (3) processes
with an appropriate choice of Χ , this result can be applied to both.
The Poisson distribution is
commonly assumed for the light generated by FWM. However, for the case of a
single photon source the goal is to try and achieve emission into single mode,
where this distribution may no longer be valid. The statistics of the photon
number.
Distribution can be classified
according to the value parameter
|
|
(4.41) |
For pulsed light sources the
Light with
Figure 4-3 :
Illustration of the statistical distribution of photons of different of the
second-order coherence parameter
Pairs generated by the nonlinear
process of spontaneous parametric down-conversion (PDC) has previously been
studied experimentally.The timing information for coincidences between the two
detectors allowed
The photon number distribution will
also clearly affect the HOM visibility that can be observed by the presence of
multi-photon pair contribution to the total state vector that will not
interfere ,
for an input state of
|
|
(4.42) |
This corresponds to the state in which one of
the two photon sources providing an input to the experiment accidentally
produces two pairs of signal and idler photons on a single laser pulse. By
taking an average for the maximum visibility expected for this state and all
the other possible combinations of states where the sources produce less than
three pairs of photons, weighted according to the probability of generating
each state, the overall HOM visibility is shown to be
|
|
(4.43) |
Where
Chapter 5
5-1 Experimental
setup
A simplified experimental setup
is shown in Figure 4-1. Femtosecond laser pulses were used as the pump field,
with the possibility of frequency tuning in a wide range. Additional spectral
selection of the output radiation using a diffraction grating made it possible
to vary the duration of the pump pulses in the range from 200 fs to 2 ps. After
passing through the fiber, the pump and idle wave were filtered out using
dichroic mirrors. The signal wave was recorded by a spectrometer, with the
possibility of working also in the monochromator mode. When conducting
measurements in the monochromator mode, a single-photon detector was installed
on the output slit of the device. We studied fiber of the SC-5.0-1040 brand
with zero dispersion near 1025 nm.
Figure 5-1 : Optical design of
the experimental setup.
The following notation is introduced in the figure: FI - Faraday insulator;
OPO - optical parametric oscillator; SP - spectrometer necessary for monitoring
and stabilizing the frequency of radiation from OPO; TDC - time - digital
converter; BPF850 - bandpass filter at 850 nm; BPF1270 - 1270 nm fiber-band
filter (3 pieces), DM950 - dichroic mirror, SPAD # 1 - single-photon detector
on a Si avalanche diode (visible range); SPAD # 2 - single-photon detector on a
Si avalanche diode (visible range); NM1064 - a 1064 notch mirror for cutting
off pumping; MS is a microstructured fiber.
5-1 Spectrometer
Measurement
а)
Figure 5-2 : Dispersion curves for the studied fiber
[Petrovnin K. V. et al. Broadband quantum light on a fiber-optic platform: from
biphotons and heralded single photons to bright squeezed vacuum // Laser
Physics Letters. - 2019 .-- T. 16. - No. 7. - S. 075401.].
a) Modeled group velocity dispersion for the main
fiber mode. The fiber parameters dcore = 4.81 μm, the air-filling ratio d / λ
is 0.45.
b) Calculated phase matching cards. The red line
corresponds to the wavelength for the idle wave, the black line corresponds to
the wavelength for the signal wave.
Figure 5-2 shows the dispersion curves for the studied fiber
[Petrovnin K. V. et al. Broadband quantum light on a fiber-optic platform: from
biphotons and heralded single photons to bright squeezed vacuum // Laser Physics
Letters. - 2019 .-- T. 16. - No. 7. - S. 075401.]. As can be seen from the
curve with a small decrease in the pump wavelength near the dispersion zero
(1026 nm), the signal lines undergo a significant shift to the anti-Stokes
region. It was in this area that the study was conducted. With a detuning of
the pump wavelength of about 1024 nm and lower, a signal of the FWM signal was
detected. A series of measurements with a step of 1 nm and at different powers
is presented in Fig. 4-3. The centers of the measured signal lines obtained
during measurements in the spectrometer mode coincide with those calculated on
the synchronism maps for the installed pumps. It should be noted that with an
increase in the pump power, the lines do not shift along the wavelength, but
only their amplification occurs, and it is nonlinear (close to the quadratic
dependence).
The camera sensitivity was calculated and the
order of the signal line power was estimated from the calculation of the camera
exposure. The camera sensitivity S = I / (P · Exp) with a fully open slit was
0.8 · 1012 (W · ms) -1. In the formula for I, these are the intensity values
that the camera gives; P is the radiation power; Exp is the exposure of the
camera. Line capacities are estimated in fractions and units of pW. This allows
you to approximately estimate the flux of signal photons, which is about
106-107 photons / s.
а)
Figure 5-3 : Experimental
characteristics of the signal line. a) Spectra of the signal line at different
pump wavelengths. b) The dependence of the signal line intensity on the pump
power. The pump wavelength is 1021 nm.
а)
Figure 5-4 : The dependence of the signal line intensity on
the pump power. The pump wavelength is 1021 nm.
At the monochromator detector,
the spectra of radiation emerging from the fiber were recorded, depending on
the pump radiation wavelength λp. The radiation power at the
fiber output was constant and amounted to 4±0,5 mW. The spectral width
5-2 Monochromator
Measurement
The introduction of radiation
directly into the entrance slit of the monochromator made it possible to avoid
aberrations and changes in the beam profile at the exit slit, due to which the
radiation was well introduced into the fiber and then into the single-photon
detector (50% for multimode and 30% for single-mode fiber). After a successful
introduction into the fiber at the exit from the monochromator, a setup was set
up to search for the signal line of the FWM when measuring the monochromator
mode (Fig. 4-1). The signal filtering scheme with dichroics and rotated notch
filters (for pump cut-out) provides attenuation at a pump wavelength of about
-70-80 dB at a pump wavelength and 30-40 dB at wavelengths from 950 nm and
higher. The input pump pulse parameters are a half-width of 2 nm, power from 1
mW -4.5 mW.
Figure 5-5 : Spectra of the
pump line from the spectrometer depending on the pump power. Signal line
spectra obtained with a single-photon detecto
The measurement was performed
without a stretcher at a pump wavelength of 1020 nm. The radiation at the fiber
exit was monitored using a spectrometer (25 ms, 20 arr.) And a power meter
(wavelength 1015 nm). The mods in the fiber were further mixed by vibration.The
radiation reflected from two dichroic mirrors (950 nm) passed through a notch
filter (1064 nm), then it was fed into a monochromator and was detected by a
single-photon detector at the exit from it. Spectra were measured on my new LV
program.
Figure 5-5 shows the spectra of the pump line from the spectrometer
depending on the pump power. The figure shows that with increasing power, the
spectral pump line begins to broaden and bifurcate due to the FSM. Figure 2
shows the spectra obtained with a single-photon detector. We also see a
noticeable broadening of the signal line and its bifurcation. The 780 nm line
is illuminated by a Ti: Sa pump laser. The asymmetry of the signal line is
associated with the use of dichroic mirrors, which additionally cut off
radiation above 950 nm.
Conclusion General
Non-linear effects, it is also necessary to
ensure that the gain fiber does not exhibit photons, that the quality of the
beam is limited by diffraction and that the thermal load is managed adequately.
One of the conventional techniques used to attenuate non-linear effects is to
decrease the length of the cavity.
However, this involves increasing the
absorption of the fiber to extract the same power and increasing the thermal
load. Another method to decrease nonlinear effects is to increase the
diameter of the core of the gain fiber. In this case, however, it becomes
difficult to design a fiber that is both single-mode (or operable in a
single-mode regime) and not sensitive to photocuring due to the constraints
imposed by phosphorus on the index profile. It is, therefore, necessary to find
a compromise between the different parameters to satisfy both the different
constraints of the laser.
Among the three processes that we have
presented in this chapter, the four-wave mixing process is at the heart of our
study: it is by this mechanism that the correlated photon pairs useful for
quantum telecommunications will be generated.
Raman scattering, on the other hand,
constitutes the very problematic of our work: as the main limitation of the
quantum performances of the sources of pairs of correlated photons based on a
fibered architecture with a silica core, this is the process we are looking for
to free us by proposing a fiber-based architecture with a liquid core. As we
mentioned earlier, the Raman spectrum of liquids is generally very different
from that of very fine silica rather than a very wide continuum, and with a
generally large Raman offset.
These particular properties will allow, by
playing on the microstructure geometry of the fiber and on the linear
refractive index of the liquid, to reject the main Raman lines of the liquid
outside the fiber transmission band (thus freeing itself of the majority of
Raman photons, which can simply no longer be transmitted in the core of the
fiber), and to position the Raman lines of lower intensity in spectral ranges
where the four-wave mixing process cannot take place (so that the Raman photons
will be generated at wavelengths different from the signal and idler photons,
and can, therefore, be filtered).
The theory of the generation of light at new frequencies by a nonlinear
interaction in a PCF was discussed, the focus there was on the regime of high
pump power. The theory relating to the underlying nature of the generated light
fields, which becomes apparent when a low peak power pulsed pump source is used
and only a single pair of signal and idler photons are produced on a given
pulse of the pump laser.
Symbol list
|
Symbol |
Definition |
Units |
|
|
Effective area |
|
|
A(z,t) |
Electric field envelope |
|
|
a |
Ray of the heart |
|
|
α |
Fund losses |
dB/km
ou |
|
|
Pump
absorption |
dB/m |
|
|
Group dispersion setting |
S2 / m |
|
c |
Speed of light in a vacuum |
m / s |
|
D |
Fiber diameter |
|
|
|
Filter bandwidth |
nm |
|
|
Separation of longitudinal modes |
Hz |
|
|
Spectral widening caused by time shift |
Hz |
|
|
Average value of the amplitude of the index modulation |
- |
|
|
Electric field |
V / m |
|
|
Vacuum permittivity |
F / m |
|
|
Modal recovery |
- |
|
|
Raman fraction |
- |
|
|
Nonlinear coefficient |
m / W |
|
|
Raman gain coefficient |
m / W |
|
|
Effective Raman gain coefficient |
m / W |
|
|
Material frequency response |
1 / Hz |
|
|
Vibrational response of the material |
1 / s |
|
H |
Planck constant |
𝑚2𝑘𝑔/𝑠 |
|
𝐼 |
Electric field strength |
W/m2 |
|
𝐿 |
Fiber length |
m |
|
𝐿𝑒𝑓 𝑓 |
Effective length |
m |
|
𝐿𝑚𝑜𝑦 |
Average length of photon path |
m |
|
𝐿𝑚𝑜𝑦,𝑔 |
Generalized average photon path length |
m |
|
|
Phase mask period |
nm |
|
𝜆 |
Wavelength |
nm |
|
𝜆0 |
Central emission wavelength |
nm |
|
𝜆𝐵 |
Bragg wavelength |
nm |
|
𝜆𝑠 |
Signal wavelength |
nm |
|
𝜆𝑝 |
Pump wavelength |
nm |
|
𝑁0 |
Concentration of active ions in the heart |
ions/𝑚3 |
|
𝑁𝑖 |
Ion population at level 𝑖 |
ions/𝑚3 |
|
𝑛 |
Number of points in the FFT |
- |
|
𝑛2 |
Nonlinear refractive index |
- |
|
𝑛 𝑐𝑜𝑒𝑢𝑟 |
Heart refractive index |
- |
|
𝑛𝑔𝑎𝑖𝑛𝑒 |
Sheath refraction index |
- |
|
𝑛𝑒𝑓 𝑓 |
Effective index |
- |
|
𝑛𝑝(𝑧) |
Photons generated according to the position |
- |
|
ON |
Digital aperture |
- |
|
𝜔0 |
Carrier frequency |
rad/s |
|
𝑃𝑜𝑢𝑡 |
Laser output power |
W |
|
𝑃𝑝 |
Pump power |
W |
|
𝑃𝑟 |
Raman Power |
W |
|
𝑃𝑠 |
Signal strength |
W |
|
𝑃𝑐𝑟 |
Critical power |
W |
|
𝑃(𝑧) |
Power distribution |
W |
|
𝜓(𝑥,𝑦) |
Modal profile |
- |
|
𝜑𝑛l |
Non-linear phase shift |
Rad |
|
𝑅 |
Reflectivity |
% |
|
𝑅(𝑡) |
Material time response |
1/s |
|
𝑅𝑅 |
Raman rejection rate |
dB |
|
𝑅𝑒𝑓 𝑓 |
Effective reflectivity |
% |
|
𝜎𝑎(𝜆) |
Effective section for absorption of active ions |
m2 |
|
𝜎𝑒(𝜆) |
Active ion emission cross section |
m2 |
|
𝜎𝑠𝑝𝑒𝑐𝑡𝑟𝑎𝑙 |
Spectral standard deviation |
nm |
|
𝑇 |
Time window length |
s |
|
𝜏 |
Life time |
s |
|
𝜏1, 𝜏2 |
Vibration damping constants |
s |
|
𝑉 |
Standardized frequency |
- |
|
𝑊𝑖𝑗 |
Transition rate stimulated between levels 𝑖 and 𝑗 |
1/s |
|
Ω |
Diameter of the fundamental mode |
𝜇m |
|
𝜒 𝑖 |
Order susceptibility tensor |
m𝑖−1/V𝑖−1 |
|
|
Annihilation operator |
- |
|
|
Creation operator |
- |
|
|
Magnetic induction operator |
- |
|
|
Magnetic induction |
- |
|
|
Number operator |
- |
Abbrviations List
|
Abbreviation |
Definition |
|
FWHM FWM C CW MCVD MFD MOPA OSA SPM PCF NA LFHP HOM XPM ZDW |
Full Width at
Half Maximum Four Wave Mixing Carbon dioxide Continuous
operation (Coutinous Wave) Modified chemical
vapor deposition Modal Field
Diameter (Mode Field Diameter) Master oscillator
for power amplification Optical spectrum
analyzer Self Phase
Modulation Photonic Crystal
Fiber Numerical
Aperture High power fiber
laser Hong-Ou-Mande Cross-phase
modulation Zero
dispersion wavelength |
References
·
Alex Robert , McMillan DEVELOPMENT OF AN ALL-FIBRE SOURCE
OF HERALDED SINGLE PHOTONS , University of Bath Department of Physics September
2011
·
Raid Daoud ,Genetic Approach Based Design of
Dispersion-free Optical Fiber ,Jan 2009
·
Kitayama, K. (2014). Light propagation in optical fibers.
In Optical Code Division Multiple Access: A Practical Perspective (pp. 65-106).
Cambridge: Cambridge University Press. doi:10.1017/CBO9781139206914.005
·
Jean-Simon, DESIGN OF A SOURCE OF ADVERTISED PHOTONS
GENERATED IN CORBIL MICROSTRUCTURED FIBER,
MONTREAL UNIVERSITY, 2010
·
Rim CHERIF, Study of Non-Linear Effects in Photonic
Crystal Fibers, University of November 7
in Carthage The Higher School of Communications of Tunis
·
Margaux Barbier, Generation of correlated photon pairs by
spontaneous four-wave mixing in liquid core-microstructured fibers, 2014.
Other :
·
Jean-Charles Beugnot. Brillouin scattering in
microstructured optical fibers. Physics. University of Franche-Comté, 2007.
·
Maxime Droques. Study of the modulatory instability
process in optical fibers presenting a periodic dispersion profile. University
of Lille, 2013.
·
S. Coen, A. Lun Chau, R. Leonhardt, J. Harvey, J. Knight,
W. Wadsworth, and P. Russell, “Supercontinuum generation by stimulated Raman
scattering and parametric four-wave mixing in photonic crystal fibers,” J. Opt.
Soc. Am. B 19, 753 (2002).
·
J. Dudley, G. Genty, and S. Coen, “Supercontinuum
generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135 (2006).
·
H. Kano and H. Hamaguchi, “Characterization of a
supercontinuum generated from a photonic crystal fiber and its application to
coherent Raman spectroscopy,” Opt. Lett. 28, 2360 (2003).
·
T. Ohara, H. Takara, T. Yamamoto, H. Masuda, T. Morioka,
M. Abe, and H. Takahashi, “Over-1000-channel ultradense WDM transmission with
supercontinuum multicarrier source,” J. Lightwave Technol. 24, 2311 (2006).
·
J. Hansryd, P. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist,
“Fiber-based optical parametric amplifiers and their applications,” IEEE J.
Select. Topics Quantum Electron.8, 506 (2002).
·
K. Wong, K. Shimizu, M. Marhic, K. Uesaka, G.
Kalogerakis, and L. Kazovsky, “Continuous-wave fiber optical parametric
wavelength converter with +40-dB conversion efficiency and a 3.8-dB noise
figure,” Opt. Lett. 28, 692 (2003).
·
Q. Lin, R. Jiang, C. Marki, C. McKinstrie, R. Jopson, J. Ford,
G. Agrawal, and S. Radic, “40-Gb/s optical switching and wavelength
multicasting in a two-pump parametric device,” IEEE Photon. Tech. Lett. 17,
2376 (2005).
·
P. Russell, “Photonic-crystal fibers,” J. Lightwave
Technol. 24, 4729 (2006). F. Benabid and P. Roberts, “Linear and nonlinear
optical properties of hollow core photonic crystal fiber,” J. Mod. Opt. 58, 87
(2011).
·
G. Agrawal, Nonlinear Fiber Optics (Academic Press,
2007), 4th ed.
·
D. Burnham and D. Weinberg, “Observation of simultaneity
in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84
(1970).
·
C. Hong and L. Mandel, “Theory of parametric frequency down-conversion
of light,” Phys. Rev. A 31, 2409 (1985).
·
S. Friberg, C. Hong, and L. Mandel, “Measurement of time
delays in the parametric production of photon pairs,” Phys. Rev. Lett. 54, 2011
(1985).
·
W. Tittel and G. Weihs, “Photonic entanglement for
fundamental tests and quantum communication,” Quantum Inf. Comput. 1, 3 (2001).
·
S. Freedman and J. Clauser, “Experimental test of local hidden-variable
theories,” Phys. Rev. Lett. 28, 938 (1972).
·
E. Fry and R. Thompson, “Experimental test of local
hidden-variable theories,” Phys. Rev. Lett. 37, 465 (1976).
·
Z. Y. Ou, C. Hong, and L. Mandel, “Violations of locality
in correlation measurements with a beam splitter,” Phys. Lett. A 122, 11
(1987).
·
Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality
and classical probability in a twophoton correlation experiment,” Phys. Rev.
Lett. 61, 50 (1988).
·
Aspect, P. Grangier, and G. Roger, “Experimental tests of
realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47, 460 (1981).
·
J. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic
quantum technologies,” Nature Photonics 3, 687 (2009).
·
J. Fulconis, O. Alibart, J. O’Brien, W. Wadsworth, and J.
Rarity, “Photonic crystal fiber source of photon pairs for quantum information
processing,” arXiv :quant-ph/0611232v1 (2006).
·
N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum
cryptography,” Rev. Mod. Phys. 74, 145 (2002).
·
T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A.
Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84,
4729 (2000).
·
Y.-H. Kim, S. Kulik, and Y. Shih, “Quantum teleportation
of a polarization state with a complete Bell state measurement,” Phys. Rev.
Lett. 86, 1370 (2001).
·
Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré,
and N. Gisin, “Distribution of time-bin entangled qubits over 50 km of optical
fiber,” Phys. Rev. Lett. 93, 180502 (2004).
·
H. Hübel, M. Vanner, T. Lederer, B. Blauensteiner, T.
Lorünser, A. Poppe, and A. Zeilinger, “High-fidelity transmission of
polarization encoded qubits from an entangled source over 100 km of fiber,”
Opt. Express 15, 7853 (2007).
·
M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda,
W. Boxleitner, T. Debuischert, E. Diamanti, M. Dianati, J. Dynes, S. Fasel, S.
Fossier, M. Fürst, J.-D. Gautier, O. Gay,
·
N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel,
H. Hübel, G. Humer, T. Länger,
·
M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N.
Lütkenhaus, A. Marhold, T. Matyus,
·
O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe,
E. Querasser, G. Ribordy,
·
S. Robyr, L. Salvail, A. Sharpe, A. Shields, D. Stucki,
M. Suda, C. Tamas, T. Themel,
·
R. Thew, Y. Thoma, A. Treiber, P. Trinkler, R.
Tualle-Brouri, F. Vannel, N. Walenta,
·
H. Weier, H. Weinfurter, I. Wimberger, Z. Yuan, H. Zbinden,
and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New
J. Phys. 11, 075001 (2009).
·
V. Scarani, H. Bechmann-Pasquinucci, N. Cerf, M. Dusek,
N. Lütkenhaus, and M. Peev, “The security of practical quantum key
distribution,” Rev. Mod. Phys. 81, 1301 (2009).
·
S. Tanzilli, W. Tittel, H. de Riedmatten, H. Zbinden, P.
Baldi, M. de Micheli, D. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum
communication,” Eur. Phys. J. D 18, 155 (2002).
·
V. Berger and J.-M. Gérard, “Sources semiconductrices de
photons uniques ou de photons jumeaux pour l’information quantique,” C. R.
Physique 4, 701 (2003).
·
R. A. Millikan, “A Direct Photoelectric Determination of
Planck's “h”,” Phys. Rev. 7, 355 (1916).
·
M. Planck, “On the Law of Distribution of Energy in the
Normal Spectrum,” Annalen der Physik 4, 553 (1901).
·
T. H. Maiman, “Stimulated Optical Radiation in Ruby,”
Nature 187, 493 (1960).
·
P. A. Franken, A. E. Hill, C. W. Peters and G. Weinreich,
“Generation of Optical Harmonics,” Phys. Rev. Lett. 7, 118 (1961).
·
C. S. van Heel, “A New Method of transporting Optical
Images without Aberrations,” Nature 173, 39 (1954
Figure References
[1] Raid Daoud ,Genetic Approach Based Design of
Dispersion-free Optical Fiber ,Jan 2009
[2] Kitayama,
K. (2014). Light propagation in optical fibers. In Optical Code Division
Multiple Access: A Practical Perspective (pp. 65-106). Cambridge: Cambridge
University Press. doi:10.1017/CBO9781139206914.005
[3] Alex Robert
, McMillan DEVELOPMENT OF AN ALL-FIBRE SOURCE OF
HERALDED SINGLE
PHOTONS , University of Bath Department of Physics September 2011
[4] Jean-Simon,
DESIGN OF A SOURCE OF ADVERTISED PHOTONS GENERATED IN CORBIL MICROSTRUCTURED
FIBER, MONTREAL UNIVERSITY, 2010
[5] Rim CHERIF,
Study of Non-Linear Effects in Photonic Crystal Fibers, University of November 7 in Carthage The
Higher School of Communications of Tunis
[6] Margaux
Barbier, Generation of correlated photon pairs by spontaneous four-wave mixing
in liquid core-microstructured fibers, 2014.