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.  and  and the refractive indices of the core and cladding materials respectively.

  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  exceeds the critical angle then total internal reflection (TIR) will occur and the light will remain confined to the fiber core. The critical angle is defined by

c = sin -1

(1.2)

   Where and are the refractive indices of the cladding and core respectively. As long as the initial launch angle  of the light is close enough to the fiber axis to allow TIR, light can propagate along the entire length of the core in this manner. In order to understand how the wave nature of light influences the guidance properties of the fiber it is necessary to solve Maxwell’s equations for this step-index fiber geometry. For a dielectric non-magnetic medium such as silica

× E =  (µ0 H)

(1.3)

                   And

× H =  (ε0 E + P)

(1.4)

  Where  and  are the electric and magnetic field vectors, and  are the permittivity and permeability of free space and  is the induced electric polarization  of the material.

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

2 (r,ω) + n2 (ω)   (r,ω) = 0

(1.5)

   Where  is the angular frequency of the light wave,  is the frequency dependent refractive index of the medium, (r,ω)  is the Fourier transform of the electric field vector  and c is the speed of light in a vacuum . The cylindrical polar coordinates ) it can be solved by a separation of variables to find the general solution for the z-component of the electric field, which is given by

z = A(ω)F(ρ)exp(im )exp(i )

(1.6)

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

+  + (n2  – 2–    ) F = 0

(1.7)

  Where  is the free-space wavenumber,  is an integer and b is the propagation constant. Each value of the integer  gives several potential values for the propagation constant . Each of these solutions represents a potential guided mode that is supported by the structure, the transverse spatial profile of light launched into a single mode of the fibre will be constant as it propagates, apart from a phase shift given by the factor  Comparing the role of b here to the simple ray optics case that was considered previously, it is apparent that the propagation constant is analogous to the component of the wavevector  in the direction of propagation. This allows an effective index of the mode, , to be defined according to

(ω) = neff (ω)

(1.8)

  Where n_eff is dependent on the angular frequency of the wave . For each potential value of the corresponding mode is designated  (similarly   modes can also be found by considering solutions for the z-component of the magnetic field . For modes to be guided by the structure, the propagation constant b must lie within the range.

  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  is the relative strength of the  th resonance occurring at angular frequency  , and the sum extends over all nearby contributing resonances. The parameters  and  are obtained experimentally and are well known for silica glass. When considering the propagation of a pulse in a medium, consisting of a spread of frequencies, it is useful to consider the group velocity at which the pulse envelope will travel, for which the group index ng can be defined as

Ng =  = c  = n( ) =

(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

=  + (  - )  +  (  - )2  + …

(1.11)

     

 Where

=       (m =1,2,…)

(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  with respect to ω) is responsible for group velocity dispersion, where

 

(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  , however, these higher order terms become important for the phase-matching of nonlinear processes close to the zero-dispersion wavelength of the fibre, where  .

  The dispersion profile of a fibre is commonly described using the dispersion parameter , which is related to the group velocity dispersion coefficient  by

 

D =

(1.15)

  at a wavelength of .  is normally given in units of ps , as this then indicates the expected temporal broadening, per nanometre of pulse bandwidth, after propagation through a kilometre of the fibre.

 

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  and are the dispersion contributions from the material and waveguiding respectively. For standard step-index fibres with a low index contrast between the core and cladding, the waveguide contribution to the dispersion is small.

nCL k0  nCO k0

(1.17)

 

  It is common to express the modes of the waveguide instead in terms of linearly polarised modes (  modes) that can be found from a superposition of the HE and EH 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 , which defines how many modes will be guided by the structure at a specific wavelength. For a given index contrast,  can be defined as

 

V = k0α

(1.19)

 

  For a wavelength of light λ, where  = 2π/λ and  is the fibre core radius. As the V value is reduced, such as by considering fibres with smaller cores or by working at longer wavelengths, the effective mode index for all the fibre modes is reduced.       Outside the core the Bessel function shape is combined with an exponential decay that dominates the mode field profile far from the core. This can be approximated with quite high accuracy as a Gaussian profile, so that in Cartesian coordinates

 

F (x, y)  exp [-(x2 + y2) / w2]

(1.20)

  

  Where  is the  width of the Gaussian profile for the fundamental mode. The consideration of the mode field diameter is also important when considering guidance in photonic crystal fibres and the dispersion properties of fibres .

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 A_eff, 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  and is defined to be

 

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 ( ), which is given by

 

LB =	(1.23)

 Where  and  are the propagation constants for the fast and slow fibre axes. The beat length is a measure of the strength of the birefringence, and is shorter for more strongly birefringent fibres. Linearly polarised light aligned with either the fast or slow axis will not experience a change in polarisation as it propagates.

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 =  log  dB/km

(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  is launched into a fibre, after propagation through a distance L the expected remaining power  is related to.

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  and the diameter of the holes . These opto-geometric parameters make it possible to define the ratio  corresponding to the proportion of air present in the fiber.

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  is given by 

=	(2-1)

  Where  is the permittivity of free space. For linear, isotropic, homogeneous materials.

P =  x	(2.2)

   Where  is the electric susceptibility of the material, related to the refractive index of the material via the relative permittivity of the material by

n =  =		(2.3)

  This linear dependence of  on the electric field , when considering the linear propagation properties of fibres. In the case of intense incident electric fields, the polarisation of the material is no longer able to respond in a linear fashion to the applied field. As the functional dependence of on the electric field intensity is generally not known,

it can be approximated by a Taylor expansion, giving

 

P = 0(x(1) .E + x(2) : EE + x(3) : EEE + …

(2.4)

     

   Where x⁽ⁱ⁾ is the  order susceptibility. For materials with inversion symmetry, such as silica glass, the second order susceptibility , as this would otherwise imply a preferred direction for the material polarisation irrespective of the direction of the applied field. Since the nonlinear response is dominated by the lowest order non-zero term, the third order  susceptibility is responsible for most of the nonlinear effects that can be observed in optical fibres, and higher order terms in the Taylor expansion can be disregarded. (  is often referred to as the Kerr nonlinearity. As the response time of (  is typically , its effect on an input light field is usually modelled as being instantaneous .

  The linear expression for the refractive index given in can be modified to account for the addition of the   is given by

 

(ω,|E|2) = n2|E|2

(2.5)

 

  Here  is the linear component of refractive index described previously, while the second term is the nonlinear contribution to the refractive index that is dependent on the intensity of the of the light in the fibre. n₂ is the nonlinear index coefficient and is related to  by

 

n2 = Re(  )    (m2/w)

(2.6)

   Where  is the component of the nonlinear susceptibility that relates to the nonlinear interaction of co-polarised waves. The consequences of the nonlinear contribution can be seen by considering the difference in the relative phase experienced for an intense pulse as it propagated through the nonlinear medium, compared to that of a lower intensity pulse where the nonlinear component of the phase shift is negligible.

A linearly polarised wave of frequency propagating in the z-direction through a material with the refractive index profile given by is described by the equation

E=  + c.c

(2.7)

Where     k_0a =  = , I |E_a|²,       and c.c. is the complex conjugate.

  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

 

Leff = [1-exp(-αL)]		(2.8)

   After propagation through a length of fibre , the additional nonlinear phase shift experienced for a high intensity pulse is therefore given by

 

= -n2k0aLeffI(t)

(2.9)

  

  This effect is known as self-phase modulation ( ), the intensity dependence of the nonlinear material response leads to a change in a light pulse as it propagates that is related to the initial pulse profile. For a pulse, where the intensity is time dependent, this leads to a time dependent shift in the instantaneous frequency across the pulse

=  = -n2k0aLeff

(2.10)

 

  Where  is the shift in the instantaneous frequency away from the central carrier frequency .Since the frequency shift is related to the slope of the pulse intensity, the effect is most pronounced for high peak power, short duration pulses. The frequency shift is positive for decreasing I, which leads to new blue-shifted frequency components being continuously generated near the trailing edge of the pulse as it propagates. Conversely, red-shifted components are generated at the leading edge.

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  becomes sufficiently large .The reduction of peak power at the central wavelength reduces the useful power available for further nonlinear processes when using such a laser as a pump source.

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:

 

-    = µ

(2.11)

 

  Where  is the linear refractive index of the medium. This equation, therefore, takes the form of an inhomogeneous wave equation, with a second non-zero member which acts as a source term. The solutions of this equation are expressed as the sum of a free wave, general solution of the homogeneous wave equation and of a forced wave, particular solution of the inhomogeneous equation, induced by the source term . We will assume below that the electric field and non-linear polarization can be represented by the discrete sum of all of their frequency components, according to

(t) =  ( n) nt	(2.12)
(t)=  )	(2.13)

 In the frequency domain, the propagation equation takes the following form:               

Δ ( ) + ( ) = µ0 NL ( )

(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 ( ) and PNL ( ) can be written in the form:

(z ) = A(z ) z	(2.15)
NL (z ) = NL (z ) F(ω)z	(2.16)

 

  Where  and    are respectively the modules of the wave vectors of the free wave and the forced wave at the frequency w, and  and  are the envelopes of the field which oscillates at the frequency w and of the non-linear polarization which radiates at the frequency .  Thus, in terms of envelopes, and within the framework of the slowly variable envelopes approximation, the nonlinear propagation equation  is written:

=  [ NL (z ) .  (ω)z

(2.17)

  With   represents the phase mismatch between the free wave and the forced wave.  In the case where  can be considered as independent of Z (parametric approximation), equation (2.11) above is very simple to solve.  Taking as initial condition ,

and recalling that the intensity of the wave at w is expressed

I(z ) = 2n0(ω) 0c | A(z )|2

(2.18)

as we obtain:

I(z )  =   | NL(ω) . |²  z) z²	(2.19)

                                        With        sin(x) = sin(x )/ (x)

  When the phase mismatch    between the free wave and the forced wave is not zero,  follows a sinusoidal evolution which can be interpreted as a succession of interference states alternately constructive and destructive between the free wave and the forced wave.  On the other hand, if the free wave and the forced wave are in phase agreement the interference is always constructive, and  increases quadratically with z (as long as we remain within the framework of the parametric approximation). The wave at frequency  can therefore only be generated efficiently if  is very weak.

  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  and will yield part of their energy to the waves at  and , whose initial intensity is very weak, or even zero. More precisely, the situation which interests us involves a single wave of high intensity (called pump wave) at the frequency , and we speak in this case of a four-wave degenerate mixture in frequency. The two waves generated and/or amplified are called signal wave (at frequency ) and complementary wave, or idler (at frequency ) .

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 . Of course, for a given pump frequency ωp, an infinity of frequency couples (ωs; ωi) makes it possible to satisfy this condition. But, as already mentioned above, the process is only effective if the phase tuning condition is also respected. This phase tuning condition is nothing other than a condition of conservation of the wave vectors, or of the pulse, as I will detail below.

  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:

(z t) = p(z t) + 8(z t) + i(z t)

(3.1)

We assume that the three waves have the same state

j(z t) = ( j ) j + c.c.)   = ( j ) k ; z- ω ,t) + c.c.)

(3.2)

With   or .

    From part to equation (3), we can calculate the expression of the complex amplitudes of the nonlinear 3 polarization  at frequencies  and . let's start with the component at

(3) ( p ; z) = є0 (3) ( ωp ; ωp ; ωp ; -ωp) |E ( p ; z)|2  E( p ; z)	(3.2.1)
+ 6є0 (3)  ( ωp ; ωp ; ωp ; -ω8) |E ( 8 ; z)|2  E( 8 ; z)	(3.2.2)
+  6є0 (3)  ( ωp ; ωp ; ωp ; -ω8) |E ( 8 ; z)|2  E( 8 ; z)	(3.2.3)
+ 6є