Titre: Mesures et Simulations D'impédance Longitudinale et Transverse pour ThomX Mots clés:Générateurs de rayons X,Accélérateurs d'électrons

Résumé:Les études sur les champs de sillage et l'impédance sont d'une importance capitale pour atteindre la qualité du faisceau de particules chargées, notamment dans le contexte des accélérateurs à fort courant crête et à haute luminosité. Leur effet sur la dynamique du faisceau a été étudié par différentes méthodes afin de réduire le budget d'impédance de l'ensemble de l'accélérateur et d'atteindre la qualité de faisceau attendue depuis de nombreuses années. En raison de la structure complexe des pièces de l'accélérateur, il n'est pas possible d'appliquer des calculs théoriques pour définir les impédances et les effets des champs de sillage. Les mesures au banc et les simulations électromagnétiques étaient nécessaires pour étudier le bilan d'impédance de l'ensemble de l'accélérateur.

En outre, les informations générales sur les mesures à fil et la conception du banc de mesures seront présentées pour les différentes mesures : à un fil, deux fils et le fil mobile pour obtenir les impédances longitudinales, dipolaires et quadrupolaires. La spécification du banc pour réduire le bruit sera brièvement expliquée. La comparaison des résultats avec la simulation électromagnétique sera présentée pour chaque pièce de l'accélérateur pour la ligne de transfert ThomX et l'anneau de stockage avec les études d'erreur. Enfin, l'effet de l'impédance totale sur l'accélérateur avec les calculs du kick factor sont résumés.

Title: Longitudinal and Transverse Impedance Measurements and Simulations for ThomX Keywords: X-Ray generation, electron accelerator,wakefields,impedance Abstract:The wakefields and impedance studies are of paramount importance to reach the charged particle beam quality especially in the context of high peak current and high luminosity accelerators. Due to the complex structure of the accelerator pieces, it is not possible for applying theoretical calculations for defining the impedances and wakefields effects. The bench measurements and electromagnetic simulations were needed for studying the impedance budget of the whole accelerator. This thesis will focus on the transverse impedances study on ThomX accelerator pieces with both bench measurements and simulations. However, the both bench measurements and simulations have its own problems which is needed to be double check the results to minimize the error. Initially, the information about the theoretical background of the collective effects especially impedances and the wire measurements will be given with the wakefield and impedance formalism. The electromagnetic simulation parameters will be studied carefully and the results will be compared with the theoretical calculations of the well known pieces. Moreover, the general information about the wire measurements and the designed setup will be presented for one wire, two wire and the moving wire measurements for obtaining the longitudinal, dipolar and quadrupolar impedances. The specification of the bench for reducing the noise will be explained briefly. The comparison of the results with the electromagnetic simulation will be presented for each accelerator piece for ThomX transfer line and the storage ring with the error studies. Finally, total impedance effect on the accelerator with kick factor calculations were shown in this work.

5

To my mother and father... This thesis would not have been possible without the support and contribution of many people.Therefore, they should all be thanked individually.

First of all, I would like to thank Frank Zimermann and Thomas Thuiller for accepting to be rapporteurs for this thesis. Also I thank Fabian Zomer, Benoit Salvant, Alexis Gamelin for their participation in the jury.

The most importantly it is a genuine pleasure to express my thanks and gratitude to Christelle Bruni who always supported me both scientifically and humanely. She always gave her best to whatever she is doing and she always guided me by having an incredible curiosity in her head and a wonderful heart inside. This work become the reality with the support of many people within the IJ-CLab.Thanks to Damien Le Guidec for his important contribution to the impedance measurement bench and also his support as a friend. I would also like to thank to Jean-Noel Cayla and Vincent Chaumat for their help in all the measurements.

All in all, thank to all the other members, doctoral and post-doctoral students of IJCLab for their support and the good times we shared. I would like to thank my friends who have assisted me in my academic and personal life. I especially thank Can who had always listened to me and been by my side when i needed.

Finally, I would like to thank my parents, Ümmühan and Yılmaz Ergenlik, who have always been there for me, supported me and kept my happiness above everything. Their love and support made me who I am and I will always be grateful for it.

i

Abstract

The wakefields and impedance studies are of paramount importance to reach the expected beam quality of the charged particle, especially in the context of high peak current and high luminosity accelerators. Their effect on beam dynamics have been studied using different methods to reduce the impedance budget of the entire accelerator, and then to achieve the expected beam quality for many years.

Due to the complex structure of the accelerator components, it is not possible to apply theoretical calculations to define the impedance and the wakefield effects.

The bench measurements and the electromagnetic simulations were needed to study the impedance budget of the entire accelerator. This thesis will focus on the transverse impedance study of the ThomX accelerator components with both the bench measurements and the simulations. However, both the bench measurements and the simulations have their own problems, necessitating the double-checking of the results to minimise the error. Initially, information about the theoretical background of the collective effects, especially impedances and wire measurements, will be given in wakefield and impedance formalism. The electromagnetic simulation parameters will be studied carefully and the results will be compared with the theoretical calculations of the wellknown components. The reliability and the limit of the simulations will be discussed.

Moreover, general information about wire measurements and their designed setup will be presented for single-wire, two-wire, and moving wire measurements to obtain the longitudinal, dipolar, and quadrupolar impedances. The specification of the bench measurements to reduce noise will be explained briefly. Comparison of the results to the electromagnetic simulations will be presented for each accelerator component of the ThomX transfer line and the storage ring, along with their error studies. Finally, the total impedance effect on the accelerator with kick factor calculations will be shown in this work.

Résumé

Les études sur les champs de sillage et l'impédance sont d'une importance capitale pour atteindre la qualité du faisceau de particules chargées, notamment dans le contexte des accélérateurs à fort courant crête et à haute luminosité. Depuis de nombreuses années, leur effet sur la dynamique du faisceau a été étudié par différentes méthodes afin de réduire le budget d'impédance de l'ensemble de l'accélérateur et d'atteindre la qualité de faisceau attendue. En raison de la structure complexe des pièces de l'accélérateur, il n'est pas possible d'appliquer des calculs théoriques pour définir les impédances et les effets des champs de sillage.

Les mesures au banc et les simulations électromagnétiques étaient nécessaires pour étudier le bilan d'impédance de l'ensemble de l'accélérateur.

Cette thèse se concentre sur l'étude des impédances transverses sur les pièces de l'accélérateur ThomX avec le déveoppement d'un banc de mesures et des simulations. Cependant, chaque méthodes a des problèmes inhérents et il est nécessaire de vérifier les résultats afin de minimiser les erreurs. Dans un premier temps, les informations sur le contexte théorique des effets collectifs, en particulier les impédances et les mesures à fils seront présentées avec le formalisme des champs de sillages et des impédances. Les paramètres de la simulation électromagnétique seront étudiés attentivement et les résultats seront comparés aux calculs théoriques des pièces bien connues. La fiabilité et les limites des simulations seront discutées.

En outre, les informations générales sur les mesures à fil et la conception du banc de mesures seront présentées pour les différentes mesures : à un fil, deux fils et le fil mobile pour obtenir les impédances longitudinales, dipolaires et quadrupolaires. La spécification du banc pour réduire le bruit sera brièvement expliquée. La comparaison des résultats avec la simulation électromagnétique sera présentée pour chaque pièce de l'accélérateur pour la ligne de transfert ThomX et l'anneau de stockage avec les études d'erreur. Enfin, l'effet de l'impédance totale sur 1 l'accélérateur avec les calculs du kick factor sont résumés.

List of Figures

1. [START_REF]Azimuthal beam cavity interaction[END_REF] The scheme of the ISR when the bunch length s b is larger than the emitted wavelength l on top and the CSR when the bunch length s b is smaller than the emitted wavelength l below. . . . . . . . . . . . . . 1.2 The general scheme of electric field of particle in non relativistic, relativistic and ultra-relativistic cases [START_REF] Apsimon | The Development and Implementation of a Beam Position Monitoring System for use in the FONT Feedback System at ATF2[END_REF]. . . . . . . . . . . . . . . . . . . . 1.3 The general scheme of the Coulomb repulsion in blue and magnetic attraction between two particles in green at the top, and the multiparticle cases at the bottom [START_REF] Wilson | Accelerator lecture 13, multi-particle effects: Space charge[END_REF]. When the b goes to 1 and g >> 1, the attractive and repulsive forces started to have same magnitude in the opposite direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The evolution of the electric field of the ultrarelativistic gaussian bunch along its path (z direction) going through the basic step in-out cavity.

1. [START_REF]Transmission lines and electronic signal handling[END_REF] The general scheme of the test and witness particle inside the beam pipe. Source particle emits EM field which cause a kick on the witness particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

The high energy photon collide with the electron which leads to a energy transfer from high energy photon to electron [START_REF] Venugopal | de broglie wavelength and frequency of scattered electrons in the compton effect[END_REF] . . . . . . .

2.2

The energy and brightness comparison of different machines. CCS is the acronym for Compact Compton Sources [START_REF] Jacquet | High intensity compact compton x-ray sources: Challenges and potential of applications[END_REF] . . . . . . . . . . . . .

2.6

The 3D scheme of the Thomx. Shown in the figure are its: (1) 8 dipole magnets (red), (2) 24 quadrupole magnets (blue) (3) 12 sextupole magnets (yellow), (4) septum, (5) RF cavity, (6) 2 kicker magnets, [START_REF] André | Control of laser plasma accelerated electrons for light sources[END_REF] Fabry-Perot cavity, [START_REF] Apsimon | The Development and Implementation of a Beam Position Monitoring System for use in the FONT Feedback System at ATF2[END_REF] Interaction Point Chamber (IP), [START_REF] Bane | An accurate, simplified model intrabeam scattering[END_REF] Feedback Transverse Kicker(FBT) and [START_REF] Bassetti | Closed expression for the electrical field of a two-dimensional Gaussian charge[END_REF] Virtual Interaction Point. . . . . . . .

3.1

The position of the source and witness particle(integration line) for obtaining the different impedance component with CST. . . . . . . . .

The wake potential results of Interaction Point Chamber with Indirect

Interface (red) and Indirect Test Beam (black) methods. The simulations were performed with a total mesh cell 1.2 • 10 6 and the Gaussian beam (s = 2 mm and charge = 1 • 10 9 C) . . . . . . . . . . . . . . . . . .

List of Tables

2.1

The main characteristic of the X-Ray beam of ThomX with respect to different applications. [START_REF] Jacquet | High intensity compact compton x-ray sources: Challenges and potential of applications[END_REF] . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

-Introduction

Particle accelerators today are indispensable tools not only in physics, but also in medicine (production of X-rays), and in the study of works of art (e.g. Louvre Museum has an accelerator). Accelerator physics is a field encompassing all aspects necessary from the design to the operation of their equipment. In addition to the technical design and realization of the accelerator components, understanding how charged particles move along the accelerator is crucial.

To ensure the proper operation of the accelerators, it is essential to be able to control the motion of the particles. Especially the resulting beam dynamics of charged particles concerning their designed parameters allow the targeted performances to be attained from the design to the commissioning phase. The beam dynamics study contains both the description of the individual evolution of a particle's position and momentum and the interaction of the particles with each other and their surroundings. The interaction of particles with the accelerator components and the particles themselves becomes an important topic since the beginning of this science called collective effects.

For that, I will use the collective effect formalism, as well as impedance, to describe the way the environment of the accelerator influences the particle beam, and the interactions of particles with each other. The aim is to explain how one particle in the beam influences the dynamics of the others. The collective effects are factors limiting the increase in the intensity of accelerators, in addition to having a great effect on the expected beam parameters and quality at the end of the accelerator complex. The beam intensity can be defined as the number of charges per unit time. However, the collective effects are difficult to analyze analytically with complex accelerator structures. The simulation programs and bench mea-surements are used to describe the collective effects for different components.

All in all, the impedance budget, which is the estimation of the impedance for each component, should be crucial for the accelerators. This thesis focuses on impedance, and expresses the resistive and geometrical effects of the wakefields in the frequency domain (see definitions below). The particles are accelerated under strong vacuum conditions to suppress the effects of collisions between the accelerated particles and the air molecules. The wakefields are the electromagnetic (EM) fields created by the particles circulating inside this kind of vacuum tube. More precisely, the optimization of the measurement method, which is called the coaxial wire method, is the core of this thesis. This method uses the equivalence of the EM field between the particle beam and a current that flows through a thin wire. The derivation of the geometric impedance effect will be performed as well, by checking the manufacturing results of the mechanical components.

In this first chapter, different collective effects and examples of limitations of such effects will be introduced, before going into details of wakefields and impedances.

General information about the different types of collective effects with theoretical models will be given, the methods to predict and compensate for the collective effects with both simulation and measurements will be explained. Afterwards, the introduction to the notions of impedance will be shown both theoretically and experimentally with wire measurements. Then, general information will be given about the ThomX project, along with the collective effects limiting this complex accelerator. The main components will be analyzed with wire measurements and simulations in this thesis.

. Collective Effects

In accelerators, there are two types of particle dynamics. The first one is the single-particle dynamics, which is used in the design stage of the accelerator.

The main purpose is to study the particle's motion inside the accelerator while neglecting the interaction with other particles or its surroundings. The second one is called "Collective Effects" which studies the multiparticle dynamics: the particles' interaction with each other and their environment [START_REF] Zimmermann | Introduction to Collective Effects in Particle Accelerators[END_REF].

Particle accelerators are complex tools both theoretically and experimentally.

Starting from the first concept of particle accelerators, which is the circulation of a beam of charged particles with a high Lorentz factor in a vacuum tube along its axis, one can already point out a theoretical difficulty. How to describe the way the beam is oriented? How to order it? How to describe precisely the behavior of the particles inside the vacuum tube? The answers lie in the so-called collective effect formalism. This describes the way in which the particle inside the beam influences the dynamics of the others. The collective effects that act on beam dynamics may lead to many instabilities. When the instabilities in accelerators are mentioned, it refers to effects that decrease the performances of the accelerators in terms of intensity, efficiency, or beam quality such as particle loss, energy spread, tune shift, etc.

There are many different kind of collective effects but the main ones related to electron particle accelerators are [START_REF] Metral | Impedance and Collective Effects[END_REF][START_REF] Ruth | An overview of collective effects in circular and linear accelerators[END_REF]:

• Space Charge Effect
• Touschek Effect and Intra Beam Scattering (IBS)

• Wakefields and Impedances

• Coherent Synchrotron Radiation (CSR)

• Ion Cloud In the following sections, all these effects are presented in more detail.

. Coherent Synchrotron Radiation (CSR)

In general particles under acceleration emit radiation. The particles feel acceleration due to the bending effect of the dipole magnets, which creates narrow cone radiation in the direction tangential to beam trajectory. The angle of the radiation cone is inversely proportional to Lorentz factor g. This is called Synchrotron Radiation (SR). In SR machines, this effect can be used to create high-energy X-rays for different applications. However, in some machines it is an undesired effect. When the electron's bunch is short, it coherently emits radiation. This coherent radiation can either increase or decrease the electron-beam energy. More precisely, emitted radiation affects only some parts of the beam by accelerating or decelerating it. This is called the Coherent Synchrotron Radiation (CSR) effect [START_REF] Novokhatski | Coherent Synchrotron Radiation: Theory and Simulations[END_REF]. Let us look at Figure 1.1: in the top case -Incoherent Synchrotron Radiation-the particles emit radiation with varied phases, however in the bottom case -Coherent Synchrotron Radiation-the particles emit radiation in phase. These effects depend on the bunch length s b and on the wavelength of the emitted radiation.

If we look at the CSR effect analytically, the total radiated field will be equal to the sum of the individual radiation emitted by each electron for the incoherent case [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF]:

E T = N Â n=1 E n (l ) = N Â n=1 E(l )e if n (1.1)
where E n (l )e if n is the emitted radiation by single electron n, f n is the phase of the field of the n th particle and l is the radiated wavelength. Indeed, for incoherent SR, the intensity of the radiation is proportional to the number of the electrons N [START_REF] Wiedemann | Particle Accelerator Physics. Graduate Texts in Physics[END_REF]. However, in the coherent case the particles are in nearly the same position due to the short bunch length which lead to radiation without a phase difference.

In the full coherence case the intensity will be proportional to N(N 1)| r(w) | 2 [START_REF] Nodvick | Suppression of coherent radiation by electrons in a synchrotron[END_REF].

The r(w) is the spectral charge density which is the Fourier transform of the longitudinal charge density as in the equation: where w is the angular frequency 2p f , and c is the speed of light.

The CSR effect can also be modeled by wakefields, which are defined by the momentum changes of the particles with respect to external fields in time domain and the impedance, which is the representation in the frequency domain. They are both explained briefly in section 1.1.5. The impedance equation of the CSR field is:

Z L (w) = Z 0 2p e ip/6 3 1/3 G ✓ 2 3 ◆ w 1/3 c 1/3 r 2/3 (1.3)
where G is the gamma function, w is the frequency, r is the bending radius, and Z 0 = 376.73W is the free space impedance. This equation allows us to calculate the CSR impedance which comes from the dipole magnet with a bending radius.

Moreover, the real part of the CSR field's impedance is related to the spectrum of the energy loss [START_REF] Chao | [END_REF], which can be extracted from Equation 1.3. Thus the power of synchrotron radiation per unit length in unit spectral interval becomes:

dP dw = e 2
4p 2 e 0 Re(Z L (w))

(1.4)
In general the CSR field can act on part of the beam itself or it can be reflected back from the beam pipe boundaries and affect the bunch tail. These effects can lower the beam energy in some parts but raise in another part. This effect increases the energy spread of the bunch and leads to bunch lengthening.

. Touschek Effect and Intrabeam Scattering

In the accelerators there is a probability of electron-electron scattering inside the bunch. These scatterings can cause momentum exchange between particles and as a result they may lead to particle losses on vacuum pipe. Multiple small-angle scatterings at a high rate are known as Intrabeam scattering (IBS).

This scattering inside the bunch brings a small momentum change which causes transverse emittance and energy spread growth. When the momentum exchange of the scattered particles are too large and cause particle losses, these scatterings are called the Touschek effect. The number of lost particles depends on the Touschek lifetime, which gives the time passed for a one-half reduction of the bunch charge.

Especially for the Compton backscattering and synchrotron light sources, the emittance growth from the intrabeam scattering can be really problematic for achieving the expected qualifications of the X-ray. The general theoretical models were derived by many scientists, such as Piwinski [START_REF] Piwinski | Intra-beam-scattering[END_REF], Bjorken, and Mtingwa [START_REF] Bjorken | [END_REF].

In the designed stage of the storage rings, the intrabeam scattering should be taken into account and the magnetic design should be performed to compensate for the IBS effect. For example, for 3 rd generation light sources in general, the emittance growth time is proportional to the g 3 [START_REF] Bane | An accurate, simplified model intrabeam scattering[END_REF]. Some light sources are approaching the diffraction limit. However these light sources will have high transverse electron density, giving rise to further consequent IBS effects. This is partially compensated by bunch lengthening to achieve the needed Touschek lifetime and to reduce the effect of the strong intrabeam scattering. For this reason, they generally use lower RF frequencies (less than common 350 MHz to 500 MHz systems) and bunch lengthening using higher harmonic cavities [START_REF] Helmholtz | bessy vsr variable pulse length storage ring upgrade of bessy ii[END_REF].

. Ion Cloud

In accelerators, one of the main issues is to achieve vacuum. As far as vacuum technologies are considered, the perfect vacuum is not possible. Generally, low level of residual gas remains inside the beam pipe even after the vacuum pumpdown. The interaction with the electron and residual gas will create electron-ion pairs. The electrons will disperse due to the Coulomb interaction with the electron beam. However, the dynamics of the residual vacuum ions inside the beam pipe can significantly affect beam dynamics and lead to many instabilities. In the proton machines like LHC, the electron cloud is also problematic and studied extensively to achieve the designed performance of the accelerator [START_REF] Zimmermann | The electron cloud instability: Summary of measurements and understanding[END_REF]. Generally, the interactions between electron beams and residual ions were studied both in theoretical and experimental sides [START_REF] Sagan | Some aspects of the longitudinal motion of ions in electron storage rings[END_REF][START_REF] Huang | Beam-Beam Effect: Crab Dynamics Calculation in JLEIC[END_REF][START_REF] Ziemann | Beyond Bassetti and Erskine: Beam-beam deflections for non-Gaussian beams[END_REF].

The beam-ion interaction may depend on many criteria such as ion trapping conditions in the potential well of electron beam, oscillation frequencies, transverse equilibrium distribution, and longitudinal collection points. The beam passing through inside the ions will create the same effect as focusing lenses do on the ions. This focusing will lead to ions being trapped. The mass of the ions will affect this trap mechanism. For each ion beam, the critical mass should be defined.

The ions that have a mass larger than the critical mass will be trapped because the lighter ions will drift too far away before having focused by the next electron bunch. These trapped ions will affect the beam and create transverse instabilities.

The residual ion dynamics can be described with the velocity kick given by the electron bunch pass. This kick was described by Sagan [START_REF] Sagan | Some aspects of the longitudinal motion of ions in electron storage rings[END_REF] for the longitudinal case and by Bassetti-Erskine [START_REF] Bassetti | Closed expression for the electrical field of a two-dimensional Gaussian charge[END_REF] for transverse case. The longitudinal kick d v s from Sagan equation can be seen in equation 1.5 and the transverse velocity kicks from Bassetti-Erskine formula can be seen in equations 1.6 and 1.7.

Dv s = [ a x e x + (hs d )(h 0 s d )] ∂ Dv x ∂ x a y e y ∂ Dv y ∂ y (1.5) Dv x = NK p p q 2(s 2 x s 2 y ) Im " w x + jy q 2(s 2 x s 2 y ) e x 2 2s 2 x + y 2 2s 2 y w (x s 2 y s 2 x + jy s 2 x s 2 y ) q 2(s 2 x s 2 y ) # (1.6) Dv y = NK p p q 2(s 2 x s 2 y ) Re " w x + jy q 2(s 2 x s 2 y ) e x 2 2s 2 x + y 2 2s 2 y w (x s 2 y s 2 x + jy s 2 x s 2 y ) q 2(s 2 x s 2 y ) # (1.7)
where w is the complex error function called Faddeeva, N is the number of electrons inside the bunch, K = 2r p c 2 v e A , r p is the classic proton radius, s x and s y are the standard deviation of Gaussian bunch distribution for the x and y axes, s d is the standard deviation of Gaussian bunch distribution for longitudinal plane, e x and e y are the emittance in x and y axis, a, b and g are the Twiss parameters, n and n 0 are the dispersion function, A is the atomic mass number of the ion. The trapping mechanism has some conditions related to the atomic mass of the electrons. The x (s x +s y ) respectively. The S b = L n is the bunch spacing. If the atomic mass numbers of the ions are larger than A c y and A c x , the ions will be trapped. Otherwise, the ions are already drifted away before the focusing of the former bunch. The solution of these equations can be used for describing the beam-ion interaction. Different numerical calculation tools can be used to see the effect of the beam-ion interaction over the beam itself.

Ion cloud problem is generally solved with clearing electrodes or clearing gaps [START_REF] Full | Detection and clearing of trapped ions in the high current cornell photoinjector[END_REF][START_REF] Poncet | Ion trapping, clearing, beam-ion interactions[END_REF]. The ion clearing electrodes are voltage applied plates for creating transverse electric field to collect ions. In some machines, the BPM (Beam Position Monitor) can also be used as clearing electrodes. The field simulations and applied voltage measurements should be performed carefully so as not to allow the coupling between the electrodes or to interfere with the BPM signal. Clearing gaps consist of empty RF buckets between some bunches, and when these are performed in accelerators, they give the ion cloud time to escape the potential created by the charged particle beam.

. Space Charge

A well-known collective effect is the space charge effect, which describes the repeal force of charged particles within the bunch. The space charge effect formulations were done in various research and expanded upon over the years. Space charge impacts different accelerators in different ways, for example in hadron colliders, they have a limiting effect on achievable brightness, whereas in electron guns and photoinjectors they impact their intensity. The electron intensity is a key parameter for many applications such as FELs, Compton machines etc. The intensity can be defined as the number of particles per unit area. In photoinjectors, this effect can be partially compensated by solenoid magnets [START_REF] Chauvin | Space-charge effect[END_REF].

The particles moving inside the beam pipe are affected by EM forces. One of these EM forces comes from self created fields which is also affecting the beam itself. Let us first take the single particle field and the movement to understand this phenomenon better.

The charge particle creates an electric field flux over the spherical Gaussian surface in accordance with Gauss law. Since the surface area is taken as 4pr 2 , the electric field created by the particle in every direction becomes E in Equation 1.8.

But if the particle is in motion, the field lines start to contract with the relativistic boost. The comparison scheme of the electric field in different cases can be seen in Figure 1.2. If we take the direction of motion in the z direction, the relation coming from Lorentz transformation in cylindrical coordinate system can be seen in equations 1.9, 1.10 and 1.11

E = q 4pe 0 r 2 (1.8) E r = qg 4pe 0 r 2
(1.9)

B q = qgb 4pe 0 cr 2
(1.10)

E z = q 4pe 0 z 2 g 2 (1.11)
where E r is the electric field in radial direction, E z is the electric field in the direc- The general scheme of electric eld of particle in non relativistic, relativistic and ultra-relativistic cases [START_REF] Apsimon | The Development and Implementation of a Beam Position Monitoring System for use in the FONT Feedback System at ATF2[END_REF].

tion of motion, B q is the magnetic field in azimuthal direction, g and b are relativistic factors, and e 0 is the vacuum permittivity. When the ultra-relativistic limit g >> 1, the electric field lines of the charge particle become purely perpendicular to direction of motion, which means that the electric field in the direction of mo-tion approaches zero and the fields remain only in the perpendicular direction. Of course, with the multiparticle case, the field line equations change in accordance with Maxwell's equation. If we look at the many particle (bunch) case, due to the velocity of the particles it is taken as a current, and the magnetic field is created in the azimuthal direction to that of the motion. Again, in ultra-relativistic cases, the field lines contract in the perpendicular direction to that of the motion. When two particles with the same charge travel in the same direction with a distance between them, they repel each other due to the Coulomb force and attract each other due the magnetic force as shown in Figure 1.3.

For the low g bunch particle case in the longitudinal direction, the only effective force is the Coulomb force, which can cause lengthening of the bunched beam called longitudinal space charge. For the ultra-relativistic cases with a perfect electric conductor pipe, the two forces cancel each other in the transverse plane.

But the beam pipe perturbs the EM fields created by the beam. The electric field creates an image charge with the same speed and the opposite sign over the surface of the beam pipe. This image charge creates an electric field and due to finite electrical conductivity it can affect the beam. This effect is called indirect space charge. All these space charges in accelerators decrease with increasing g which can be seen in Equation 1.11.

The space charge can be effective on the beam parameters such as tune, bunch length, emittance etc. The number of oscillations in transverse in one turn of a circular accelerator can be defined as the betatron tune. It is a crucial parameter for circular accelerators because its unwanted change will affect the oscillations which can eventually lead to particle loss. Also, especially in the linear acceleration and transport parts, the space charge can distort the phase space distribution thus contributing to emittance growth (and energy spread) [30]. The tool for defining how a particle in the beam influences the dynamics of the others is called the wakefield formalism. There are different kinds of wakefields observed in accelerators. The first one is called the geometric wakefield, which can be seen when the particle evolves in a discontinuous geometry. The second source of wakefields is due to the fact that the walls of the accelerator tube have a finite conductivity, and we have a resistive term that is added to the continuity equations (e.g. empirical law about charge conservation.) Also the wakefield and impedance formalism can be used in Coherent Synchrotron radiation effect and ion cloud, which were explained in previous sections 1.1.1 and 1.1.3. The wakefield formalism contains the total energy loss of the whole particle with the line distribution l (z) which lead to wake function in time domain. The Fourier transform of the wake function -frequency domain representation of the energy losswill give the impedance. The theoretical background of the wake function and impedance formalism will be given in Section 1.2.

. Wakefields and Impedances

Let us take the wakefields one by one. First of all, imagine a space full of people with them creating a line moving with constant speed through the space. When there is a discontinuity in the space -a narrowing in the road-the line will break down and people begin to feel kicks from each other trying continue on their way. This is similar to the effect our particles experience when they detect a discontinuity in their path. Due to the fields created by the particles, every particle feels a momentum kick with respect to their position. Also, the head of the line will move with their initial speed while the back of the line slows down. This is the cause of head-tail instability in the accelerators. The solution of the Haissinski equation, which defines the longitudinal density with the self created fields of the beam, can

give the resulting bunch lengthening [START_REF] Haissinski | Exact longitudinal equilibrium distribution of stored electrons in the presence of self-fields[END_REF]. These wakefields in accelerators are geometric wakefields or geometric impedances, which come from the change in the beam pipe structure and lead to instabilities [START_REF] Wanzenberg | Impedances and Instabilities[END_REF][START_REF] Heifets | Coupling impedance in modern accelerators[END_REF][START_REF] Zotter | Impedances and wakes in high-energy particle accelerators[END_REF][START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF]. Let us take a basic cavity called step in-out, which is a pillbox cavity with step transition at both ends. In Figure 1.4, when the beam passes through the step in-out cavity, each bunch will create an EM field behind, which may be trapped due to the geometry of the accelerator component. This EM field is one example of wakefields which affects the beam itself and can cause deflections, defocusing of the beam, particle loss etc.

The second wakefield is the resistive wall wakefield. Let us consider an ultrarelativistic beam which travels inside a perfect electric conducting pipe. All the EM field lines will be purely perpendicular to the direction of motion due to the relativistic contraction as explained in the space charge section. So the electric and magnetic fields will cancel each other and the particles will not feel any force.

This mechanism was explained in the former chapter with the Figure 1.3. Unfortunately, the materials used in beam pipes are far from being smooth or perfectly conductive. The beam moving inside the vacuum pipe will induce an EM field and create an image charge on the vacuum pipe with the skin depth. Skin depth is the inward distance which can be penetrated by an EM field through a metallic surface. When it is not a perfectly conductive electrical pipe, the created image charge's velocity and field will be different from the beam's due to resistive losses.

The resistive wall wakefield will be increased with the frequency of the impedance [START_REF] Wiedemann | Particle Accelerator Physics. Graduate Texts in Physics[END_REF].

Impedance studies are important for defining the dynamics of charged particle beams. Furthermore, they provide an approach to manipulate the phase space of the electron beam and comparative results for diagnostic studies. This thesis will focus on the geometric impedance model in transverse axis especially.

. Impedance

As mentioned in the former subsection, impedance is the representation of the wakefields in the frequency domain. The impedances are easy to use for analytical calculations or with frequency dependent objects in some cases. The impedance terms can be divided into two: longitudinal and transverse impedances.

The longitudinal impedance does not depend on the transverse position of the particles. Only the effects of the beam propagation axis are taken into account.

These effects can cause energy loss of the beam or create energy spread inside the bunch which can lead to bunch lengthening.

When we come to the transverse impedance, it can be split in two terms: the dipolar component and the quadrupolar one. Dipolar terms related to transverse momentum kick have effects on every particle as a result of transverse deflecting field. Dipolar impedance causes coherent effect which drives the coherent instabilities. Quadrupolar terms also give a momentum kick resulting in the focusing or defocusing field effect. Quadrupolar impedance causes incoherent effect which detunes the single particles dynamics. The introduction of source and witness particles, needed to understand these effects, will be introduced in the next section.

Each component have a direct impact on the beam dynamics. The theoretical background and formalism will be briefly explained in this section.

. General Theory of Longitudinal Impedance

In general electromagnetic theory, the interaction of the beam with its surroundings is described with the wake function in time domain and impedance in the frequency domain. The effect of wake fields come from the solution of Maxwell's equation with beam specifications and boundary conditions.

Let us take a witness particle w and a source particle s that will allow us to trans- The general scheme of the test and witness particle inside the beam pipe. Source particle emits EM eld which cause a kick on the witness particle.

late this effect analytically. A particle source is travelling along the z longitudinal propagation axis inside the beam pipe with charge q s and a transverse coordinate (x s , y s ) and a witness particle is travelling with a charge q w and a transverse coordinate (x w , y w ) as shown in Figure 1.5. The source particle creates behind it an oscillating field which acts as a force on the witness one. So, the source particle will modify the EM field felt by the witness particle. This will induce a variation of the momentum of the latter, due to the Lorentz force F L = q( Ẽ +ṽ ⇥ B). It is then possible to define the wake function W as the integral of the work of the Lorentz force applied to the witness particle, on all the previous positions of the source particle. The longitudinal wake function has a unit of V /C and the transverse wake function has a unit of V /Cm.This function can be decomposed on the longitudinal (L) and transverse (?) planes of the beam motion as:

W L = Z • • dt 1 q E L (r source , r test , vt z,t)
(1.12)

W ? = 1 q Z • • dt(E + v c ⇥ H) ? (r source , r test , vt z,t) (1.13)
Where v is the velocity of the particles, q is their charge, and r is the position in the transverse plane. Moreover, s represents the longitudinal position of the test particle and z the longitudinal distance between the two particles. Finally t

W L (x s , y s , x w , y w , vt) = 1 q s q w Z • • F L z(x, y, s, x s , y s , vt)dt (1.14)
where

F L z is the longitudinal component of the Lorentz force, velocity v = b c
where c is the speed of light and b is the relativistic factor. The wake function depends on the transverse offset of the source and witness particles.

The series expansion can be applied to the wake function:

W L (u s , u t , z) = W L (0, 0, z) +W L (0, u t , z)u t +W L (u s , 0, z)u s + .... (1.15)
The longitudinal wake function can be defined as the zero th order term of this expansion. This means that the longitudinal wake function does not depend on the transverse position of the source and witness particles:

W L (z) = W L (0, 0, z) (1.16)
Wake function notation is really useful for the simulations and beam dynamic models, but for real time measurements the frequency domain representation is more practical. It is however much easier to work in the frequency domain and to not consider the wake function directly, but its Fourier transform instead, as a transfer function that we call impedance. This representation is called beam coupling impedance and it can be written as the Fourier transform of the wake function:

Z L (u s , u t , z) = Z • • W L (x s , y s , x w , y w , vt)e jwz v dz (1.17)
where j is the imaginary unit and w = 2p f is the angular frequency conjugate variable. One can also define the impedance directly from the frequency domain.

The unit of the longitudinal impedance becomes W.

In the beam dynamics the wake fields are widely effective. There are two different kinds of impedance: broad-band and narrow-band impedances. Broad band impedances occur when the wakefield vanishes just after the bunch, meaning that it is only effective on the coupled bunch collective effect. On the other hand, in narrow-band impedance the time window is much bigger than the bunch distances in which the coupling between two successive bunches can result. This means that the wakefield is longer than the distance between two bunches which leads to coupling of the wakefields created by the two different bunch.

Loss Factor

The concept of loss factor is important in the impedance studies to describe the effect on beam dynamics. The loss factor is used for the energy loss calculation of the bunch with total charge Q. It can be written as:

k L = Z • • W L (0, 0, z)r(z)dz (1.18)
where W L (z) is the longitudinal wake function and r(z) is the longitudinal charge density. It can also be written as a function of the impedance as:

k L = 1 p Z • • Re[Z L (w)]| r(z)| 2 dz (1.19)
The Im[Z(w)] does not affect the loss factor; only the real part of the impedance should be considered. Then the overall energy loss equation becomes [START_REF] Ferrario | [END_REF]:

d E = k L Q 2 (1.20)

Effective Impedances

When single bunch dynamics which can be seen in broad-band impedances, the effective impedance must be mentioned. Impedance is especially needed for calculating the tune shifts, and even for loss factor [START_REF] Chao | [END_REF]. For the broadband impedance, the expected impact are bunch lengthening, phase shift, and energy spread growth. These effects depend on the loss factor, the Fourier transform of the bunch spectrum, and the effective impedance. The general definition of the effective impedance is [START_REF] Smaluk | Impedance computations and beam-based measurements: A problem of discrepancy[END_REF]:

Z L w = Â i=• i= • Z L (w 0 ) w 0 h l (w 0 ) Â i=• i= • h l (w 0 ) (1.21)
where h l (w 0 ) is the spectral density which is equal to | r(z)| 2 , w 0 = w 0 p + lw s where w s is the synchrotron frequency, w 0 the revolution angular frequency, l depends on type of the oscillation, and p is the momentum. The spectral density is different for each type of the beam. As a Gaussian beam, the spectral density is equal to:

h l (w) = ws z c 2l e w 2 s z 2 c 2 (1.22)
where s z is equal to standard deviation of the Gaussian bunch. From the effective impedance, longitudinal frequency shifts then become [START_REF] Chao | [END_REF]:

d w L = i 2p G(m + 1/2) 2 m (m 1)! N B e 2 h T 0 Eb 2 w s s 3 t Z L w (1.23)
where s t is the RMS bunch length in time, m is the azimuthal mode number, E is the beam energy, and T 0 is the revolution period.

. General Theory of Transverse Impedance

For transverse impedance calculation let us again take the source and witness particle in Figure 1.5 with general off-axis position. The transverse wake function can also be written from Lorentz force FL = q( Ẽ +ṽ ⇥ B):

W ? (x s , y s , x w , y w , vt) = 1 q s q w Z • • F Lxy (x, y, s, x0, y0, vt)dt (1.24)
where F Lxy is the transverse component of the Lorentz force. The series expansion also applies to transverse wake function in the offset of source and witness particle as in the equation:

W ? (ũ s , ũw , z) = W T (0, 0, z) + -? W ? (0, ũw , z)ũ w + -? W ? (0,ũ s , z)ũ s + .... = W 0 (z) +W Q (ũ w , z)ũ w +W D (ũ s , z)ũ s + .... (1.25)
where W T is the transverse component when the source and witness particles are at the center, W D is dipolar or driving term, and W Q is quadrupolar or detuning term. Although there is no transverse effect when both particles are at the center, all terms except the first one vanish. If we split the transverse wake function equation into two transverse planes x and y, the equation becomes:

W x = W x Q (x w , z)x w +W x D (x s , z)x s W y = W y Q (y w , z)y w +W y D (y s , z)y s (1.26)
The summation of these two equations will give the total transverse wake function (W T ). As mentioned before, the dipolar component depends only on source's own transverse position which directly gives a global offset, however the quadrupolar term only depends on the witness particle's position which acts as focusing or defocusing effect. The dipolar term will lead to a global change of direction of the bunch, it will act as a dipole kick, while the quadrupolar term will "detune" it, leading to a defocusing or focusing term as a quadrupole kick, for example. All in all, for symmetric structures especially, the dipolar terms are the dominant ones and the quadrupolar effect generally leads to zero. This is why usually the only component taken into account is the dipolar one. But accelerators currently use mostly asymmetric structures. This leads also to different impedances in horizontal and transverse directions. The generalized transverse impedance is the sum of the dipolar and quadrupolar impedances in both planes.

Similar to the longitudinal wake function, the transverse wake function can be written in frequency domain. The Fourier transform of the transverse wake function will give the transverse impedance as:

Z ? (ũ s , ũw , z) = j Z • • W ? xs , xw ,ỹ s , ỹw , vt)e jwz v dz (1.27)
Also it can be expanded in series as:

Z x = Z x Q (x w , z)x w + Z x D (x s , z)x s Z y = Z y Q (y w , z)y w + Z y D (y s , z)y s (1.28)
where Z Q is quadrupolar transverse impedance and Z D is dipolar transverse impedance. It should be noted that the unit of longitudinal impedance is W whenever the dipolar and quadrupolar impedances are expressed in W/m.

As long as only single particle dynamics are concerned, we use wake function to define the interaction. However in accelerators the particles are defined as a bunch with charge density l (z). When the particle distribution effect on witness particles are considered, the equations can be written in terms of wake potentials W P . Wake potential can be written as the convolution of the wake function with charge density as:

W ? P = W ? ⇤ l (z) = Z • • l (z 0 )W ? (z z 0 )dz 0 (1.29)

Panofsky-Wenzel Theorem

The Panofsky-Wenzel Theorem defines the general relation between the longitudinal and transverse wakefields. More explicitly, the derivation of the transverse wake potential with longitudinal component or the longitudinal wake potentials gradient will give the transverse wake potentials. The derivation of the Panofsky-Wenzel Theorem from Maxwell's equations was studied in numerous works [START_REF] Vaganian | The Panofsky-wenzel Theorem And General Relations For The wake Potential[END_REF].

This theorem is of great importance in impedance calculations. Both for analytical and numerical calculations, the relation between transverse and longitudinal wakefield-impedances are derived from this formula, as in the case for EM simulation programs, for example. Even for some cases, the transverse impedance is directly calculated from longitudinal impedance through Panowsky-Wenzel relation. The wake function application of the Panofsky-Wenzel Theorem is [START_REF] Dohlus | An introduction to wake fields and impedances[END_REF]:

∂ ∂ s w x = ∂ ∂ x w L (1.30) ∂ ∂ s w y = ∂ ∂ y w L (1.31) ∂ ∂ x w y = ∂ ∂ y w x (1.32)
where w x and w y are the wake function in x and y axes respectively, and x and y

give the position of the test particle in transverse and s in longitudinal.

In general the theorem leads to transverse gradient of the integrated longitudi-nal field as in the equation:

w(r, x, y, s) ? = -? Z s • w L (r, x, y, s 0 )ds 0 (1.33)
where r is the position of the source particle.

Kick Factor

In the transverse plane the equivalence of the loss factor is called kick factor. It describes the transverse kick that a particle experiences due to the impedance.

The kick factor can be written with impedance as:

k T = 1 p Z • • Im[Z(w)]| r(z)| 2 dz (1.34)
Transverse case only effective impedance is the Im[Z(w)] in the kick factor. The particles inside the circular accelerator perform betatron oscillation which is the oscillations of the particles with respect to the equilibrium orbit. This behavior obeys the equation of motion and the frequency of the oscillations can be defined as the tune. However, instabilities and collective effects can lead to tune shifts which can be extracted from kick factor.

. Impedance Simulations

The best way to create impedance budget is to utilize simulation programs to solve 2D and 3D structures. The impedance calculations depend on differential equations which cannot be solved analytically in complex structures. However, some simple structures can be solved. The most widely known is the step in/step out, which is introduced at the end of this section. Otherwise, the only possible solution for complex structures is to solve them using numerical methods. The numerical methods are the mathematical tools that are designed to solve numerical problems with proper approximations. The simulation tools become important due to the increasing complexity of the accelerator components, the beam quality, and stability requirements. Especially in the design part of the accelerators, this kind of simulation tools are widely used to reduce the total impedance. Indeed some effects can be suppressed with a proper design. Additionally, these simulations are important for their usage in beam dynamics studies and in the understanding of the accelerator performances.

There are different programs and codes available for use in impedance calculations. The CST [START_REF]CST Particle Studio[END_REF] and HFSS [START_REF] Hfss | [END_REF] are useful as commercial products for EM and wakefield simulations, while ABCI [START_REF]Azimuthal beam cavity interaction[END_REF] and MAFIA [START_REF] Chou | Wakefield and impedance studies of a liner using mafia[END_REF] can be used for 2D and 3D structures simulations. The GdFidl [START_REF] Gdfidl | [END_REF] code is suitable for parallel computing. But in general we can catalogue the codes into two groups as time domain and frequency domain simulation programs.

Time domain simulations are performed by solving the Maxwell's equations of EM fields created within the structure in the time domain, using different methods of numerical solutions to differential equations. The time domain simulations generally give wake potential and the wake function or impedance can be extracted by convolution or Fourier transform. On the other hand, the frequency domain solvers directly calculate beam coupling impedance in frequency spectrum with many different numerical methods.

These two methods have their own advantages and disadvantages. The frequency domain solvers generally work in two ways: eigenmode simulation or simulating the coaxial wire method. Eigenmode simulation can only be used in strong resonant objects with the resonant modes. On account of the fact that only impedances that are in resonant frequency of the object can be simulated with eigenmode simulations, only the narrow-band strong impedance peaks can be found using them. Also, eigenmode simulations can be performed with simulating measurements setups. These simulations include copying wire measurements, which means that S parameters are simulated by adding a wire inside, and then adding waveguide ports an each end of the device as a replacement for matching connections. This method is not very useful for encountering the same problems in bench measurements or with the computational problems. This is why we did not prefer to use eigenmode simulations in impedance calculations.

Time domain simulations are the most practical. CST, MAFIA, and Gdfidl use time domain, simulating the beam directly inside the object. These are the most suitable options for impedance simulation due to their wide usage in various components and using the particle beam in the simulations. Different programs have their own additional parameters to increase the efficiency of the computational power and the accuracy of the results. In general the working procedure of time domain simulations is as follows:

• Firstly, for different kind of numerical solving methods, the mesh type and mesh parameters were set and checked on the structure. The meshing can be changed with respect to the complexity of the structure. The parameters will be defined briefly for each specific tool in the next section.

• The definition of the source signal is set with respect to the beam profile.

Generally in time domain solvers, the parameters which can be settled are the longitudinal bunch density, charge, and velocity. Moreover, the path of the witness particle (called the integration line) should be set.

• The wakelength should be chosen carefully to put a limit to the distance between the source and the test particle. It will also be effective in the spectrum of the impedance. In general, working with High-Q resonated components or in the low frequencies require longer wakelengths.

• The method for computing wake potential and the boundary condition depends on the simulation tool. Each have an impact on the accuracy of the simulation results. It will be detailed in the next section for each specific tool.

• After the simulations performed, it will be needed to deconvolve the wake potential from the source signal to obtain the wakefunction. In some tools, an external FFT (Fast Fourier Transform) algorithm is also needed to calculate impedance in frequency domain.

These are the general procedure for time domain simulations and they can differ with respect to the product used. The impedance simulations can be seen as straightforward, yet there are always difficulties to setting helpful parameters.

Wrongly set parameters can lead to false simulation results. They can also prevent the computational power needed for simulation, especially for complex objects, from being achievable. The simplification of the model is also needed since, even with the simplification, the simulations are time consuming.

Other than this, the simulation programs generally return the wake potential as an output, not the wake function or impedance itself. The deconvolution should be performed with the bunch profile to obtain the wake function. The spectrum is limited by the bunch spectrum and it generally depends on the computing capabilities. As we take into account the computational time and precision of the simulation programs, the study of the impedance in accelerators should also be derived experimentally.

The impedance calculation of the ThomX project elements will be studied by both bench measurements and simulation tools for these reasons. The CST particle studio was chosen as the simulation tool. The CST is a well-known commercial product with extensive support options. Additionally it uses the beam itself directly for simulations and creates the impedance spectrum automatically. The brief information about the simulation parameters will be given in Chapter 4.

. Impedance Measurement Techniques

Due to the reliability of the impedance simulations mentioned in the former section and the impossibility of analytic calculations for complex structures, impedance measurement is crucial for crosschecking the simulation results and creating the impedance budget. Bench-top impedance measurements were preferred in our research.

There are two ways to define the impedance of the accelerator structure: the bench top experiments and beam based methods. The beam based methods are used frequently, but only after the commissioning of the accelerator to check the impedance budget. On the other hand, solely frequency based impedance measurements are problematic to obtain with beam based measurements. The parameters of the kick-loss factor, effective impedance, bunch spectrum etc. can be calculated from the beam based measurements and it can be helpful to characterize the effect on beam dynamics [START_REF] Smaluk | Beam-based Impedance Measurement Techniques[END_REF].

Longitudinal impedance measurements are more straightforward and can be performed with one wire. Furthermore, the transverse impedance measurements can be done with wire measurements by combining two-wires and moving-wire methods. Both measurements are needed to fully retrieve every term of transverse impedance.

. Theory of the Wire Measurements

Transverse impedance measurements can be done with wire measurements by combining the two-wire and the moving wire methods. Both measurements are needed to fully retrieve all the transverse impedance terms. To emphasize the measurement process, it is better to use a different formalism. Let us begin by defining the generalized longitudinal impedance from EM fields and m-th (m = 0, 1, 2,...) order current density J m propogating in the z-direction on a wire [START_REF] Tsutsui | On single wire technique for transverse coupling impedance measurement[END_REF][START_REF] Day | Simulations of coaxial wire measurements of the impedance of asymmetric structures[END_REF]:

J = Id (x x 0 )d (y y 0 )e j(wt kz) (1.35) = I 2pa • Â • e jm(Q Q 0 ) e j(wt kz) = 1 (1.36) = • Â • a m e jmQ 0 J m (1.37)
where a is the displacement of the wire from the central axis, q is the azimuthal angle of the wire displacement, and x 0 and y 0 are:

x 0 = a cos(Q 0 ) (1.38) y 0 = a sin(Q 0 ) (1.39)
The impedance coming from the current density and the electric field becomes as follows:

Z = Z 0,0 + ae jq (Z 1,0 + Z 0, 1 ) + ae jq (Z 0,1 + Z 1,0 ) +a 2 e 2 jq (Z 2,0 + Z 1, 1 + Z 0, 2 ) + a 2 (Z 1,1 + Z 1, 1 ) +a 2 e 2 jq (Z 0,2 + Z 1,1 + Z 2,0 ) + O(a 3 ) (1.40)
where Z m,n (m, n = 0, ±1,...) is the impedance with n-th order azimuthal components, and Z 0,0 is the longitudinal term measured on the axis. If we apply Panowsky-Wenzel Theorem to the generalized longitudinal impedance and ignore the higher order and coupling terms, the transverse vertical and horizontal components in Cartesian coordinate system can be found using Equations 1.41 and 1.42.

Z x = x (Z dip x Z quad x ) (1.41) Z y = y (Z dip y + Z quad y ) (1.42)
where x and y are the offset values of the wires from the center. If we take measurements at x = a when y = 0 and Q = 0, (x, y, Q) = (a, 0, 0), dipolar and quadrupolar component impedances can be written as:

Z = Z 0,0 + a 2 (Z 1,1 + Z 1, 1 + Z 1, 1 + Z 2,0 + Z 0, 2 Z 0,2 Z 2,0 ) (1.43) = Z 0,0 + a 2 (Z dip x Z quad x ) (1.44)
Also for at y = a when x = 0 and Q = p/2, (x, y, Q) = (0, a, p/2):

Z = Z 0,0 + a 2 (Z dip y + Z quad y ) (1.45) Z dip x/y = 1 k [Z 1,1 ± Z 1, 1 ± Z 1,1 + Z 1, 1 ]
(1.46)

Z quad x/y = 2 (Z 0,2 + Z 0, 2 ) k (1.47)
where Z dip x/y is the dipolar impedance, Z quad x/y is the quadrupolar impedance, w is the angular frequency, c is the speed of light, and k = w c is the wave number. The wire scan of the longitudinal measurements using different offset values will give us the results of dipolar and quadrupolar components by applying a polynomial fit to Equations 1.45 and1. [START_REF] Helmholtz | bessy vsr variable pulse length storage ring upgrade of bessy ii[END_REF]. The results will be used in the Equations 1.48 and 1.49.

Z x = Z dip x Z quad x (1.48) Z y = Z dip y + Z quad y (1.49)
Two wires separated by a distance D and driven with opposite phase will give the impedance in Equation 1. [START_REF] Henke | Impedances of a Set of Cylindrical Resonators with Beam Pipes[END_REF]. For structures that have both top/bottom and left/right symmetry, the dipolar component can be measured directly from Two Wire Measurements(TWM), which will be explained in detail in Chapter 5. Furthermore, Moving Wire Measurements (MWMs) will give the sum of the longitudinal, dipolar, and quadrupolar components as in Equation 1.40; and the quadrupolar component in Equation 1.47 can be extracted by taking two-wire and on-axis measurements.

-ThomX Project

. Light Sources

Accelerator physics is a field of science that is rapidly expanding all over the world due to accelerators' diverse usage in many different areas. As their usage expands, so does the quest for technological advances in accelerator science. One such usage of accelerators is producing X-Rays. X-Rays are radiated photons with high energy ranges, from 100 eV to 100 keV.

The discovery of the X-Ray was made by Wilhelm Röntgen in 1895. He won the Nobel prize in 1901 for his in-depth studies and the resulting discovery of the X-Rays. In the mid-1890s, he was studying the phenomenon of luminescence in cathode ray tubes. He was working with an experimental setup consisting of two electrodes (anode and cathode) placed inside a hollow glass tube called a "Crookes tube." Electrons detached from the cathode hit the glass before reaching the anode, creating flashes of light called fluorescence. In 1895, he changed the experiment a little by covering the tube with black cardboard and darkening the room so as to be able to understand how the light transmittance worked, and then repeated the experiment. From a distance of 2 meters from the test tube, he noticed a glow in the paper wrapped in barium platinocyanide. He repeated the experiment and observed the same result each time. He described it as a new sort of ray that could pass through a thin surface and named it the "X-Ray", using the letter X, which symbolizes the unknown in mathematics.

Later, these rays were dubbed "Röntgen rays". As it was observed in the experiment, the X-Rays are able to pass through thin matter due to their low wavelength, ranging from 10 pm to 10 nm. It allows X-Rays to be utilized in many applications, from medical to cultural heritage studies, all over the world [START_REF] Uda | X-rays for Archaeology[END_REF][START_REF] Wagner | Imaging with x-rays[END_REF][START_REF] Hendee | History and status of x-ray mammography[END_REF][START_REF] Casali | X-Ray Tomography[END_REF]. Due to this wide range of utilization, different kind of x-ray sources have been discovered over the years. The easiest means of production for X-Rays are crook tubes, yet it is unfortunate that they produce low-intensity rays. To increase the intensity of the X-Rays, accelerator-based photon sources have been invented; such as synchrotron light sources, free electron lasers, and Compton backscattering machines.

Synchrotron lights are generated when the accelerated particles encounter strong magnetic fields. Synchrotron light sources are circular accelerators that generate high-intensity and low-emittance lights. The new generation free-electron lasers are linear accelerators that generate X-Rays by means of undulators providing coherent amplified emission. These two devices generate high-intensity X-Rays, however, they also need high-cost infrastructure and a huge area, anywhere between one hundred square metres to several square kilometres. The solution to the high-intensity with high size problem comes from the compact Compton Backscattering Sources (CBS). The CBSs create high-energy photons that gain energy from low-energy photons by interacting with relativistic electrons. The Compton scattering is the scattering of an electron and a photon resulting from the collision of the photon of high-energy X-rays with the free electrons as in Figure 2.1.

The energy is transferred to electron in this occurrence. In the backscattering process however, the accelerated electron collides with the photon and transfer its energy to it. It is one of the most efficient photon energy amplifiers, and has a gain factor proportional to g 2 . The Compton backscattering light sources consisting of differing structures are also utilized in many fields all over the world [START_REF]Compton scattering for spectroscopic detection of ultra-fast, high flux, broad energy range x-rays[END_REF][START_REF] Deitrick | High-brilliance, high-flux compact inverse compton light source[END_REF][START_REF] Madey | Van de graaf-based 13.5 nm inversecompton light source[END_REF][START_REF] Pogorelsky | Converting conventional electron accelerators to high peak brilliance compton light sources[END_REF]. The brightness and energy comparisons of the light sources can be seen in Figure 2.2. The brightness of the CBS is much larger than those of the X-Ray tubes. The same energy scale can be achieved with CBSs, however, they have lower brightness values compared to that of the synchrotron light sources. The high energy photon collide with the electron which leads to a energy transfer from high energy photon to electron [START_REF] Venugopal | de broglie wavelength and frequency of scattered electrons in the compton effect[END_REF] The impact of these facts is felt when the area of tens of square metres needed for the CBSs are compared to the hundred square metres to several square kilometres needed for the synchrotron light sources. Thus small-scale infrastructures help democratize the access to these kind of sources for hospitals, art museums, and laboratories that need them.

. General Overview

ThomX is a Compact Compton Backscattering light source, built on the campus of Paris-Sud University in Orsay under the Irene-Joliot Curie Laboratory (IJCLab), which used to be 5 separate labs including LAL, now merged into one [START_REF] Variola | The ThomX Project[END_REF].

The electron storage ring with the energy of 50 MeV, charge of 1 nC, and a short duration of a few ps is sensitive to collective effects. The study of collective effects and beam dynamics is crucial to maximise the X-Ray flux. ThomX can be used in various areas with different characteristics. The storage ring decreases the bunch spacing in order to produce a high repetition rate X-ray source, which is crucial Depending on its value and stored charge in the storage ring, various applications can use ThomX X-Ray source. ThomX project's characteristics, design, and parameters will be thoroughly explained in the next section.

ThomX machine stands out amongst others of its ilk with its high flux and compactness. The expected photon rate's range is between 10 12 10 13 photons per second, and the area of the machine is a meager 70m 2 , which can be seen in Fig- this machine is found to be suitable for a wide range of applications. For example, the compact area makes ThomX operable in hospitals, museums, and laboratories

. ThomX electron source and linac

The photoinjector has been chosen as the electron source because of its capability to generate a very low emittance electron beam, without needing an additional prebunching cavity or a buncher, as the thermionic one does. In general, the photoinjector is one type of the many particle sources that are capable of generating bright electron beams with the photoelectric effect. The photocathode is shot with a laser pulse, which leads to the emission of an electron moving in the direction of the accelerating section, as shown in Figure 2.4. The ThomX photoinjector consists of a 3 GHz 2.5-cell RF gun, a magnesium or copper photocathode, a laser system, and solenoid magnets. The laser has a wavelength of 260 nm and energy of 100 µJ per pulse, with pulse duration of around 2-5 ps to match the RF frequency. The laser has been placed outside the shielded area and the light beam should be able to travel for about 20 m.

When the laser beam collides with the photocathode, standing wave EM RF fields are applied to accelerate the generated electrons. Electrons are focused by the solenoid-generated magnetic field at the end of the photoinjector, to com- pensate for the increase in the emittance as a result of the space charge. In fact, a characteristic of this sort of source is the capability for the electron beam to be accelerated from rest to relativistic energies in merely a few centimetres. Since its contenders suffer from the space charge effect at the charge of 1 nC, the photoinjector is exceedingly advantageous to use. The entire design and positioning of the solenoid magnets are determined to find a compromise between the compactness of the machines with the emittance growth. Electrons enter the linac section -which has a gradient of 12.5MV m 1 to achieve the energy of 50 MeV upon exiting. The Linac section has been chosen for its similarity in design to one of the pre-injectors of LEP at CERN, and was designed at IJCLab (formerly known as LAL in the 1980's.) The RF cavity in the accelerating section needs

. Transfer line and extraction line

After the linac section, the beam reaches the storage ring through the transfer line (TL). The TL can direct the beam to either extraction line or the storage ring.

For this, the beam is rotated by 180 with 8 dipoles in the transfer line. The TL optics should be arranged from the parameters of the beam exiting the linac section to the storage ring. As the parameters in the ring will directly depend on those at the output of the linac, the transfer line must remain versatile to accommodate the properties of the beam at the output of the linac, especially the twiss parameters, which depends on the linac tuning. To do so, TL has 7 quadrupoles.

In the first straight path, the beam may be dumped after going through different beam diagnostics. A quadrupole triplet serves here to measure the twiss parameters of the beam at the linac exit thanks to the well known 3-gradient method.

Then, the beam can be bent by 90 with two nearby dipoles and sent to a diagnostics station which will allow the measurement of the beam energy and its This kind of matching is generally performed with the optic codes. This matching was performed with upgrades I made on the CODAL program. Let us first introduce tracking codes and then the CODAL code which was used for ThomX. The construction of a tracking code is a totally local problem which means that each element is defined by a local Hamiltonian. The Hamiltonian's reference system has to be chosen with respect to the geometric needs. These geometric needs contain the general forms of the elements and the magnetic fields. This is why the solution of the Hamiltonian should depend on the type of magnetic element which means the need for approximations or exact results can be changed from one element to another [START_REF] Nadolski | Application de L'Analyse en Fréquence à L'Étude de la Dynamique des Sources de Lumière[END_REF]. The first version of the code was created by Alexandre Loulergue from SOLEIL, and it depends on the solution of the Hamiltonian with each element of the lattice respectively. For example exact integration is applied for the drifts or dipole magnets but some approximations are performed for quadrupoles or sextupoles.

The first version of CODAL has many different modules for calculating the result of the collective effect on beam dynamics such as CSR, space charge, geometric wakefield etc. This version of the code was used in numerous research [START_REF] Khojoyan | Transport studies of lpa electron beam towards the fel amplification at coxinel[END_REF][START_REF] André | Control of laser plasma accelerated electrons for light sources[END_REF].

The first upgrade of the CODAL was done by Illya Drebot, and Alexis Gamelin who studied the beam dynamics in the transient microbunching regime [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF] for ThomX. The microbunching instability is the emanation of variations in the density of the electrons in the beam with time; this leads to Coherent Synchrotron Radiations (CSR) which have the range of wavelengths shorter than the bunch length.

Full start to end 6D tracking can be performed by combining CODAL with the AS-TRA code. The tracking in the photoinjector and linac is performed with a program called ASTRA [START_REF] Flöttmann | ASTRA particle tracking code[END_REF]. Output from ASTRA dealing with the 6D particle coordinates are read from CODAL to perform the tracking in the transfer line and the ring. Then, the program simulates the entire accelerator and performs 6D phase space tracking from injection to storage ring.Also the program itself is adapted to create 6D phase space of the beam with respect to planned input parameters. The results were checked with different programs such as MADX [START_REF] Deniau | The mad-x program[END_REF] for beam creation.

As said before, the quadrupole value set of the transfer line have an importance to match the Twiss parameters from the linac to the storage ring. The extraction line must then be adjusted to lead the beam to the beam stop whether it comes from the ring or the transfer line. So, this work of matching is important for future operation of the accelerator. Three different options are now available in CODAL.

The matching and tracking can be performed with respect to the storage ring or the extraction line, a third option is to perform the matching with respect to the storage ring, and the tracking with respect to extraction line.

. Storage ring

In the storage ring, the single electron beam is then stored for 20 ms which corresponds to the bunch repetition rate in the linac. In fact, due to the fast damage of the electron properties due to the Touscheck lifetime and due to the collective effects, the storing time is limited. Such behavior is called pulsed mode storage ring, for which the synchrotron damping time is almost negligible compared to the storage time. Such a mode ensure quasi constant X-ray fluxes. provide strong and fast magnetic force to make the beam straight and well-placed with respect to the ring's orbit. After the beam is arranged well, the beam begins to spin inside the 18 metre-long storage ring for 20ms. The storage ring has 8 dipoles to spin the beam, and 24 quadrupoles and 12 sextupoles to arrange its expected parameters. The Compton Backscattering interaction will occur at the Interaction Point (IP) [START_REF] Apsimon | The Development and Implementation of a Beam Position Monitoring System for use in the FONT Feedback System at ATF2[END_REF] with the laser light which is stored in Fabry-Perot cavity [START_REF] André | Control of laser plasma accelerated electrons for light sources[END_REF]. The lattice has been designed to minimize the beam size at the IP with the parameters in Table 2.4, as the X-ray flux directly depends on them. Accelerator physics is a field of science encompassing every aspect necessary for the design and operation of the equipment and understanding the resulting dynamics of charged particles with respect to their design parameters. The interaction of particles with the accelerator's components and the accelerator itself becomes an important topic since the beginning of this science. Especially for the compact Compton Backscattering light sources, to be able to understand these interactions and their impact on the beam parameters are crucial to have the desired X-Rays. This interaction is defined as "collective effects" which come up with many different types such as geometric and resistive wakefields, space charge, ion cloud etc. These effects increase as the number of particles increases. ThomX

. Collective Effects in ThomX

nominal is foreseen at 1 nC into a short piscosecond bunch length, reaching a half kA peak current. As a result for the storage rings, the main limiting factors for the beam current are instabilities and collective effects as described in section .

The first main source of instabilities is Coherent Synchrotron Radiation (CSR).

For the ThomX storage ring, the CSR effect and beam dynamics were studied in former works [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF]. This one leads to a limiting current operation, which can be improved with the tuning of the linac.

The other well-known collective effect is the space charge effect. The space charge effect in accelerator decreases with increasing energy. In ThomX, the space charge is mostly effective in the injection and the linac, due to low g as shown in [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF].

The other two collective effects which can be seen in ThomX are the Intra Beam Scattering(IBS) and the Touschek effect which deal with the design parameters.

In the accelerators, there is a probability of electron-electron scattering inside the bunch where the energy is lower than 1 GeV .

These scatterings can cause a momentum exchange between particles and as a result can cause particle losses. This effect is called the Touschek effect. The Touschek lifetime is estimated around 10 s. It defines the loss rate of the particles into the ring. On the other hand, the IBS refers to the electron-electron collisions which are in a high rate, but do not lead to beam loss. But these collisions inside the bunch bring about a small momentum change which cause energy spread and emittance growth. As the brightness of the radiation is intrinsically dependent on these parameters, the storage time was limited to 20ms. During this time, the emittance increases by a factor of 3 to 10 (depending on the lattice and ye bunch parameters) and the energy dispersion increses by 30%.

The wakefield and impedance study is crucial for ThomX storage ring. As a geometric impedance source, the ThomX accelerator has BPM's, bellows, RF cavity with tapers, FBT, kickers, dipole chambers, and a septum. For each component, the impedance budget should be studied and the production should be checked with impedance measurements. Especially, loss and kick factors directly affect the beam dynamics and cause many instabilities. The beam manipulation should be performed with respect to these studies. the wire: it does not create any additional fields as it was the case in the analytic scheme. However, the mesh should be homogeneous to avoid any reflections between the meshes, especially in the direction in which the beam is travelling.

-Choice of Impedance

• Boundary Condition: The boundary conditions define the states of the areas at the entry, the exit, and the surroundings of the component that is being simulated. There are several modes for this, such as "open," which makes the boundary plane act as a perfectly matched microwave absorbing material, and will guarantee that this plane is seen as open space. For the "electric" mode, the boundary acts as a perfect ground plane placed. The boundary conditions must be in "open" mode in the direction perpendicular to the beam for it to be able to absorb all modes and prevent nonphysical reflections.

Depending on the state and the geometry, the boundary conditions on the other planes can be in "open," "magnetic," or "electric" modes. The main purpose of the boundary conditions is to prevent undesirable reflections, and when necessary, to provide lossless transition for the EM fields.

• Integration Method: It defines the integration method to calculate the wake potential. There are 3 different ways that should be applied to different components:

a) Direct
This method depends on directly integrating the EM force in Equation 1.14 along the integration line. This method is well known to be the least accurate one. Comparisons between methods were performed, and will be presented in the section 3.2.

b) Indirect Test Beams

This is the most accurate integration method. However, it can only be used for ultra-relativistic electron beams. The beam tube's cross section at the entry boundary should be identical to the cross section at the exit boundary.

In addition to this, the structure must be convex, which basically means that the cross section of the part between the beam tubes must be larger than the tube's cross section. Consequently, this integration method cannot be applied to a collimator in general.

c) Indirect Interfaces

If the cross section of the beam tube is nonidentical at the entry and the exit boundaries, one can choose this method for ultra-relativistic beams. Concave structures cannot be treated with the Indirect Test Beams method, but with the Indirect Interfaces method.

• The Background Settings: The external (non-component) space should be defined. The background can be set as a specific material or as vacuum.

• Symmetry Planes: When the component model is symmetrical and the wake field type allows us to use the symmetry planes, it will save us computational time.

. Setting of the Parameters of CST Simulations

Some of the simulation parameters were previously set with the predefined conditions. However, others have to be optimised before being set. Preliminary simulations should be performed to tune them and to obtain maximum accuracy and efficient simulation time. This work is fundamental, as complex structures are involved in ThomX for transverse impedance budget evaluation. This complexity is naturally accompanied by simulations taking a long time. It is then crucial to optimise the ratio of computation time to precision of the output wake potential.

In the beginning of ThomX impedance simulations, we used transmission line injection scheme to avoid nonphysical longitudinal electric fields. The effect of the simulation parameters -number of mesh cell and homogeneity of the meshwas checked as the first part of the test simulations. As the integration method for the simulation of ThomX pieces, it was decided to use the Indirect Test Beams.

Mesh cell per wavelength defines how many meshes that will be put with respect to wavelength. The homogeneity can be achieved with mesh equilibrium ratio which defines the proportionality of size difference of two neighborhood mesh cells.

. Mesh Cell

For calculating to EM field of the entire structure, the volume was divided into meshes for which the EM fields were calculated separately and interpolated or extrapolated afterwards using the numerical methods. In order to be able to have precise and reliable calculation results, the mesh parameters should be set properly. This is only made possible by doing some testing to see the effects of each parameter. The parameters used in CST are the Mesh cell per wavelength, the mesh equilibrium ratio and the total number of mesh cells. Along with the simulation precision, these parameters also affect the simulation time and needed computational power. For this reason, numerous tests were performed. Mesh cell per wavelength was tweaked between 8 and 25 to get 1 • 10 5 to 1 • 10 7 number of cells and smooth mesh equilibrium ratio was changed between 1.05 and 4 to see the homogeneity.

. Mesh Cell Equilibrium Ratio

The simulations were performed within step in-out cavity, which are the simplest cavities, widely used in accelerators and also are well-known devices. The simulation duration is significantly lowered and gives us the ability to compare the results with analytical calculations.

For transmission line injection scheme, the mesh should be homogeneous in the longitudinal direction in which we need the smooth mesh equilibrium ratio as small as possible. The smooth mesh equilibrium ratio was changed between 1.05 and 4 for step in-out cavity. As a result, for this structure, a significant change in the simulation duration has not been observed. Even as the smooth mesh equilibrium ratio increased, some frequency peaks were lost in the impedance spectrum. So the smallest value (1.05) was used for simulations to get maximum mesh homogeneity and the best results.

. Mesh Cell Per Wavelength

In the test to get the optimum ratio of mesh cell per wavelength, the mesh numbers increased in each step as expected. However, the simulation time has also increased with the increasing number of meshes. The tests were performed using basic step in-out simulation with the wavelength of 1 • 10 3 and the mesh equilibrium ratio of 1.05. The step in-out structure is a cylindrical cavity with two small cylindrical pipes attached to the end-caps. The geometry will be explained in more detail in Section 4.1. With the results obtained by using different numbers of mesh cells per wavelength, as shown in Table 3.1, the total number of mesh cells increases significantly, which leads to increased simulation duration. Also, impedance peaks were analysed for each value, and it was seen that after 15, the resolution of the peaks were the same for each step in-out cavity.

. Computing Time

For the EM simulation, it is important to have balance between simulation time and the accuracy. The optimal number changes with respect to the complexity of the structures. For example, for the pumping port a mesh cell per wavelength of 15 is enough however due to its electrode the bpm has at least 20. This is why, the optimal mesh cell per wavelength of 15 to 20 was found for ThomX structures. shown (others set to the wavelength of 1 • 10 3 and the mesh equilibrium ratio of 1.05.). After the 20 mesh cell per wavelength all the simulation results were the same for step in-out cavity. The simulations were performed using a 6-core, 12-processor Intel Xeon 2.10 GHz computer, with 128 GB of RAM.

. Integration Methods

The integration method test simulations begin with electron beam longitudinal sigma being 4 mm to get the beam frequency to be around 20 GHz, the relativistic beta is 1, and the charge is 1 • 10 9 C. For the test, the longer beam longitudinal sigma was used to lessen the simulation time. The frequency of the simulation is between 0 to 25 GHz with a wavelength of 5000 mm.

To see the Indirect Interface (II) and Indirect Test Beam (ITB) methods' differences, basic simulations were performed using both the step in-out structure and the Interaction Point Chamber (IPC). The IPC will be studied thoroughly in the next section, however, there are two reasons why this assembly was chosen for this test. Firstly, IPC contains two electrodes which can be problematic for the meshing in CST. Also it gives an opportunity to simulate the object using both methods, unlike the other accelerator components of the ThomX. The results were the same in step in-out structure with a high mesh number. However, it is not possible to have the same mesh performance with more complex structures like the BPM or the FBT due to the computational power restrictions. When the simulations were performed with the II method, they gave analogical resonance peaks and non physical deviations on the wake potential with the sharp edge flat structure and also with the ThomX components. Both II and ITB can be applied to IPC and they are also complex enough the check the results with CST. In Figure 3.2 the wake potential results of IPC with II method has non-physical deviation because the II method cannot be applied when the mesh is not proper.

C)

impedance spectrum becomes physical.

Consequently, the structures suitable for use of the ITB method were simulated by using them. The rest were simplified to be simulated using the II method to avoid the meshing issues.

. Conductivity

In order to highlight the resistive wakefield effects, the electrical conductivity of the material was changed after finding the optimal values for the mesh cell per wavelength and the smooth mesh equilibrium ratio. The longitudinal resistive wall impedance is inversely proportional with the electric conductivity [START_REF]CAS -CERN Accelerator School : 5th Advanced Accelerator Physics Course[END_REF] as shown in Equation 3.2:

∂ Z k ∂ z = 1 + i 2pb r wZ 0 2cs c F (3.2)
where s c is the electric conductivity of the material, w is the frequency of the impedance, F is a form factor depending on the pipe's cross section (can be taken as 1 for rectangular pipe longitudinal impedance), b is half-height of the shape and, Z 0 is the vacuum impedance.

For the simulations, a rectangular box was chosen to investigate only the conductivity phenomena while avoiding the geometric impedance. Indeed, the rectangular box has no geometric impedance component but only a resistive wall with a finite conductivity material. The rectangular box was constructed with a perfect electrical conductor and 3 different sorts of copper with electrical conductivities of

-Longitudinal impedance

. Principle of the longitudinal impedan measurements

The coaxial wire method is used to measure the longitudinal and transverse impedances of accelerator components. The method was first described by Sands and Rees in 1974 [START_REF] Sands | A Bench Measurement of the Energy Loss of a Stored Beam to a Cavity[END_REF] and the technique was studied and detailed by Gluckstern and Vaccaro [START_REF] Vaccaro | Coupling impedance measurements: an improved wire method[END_REF][START_REF] Gluckstern | Analysis of Coaxial Wire Measurement of Longitudinal Coupling Impedance[END_REF] later on. The initial longitudinal measurements for ThomX parts were explained briefly in a previous works [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF][START_REF] Gamelin | Longitudinal impedance and coherent synchrotron radiation models for thomx storage ring[END_REF]. However, further measurements were done on longitudinal components to minimise errors and increase reliability, and to fill the gaps in former works done for ThomX storage ring components. The general scheme of the setup is simple and can be seen in Figure 4.1.

The wire passing through the Device Under Test (DUT) is connected to the Vector Network Analyzer (VNA) with a matching network to measure the S parameters.

In theory, the EM field distribution of the ultra-relativistic charged beam is similar to the fields in a Transverse Electromagnetic (TEM) line [START_REF] Kroyer | Longitudinal and transverse wire measurements for the evaluation of impedance reduction measures on the mke extraction kickers[END_REF]. So, the impedance measurements can be done with a single wire, or two wires that can be used in TEM mode. TEM mode is the radially symmetrical EM field mode, which means that the EM field is perpendicular to the field propagation. For example, the EM field equations for TEM mode coaxial wire (Eqn.4.1) and the beam (Eqn. 4.2) are [START_REF] Niedermayer | Determination of beam coupling impedance in the frequency domain[END_REF]: respectively; where E r is the electric field in radial direction, H f is the magnetic field in azimuthal direction, r is the radial position, w is the frequency, and Z 0 is the vacuum impedance.

E r (r, w) = Z 0 H f (r, w) = constant 2pr exp( j w c z) (4.1)
In general, simulating the beam field in the TEM mode of a coaxial wire is only possible below the first higher order mode cut-off. The longitudinal impedance measurements are more straightforward compared to transverse ones [START_REF] Caspers | Impedance determination from bench measurements[END_REF]. The wire is inserted into to the DUT and the scattering parameter S 21 , which gives the ratio of the output power to the input power, is measured. More information and mathematical explanation about the S parameters will be given in Section 4.1.1.

The different approximations for calculating the impedance from bench measurements can be found in [START_REF] Caspers | Impedance determination from bench measurements[END_REF][START_REF] Hahn | Coaxial wire measurements of the longitudinal coupling impedance[END_REF][START_REF] Jensen | An improved log-formula for homogeneously distributed impedance[END_REF]. The most widely used formula to obtain impedance from S parameters is the log formula, which is mostly for distributed impedance but can also be applied for lumped impedances. The longitudinal beam coupling impedance Z can be calculated by using the log formula:

Z = 2Z L ln( S21 DUT S21 REF ) (4.3)
where Z L is the line impedance (described in Section 5.2), S21 DUT is the measured S parameter of the DUT, and S21 REF is the measured S parameter of a reference component. The condition of Z >> Z L is essential for the log formula. The reference component should be measured to disentangle the defaults of the setup and the geometry of the beam pipe cross section. In our case, the vacuum pipe of ThomX was chosen.

. RF Network and S Parameters

The S parameters are currently used in the field of microwaves to describe the Although the formalism of the S parameters is applicable for all frequencies of EM waves. As a consequence, it is very useful for a wide range of impedance measurements. These parameters depend on the measured frequency and can be measured using well-known vector network analyzers (VNA). They are generally represented in matrix form that can be manipulated with linear algebra laws.

Let us consider an amount of power b i delivered in the i = 1, . . . , N ports, and the received power called a j from the j th port. Then, the scattering parameters S i j linked these two values as follows:

b i a j = S i j (4.4)
The scattering matrix as shown in Figure 4.2 can be determined by VNA. Prior to the measurements, the VNA should be calibrated to lessen the effect of the cables. The goal of the matching is to cancel the non-diagonal terms of the matrix out. Also, semi-rigid cables should be chosen to minimize the phase drift of the incoming signal. For the measurements, it is important to make a matching of the impedance of the setup to perform S-parameter measurements for greatest reliability. The impedance matching procedure and effects will be presented in the next section. The impedance of the VNA (Z 0 = 50W) must be matched with the line impedance for the EM wave to propagate through the wire up to the VNA without reflection.

. Line Impedance

The characteristic impedance of a transmission line can be described as the form of permeability (transmissivity of the em fiels) of the medium. It plays a role

similar to what we observe with sound waves or EM waves. When a wave crosses the border between two different media, part of its energy cannot be transmitted from one medium to another and travels back in the other direction. In a transmission line, it corresponds to the impedance that could be measured at its terminal if it had an infinite length. This is the reason why transmission lines should be "terminated" by loads which correspond to this line impedance. Thus, the signal is lost in a load which gives the same effect as the line continuing to infinity and the signal not being reflected.

The line impedance, in our experimental setup, can be obtained in two ways.

First it can be simulated through CST and secondly it can be calculated from the analytic formula for the case of round chambers. The simulations were done before [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF] with CST Microwave Studio with various wire diameters such as 0.2 mm, 0.6 mm, 1.0 mm, 1.4 mm, 1.8 mm and 2.2 mm and the relative error between the simulation and analytical calculation of the round pipe with an equivalent diameter of 30.9 mm. The relative error was found as 0.02 % for 0.6 to 1.0 mm wires.

. Longitudinal Impedance Analytical Expression Related to Bench Characteristics

The equation for line impedance is [START_REF] Kroyer | Longitudinal and transverse wire measurements for the evaluation of impedance reduction measures on the mke extraction kickers[END_REF]:

Z L = Z v 2p ln( D d ) (4.5)
where Z v is the vacuum impedance which is 376.73 W, D is the diameter of the beam pipe, and d is the diameter of the wire.

Being able to get a high impedance value, meaning having the wires as thin as possible, is important to achieve the best performance of acting of the beam as an ideal current source with minimizing the field disturbing effect of the wire.

In the most general way, the impedance components should be smaller than the line impedance. However in reality, the accelerator components have a line impedance in a range of a couple of hundred Ohms, unfortunately. Also, it is more preferable to work with a low line impedance to measure small impedances due to the increase in the attenuation on the wire with smaller wire diameters. For these reasons, achieving the optimum wire diameter is important for wire mea-surements.

The measurements were performed with wires of diameters 0.1 mm, 0.5 mm and 1.0 mm. As can be seen in Figure 4.3, the measured amplitude of the S parameters on each resonance frequency decreases as the frequency increases, meaning higher attenuation. Let us look closely at one of the troughs in the measurement in the graph shown in Figure 4.4. There are two close peaks for three different wire measurements, however, the trough amplitude gets smaller with increasing wire diameter. The second trough cannot be observed for 0.1 mm-wire measurements even though it is a strong resonance trough. So for our case using 1 mm wire is important to observe all troughs. The attenuation on the wire creates a significant difference between the beam and the wire measurements. The quantity of this attenuation also depends on the impedance type. There are two impedance types called distributed and lumped impedance. The difference between these is their distribution along the z axis. As can be seen in Equations 4.6 and 4.7, the lumped impedance is localized to an area, but the distributed is equally dispersed along the accelerator component.

∂ Z lumped (w, z) ∂ z = Z total (w)d (z z 0 ) (4.6)
∂ Z dist (w, z) ∂ z = Z total (w) l (4.7)
where w is the frequency and l is the length of the accelerator component. The lumped impedance contains geometric impedance and the distributed one contains the resistive wall impedances. In the realisation of the accelerators, the components generally have them both. There are different kinds of formulas to calculate them separately [START_REF] Sands | A Bench Measurement of the Energy Loss of a Stored Beam to a Cavity[END_REF][START_REF] Hahn | On coaxial wire measurements of the longitudinal coupling impedance[END_REF][START_REF] Jensen | An improved log-formula for homogeneously distributed impedance[END_REF][START_REF] Vaccaro | Coupling impedance measurements: an improved wire method[END_REF][START_REF] Karantzoulis | An overview on impedances and impedance measuring methods for accelerators[END_REF], however they will give differing results compared to each other. So a new formula that gives a correct result for both should be applied for measurement. This leads us to the formula in Equation 4.3.

. Measurement Bench

The longitudinal measurements were performed before my thesis for some ThomX components [START_REF] Gamelin | Longitudinal impedance and coherent synchrotron radiation models for thomx storage ring[END_REF]. However, there was an electrical connection problem which created additional resonances around 1.5 and 6 GHz in the old setup. Also the electrical connection problems create some deviations in the measurements which lead to additional errors. Furthermore, the bench should be adapted to the two-wire and moving-wire measurements to be able to deduce the transverse in addition to the longitudinal one. The new bench was proposed to be designed to solve the electrical connection problems, minimising the error and adapting it to transverse measurements. The two setups can be seen in Figure 4.5.

In the old bench, the RF connector shield was movable, and the connection of the surface of end-caps and DUT was not fit perfectly as can be seen in Fig- ure 4.5a. As opposed to the old bench, the shell was designed to be completely closed to shield the RF connectors properly, and a small screw was added to make a good electrical connection with the end-caps as in Figure 4.5b. The half-shell RF shielding flange at the connecting point components was changed to full-shell, enclosing the connectors completely. This reduces the additional impedance noise around 1.5 and 6 GHz with the RF connector shell. Also, the error on the measurement was significantly reduced from 4 5 dB to below 1 dB with the new setup as shown in Figure 4.6. In the literature, due to small amplitude and high noise values, the measurements were not performed widely; and the performed ones has a high noise in the measurements [START_REF] Kroyer | Longitudinal and transverse wire measurements for the evaluation of impedance reduction measures on the mke extraction kickers[END_REF]. The comparison tests were done with the bellows, which has strong impedance peaks.

. Alternative Method for Wire Measurements: Resonator Coaxial Wire

Method

The resonator coaxial model can be used to determine the real part of the longitudinal impedance, and also the line attenuation as an alternative. As mentioned before, there is an attenuation on the impedance due to the wire. This method allows us to determine the amount of attenuation we will experience. The same setup for the wire measurements can then be used by changing the transmission line matching with capacitive coupling, shown in Figure 4.7 [START_REF] Kroyer | Longitudinal and transverse wire measurements for the evaluation of impedance reduction measures on the mke extraction kickers[END_REF]. The capacitive coupling at the connection to the port allows the DUT to resonate at the frequencies as in Equation 4.8.

f = nv 2L (4.8)
where L is the length of the DUT, v is the velocity of the wave, and n is the number of the resonance. This method enables us to measure line attenuation precisely and to get rid of the matching problems and the mechanical errors, like calibration problems or electrical connections. However, this method only measures the real part of the longitudinal impedance. There is also another method used which consisted in short circuit to port 2. This method changes the frequency as in Equation 4.9.

f = (2n 1)v 4L (4.9)
The line attenuation can be calculated from the Q factor of the DUT resonance

(Q 0 ) in isolation which has a relation Q 0 = Q L (1 + k)
where Q L is the Q factor of the DUT with the coupling and k is the coupling coefficient. The relation of the line attenuation (a) with the Q factor and frequency is shown in Equation 4.10:

a = p v f Q 0 (4.10)
The equation gives not only the beam impedance attenuation but also the attenuation due to the insertion of the wire inside the pipe. If only the attenuation due to the inserting wire is needed, it can be found from the Equation 4.11 [START_REF] Day | Measurements and Simulations of Impedance Reduction Techniques in Particle Accelerators[END_REF].

a w = p pr w e f 1 dln(D/d) (4.11)
where r w is the wire resistivity, e is the permittivity of the wire material, d is the diameter of the wire, and D is the diameter of the DUT. As can be observed in Equation 4.8, the measurements' frequency span is limited by the length of the DUT.

. The Coaxial Wire Measurement

Limit -High Q-Factor Impedances

For the accelerator components such as cavities that have a high quality factor impedance (Q>100), it is not suitable to use the wire method due to the perturbation at the boundaries especially under the cut-off frequency of the beam pipes.

For the longitudinal case, it was proven that the wire method gives inaccurate results for high quality factor objects like cavities [START_REF] Panniello | The stretched wire method: A comparative analysis performed by means of the mode matching technique[END_REF]. With the beam, the losses do not contribute to the propagation below the cut-off frequency, the only loss comes from the trapped modes of the cavity. However, putting a wire shifts the cut-off frequency to zero and allows the propagation losses which create strong impedances for all frequencies. The longitudinal impedance for high quality factor components was investigated before and the unreliability of the wire method was well described for pillbox cavity as an example [START_REF] Panniello | The stretched wire method: A comparative analysis performed by means of the mode matching technique[END_REF].

All in all, the wire creates additional impedance peaks which were not seen by the beam. However, in the ThomX TL there are many bellows which can be seen as a pillbox with multiple corrugations. For the storage ring part, the impedance peaks are suppressed with RF fingers, however, for the transfer line there are bellows without fingers. Even if it is one pass in the transfer line, it is important to define the effect of these bellows on the impedance budget of ThomX. Therefore, the investigation of the pillbox cavity with wire measurements and also analytically are important to build a suitable model for the bellows.

. Theory and Simulation Comparison

The calculation of an analytic solution for the impedance of a simple object -like the step in-out cavity-is possible. These calculations and their comparison with the simulation and measurements allow us to test the reliability of the measurements and the simulation parameters.

Let us first take the step in-out cavity of length g, larger radius R, and step radius b as shown in Figure 4.8. If it is made of a perfect electric conductor material, the longitudinal impedance of step in-out will be like in Equations 4.12 and 4.13 [START_REF] Gluckstern | Analytic methods for calculating coupling impedances[END_REF]. However, to solve these equations, we have to first solve the coupled Equation 4.14 with using Equations 4.15 and 4.16.

Z L (w) = Z g 0 f (z 0 )e j w v z dz 0 Q (4.12) Z L (w) Z 0 = jc wb 2 I 2 0 (x) Z g 0 g(z 0
)e jkz dz 0 (4.13)

e ikz = Z g 0 [K p (z z 0 ) + K c (z z 0 )]g(z 0 ) dz 0 (4.14)
f (z 0 ) = jcQZ 0 wbI 0 (x) g(z 0 ) (4.17)
where h l and e l represent orthonormal bases of the eigenmode vectors of the cavity, I 0 is the Bessel function of the first order 0, x = wb vg , Q is the charge, and Z 0 is the vacuum impedance. To be able to solve these equations, the cavity modes e l , h l within the cavity should be solved first, satisfying the boundary conditions.

w l = c q (x 0l /R) 2 + k 2 l k l = lp g (4.18)
where x 0l is the l th Bessel function of order 0 which allows us to extract h l from the Maxwell-Faraday equation.

h l (z) = c w l (k l (AJ 1 (u) + BN 1 (u))cos(k l z) q (w l /c) 2 k 2 l (CJ 0 0 (u) + DN 0 0 (u))) (4.19)
where N 0 and N 1 are the zeroth order Neumann function, J 0 is the Bessel function of the first order 0. All these equations lead us to f (z).

m(z) = Z • • f (z 0 )K p (z z 0 )dz 0 (4.20) m(z) ⌘ QZ 0 2pbJ 0 (x) e j w v z Z g 0 f (z 0 )K c (z, z 0 )dz 0 (4.21) f (z) = Z • • m(n) Kp (n) e jnz dn 2p (4.22)
From the Equation 4.18 the resonance impedance peak frequencies can be found. It can be used as a first step to check the measurements' reliability. Also, these derivations allow us longitudinal impedances for the entire spectrum to be created.

The analysis started with the step in-out cavity with a radius of 101 mm and length of 220 mm. Firstly the resonance peak frequencies (due to the trapped modes) were calculated from the Equation 4.18. And then the same step in-out cavity was measured using wire measurements and simulated with CST Particle

Studio. The resonant peaks were expected from the analytical calculation and simulations as in the Table 4.1. However, as mentioned in the former section, the step in-out cavity has a high Q-factor impedance, which means that we will see the contribution of the propagation losses. So for all modes from the trapped and propagation will be seen in both simulation and measurements as shown in Figure 4.9 and Table 4.2, which contains the resonant peak frequencies. The huge deviation between the measurement results and the CST simulations was expected due the wire. As the comparison of the measurements and simulations, the results of 0.1 mm were chosen to minimize the effects of the inserted wire. As can be seen in Figure 4.10, when the wire diameter gets bigger, more peaks are observed due to the fact that the wire perturbs the field inside the component. For the strong resonant peaks and low frequencies, the higher line impedance will compensate the effect on the wire attenuation. However, if the amplitude of the peaks is lower, some peaks cannot be observed in higher frequencies, as observed in our measurement with 0.1 mm wire in 5.99 GHz.

. ThomX Components'

Longitudinal Impedance

. Reference Longitudinal Measurements

The ThomX reference component is non-cylindrical and has the diameters of 28 mm vertically, and 40 mm horizontally, as shown in Figure 4.11. The reference measurements were performed 5 times to minimize human error. The mean of these measurements and the standard deviation can be seen in Figure 4.12. The standard deviation is below 1 dB for frequencies of up to 5 GHz. The main reason for errors at higher frequencies is the loss due the resistors et such a frequency.

. Fast Transverse Feedback -FBT

In order to suppress beam instabilities with a fast response, transverse feedback kicker is used. It applies to the beam a counter-kick to damp coherent transverse motion, bunch by bunch and turn by turn. The transverse motion ids detected by an other system such as BPM. The FBT of ThomX was designed to have a minimum coupling impedance with capacitive height 11 mm [6]. The 3D CST model of the transverse feedback kicker has been studied in former works [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF]6]. After the production and the assembly, signal tests were done for the feed-throughs with VNA and compared with the design values. It was comparable with the simulations [6]. The picture of the manufactured FBT can be seen in Figure 4.13.

The longitudinal measurements of the FBT were performed to check the accuracy of former works and simulation results. Also, the longitudinal impedance would need to be used in moving-wire method. On the other hand, the longitudinal result comparison and error propagation with the moving-wire method will be 91 The measurements are also important to check the manufacturing quality. The capacitive gap and the height can cause a frequency shift in the impedance domain. The assembly process of the FBT is tricky. The copper foil was placed with hand in a really small opening. So, the error on the displacement of the copper foil is high and should be checked with the measurements. If the frequency peaks position measured and simulated are in agreement, it implies that h has been correctly placed as there is a dependence between both. These measurements were also needed as input data for other measurement methods. They will also be compared with other methods: the measurements in Chapter 6 which can be performed both 1 to 6 GHz and 0 to 2.5 GHz, but the measurements in Chapter 5

can only be performed between 0 to 2.5 GHz. This is the reason for performing the longitudinal measurements up to 2.5 GHz, even though we were capable of performing them up to 6 GHz.

The simulations were performed with proper matching at the feedthrough, however, the amplitude was too small to be cross-checked with the measurements.

That is why the measurements were performed without matching at the feedthrough.

The comparison of the results in Figure 4.15 show that the capacitive gap and the height were assembled correctly with respect to the design parameters. Six impedance peaks were seen in both the simulations and the measurements, and they are in the same exact frequency span.

. Bellows Longitudinal Measurements

In this section, the longitudinal measurement (using one-wire method) of the bellows with and without RF fingers will be presented. The comparison with the moving-wire method will be done later, in Chapter 6.

Bellows are widely used in accelerators to compensate for thermal expansions and contractions. They are also given sufficient tolerance to compensate for misalignment. The geometry of the bellows has cavity-like structures with numerous corrugations. The designed CST model of the ThomX bellows can be seen in figure 4.16. Due to their geometry, bellows tend to create high impedances and trig-ger higher order modes. This behavior can create beam instabilities and heats.

The RF fingers can be used to shield RF fields to reduce impedances. These RF fingers are also movable in the beam direction to handle the stretch of the bellows.

The measurements and simulations are quite important to check efficiency of the RF fingers. The physical understanding of the impedance plots can be achieved by approximating with some analytical formula without running the impedance simulation.

This approach can also help us to check the reliability of the simulations especially above the cut off frequency. In literature many theoretical approximations were studied with the bellows [START_REF] Of | IEEE Particle Accelerator Conference[END_REF][START_REF] Beam | Proceedings of the Workshop on the RHIC Performance at Brookhaven National Laboratory[END_REF][START_REF] Shobuda | Analytical evaluations of coupling impedances of resistive and magnetic bellows[END_REF]. Around the cut-off frequency, the impedance measurements are not valid and the simulations have problems due to the mesh size of the small and close corrugations. The one of the analytical approximation is assuming a rectangular corrugation of the bellows having depth D, half width g and beam pipe radius b which was studied in [START_REF] Henke | Impedances of a Set of Cylindrical Resonators with Beam Pipes[END_REF] and simplified when g/b << 1 [START_REF] Of | IEEE Particle Accelerator Conference[END_REF]:

Z(w) = gZ 0 pbI 2 0 ( b)D (4. 23
)
where k = w/c, Z 0 = 377, I 0 is the modified Bessel function of order zero, b = bk/gb , b and g are the relativistic velocity parameters of the beam particles. D is simplified to [START_REF] Of | IEEE Particle Accelerator Conference[END_REF]:

D = j cot (kD) s  s=1 2kg q k 2 b 2 j 2 0s + •  s=S+1 2 jkg q j 2 0s k 2 b 2 (4.24)
where j 0s is the s-th zero of the Bessel function J 0 . The zero points of the imaginary part of the D function will give us the resonant peaks. When we neglect the summation terms the resonances will be seen in kD = p/2, 3p/2, ... .For our model the D = 1.5 cm leads to resonant frequencies at 5 GHz,15 GHz...

Compared to former work [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF], we explored the role of the wire diameter as it has a direct impact on the line impedance and frequency precision it will be presented in next sections.

First, I will go through detailed simulations and measurements without the RF fingers. The RF finger is a metallic component which surrounds the beam and prevents it from seeing the bellows corrugations. The design was optimized to fit inside the bellows well, as shown in Figure 4.17.

The reason for measuring without the RF fingers is threefold: i) the strong impedance peaks of the bellows allow the reliability of the measurements to be verified, ii) its ability to confirm the reliability of the RF fingers, and ii) the fact that there are many RF fingerless bellows in the Linac and TL of the ThomX, the impedance of which could have effects on the beam in its sectors of the accelerator.

Firstly, the longitudinal impedance measurements were performed with a 1 mm wire. The longitudinal impedance measurements and simulations were detailed in the former work [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF]. The measurements were only performed on axis for checking the new setup which is adapted to TWMs and MWMs. Also the measurements with different wire diameters were performed to see the impact of the wire diameter on the measurements.

Many sharp peaks and high impedance are expected with the bellows with many 97 Figure 4.17: The 3D designed of the bellows with RF Fingers corrugations [START_REF] Of | IEEE Particle Accelerator Conference[END_REF]. However, it is impossible to find the impedance results of the bellows through analytic calculations. The measurements and simulation were needed to describe the frequencies and the amplitude of the bellows impedance peaks. However below the cut-off frequency due the analytical calculations with our model the peaks should be observed at 5 GHz and 15GHz. But of course in reality the impedance peaks were observed even below the cut off frequency due to the step in-out cavity nature of the bellow pieces.

Fırst it was started with the CST simulations. In the simulations without RF fingers [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF] four peaks at 2.08GHz, 3.51GHz, 4.38GHz and 4.89GHz were observed. After, the simulation results are cross-checked with the wire measurements. The measurement with 1mm wire diameter has been performed as shown in Fig. 4.18. The only two peaks are measured at 2.2GHz and 4.3GHz which is consistent with the former measurements and simulations which can be seen in figure 4.19 [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF]. There is a maximum deviation on the measurements about 5% which is expected due to the movable nature of the bellows, the uncertainty on the 5 measurements and mechanical tolerance. The measurements were also performed with 0.1mm wires. In this case, the four peaks are also measured at 2.2GHz, 3.6GHz, 4.3GHz and 4.6GHz contrary to the 1mm wire case. We find again a frequency shift of around 5 %. There is deviation in frequency span with 1mm and 0.1mm wire measurements and simulations. For 0.1mm measurements deviation is less than the 1mm wires which can be seen in figure 4.19, first peak slightly deviate to 2.14GHz, the second peak is also at 3.51GHz, third peak is deviated to 4.30GHz and the fourth peak can not be seen in measurements.

One hypothesis of these shifts can be the variable geometry of the real below. It is quite expected for comparison between wire measurements and simulation. The resonant frequencies of the bellows depend on the beam pipe diameter, corrugations number, depth and half width of the corrugations [START_REF] Of | IEEE Particle Accelerator Conference[END_REF]. Below cut-off frequency, due to its higher order and trapped modes and cavity-like behavior of the designed bellows, the accuracy of analytical calculations cannot be relied upon.

However, in successive measurements, the bellows' geometry was kept the same by using fixing components for all every wire diameter measurement. So, the observed effects between 1mm and 0.1mm wire measurements are mostly due to the wire diameter and are not due to the modification of the geometry. Moreover the wire inside the pipe distorts the EM field which can cause frequency shifts [START_REF] Belver-Aguilar | Transverse impedance measurements and dc breakdown tests on the first stripline kicker prototype for the clic damping rings[END_REF] as discussed in section 4.1.3.

The measurements with 4 different wire diameters were performed to understand the frequency shift with respect to the wire diameter.

As a first observation, in general the amplitude is much smaller than the simulation results. This loss comes from the resistors and the attenuation due to the thickness of the wire [START_REF] David Large | Broadband Cable Access Networks[END_REF]. This was also observed with the different wire measurements in the Section 5.2. The current density is also important to measure the signal from the wire. Also resistance value of the carbon resistors is strongly frequency dependent. this creates additional noise and power loss especially in higher frequencies [START_REF] Hayt | Engineering Electromagnetics[END_REF].

As a last step for checking the reliability of RF fingers the simulation and measurements were performed with RF finger (see Figure 4.20). The CST simulation results show that the RF fingers have removed all the impedance peaks. In the measurements, it has big uncertainty due to the movable geometry of the bellow and the geometry of the RF fingers inside. There is small oscillation around zero which comes from the measurement uncertainty however it successfully suppressed all the high impedance peaks. the BPM4 and BPM8 can be seen, the four electrodes in BPM4, BPM6 and BPM8 are used for the beam diagnostic however the other electrodes in BPM6 and BPM8 are used for additional jobs such as ion clearing, longitudinal feedback and transverse feeback with FBT. The metal outside the BPM is stainless steel and the buttons and pick ups were made with molybdenum. The design and the impedance matching are crucial for the impedance simulations. Unfortunately, the electrodes were bought as ready to use by a private company and they do not allow us to access the actual design. In the simulation, the classic design was used for button BPMs however checking the results with the measurement is crucial in this situation. In the former works [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF], we showed that the impedance amplitude is proportional to the number of the electrodes. The EM field of the moving beam inside the BPM can be trapped between the BPM button and the button housing. Also the EM fields can propagate inside dielectric materials, can be reflected and thus affect the beam. The trapped modes, mechanisms and frequency were searched in the former works [START_REF] Gamelin | Collective effects in a transient microbunching regime and ion cloud mitigation in ThomX[END_REF] which lead to Equation 4.25 [START_REF] Duarte | Design and impedance optimization of the sirius bpm button[END_REF].

f = c 2p p e r s ✓ 2m r p + r h ◆ 2 + ✓ p p t c ◆ 2 (4.25)
where e r is the relative dielectric permittivity, r p is the pin radius, m is azimuthal index, p is the longitudinal mode number and t c is the dielectric thickness. For our model, the trapped mode frequencies are 8.642GHz, 17.284GHz and 25.926GHz. The trapped modes of the BPM (see Figure 4.23) are only seen after the 8GHz which is over our measurement limits. But if we look at the S parameter measurements of the BPM and reference component (see Figure 4.22), no difference can be seen up to 6 GHz as expected. The measurements were performed only for BPM4 and it is valid for the other types which was checked by the simulations as in the Figure 4.23.

. Pumping Port

The pumping ports are the accelerator components whose aim is to provide connection between the beam pipe and the vacuum pumps. As shown in Figure 4.24 the pumping ports contain a circular opening for the pump connection perpendicular to the typical ThomX cross-section through the beam direction. Inside the pumping port there are grids longing through the beam direction. There are different types of pumping ports inside the ThomX, some straight and some of them bended. The measurements and simulations were performed only for the straight piece and it was assumed that the results of the straight and bended components were the same due to the similarity of the grids.

104

. Other Components

Unfortunately, due to the planning delay of ThomX project, impedance measurements have not been a priority and the other components have not been mea-105 sured despite some of them dominating the impedance budget such as septum or kickers. Only the simulations were performed for these objects. The longitudinal simulations of them were shown in the previous thesis of Alexis Gamelin [34].

106 5 -Two Wire Measurements

. Measurement Description

Dipolar and quadrupolar impedance components are important to define the effects given to the beam as offsets and the focusing or defocusing effects, explained briefly in Section 1.2. This gives us the ability to evaluate the related tune shift and additional needs for orbit correction.

Defining these components is problematic both with the measurements and simulations. Generally, these components are studied with the simulation programs such as CST Particle Studio, however for the complex structures, one can face many problems due to the meshing. Checking simulations with measurements were crucial, especially for the dipolar components.

The measurement setup can have two issues to solve, (i) the signal is too low to be measured and it is generally below the systematic noise and (ii) the wire spacing should be small and exact to have reliable results. The solution to the first one is to lower the noise as much as possible by designing the bench with good electrical contact and using metal shielding with RF foams to suppress signals from the environment. For the second issue, the bench was designed for the wires to be stretched easily with the precise offset, which will be briefly explained in Section 5.4. Also, the tension on the wires affects the measurement results, and this new bench allows us to have a similar strength in all measurements.

The other problem effective on the measurements is the wire diameter, for which a trade-off has to be made. The wire diameter should be chosen carefully to be able to find a balance between having it as small as possible to be in the range of wire measurement approximation, which is explained in Section 1.2.5, and having the wire with a diameter as large as possible to minimize the attenuation on it, and then to maximise the signal level. All these modifications allow us to expand the frequency range of the measurements and to be able to measure even the smallest signals. The dipolar impedance comes from the fringe fields of wakefields, which makes them too small to measure. Simulation programs, including the CST, need massive computational power to be able to define the correct boundary conditions and to create the correct model. Since such power is not available, the results cannot be fully relied upon. Also all these problems grow with increasing frequency as mentioned before, which lead to further difficulties in applying it to high frequencies. In this work, with multiple improvements on the measurements' setup and procedure, determination of the impedance up to 6

GHz was achieved with detailed error study.

The most known method for transverse impedance is the two-wire method, which contains two wires driven with opposite phases, giving only the dipolar contribution of the impedance.

The other method is the moving wire method (MWM), which gives the sum of the dipolar and quadrupolar components. The two wire measurements method (TWMs) is crucial to define the dipolar component alone separately. Then, it also allows to extract the quadrupolar component from MWMs. This enables to calculate effect on beam dynamics of each transverse components having a different impact on the beam. The moving wire measurements were also performed and presented in Chapter 6. The method is based on stretching one wire with a set offset and measuring the S parameters. Measurement setup, impedance matching, and the S parameters will be explained briefly in the following sections.

The last method that can be used for transverse measurements is the coil measurements. This method is convenient to use as it can detect extremely small signals that cannot be measured by the TWM at low frequencies. Two wires are simply replaced by a multi-turn coil in the experiment. It will magnify the induced voltage by the number of turns of the coil. However, in our specific accelerator component, this method makes it insufficient to use due to the frequency limitation, which is generally around 1 MHz. On the other hand, instrument noise and the temperature scroll of the coil resistance limit the accuracy of the measurements even further.

In this chapter of the thesis, the results with the TWM and CST simulations for the dipolar component will be presented. This method has been investigated a few times in the literature, but mainly at lower frequencies or high ripples due to residual mismatch [START_REF] Kroyer | Longitudinal and transverse wire measurements for the evaluation of impedance reduction measures on the mke extraction kickers[END_REF][START_REF] Niedermayer | Bench measurements and simulations of beam coupling impedance[END_REF]. The main problems of the TWMs are temperature drifts and noise. The temperature drift problem can be solved with carbon resistors, however, due to their high inductance, they can cause residual mismatch, especially in high frequencies. Some research show that longer setups give more precise results in TWMs [START_REF] Niedermayer | Bench measurements and simulations of beam coupling impedance[END_REF]. So the bench is designed to be larger to minimize the size effect and the electrical connections were studied to minimize the reflections.

The improvements allow us to reduce the noise and perform the measurements precisely even for the high frequencies. The bench and the improvements will be presented in the following sections.

. Line Impedance

Take the sound waves as an equivalence of the EM waves. When the sound waves enter a medium from a different medium, some part of the sound waves' energy will not be transmitted and reflected back to the first medium. The same phenomenon will appear if the measurement device is not matched to the transmission line. Let's look at Figure 5.1; as an incident signal, the step function with a voltage v 0 was sent from the first port of the transmission line with a line impedance Z and for different cases when the matching resistance R = •, R = 0, R > Z, R < Z, and R = Z. For all the cases except for the perfect matching R = Z, the signal will be reflected from the end and will interfere with the signal. This interference will create a complex signal and suppress the expected signal, which is the purpose of the measurements. The line impedance of a transmission line can be defined as the permeability of the medium. So, the same behavior which can be observed for the EM waves inside the transmission line, can be defined as impedance. This impedance can be measured in its terminals -the end points of the transmission line-if the line has a finite length. All in all, the impedance matching, which means the termination with suitable loads, are important to transmit the signal without reflections which can interfere with the expected signal.

The line impedance value calculations were done differently for different struc-tures and measurement setups because the line impedance depends on the measurement device's geometry. For wire measurements, there is an equation to calculate line impedances, derived from capacitance and inductance values, which will be related via the wire diameter and beam pipe diameter in the wire measurement case. For TWMs, if the space between the wires (D) is kept to a minimum, the effect of the beam pipe on the line impedance calculation will be negleted; since D will approach zero, thus making the line impedance due to the beam pipe approach zero, as can be seen in Equation 5.1. So the only matching that should be done is to utilize a measurement device with 50 ohms of resistance. But for checking the effect of the space between wires the line impedance equation for two wires is [START_REF] Kroyer | Longitudinal and transverse wire measurements for the evaluation of impedance reduction measures on the mke extraction kickers[END_REF]:

Z L = 120 arccosh D d (5.1)
where d is the wire diameter and D the spacing. The matching for TWM should also be done with the device itself along with its end caps used in the measurement setup, which will be explained one by one in the following sections. Figure 5.2 shows the general measurement setup of the TWM which has two wires stretched with a distance D inside the DUT, connected to the VNA with the matching resistors and hybrid couplers. The resistive matching at the end of the line should be applied according to the measurement device (e.g. VNA) which usually has the value of Z 0 = 50 W. It also has to be summed with half of the line impedance [START_REF] Kroyer | Longitudinal and transverse wire measurements for the evaluation of impedance reduction measures on the mke extraction kickers[END_REF]. However, in the measurements with a small D, as can be seen from Equation 5.1, only the 50 Ohm matching can be enough. When the matching is done with resistors, the total value of the resistive matching (R s ) should be equal to:

R s = Z L /2 Z 0 (5.2)
The matching means that the reflection of EM wave is cancelled, meaning the wave is fully transmitted.

The simplest way to match the transmission line to the experimental device is to use carbon resistors located at each end of the transmission line. However, the basic resistors are not perfectly suitable for use in the entire range of the frequencies. In fact, they have an inductance which could create mismatch at higher frequencies (up to 3.5 GHz). To solve this problem, a 10 dB attenuator which is matched to 50 W and does not have an inductance, can be used at each end of the setup. Unfortunately, the level of signals we are aiming to measure is very low.

Additional attenuators can suppress the non-linear noise in the base of impedance measurements versus frequency, but at the expense of the output signal level. So we have to notice there, that these attenuators can be used only if the signal level is sufficient due to the attenuation they bring to the measurements.

. Simulations for Wire Spacing

quadrupolar components of the wake potential, and each result of the Indirect Test Beam Method was in accordance with the results in the former study. Therefore, the wire spacing to be used in the wire measurement was decided to be 10% to 20% of the beam pipe diameter. One important result of the simulations was that when the Indirect Interface Method was used, the wake potential function changed and its amplitude increased (see Figure 5.4). After seeing the results of the simulations, it was decided to use the Indirect Test Beam method for suitable accelerator components. The components not suitable were simulated by Indirect Interface method, but the possible errors on the simu-114 lations were noted.

. Descriptions of the Measurement

Bench

The TWMs were performed by stretching two wires with a small distance (D), and by driving the signal between them with opposite phase as it was explained briefly in Section 1.2.5. The measurement setup consists of two end caps, as can be seen in Figure 5.5. These two components attach to each end of the DUT, allowing us to use the same setup for every component of the accelerator, regardless of length.

Each of These two endcaps have two SMA connectors on both ends that connect the wires to the VNA. During the production of the test bench, a 3D printed plate was placed inside the endcaps to be able to retain a specific D. The reasons for the use of 3D printed components are threefold: i) increasing the precision of D,

ii) ease of printing in a wide variety of shapes, and iii) the 3D print material being an electrical insulator ineffective to the measurement results.

The 3D printed plates have the multiple holes bored into them with specific offsets, as can be seen in Figure 5.6. 3D printed pins are placed into these holes, and the wires are bent around them to have a precise distance between them.

Six pins per endcap were placed into the board, positions as follows: Three pins with 1 mm distance between them at 4, 5 and 6 mm, a fourth one at 6.5 mm, a fifth one at 10% of the vertical diameter of the beam pipe off centre, and the last one, which held the wire, at the centre. The reasons to start at 4 mm were: 1) to stay in the range of 10% to 20% of the beam pipe diameter, and 2) make the bench measurement setup also applicable to MWMs. The manufacturing of the pin holes is precise for diameters of up to 100 µm. When all pins are removed, the distance between the wires is 1 cm. This lends us the ability to use the same setup for both the TWMs with Ds of 5 mm and 1 cm, and the MWMs with 5 different offsets with 1 mm between them, due to the space required by the connectors exceeding what is needed for the measurements. As connectors, the SMA types were chosen due to their small size and their high efficiency transmitting the signal from DC to tens of GHz.

Two problems arise, as mentioned in Section 1.2.5, one of them was the imperfections of the electrical connection and the other the environmental noise from the connectors, both resulting in strong noise and reflections. This was why the bench was redesigned using the intertwined metallic structures of components #1 and #2 shown in Figure 5.7. The face of Component #1 fits the surface of the DUT and is fitted with 8 bolts to guarantee contact. Three different sizes of the Component #2 were designed to fit every accelerator parts of ThomX.

For the environmental noise problem, small shielding were manufactured and placed to cover the connectors. The RF foam was added on the ends of the 3D printed components and shielding to absorb the unwanted waves.

. 4-Port 180 Hybrid Couplers

In the experiment, 180 hybrid couplers were used to obtain phase opposition between the signal propagated by two wires [START_REF] Rouphael | Chapter 2 -microwave network design and analysis[END_REF]. The general schematic of the The intertwined metallic structures of #1 and #2 were designed to achieve good electrical connection between the DUT and the measurement setup.

180 Hybrid coupler can be seen in Figure 5.8. The hybrid coupler is a lossless and matched 4-port device which splits signals into two equal ones. Each signal is attenuated by 3 dB. The power ratio of the power in and power out is given by the scattering matrix of the hybrid coupler, which can be seen in Equation 5.3. In this work, two different Hybrid Couplers were used, since no Hybrid Coupler covers the entire range of frequencies. One is the 180 Hybrid Coupler with a frequency range of 0 to 2.5 GHz, and the other has a frequency range of 1 to 20

GHz. Both have 3 dB attenuation. The choice of when each one was used was made according to the simulation results.

. Transverse Impedance Calculation

TWMs are performed by driving two signals on the wires with opposite phase.

The S 21 parameter can be measured with a 4-port VNA, or using 4-port splitters and combiners with a 2-port VNA. For our measurements, 4-port 180 Hybrid Couplers which were described in the above subsection 5.4.1 were used. The wire spacing should be much smaller than the diameter of the beam pipe diameter in order to apply the approximation of an infinite length transmission line to ensure proper impedance matching. Unfortunately, to have a measurable effect, a conflict must be addressed. The distance must be large enough to measure the EM interaction, but also small enough to remain within the approximation range. The wires' spacing was chosen to optimal between 10% and 20% of the beam pipe diameter, in accordance with former studies [START_REF] Zanini | Electromagnetic Simulation of CERN Accelerator Components and Experimental Applications[END_REF].

The impedance can be calculated, as for the longitudinal case, using Equation 5.4 from the measured S 21 . The dipolar component of the transverse impedance of the vertical or the horizontal plane becomes [START_REF] Kroyer | Longitudinal and transverse wire measurements for the evaluation of impedance reduction measures on the mke extraction kickers[END_REF]:

Z T = cZ 2p f D 2 (5.4)
where c is the speed of light, f is the frequency and Z is the impedance which was calculated from the log formula.

. Reference Measurements

The geometry of the basic vacuum pipe has also an effect on the impedance measurements. To extract the longitudinal and transverse impedance values, only the basic vacuum pipe, named hereafter as a reference pipe, should be measured.

As is the case with the other accelerator components, it is necessary to perform S 21 measurements for the reference component. It will be used in Equation 4.3 as S 21 re f to reckon with the errors coming from the setup and the pipe geometry.

The straight beam pipe of the ThomX was used as a reference section. In order to be able to confirm that the pipe length is not effective in the reference measurements, two different lengths were used for the reference section.

We also performed the reference measurement with each wire diameter and wire spacing, with all the setup components such as hybrids, matching resistors etc. The measurements were also repeated 5 times and the resulting error fluctuation were calculated. The decreasing power loss with the decreasing frequency is due to the power lost in the carbon resistors, which changes along with the frequency.

. FBT

The FBT measurements were performed with TWM to determine the dipolar component of the transverse impedance. The measurements were performed with two different wire spacing to evaluate the effect of the higher order terms and to also check the error. Same as the one wire, the measurements were performed five times for each wire offset. The distances were chosen as 5 mm and 1 cm which are as small as possible to stay within a regime of weak perturbation of the fringe field [START_REF] Kroyer | Longitudinal and transverse wire measurements for the evaluation of impedance reduction measures on the mke extraction kickers[END_REF].

The measured S parameters were used in Equation 4. So, the difference in the amplitude of horizontal for two wire is only in the first peak and it is around 10% due to the contribution of the higher order modes. As mentioned in the former section the higher order modes contribution is increased with the increasing distance between the wires.

As a last step, the measurement results were checked with the CST simulations which gave comparable results, especially in the frequency span (see red plots in fig. 5.11 and 5.12). The results of the TWMs and CST simulations in the whole frequency span can be seen in Figure 5.11 and Figure 5.12. The frequency shift is around 0.05 and 0.1 GHz (15%), which is acceptable in our case. These frequency shifts are due to the presence of the wire detuning the resonance and to non-ideal properties of the resistors and cables [START_REF] Belver-Aguilar | Measurements and laboratory tests on a prototype stripline kicker for the clic damping rings[END_REF][START_REF] Belver-Aguilar | Transverse impedance measurements and dc breakdown tests on the first stripline kicker prototype for the clic damping rings[END_REF]. Also for both cases, the proper CST simulations need too much computational power due to their complex geometry.

Furthermore, it was not really possible to apply geometry manipulations that were mentioned in Section 3.2. The simulation results could not be performed with better parameters because the computation time already spanned over four weeks.

Also, since inserting a wire will result in strong field distortions, they lead to shifts in the frequency spectrum [START_REF] Salvant | Building the impedance model of a real machine[END_REF].

ThomX ring has a small aperture, contributing further to nonlinear effects. The differences in amplitude and the shifts in frequency also can be explained with the power loss in the resistors and the attenuation due to the thickness of the wire [START_REF] David Large | Broadband Cable Access Networks[END_REF]. The resistors used for matching will bring additional noise in high frequencies because of the current density leading to additional noise and power loss [START_REF] Hayt | Engineering Electromagnetics[END_REF]. This is the main reason why we cannot observe the two signals above 1.5 GHz, which are empirically observed in the simulation. The simulations were performed with 15 mesh cell per wavelength, 5000 mm wakelength, 5 mm distance and 2 mm beam sigma.The simulations were performed with proper matching at the feedthroughs and the measurements were performed without the matching resistors at the feedthroughs, due to the results being of too low an amplitude. The simulations were performed with 15 mesh cell per wavelength, 5000 mm wakelength, 5 mm distance and 2 mm beam sigma.The simulations were performed with proper matching at the feedthroughs and the measurements were performed without the matching resistors at the feedthroughs, due to the results being of too low an amplitude.

. Bellows

The bellow measurements were performed with TWM to determine the dipolar component of the transverse impedance. The distance was chosen as 1 cm [START_REF] Kroyer | Longitudinal and transverse wire measurements for the evaluation of impedance reduction measures on the mke extraction kickers[END_REF]. If we simplify our model to step in-out structure, which can be considered as a bellow with only one corrugation, strong dipolar impedance peaks are expected [START_REF] Of | IEEE Particle Accelerator Conference[END_REF]. As for the longitudinal impedance case, it cannot be defined using only analytical calculations due to presence of tens of small corrugations. CST simulations were performed to find high resonant impedance peaks first, and then their validity was checked with the TWMs with 1 mm wires.

The simulations should be performed along both axes. In the horizontal case, all four peaks in the CST simulations were also seen in the measurements with acceptable frequency shifts, which can be seen in Figure 5.13. The first peak on the simulation at 2.69 GHz is slightly shifted to 2.92 GHz on the measurements. It is around 4 % error on the frequency span. The second at 4.19, the third at 4.77,

and the fourth at 5.33 GHz are shifted around 0.08, 0.01 and 0.1 GHz in the measurements. In the vertical case, the measurements are also comparable with the CST simulations as in Figure 5.14. In the vertical simulations, due to the closeness of the integration line to the component walls in the entry, the simulation results

are not as accurate as they were in the horizontal ones. Unfortunately, the mesh number needed to have the same accuracy is very much above our computational power. The peaks at 3.22 GHz and 3.89 GHz are slightly shifted to 3.20 GHz and 3.86

GHz on the measurements. The third peak at 4.35 GHz is shifted to 4.14 GHz, having the biggest error at 4.6%. This level of deviation is expected due to the thickness of the wire, movable geometry of the bellows and the mechanical tolerance of the manufacturing.

In the results there is a huge base noise, even for reference and FBT measurements. The noise filtering function of MATLAB was used to smooth the data to achieve better results, especially for broadband resonance peaks. All graphs were created using a noise filtering function.

. BPM

The analytical calculations and impedance estimations of ThomX's BPM button are difficult. First, the button's small housing cavity contributes to the impedance.

And second, the electrode parts have been bought from a private company, making their actual designs nearly impossible to access. This is why the only way to correctly estimate transverse impedance budget is to compare the wire measurements with CST simulations. After the commissioning of the ThomX accelerator, the results can be compared with beam data.

The TWMs were performed with 1cm and 5mm wire distances with the 1mm wire diameter. The simulation results led to the decision to the use of a hybrid coupler with a frequency of 1 to 6 GHz. The hybrid coupler is unreliable at the boundaries.

The impedance results are presented in both horizontal and vertical planes as in Figure 5.17 and Figure 5.18 respectively. The negative impedance seen both the vertical and the horizontal measurements are the effects of the hybrid coupler and the bench setup. This noise also has a huge deviation for each measurement, which leads to significant measurement errors.

For the total definition of the transverse impedance, measurements were performed in both the horizontal and the vertical planes. The maximum error on the measurements around 2 dB in Figure 5.19 which come up to around 4 W/mm as a dipolar impedance. In Figure 5.17, the simulations show only one peak at 8.4 GHz, which is the out of the measurements' range. As it was calculated in Section 4.5.4

these peaks come from the trapped modes of the electrode. Along the vertical axis, there is only one peak seen in the simulation, also at 8.44GHz. Unfortunately, we cannot see such range empirically, as shown in Fig-

. Pumping Port

The pumping port, designed to be used with near-zero impedance values, was used in both simulations and measurements. The manufacturing of the grids should be checked with measurements for any possible production errors, especially for the grids, mentioned in Section 4.5.5. The measurement and simulation results in figures5.20 and 5.21 show that there is around zero impedance for both vertical and horizontal dipolar impedance measurements and simulations. There is a small offset in the simulation results around 0.6 W which comes from the resistive wall impedance, due to the material which has a finite electrical conductivity. The total effect of the kick factor for both the transverse component for the

. Dipolar Impedance Budget

-Moving Wire

Measurements

In this chapter, the moving wire method (MWM), which gives access to the longitudinal and quadrupolar components, will be presented. For this method, a single wire is placed with an offset along the structure that has to be measured. For simplicity's sake, the offset is set either in the horizontal (x) and in the vertical (y) direction. The general concept for these measurements, which supposes asymmetric structures, will be introduced here.

. Moving Wire Method

The general formalism of the moving wire is similar to the on-axis (longitudinal) measurements. The symmetric accelerator structures' total impedance leads to:

Z T = Z L + Z 1x x 2 + Z 1y y 2 (6.1)
where Z L is the longitudinal impedance measured at the center, Z 1x and Z 1y relate to transverse impedance, measured with an offset of x and y along the x and y axes, respectively. Let us look at Figure 6.1: On the left, measurements were performed when x = y = 0 and the only remaining component is the longitudinal one (black line). Yet were we to put an offset in only one direction, for example x, the equation would become: The dipolar and quadrupolar terms depend on these two coefficients as follows:

Z T = Z L + Z 1x x 2 (6.2)
Z x = Z dipx Z quadx = c 2p f Z 1x (6.3)
Z y = Z dipy + Z quady = c 2p f Z 1y (6.4)
Z dipx and Z dipy are the dipolar components of the transverse impedance which can be measured by the TWM. The Z quadx and Z quady are the quadrupolar components of the impedance and can be determined from the one wire measurements with an offset. As in the Equation 1.47 the quadrupolar values in x and y axis will be the same in amplitude but has the opposite sign. If we take the sum of these two equations we will find the total transverse impedance which will eliminate the quadrupolar component and equation become:

. Error Estimation on Moving Wire

Measurements

Each of the MWMs and TWMs were performed five times, and for each one, the human error that comes from stretching and soldering the wire was below 1 dB. Although after the standard deviation calculation from impedance formula the standard deviation had maximum value at the peaks around 20 W. The standard deviation formula from the impedance equation is:

s Z = 2Z L s s S 21DU T 2 S 21DUT 2 + s S 21Re f 2 S 21Re f 2 (6.6)
On the other hand, for MWMs it is tricky to incorporate the measurement fit's error into the total error propagation. A linear regression can be applied to solve this sort of problem. For the MWMs along the x axis, the equations become:

Z(x, f ) = 2Z L ln( S21 DUT S21 REF ) (6.7) Z(x, f ) = Z Long + Z 1x x 2 (6.8)
With another notation, they can be rewritten as:

X = 0 @ 1 x 2 1 A (6.9) and Y (x, f ) = Z = ✓ 2Z L ln( S21 DUT S21 REF ) ◆ (6.10) and b ( f ) = 0 @ Z Long Z 1x 1 A (6.11) Y ( f ) = X(x) T b ( f ) (6.12)
These formulae are the linear regression for which the coefficients b ( f ) can be calculated by a polynomial fit. To estimate b ( f ), the mean values were not calculated, the five measurement data for each case were used instead. This means that for m number of frequency f , we have m ⇥ 5 ⇥ 5 different values for Y ( f ). All in all, the error can be estimated directly from the fit function. As an example the error calculation and the graphs will be shown for the bellows in Section 6.4.

. Descriptions of the Measurement Bench

The general scheme of the measurement bench used for the MWMs has previously been explained, as it was the same one that was used for the longitudinal and TWM. The line impedance calculations also were the same as the longitudinal measurements ones. The same resistance value was used, and the same calibration was performed with the whole setup.

The MWMs were performed with five different offset values; the x and y being 0 mm, 4 mm, 5 mm, 6 mm, and 6.5 mm. The maximum offset here corresponds to 23% in the horizontal and 16% in the vertical axes, respectively.

. FBT

The MWMs were performed to FBT with five different offsets: x and y were taken to be 0 mm, 4 mm, 5 mm, 6 mm and 6.5 mm. The real part of the total impedance spectrum from the moving wire with different offsets can be seen in Figure 6.3. Seven narrow-band impedance peaks can be seen in the figure, which are the composition of the longitudinal, dipolar and quadrupolar components of the impedances. These measurements were taken without the perfect impedance matching on the FBT electrodes. These narrow-band impedances were expected without the use of impedance matching or with the connection problems. It allowed us to compensate for the attenuation of the amplitude of the signal that was due to the wire's diameter. For the ThomX accelerator, it is important to check the impedance budget with and without impedance matching especially, to be able to compare them with the future results, which will be obtained via the accelerator after having commissioned it.

To better understand and highlight the different impedance contributions, let us look closely at the first peak in Figure 6.4. In fact, since the first peak frequency is the lowest, it bows least to the noise stemming from the resistance, besides being the highest peak. As a consequence, the different contributions can be seen better. As explained in Equation 6.1, the transverse impedance spectrum from moving wire is the sum of the longitudinal impedance and the contribution coming from the off-axis wire value. Also, in the same figure, we superpose the longitudinal impedance onto the moving wire data. As expected, the longitudinal data (taken for a zero offset) fits the main peak, and highlights the additional contributions coming from the transverse components. Let us go through a deeper analysis to extract all the transverse impedance components. First step is to extract the Z 1x value with a polynomial fit as y = ax 2 + b, as described by Equation 6. and Z y from Equation 6.1. Prior to the fit, in some cases, applying a noise filter can be useful to extract the real small signal which was hidden by the background noise. It should also be noted that the merged effect of the two different measurements' background noise (for example the component and the reference) can create additional signals in the overall results.

As an example, let us look at the signal at 0.2520 GHz for 5 different offsets as shown in Figure 6.5. The fit function (y = a ⇤ x 2 + b) was applied to the data, and the coefficient a and b were subtracted from the results. The same process was performed for all frequencies. Prior to performing the fit function generated by MATLAB on the moving wire data, the noise filtering was applied on the data to minimize the error and maximize the amplitude of the results. The noise filtering function of medfilt1(x,n) is applied to the data which puts a nth-order median filter to the input vector(x) [START_REF] Pratt | Introduction to digital image processing[END_REF]. The comparison of longitudinal impedance from moving wire data fit and on-axis measurements can be seen in Figure 6.6. These results show the reliability of the performed fit and moving wire measurements.

The FBT measurements for the TWM were performed with offsets of 5 mm and In Figure 6.9, the first peaks of all impedances from the wire measurements can be seen. The dipolar and quadrupolar impedances are at the same frequency.

On the other hand, the longitudinal impedances peak have a difference around 10 MHz. This method allows us to even extract the transverse data up to several MHz of difference. As a last step, CST simulations were performed for each case. The entire spectrum for longitudinal impedance and transverse impedance of the vertical plane can be seen in Figure 6.10. The results in the frequency span were comparable with the measurements. However, because of the computational power needed, the simulation results' precision is low. The difference in the frequency span between the measurements and the simulation mostly came from the capacitive gap difference in the realisation of the FBT. The expected error on the manufacturing is around 1 mm which has a shift effect on the frequency of around 50 MHz. Since the amplitude of the signal grows as the offset value does, a problem arises in that the amplitude is below the uncertainty value, lowering the reliability of the measurement. For that reason, error propagation is crucial due to the small aperture of the ThomX beam pipe.

. Bellows

The distance should be big enough to measure the EM interaction, at the same time small enough to remain within the approximation range [START_REF] Kroyer | Longitudinal and transverse wire measurements for the evaluation of impedance reduction measures on the mke extraction kickers[END_REF]. In fact, the impedance formula is deduced from a Taylor expansion over the distance. Previous research by Zanini [START_REF] Zanini | Electromagnetic Simulation of CERN Accelerator Components and Experimental Applications[END_REF] has found the wire spacings to optimally have a value between 10% and 20% of the diameter of the beam pipe. The measurements were taken on-axis, and with 4 mm, 5 mm, 6 mm, and 6.5 mm offsets. The fit was performed with all the offset values, with 5 measurements per offset, 25 measurements in total.

Polynomial fit to MWMs will give the longitudinal and total transverse impedance components. The total transverse impedance is the sum of the dipolar and quadrupolar impedances. The TWMs can be used to extract the quadrupolar impedance from the MWMs.

As it was the FBT case, the longitudinal impedance of the bellows can also be In the quadrupolar impedance simulations, three peaks were observed at 2.09 GHz 3.43 GHz and 4.40 GHz. In the MWMs, only the first two peaks were seen at 2.19 GHz All the CST simulation results were plotted with the absolute value. The opposite signs of the quadrupolar impedance of the bellows in the real and imaginary parts were expected in both planes, which is shown in figures 6.15 and 6.16. As can be seen, the quadrupolar impedances have different amplitude due the elliptical shape of the beam pipe but they have the opposite signs at the same frequencies.

Secondly, in the vertical plane, the simulations needed more meshes due to the closeness of the integration line and the beam to the sides of the component in the entry and out sides. Unfortunately, the needed mesh specification to obtain better accuracy exceeded the computational power available during this thesis. The results were presented with the best specifications we were able to achieve. After the fit was applied to MWM data, the simulation result peaks shifted to 2.09 GHz from 2.17 GHz, 3.49 GHz from 3.43 GHz, and 4.43 GHz from 4.40 GHz in measurements for quadrupolar impedances respectively, as shown in Figure 6.14. However, there were additional peaks at 2.92 GHz , 4.10 GHz and 4.17 GHz, which were probably the contribution of the dipolar component. However, the simulation results were not precise due to the numerical errors on the simulations. The other peaks were also observed in TWMs, these are the dipolar impedance contribution. The noise in the high frequencies comes from the resistors and the bench itself. The resistors can cause additional noise and attenuation of the signal.

The error calculations of the MWMs and the fit were done according to the procedure explained in Section 5.1. In both the x and y planes, the error on the measurements were 60 W and 80 W maximum, respectively. This standard deviation is the error on total impedance measurements as was defined in Equation 6.1. The standard deviation on vertical impedance measurements and t

. BPM

The study of the quadrupolar impedance of BPM started with CST simulations.

It was expected to have a really low, practically non-measurable effect. The simulation results can be seen in Figure 6.19. The BPM MWMs had a strong noise issue, which can be seen in Figure 6.20. The black line in the figure is the reference measurements and the others are the S parameter results with different wire offsets. The oscillations on the S parameters' results were different than the reference measurements, which led to strong noise on the impedance calculations, up to 100 W as shown in Figure 6.21. The noise on the BPM measurements was supposed to come from the effect of the electrodes and the geometry of the BPM buttons.

. Pumping Port

As in the section 5.9, the total transverse impedance was expected to be null.

In other words, the quadrupolar impedance component should be zero. As can be seen in Figure 6.22 there were no narrow-band impedances, neither in simulation nor the measurements. In the simulations, only the small negligible contribution of the resistive wall impedance can be seen.

. RF Cavity and Tapers

The RF cavity for the ThomX was chosen as the ELETTRA type single-cell copper cavity in 500 MHz to have quite good compromise for the higher order mode (HOM) impedances and space restraint [START_REF] Massarotti | Status report on the elettra rf system[END_REF]. The cavity was connected to the tapers to adapt its cross-section to that of ThomX. For the high quality factor cavities, the transverse impedance was dominated by the higher order modes of the cavities.

The HOM calculations are generally performed in frequency domain simulation such as CST or HFSS, and the results can be checked with wire measurement.

However, it is crucial to compute the impedance and wakepotential with the CST for adding it the general impedance model. The Table 6.1 show the dipole modes of the cavity which was simulated with CST MWM and the measured results with using a vector network analyzer (VNA) in transmission mode [START_REF] El Khaldi | Simulations and rf measurements of the fundamental and higher order modes of the thomx 500 mhz cavity[END_REF].

The simulations of the dipolar impedance that were performed with CST Particle Studio was compared with the measurements and plotted in Figure 6.24. The impedance calculated by CST caused numerical errors determined by the mesh size. Also, the RF cavity which was measured with tapers led to small shifts in the frequency. The first dipole mode is exactly the same when we performed a simulation with CST Particle studio. With the increasing frequency, some HOMs were suppressed due to insufficient meshing and also an the effect of the tapers. The simulations were started with dipolar component in both planes initially.

Both horizontal and vertical simulations can be seen in Figure 6.26 and Figure 6.27 respectively. In both planes the dipolar impedance peaked below 10W, and especially for vertical plane it was negligible. The impedance peaks in the IP chamber come from the trapped modes of two electrodes.

The secondary simulations were performed with the quadrupolar component in both planes. Both horizontal and vertical simulations of quadrupolar impedance can be seen in Figure 6.28 and Figure 6.29 respectively. In the vertical plane the quadrupolar impedance peaked below 10W, however, in the horizontal component the fourth peak was around 15w. All impedance peaks in the IP chamber came from the trapped modes of two electrodes also in quadrupolar impedance. The frequency was also comparable with the dipolar component.

. Septum

The aim of the septum is to shield the beam from the DC magnet when the beam is passed to the extraction line from the transfer line. As can be seen in Figure 6.30, the septum is a non-symmetrical object. The shielding was provided by the thin mu-metal layer which is shown in the figure as dark blue color. The mu-metal has an exceptionally high relative magnetic permeability to not allow the magnetic field pass trough it. It allowed us to protect the beam which was already stored inside the ring.

Due to its complex geometry and the non-symmetrical axis, it is remarkably hard to study impedance for the septum. The first difficulty comes from the fact that the geometry gets narrower inside and creates a shape like step in-out cavity along one axis. It also has tapers in the frontends. The second point is that the septum has pumping grids on one side, and the other sides are covered with mu-metal shielding. The mu-metal magnetic permeability is frequency dependant, which creates impedance differences with respect to the frequency. Unfortunately, only the low frequency (DC to 60 Hz) data is provided by the supplier.That is why a measurement of this complex piece would have been preferable. Unfortunately, it was not compatible with the ThomX mounting schedule.

The CST simulations should be performed with respect to the change of mumetal magnetic permeability. While possible in theory, due to the computer power needed, it is not achievable in practice. As known from the research of the magnetic permeability of mu-metal, its value decreases rapidly with increasing frequency [START_REF] Williams | Chapter 15 -shielding[END_REF]. Therefore, for the simulations the relative permeability was taken as 1.

On account of the non-symmetric nature of the septum, even when particles are passing through the exact center, they will feel a horizontal kick. This kick pushes the particle in the horizontal plane towards to the pumping port grid.

Figures6.31 and 6.32 show us the dipolar and quadrupolar CST impedance simulations of the septum in both planes. In the simulation, the impedance base evolves from 5W to 15W for dipolar, and 10W to 15W for quadrupolar, which is due to the effect of the tapers at the front-ends. The narrow band peaks come from the step in-out like structure inside the septum.

. Total Impedance Budget

The vertical quadrupolar impedance simulation results of the all components can be seen in Figures 6. [START_REF] Gamelin | Longitudinal impedance and coherent synchrotron radiation models for thomx storage ring[END_REF] Table 6.2 shows the calculated quadrupolar kick factors for each component in both planes. Table 6.3 shows the calculated dipolar component which cannot be measured with TWMs.

If we take all the components for the entire storage ring, the total horizontal and vertical kick factors for dipolar impedance will be 8.8e 4 V/pC/m and 1.1e 3

V/pC/m respectively. Also, the total horizontal and vertical quadrupolar kick factors are 3.976e 3 V/pC/m and 3.3986e 3 V/pC/m, respectively.

The Accelerator Component Kick Factor (V/pC/m) Bellows -Vertical

3.81e 4

Bellows -Horizontal Septum -Vertical

2.71e 6

Septum -Horizontal

2.83e 5

Cavity+Tapers -Vertical

9.14e 5

Cavity+Tapers -Horizontal

7.43e 5

Table 6.2: The calculated quadrupolar kick factor of di erent accelerator components.

The Accelerator Component Kick Factor (V/pC/m) IP -Vertical

3.46e 6

IP -Horizontal

2.57e 6

Septum -Vertical

2.70e 6

Septum -Horizontal

2.83e 5

Cavity+Tapers -Vertical

2.88e 5

Cavity+Tapers -Horizontal

2.77e 5

Table 6.3: The calculated dipolar kick factor of di erent accelerator components.

-Conclusion

In this thesis, the beam coupling impedance studies with wire measurements and electromagnetic field simulations were presented for the ThomX accelerator structures. The main problems about the measurements and simulation were examined and many different precautions were taken to reduce the results' error.

The CST simulations were performed for each piece, and the models were studied carefully according to the computational restrictions. The results were compared with the basic structures and former works. The accuracy and function of the simulation parameters have been discussed in detail to find the balance between the computational power and the measurement accuracy. The cavity-like structure cases were studied deeply, and the limits with errors were set.

All results were checked with longitudinal, two wire and moving wire measurements. Even small impedances can be measured with wire measurement setup thanks to the upgrades performed to reduce the noise level. The first measurements were performed for longitudinal impedance for two reasons: Firstly, the new bench had to be checked, and the reliability had to be compared with the former longitudinal impedance studies. Secondly, the main error on the moving wire results generally came from the performed fit. That is why the longitudinal results were compared with the moving wire's longitudinal results to cross-check the efficiency of the performed fit. Two different methods were used to study the transverse impedance: Firstly, two-wire measurements were performed to all accelerator parts to obtain the dipolar impedance. Afterwards, the moving wire measurements were performed, which gave us the total transverse impedance by performing fit to the results. When the dipolar impedance of the two-wire measurements were extracted from the moving wire results, the quadrupolar impedance inant depuis le début de cette science appelée effets collectifs.

Dans cette thèse, j'utiliserai le formalisme des effets collectifs, ainsi que l'impédance, pour décrire la manière dont l'environnement influ-

6. 5

 5 BPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Pumping Port . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 RF Cavity and Tapers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Interaction Point (IP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Septum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Total Impedance Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion

2. 3

 3 The layout of Thomx. The ThomX contains 6 parts;photoinjector, linac, transfer line, extraction line, storage ring and the X-Ray line-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. 4

 4 photocathode, a laser system, and solenoid magnets. . . . . . . . . .

2. 5

 5 The Twiss Parameters comparison of the transmission line the storage ring and the transmission line and the extraction line (TL & EL) with the matching with respect to storage ring. The grey line represents the TL+storage ring lattice and the purple dash line represents the TL+extraction line lattice. . . . . . . . . . . . . . . . . . . . . . . . .

3. 3

 3 Zoom to the BPM4 button in CST mesh view. The simulation was performed with mesh cells per wavelength of 15, wakelength of 5000, with Gaussian beam (s = 2 mm and charge = 1 • 10 9 C) . . . . . . . . .

3. 4

 4 Wake Potential and Impedance results of step in-out cavity in onaxis simulations. The simulations were performed with 46 • 10 6 total mesh cells, wakelengths of 1000, and Gaussian beams (s = 2 mm and charge = 1 • 10 9 C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 5

 5 The longitudinal impedance simulation results of the rectangular box constructed with the sorts of copper with 5.8 • 10 5 S/m (black), 5.8 • 10 6 S/m (blue), 5.8 • 10 7 (red) S/m of electrical conductivities and a perfect electric conductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The scheme for the longitudinal wire measurements. The thick black line represents the wire. . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. 2

 2 The incident and received power to the DUT with s parameter matrix 4.3 S parameter measurements of step in-out with 0.1 mm, 0.5 mm and 1 mm wires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. 4

 4 Close-up of S parameter measurements of figure 4.3 of step in-out with 0.1 mm, 0.5 mm and 1 mm wires. . . . . . . . . . . . . . . . . . . 4.5 The photo of the old and new designs with additional connections. . .

4. 6

 6 The standard deviation comparison of the bellows longitudinal measurements with 1 mm wire. The blue line shows the standard deviation with the old bench and the red line shows the standard deviation with the new bench. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. 7

 7 The general experimental setup of the resonance measurements. On the left, both sides have capacitive coupling, and on the right, one side is short circuited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 The scheme of the step in-out cavity of length g, larger radius R, and step radius b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. 9

 9 The comparison of step in-out cavity CST simulations and measurements using 0.1 mm wire. . . . . . . . . . . . . . . . . . . . . . . . . . .

4 .

 4 10The comparison of step in-out cavity impedance measurements using 0.1 mm, 0.5 mm, and 1 mm wires. . . . . . . . . . . . . . . . . . . . . .

4 .

 4 11Up: The cross-section scheme of the ThomX reference component. Down: The manufactured reference component of ThomX with the measurement setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 .

 4 12Up: The mean of the 5 measurements performed with 1 mm wire for the reference component. Down: The standard deviation of 5 measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 .

 4 13The test bench of ThomX with FBT. . . . . . . . . . . . . . . . . . . . . .

4 .

 4 14Front and side view of the design model of the transverse feedback kicker. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.15The comparison of the FBT longitudinal impedance measurements with 1 mm wire with the CST simulations. The CST simulations were performed with a wavelength of 5000 GHz, 15 mesh cells per wavelength with open boundary at the beam direction and electric at the others.The simulations were performed with proper matching at the feedthroughs and the measurements were performed without the matching resistors at the feedthroughs, due to the results being of too low an amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 .

 4 16The simplified model of bellows for CST simulations. . . . . . . . . . . 4.17The 3D designed of the bellows with RF Fingers . . . . . . . . . . . . .

4 .

 4 18Longitudinal impedance which is measured on axis with 1 mm wire for bellows without RF fingers. . . . . . . . . . . . . . . . . . . . . . . . . 4.19The comparison of the CST simulation, 1mm wire measurements, 0.1mm wire and moving wire measurement fit results for the bellows without RF fingers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.20The longitudinal wire measurements of the bellows with RF finger (black) and the CST simulations (red). . . . . . . . . . . . . . . . . . . . 102 4.213D design of the BPM4 and BPM8 . . . . . . . . . . . . . . . . . . . . . . 103 4.22The S parameters measurement of BPM4 and reference component with 1 mm wire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.23The simulated real part of the impedance for BPM4, BPM6 and BPM8. 105 4.24The 3d design of the pumping port . . . . . . . . . . . . . . . . . . . . . 105 5.1 The diagrams of the step function signal where the r is equal to reflection coefficient. Left: The different cases of a reflection from propagating step function through transmission line with different matching in the end cap. Right: The scheme of the complex final signal which comes from the multiple reflections. [5] . . . . . . . . . . . . . . 5.2 The diagram of the measurement setup. Two wires are stretched through the DUT, matched with the resistors, and connected to the VNA for measuring the S parameters. . . . . . . . . . . . . . . . . . . . 5.3 The CST model of the rectangular pipe used in TWMs (20 mesh per wavelength and mesh equilibrium ratio is 1.05) . . . . . . . . . . . . . 5.4 The wake potential results of the rectangular pipe with II and ITB methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. 5

 5 The 2D design picture of the whole measurement setup . . . . . . . . 5.6 2D Design of the wire streching section. . . . . . . . . . . . . . . . . . .

5. 7

 7 The mechanical drawing and exploded view of the measurement bench. The intertwined metallic structures of #1 and #2 were designed to achieve good electrical connection between the DUT and the measurement setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 The representative picture of 180 Hybrid Coupler. . . . . . . . . . . . . 5.9 The phase difference test of the 180 Hybrid Coupler. . . . . . . . . . . 5.10Top: The mean of the 5 measurements of reference component. Bottom: The standard deviation of the measurements. . . . . . . . . . . . 5.11The comparison of the vertical TWMs with CST simulations for the FBT. The simulations were performed with 15 mesh cell per wavelength, 5000 mm wakelength, 5 mm distance and 2 mm beam sigma.The simulations were performed with proper matching at the feedthroughs and the measurements were performed without the matching resistors at the feedthroughs, due to the results being of too low an amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 .

 5 12The comparison of the horizontal TWMs of FBT with CST simulations. The simulations were performed with 15 mesh cell per wavelength, 5000 mm wakelength, 5 mm distance and 2 mm beam sigma.The simulations were performed with proper matching at the feedthroughs and the measurements were performed without the matching resistors at the feedthroughs, due to the results being of too low an amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.13Horizontal TWMs of the bellows without the RF fingers,performed with the 1 mm wires compared with the CST simulation results. The simulations were performed with 15 mesh cell per wavelength, 5000 mm wakelength and 2 mm beam sigma. . . . . . . . . . . . . . . . . . . . . 125 5.14Vertical TWMs of the bellows without the RF fingers, performed with the 1 mm wires compared with the CST simulation results. The simulations were performed with 15 mesh cell per wavelength, 5000 mm wakelength and 2 mm beam sigma. . . . . . . . . . . . . . . . . . . . . 125 5.15Vertical TWMs performed with the 1 mm wire on the bellows equipped with RF fingers. The measurements were performed with a 5 mm offset.126 5.16Horizontal TWMs performed with the 1 mm wire on the bellows equipped with RF fingers. The measurements were performed with a 1 cm offset.127 5.17Horizontal TWMs performed with the 1 mm-diameter wire compared with the CST simulation results of BPM4. Red: TWMs of BPM. Black: CST simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.18Vertical TWMs performed with the 1 mm-diameter wire compared with the CST simulation results of BPM4. Red: TWMs of BPM. Black: CST simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.19The standard deviation of the BPM TWMs(S21) both horizontal and vertical plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.20The vertical impedance measurements for pumping ports with the 1 mm-diameter wire and D = 5 mm compared with CST simulation results. The simulations were performed with a 5 mm offset, 1000 mm wakelength and 20 mesh cells per wavelength. . . . . . . . . . . . 5.21The horizontal impedance measurements for pumping ports with the 1 mm-diameter wire and D = 5 mm compared with CST simulation results. The simulations were performed with a 5 mm offset, 1000 mm wakelength and 20 mesh cells per wavelength. . . . . . . . . . . . 5.22The real part of horizontal dipolar impedance simulation of different components placed in the storage ring. The simulations were performed with 1 cm offset, 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.23The imaginary part of dipolar impedance simulation of different components placed in the storage ring. The simulations were performed with 1 cm offset, 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.24The real part of horizontal dipolar impedance simulation of different components placed in the storage ring. The simulations were performed with 5 mm offset, 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.25The imaginary part of dipolar impedance simulation of different components placed in the storage ring. The simulations were performed with 5 mm offset, 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 .

 5 26The total sum of vertical dipolar impedance simulation of all components(4 BPM4s, 4 BPM6s, 4 BPM8s, 2 IP chambers, 15 Pumping Ports, 18 Bellows with an RF finger, 1 Septum, 1 Transverse Kicker, and 2 Cavity+Tapers) placed in the storage ring. The simulations were performed with 5 mm offset, 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.27The total sum of horizontal dipolar impedance simulation of all components(4 BPM4s, 4 BPM6s, 4 BPM8s, 2 IP chambers, 15 Pumping Ports, 18 Bellows with an RF finger, 1 Septum, 1 Transverse Kicker, and 2 Cavity+Tapers) placed in the storage ring. The simulations were performed with 1 cm offset, 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The comparative scheme of the moving wire and on axis measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The mean value of the reference measurements with different wire offset values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The comparison of the moving wire measurements of FBT with different offsets. The impedance is the total one deduced from Equation 5.1143 6.4 Enlarged picture for the first peak of the Figure 6.3. The cyan line (0 mm) is the longitudinal impedance. . . . . . . . . . . . . . . . . . . . . 6.5 Blue stars are the data of the five measurements as were represented in Figure 6.3 with different offsets at 0.2520 GHz, and the red line is the fit function of a ⇤ x 2 + b, where a is equal to 1.39 W and b 15.2729 W/mm 6.6 The longitudinal impedance measurement comparison of the one extracted from the fit to the moving wire and on-axis for FBT. . . . . . . 6.7 The sum of the dipolar impedances in both axes for the MWM (plain black line) as in Equation 6.5, and the summation of the TWM (plain red line) as in figures 5.11 and 5.12 for FBT. . . . . . . . . . . . . . . .

6. 8

 8 The quadrupolar impedance of FBT from MWM data. The results were taken by extracting the dipolar impedance results from total transverse impedance as in equations 6.3 and 6.4. . . . . . . . . . . . . . . 6.9 The superposition of the first peaks of all FBT impedances. The longitidunal impedance has the unit of W and the transverse impedances have the unit of W/mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10The CST simulations of all FBT impedances. The simulations were performed with 5 mm offset, 1000 mm wakelength and 12 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11The horizontal transverse impedance results(dipolar+quadrupolar) of the bellows performed with 1 mm wire with 5 different offsets and extracted from MWM data by fitting the parabola. . . . . . . . . . . . . 6.12The total vertical transverse impedance data which is extracted from moving wire measurements by polynomial fit . . . . . . . . . . . . . . 6.13Superposition of the dipolar and quadrupolar horizontal impedance of bellows obtained with the CST simulations with MWMs. Black: Dipolar impedance CST simulations, Pink: Quadrupolar impedance CST simulations, and Blue: Total transverse impedance extracted from the MWMs. All the simulations were performed with 1 cm offset, 1000 mm wakelength and 20 mesh cells per wavelength. . . . . . . . . . . . 6.14Superposition of the dipolar and quadrupolar vertical impedance of bellows obtained with the CST simulations with MWMs. Black: Dipolar impedance CST simulations, Pink: Quadrupolar impedance CST simulations, and Blue: Total transverse impedance extracted from MWMs. All the simulations were performed with 5 mm offset, 2000 mm wakelength and 25 mesh cells per wavelength. . . . . . . . . . .6.15The real part of the bellows' CST simulations in both planes with 5mm offset in vertical and 10 mm offset in horizontal plane. . . . . . . . . .

6 .

 6 16The imaginary part of the bellows' CST simulations in both planes with 5mm offset in vertical and 10 mm offset in horizontal plane. . . . 153 6.17The standard deviation on horizontal impedance measurements and fit154 6.18The standard deviation on vertical impedance measurements and fit 154 6.19The quadrupolar impedance CST simulations of BPM4. The simulations were performed with 1 cm offset for the horizontal and 5 mm offset for the vertical, with 1000 mm wakelength and 20 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.20The S parameter results of BPM with different wire offsets. . . . . . . . 155 6.21The total impedance result of BPM with different wire offsets. . . . . . 155 6.22Vertical impedance measurements of pumping port with a 1 mm diameter wire which is extracted from the wire offset of 0 mm, 4 mm, 5 mm, 6 mm and 6.5 mm with polynomial fit and compared with CST simulation results of dipolar and quadrupolar impedances. The simulations were performed with 5 mm offset, 1000 mm wakelength and 20 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . 156 6.23The CST model of the cavity with tapers . . . . . . . . . . . . . . . . . . 157 6.25The CST model of Interaction Point center. . . . . . . . . . . . . . . . . 159 6.26The dipolar impedance CST simulations of the IP chamber in the horizontal plane. The simulations were performed with 10 mm offset, 1000 mm wakelength and 12 mesh cells per wavelength. . . . . . . . 160 6.27The dipolar impedance CST simulations of the IP chamber in the vertical plane. The simulations were performed with 5 mm offset, 1000 mm wakelength and 12 mesh cells per wavelength. . . . . . . . . . . . 160 6.28The quadrupolar impedance CST simulations of IP chamber in horizontal plane. The simulations were performed with 10 mm offset, 1000 wakelength and 12 mesh cells per wavelength. . . . . . . . . . . 161 6.29The quadrupolar impedance CST simulations of IP chamber in vertical plane. The simulations were performed with 5 mm offset, 1000 wakelength and 12 mesh cells per wavelength. . . . . . . . . . . . . . 6.24Up: The frequency spectrum of the dipolar impedance component of the RF cavity. Down: Enlarged section of the spectrum between 0.7 GHz and 2 GHz. The black line is the dipolar impedance which is calculated in CST PS with 2 mm bunch. The red lines represent the measured dipolar HOM frequencies. . . . . . . . . . . . . . . . . . . . . 6.30The CST model of septum. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.31The dipolar impedance simulation of the septum in both planes. The simulations were performed with 5 mm offset for the vertical and 1 cm offset for the horizontal, with 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.32The quadrupolar impedance simulation of the septum in both planes. The simulations were performed with 5 mm offset for the vertical and 1 cm offset for the horizontal, with 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.33The real part of horizontal quadrupolar impedance simulation of different components placed in the storage ring. The simulations were performed with 1 cm offset, 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.34The imaginary part of quadrupolar impedance simulation of different components placed in the storage ring. The simulations were performed with 1 cm offset, 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.35The real part of horizontal quadrupolar impedance simulation of different components placed in the storage ring. The simulations were performed with 5 mm offset, 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 .

 6 36The imaginary part of quadrupolar impedance simulation of different components placed in the storage ring. The simulations were performed with 5 mm offset, 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.37The total sum of vertical quadrupolar impedance simulation of all components(4 BPM4s, 4 BPM6s, 4 BPM8s, 2 IP chambers, 15 Pumping Ports, 18 Bellows with an RF finger, 1 Septum, 1 Transverse Kicker, and 2 Cavity+Tapers) placed in the storage ring. The simulations were performed with 5 mm offset, 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.38The total sum of horizontal quadrupolar impedance simulation of all components(4 BPM4s, 4 BPM6s, 4 BPM8s, 2 IP chambers, 15 Pumping Ports, 18 Bellows with an RF finger, 1 Septum, 1 Transverse Kicker, and 2 Cavity+Tapers) placed in the storage ring. The simulations were performed with 1 cm offset, 1000 mm wakelength and 15 mesh cells per wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 1

 1 Simulation time of the step in-out cavity with respect to the parameters shown (others set to the wavelength of 1 • 10 3 and the mesh equilibrium ratio of 1.05.). After the 20 mesh cell per wavelength all the simulation results were the same for step in-out cavity. The simulations were performed using a 6-core, 12-processor Intel Xeon 2.10 GHz computer, with 128 GB of RAM. . . . . . . . . . . . . . . . . . . . .4.1The analytically calculated frequencies (obtained by using Equation4.18), CST simulations, and measurement results of resonant peaks. . . . .4.2The frequency of resonant peaks measurements and Cst results. . . 5.1 The calculated dipolar kick factor of different accelerator components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.1 The dipole modes of the cavity which was simulated with CST MW and measured results [28]. . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.2 The calculated quadrupolar kick factor of different accelerator components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.3 The calculated dipolar kick factor of different accelerator components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Figure 1 . 1 :

 11 Figure 1.1: The scheme of the ISR when the bunch length s b is larger than the emitted wavelength l on top and the CSR when the bunch length s b is smaller than the emitted wavelength l below.

  2s y (s x +s y ) and Nr p S b 2s

Figure 1 . 2 :

 12 Figure 1.2: The general scheme of electric eld of particle in non relativistic, relativistic and ultra-relativistic cases[8].

Figure 1 . 3 :

 13 Figure 1.3: The general scheme of the Coulomb repulsion in blue and magnetic attraction between two particles in green at the top, and the multiparticle cases at the bottom [87]. When the b goes to 1 and g >> 1, the attractive and repulsive forces started to have same magnitude in the opposite direction.

Figure 1 . 4 :

 14 Figure 1.4: The evolution of the electric eld of the ultrarelativistic gaussian bunch along its path (z direction) going through the basic step in-out cavity.

Figure 1 . 5 :

 15 Figure 1.5: The general scheme of the test and witness particle inside the beam pipe.Source particle emits EM eld which cause a kick on the witness particle.

  represents the time and (E, H) is the field produced by the source particle. For the longitudinal impedance, only electric field component is valid because the electric field has only the longitudinal component of the Lorentz force. The longitudinal wake function as a function of the coordinate and the longitudinal Lorentz force can be written as:

Figure 2 .

 2 Figure 2.1:The high energy photon collide with the electron which leads to a energy transfer from high energy photon to electron[START_REF] Venugopal | de broglie wavelength and frequency of scattered electrons in the compton effect[END_REF]

Figure 2 . 2 :

 22 Figure 2.2: The energy and brightness comparison of di erent machines. CCS is the acronym for Compact Compton Sources [49]

ure 2 . 3 .

 23 When the low cost and compactness of ThomX are taken into account,

Figure 2 . 3 : 10 %

 2310 Figure 2.3: The layout of Thomx. The ThomX contains 6 parts;photoinjector, linac, transfer line, extraction line, storage ring and the X-Ray line-.

  storage ring serves to store the beam and dispose of a 16 MHz interaction point for compton back scattering process. Once stored the beam is ejected into the extraction line

Figure 2 . 4 :

 24 Figure 2.4: 3D representation of the ThomX photo injector and linac. The photo injec-tor is consist of 3 GHz 2.5-cell RF gun, a magnesium or copper photocathode, a laser system, and solenoid magnets.

  dispersion. A collimator is also available to cut-off the off-momentum particles by cutting the beam's tails off. The last bending by 90 deg direct the beam to extraction line. The beam is directed to the storage ring with an additional 13 mrad deflection by turning on the injection dipole. As mentioned, the beam from the linac should be matched transversely to the storage ring and to the extraction line. The quadrupole values should be arranged to transfer the beam from the transfer line to the storage ring or extraction line.

Figure 2 . 5 :

 25 Figure 2.5: The Twiss Parameters comparison of the transmission line the storage ring and the transmission line and the extraction line (TL & EL) with the matching with respect to storage ring. The grey line represents the TL+storage ring lattice and the purple dash line represents the TL+extraction line lattice.

Figure 2 .

 2 Figure 2.5 shows the optical functions of the transfer line for both the ring and the extraction line. The plain lines represent the optical functions of the transfer line and the first turn in the ring taking into account the septum and the kicker (needed for on axis injection into the ring. The dipoles are represented as rectangles while the quadrupoles are represented by lenses at the bottom of the figure. The structure shown is that of the transfer line followed by the ring. The structure of the extraction line is not shown. The first triplet corrects the high divergence of the beam from the linac. At 11.5 m the injection dipole in the ring is shown. At this point, the dispersion function remains constant when it goes into the extraction line (black line, injection dipole switched off). The black line represents the dispersion function when the injection dipole is switched on. We, then, recognise the typical double achromat structure of the storage ring. The dispersion function is cancelled after one turn in the ring. After setting the transfer line for injection into the ring, the quadrupoles of the transfer line are adjusted to cancel both the dispersion function before the hitch, and to have beam dimensions suitable for the hitch. The dashed lines are representing the optical functions in the extraction lines.

Figure 2 .

 2 Figure 2.6 shows the storage ring in detail. The beam is injected into the storage ring through the kickers (6) and septum (4). The septum is needed to bend the beam into the ring with the strong magnetic field it generates, and then the kickers

Figure 2 . 6 :

 26 Figure 2.6: The 3D scheme of the Thomx. Shown in the gure are its: (1) 8 dipole magnets (red), (2) 24 quadrupole magnets (blue) (3) 12 sextupole magnets (yellow), (4) septum, (5) RF cavity, (6) 2 kicker magnets, (7) Fabry-Perot cavity, (8) Interaction Point Chamber (IP), (9) Feedback Transverse Kicker(FBT) and (10) Virtual Interaction Point.

  mentioned in Section 1.2.3, the EM simulation program CST needs many different parameters to be arranged to get proper results. The wakefield solver is used in the simulation which solves the Maxwell equations in the time domain. Then the resulting EM fields are integrated through the integration line with numerical methods. The numerical integration is performed on hexahedral mesh of component in space with specific time step related with the simulation frequency. The type, size, and number of the mesh change the simulation quality; since the field is calculated in each mesh, and then interpolated or extrapolated to find overall fields.Let us introduce the parameters needed to tune the simulation, to optimise time computation, and precision on the wakefield:• Gaussian Beam Sigma: The standard deviation of the temporal Gaussian distributed electron bunch in longitudinal axis z have to be set to calculate impedance from wake potential and also wakefield calculations in equations 1.29 and 3.1.

Figure 3 . 1 :

 31 Figure 3.1: The position of the source and witness particle(integration line) for obtaining the di erent impedance component with CST.

Figure 3 . 2 : 3 . 2 . 6 .

 32326 Figure 3.2: The wake potential results of Interaction Point Chamber with Indirect Interface (red) and Indirect Test Beam (black) methods. The simulations were performed with a total mesh cell 1.2 • 10 6 and the Gaussian beam (s = 2 mm and charge = 1 • 10 9 C)

Figure 3 . 4 :

 34 Figure 3.4: Wake Potential and Impedance results of step in-out cavity in on-axis simulations. The simulations were performed with 46 • 10 6 total mesh cells, wakelengths of 1000, and Gaussian beams (s = 2 mm and charge = 1 • 10 9

5. 8 •

 8 10 5 , 5.8•10 6 , and 5.8•10 7 S/m. The expected resistive wall longitudinal impedance for rectangular box is zero as confirmed by simulation in Figure 3.5. Also, resistive wall impedance increased with the decreasing electrical conductivity as expected.

Figure 3 . 5 :

 35 Figure 3.5: The longitudinal impedance simulation results of the rectangular box constructed with the sorts of copper with 5.8 • 10 5 S/m (black), 5.8 • 10 6 S/m (blue), 5.8 • 10 7 (red) S/m of electrical conductivities and a perfect electric conductor.

Figure 4 . 1 :

 41 Figure 4.1: The scheme for the longitudinal wire measurements. The thick black line represents the wire.

  E r (r, w) = Z 0 H f (r, w) =

  behavior of networks according to the input signals. Many electrical properties can be expressed using the S parameters, such as gain, reflection losses, standing wave ratio, and reflection coefficient. The measurements of these parameters show how signals are applied to a transmission line. Also, they show how the signals change when they encounter a discontinuity caused by the insertion of an electronic component on the line. In general, S parameters are the representation of how much power is lost or gained throughout the transmission line.

Figure 4 . 2 :

 42 Figure 4.2: The incident and received power to the DUT with s parameter matrix

Figure 4 . 3 :

 43 Figure 4.3: S parameter measurements of step in-out with 0.1 mm, 0.5 mm and 1 mm wires.

Figure 4 . 4 :

 44 Figure 4.4: Close-up of S parameter measurements of gure 4.3 of step in-out with 0.1 mm, 0.5 mm and 1 mm wires.

Figure 4 . 5 :Figure 4 . 6 :

 4546 Figure 4.5: The photo of the old and new designs with additional connections.

Figure 4 . 7 :

 47 Figure 4.7: The general experimental setup of the resonance measurements. On the left, both sides have capacitive coupling, and on the right, one side is short circuited.

Figure 4 . 8 :w 2 l

 482 Figure 4.8: The scheme of the step in-out cavity of length g, larger radius R, and step radius b

Figure 4 . 9 :

 49 Figure 4.9: The comparison of step in-out cavity CST simulations and measurements using 0.1 mm wire.

Figure 4 . 10 :

 410 Figure 4.10: The comparison of step in-out cavity impedance measurements using 0.1 mm, 0.5 mm, and 1 mm wires.

Figure 4 . 11 : 5 S21Figure 4 . 12 :

 4115412 Figure 4.11: Up: The cross-section scheme of the ThomX reference component. Down: The manufactured reference component of ThomX with the measurement setup.

Figure 4 . 13 :

 413 Figure 4.13: The test bench of ThomX with FBT.

Figure 4 . 14 :

 414 Figure 4.14: Front and side view of the design model of the transverse feedback kicker. 93

Figure 4 . 15 :

 415 Figure 4.15: The comparison of the FBT longitudinal impedance measurements with 1 mm wire with the CST simulations. The CST simulations were performed with a wavelength of 5000 GHz, 15 mesh cells per wavelength with open boundary at the beam direction and electric at the others.The simulations were performed with proper matching at the feedthroughs and the measurements were performed without the matching resistors at the feedthroughs, due to the results being of too low an amplitude.

Figure 4 . 16 :

 416 Figure 4.16: The simpli ed model of bellows for CST simulations.

Figure 4 . 18 :

 418 Figure 4.18: Longitudinal impedance which is measured on axis with 1 mm wire for bellows without RF ngers.

Figure 4 . 20 :

 420 Figure 4.20: The longitudinal wire measurements of the bellows with RF nger (black)and the CST simulations (red).

Figure 4 . 21 :

 421 Figure 4.21: 3D design of the BPM4 and BPM8

Figure 4 . 22 :

 422 Figure 4.22: The S parameters measurement of BPM4 and reference component with 1 mm wire.

Figure 4 . 23 :

 423 Figure 4.23: The simulated real part of the impedance for BPM4, BPM6 and BPM8.

Figure 4 . 24 :

 424 Figure 4.24: The 3d design of the pumping port

Figure 5 . 1 :

 51 Figure 5.1: The diagrams of the step function signal where the r is equal to re ection coe cient. Left:The di erent cases of a re ection from propagating step function through transmission line with di erent matching in the end cap. Right: The scheme of the complex nal signal which comes from the multiple re ections.[START_REF]Transmission lines and electronic signal handling[END_REF]

Figure 5 . 2 :

 52 Figure 5.2: The diagram of the measurement setup. Two wires are stretched through the DUT, matched with the resistors, and connected to the VNA for measuring the S parameters.

Figure 5 . 4 :

 54 Figure 5.4: The wake potential results of the rectangular pipe with II and ITB methods.

Figure 5 . 5 :Figure 5 . 6 :

 5556 Figure 5.5: The 2D design picture of the whole measurement setup Figure 5.6: 2D Design of the wire streching section.

Figure 5 . 7 :

 57 Figure 5.7: The mechanical drawing and exploded view of the measurement bench.

3 )

 3 Port 1 is called S port, which splits the signal into two with the same phase, and Port 4 is D port, which splits the signal into two with opposite phases. For the two input signals from Ports 2 and 3, the D Port will give their sum with an opposite phase which correspond to a difference , and the S port will give the summation of the two signals with the same phase. As can be understood from the S matrix of the hybrid, the ports have no reflections.For TWMs the D Port was used for splitting and combining the signals. Tests were done to verify the specifications of the Hybrid Couplers. The 3 dB attenuation was observed continuously in the frequency range of 1 GHz to 6 GHz, without any troughs or peaks. When the D port was used, the phase difference results between Port 2 and Port 3 can be seen in Figure5.9. The phase difference between Port 2 and Port 3 is 180 as expected. When using TWM, it is essential to include Hybrids and matching resistors in the VNA Calibration.

Figure 5 . 8 :

 58 Figure 5.8: The representative picture of 180 Hybrid Coupler.

Figure 5 . 9 :

 59 Figure 5.9: The phase di erence test of the 180 Hybrid Coupler.

Figure 5 . 10 :

 510 Figure 5.10: Top: The mean of the 5 measurements of reference component. Bottom:The standard deviation of the measurements.

Figure 5 .

 5 Figure 5.10 shows the mean value of the five measurements performed with two different wire spacing and their associated errors. The spacing between the wires is not as effective as expected on the reference measurements. The standard deviation is below 0.5 dB up until the cut-off frequency (5.4GHz) of the component.

  3 and then in Equation 5.4 to calculate the dipolar impedance of the FBT.The experimental vertical and horizontal dipolar impedances for the FBT are displayed in Figure 5.11 and 5.12 respectively, in blue color for D= 5 mm and in black for D= 1mm. For the vertical dipolar impedance measurements, one can see three impedance peaks in Figure 5.11 at 0.25 GHz, 0.75 GHz and 1.19 GHz for D=5 mm; and at 0.25 GHz, 0.76GHz and 1.19 GHz for D=10 mm. The frequency and the amplitude were found to be nearly the same after the calculations for both 5mm and 1 cm in the horizontal case. However, due to the smaller diameter in the vertical plane, the measurements that were performed with a 1 cm distance have bigger amplitudes due to the higher order contributions. The amplitudes measured along the vertical plane are 112.15 W/mm, 10.41 W/mm, and 7.78 W/mm and amplitudes measured along the horizontal plane are 26.15 W/mm, 10.59 W/mm, and 8.03 W/mm. The differences in the magnitude of the first peak of the horizontal and vertical measurements are of the order of 5 W/mm and 48 W/mm for 5 mm and 1 cm wire spacing respectively.

Figure 5 . 11 :

 511 Figure 5.11: The comparison of the vertical TWMs with CST simulations for the FBT.The simulations were performed with 15 mesh cell per wavelength, 5000 mm wakelength, 5 mm distance and 2 mm beam sigma.The simulations were performed with proper matching at the feedthroughs and the measurements were performed without the matching resistors at the feedthroughs, due to the results being of too low an amplitude.

Figure 5 . 12 :

 512 Figure 5.12: The comparison of the horizontal TWMs of FBT with CST simulations.The simulations were performed with 15 mesh cell per wavelength, 5000 mm wakelength, 5 mm distance and 2 mm beam sigma.The simulations were performed with proper matching at the feedthroughs and the measurements were performed without the matching resistors at the feedthroughs, due to the results being of too low an amplitude.

Figure 5 . 13 :Figure 5 . 14 :

 513514 Figure 5.13: Horizontal TWMs of the bellows without the RF ngers,performed with the 1 mm wires compared with the CST simulation results. The simulations were performed with 15 mesh cell per wavelength, 5000 mm wakelength and 2 mm beam sigma.

Figure 5 . 15 :

 515 Figure 5.15: Vertical TWMs performed with the 1 mm wire on the bellows equipped with RF ngers. The measurements were performed with a 5 mm o set.

Figure 5 . 16 :

 516 Figure 5.16: Horizontal TWMs performed with the 1 mm wire on the bellows equipped with RF ngers. The measurements were performed with a 1 cm o set.

Figure 5 . 17 :

 517 Figure 5.17: Horizontal TWMs performed with the 1 mm-diameter wire compared with the CST simulation results of BPM4. Red: TWMs of BPM. Black: CST simulations.

Figure 5 . 18 :

 518 Figure 5.18: Vertical TWMs performed with the 1 mm-diameter wire compared with the CST simulation results of BPM4. Red: TWMs of BPM. Black: CST simulations.

Figure 5 . 19 :

 519 Figure 5.19: The standard deviation of the BPM TWMs(S21) both horizontal and vertical plane .

Figure 5 . 20 :

 520 Figure 5.20: The vertical impedance measurements for pumping ports with the 1 mmdiameter wire and D = 5 mm compared with CST simulation results. The simulations were performed with a 5 mm o set, 1000 mm wakelength and 20 mesh cells per wavelength.

Figure 5 . 21 :

 521 Figure 5.21: The horizontal impedance measurements for pumping ports with the 1 mm-diameter wire and D = 5 mm compared with CST simulation results.The simulations were performed with a 5 mm o set, 1000 mm wakelength and 20 mesh cells per wavelength.

and 5 .

 5 [START_REF] Day | Measurements and Simulations of Impedance Reduction Techniques in Particle Accelerators[END_REF] and the vertical dipolar impedance simulation results of the all components can be seen in Figures 5.[START_REF] Deitrick | High-brilliance, high-flux compact inverse compton light source[END_REF] and 5.25. As long as the bellows were used with RF fingers, the largest contribution to the dipolar impedance comes from FBT in the horizontal and BPM in the vertical directions. In the storage ring, there are 4 BPM4s, 4 BPM6s, 4 BPM8s, 2 IP chambers, 15 Pumping Ports, 18 Bellows with an RF finger, 1 Septum, 1 Transverse Kicker, and 2 Cavity+Tapers. Taking the impedance of each of the above-mentioned components into account, the collective dipolar impedances in the vertical and the horizontal directions are shown in Figures 5.26 and 5.27: The kick factor of the transverse impedances were calculated as shown in Equation 1.34. The results are the kick factors of one element. The calculation of the kick factor allows us to compute the tune shift.

Figure 5 . 22 :

 522 Figure 5.22: The real part of horizontal dipolar impedance simulation of di erent components placed in the storage ring. The simulations were performed with 1 cm o set, 1000 mm wakelength and 15 mesh cells per wavelength.

Figure 5 . 23 :

 523 Figure 5.23: The imaginary part of dipolar impedance simulation of di erent components placed in the storage ring. The simulations were performed with 1 cm o set, 1000 mm wakelength and 15 mesh cells per wavelength.

Figure 5 . 24 :

 524 Figure 5.24: The real part of horizontal dipolar impedance simulation of di erent components placed in the storage ring. The simulations were performed with 5 mm o set, 1000 mm wakelength and 15 mesh cells per wavelength.

Figure 5 . 25 :Figure 5 . 26 :

 525526 Figure 5.25: The imaginary part of dipolar impedance simulation of di erent components placed in the storage ring. The simulations were performed with 5 mm o set, 1000 mm wakelength and 15 mesh cells per wavelength.

Figure 6 . 1 :

 61 Figure 6.1: The comparative scheme of the moving wire and on axis measurements.

Figure 6 . 2 :

 62 Figure 6.2: The mean value of the reference measurements with di erent wire o set values.

Figure 6 . 3 :

 63 Figure 6.3: The comparison of the moving wire measurements of FBT with di erent o sets. The impedance is the total one deduced from Equation 5.1 .

2 .Figure 6 . 4 :

 264 Figure 6.4: Enlarged picture for the rst peak of the Figure 6.3. The cyan line (0 mm) is the longitudinal impedance.

Figure 6 . 5 :Figure 6 . 6 :Figure 6 . 7 :Figure 6 . 8 .

 65666768 Figure 6.5: Blue stars are the data of the ve measurements as were represented in Figure 6.3 with di erent o sets at 0.2520 GHz, and the red line is the t function of a ⇤ x 2 + b, where a is equal to 1.39 W and b 15.2729 W/mm 2

Figure 6 . 8 :

 68 Figure 6.8: The quadrupolar impedance of FBT from MWM data. The results weretaken by extracting the dipolar impedance results from total transverse impedance as in equations 6.3 and 6.4.

Figure 6 . 9 :

 69 Figure 6.9: The superposition of the rst peaks of all FBT impedances. The longitidunal impedance has the unit of W and the transverse impedances have the unit of W/mm

Figure 6 . 10 :

 610 Figure 6.10: The CST simulations of all FBT impedances. The simulations were performed with 5 mm o set, 1000 mm wakelength and 12 mesh cells per wavelength.

Figure 6 . 11 :

 611 Figure 6.11: The horizontal transverse impedance results(dipolar+quadrupolar) of the bellows performed with 1 mm wire with 5 di erent o sets and extracted from MWM data by tting the parabola.

Figure 6 . 12 :

 612 Figure 6.12: The total vertical transverse impedance data which is extracted from moving wire measurements by polynomial t

Figure 6 . 13 :

 613 Figure 6.13: Superposition of the dipolar and quadrupolar horizontal impedance of bellows obtained with the CST simulations with MWMs. Black: Dipolar impedance CST simulations, Pink: Quadrupolar impedance CST simulations, and Blue: Total transverse impedance extracted from the MWMs.All the simulations were performed with 1 cm o set, 1000 mm wakelength and 20 mesh cells per wavelength.

Figure 6 . 14 :

 614 Figure 6.14: Superposition of the dipolar and quadrupolar vertical impedance of bellows obtained with the CST simulations with MWMs. Black: Dipolar impedance CST simulations, Pink: Quadrupolar impedance CST simulations, and Blue: Total transverse impedance extracted from MWMs. All the simulations were performed with 5 mm o set, 2000 mm wakelength and 25 mesh cells per wavelength.

Figure 6 . 15 :

 615 Figure 6.15: The real part of the bellows' CST simulations in both planes with 5mm o set in vertical and 10 mm o set in horizontal plane.

Figure 6 . 16 :

 616 Figure 6.16: The imaginary part of the bellows' CST simulations in both planes with 5mm o set in vertical and 10 mm o set in horizontal plane.

Figure 6 . 17 :Figure 6 .

 6176 Figure 6.17: The standard deviation on horizontal impedance measurements and t

Figure 6 . 19 :

 619 Figure 6.19: The quadrupolar impedance CST simulations of BPM4. The simulations were performed with 1 cm o set for the horizontal and 5 mm o set for the vertical, with 1000 mm wakelength and 20 mesh cells per wavelength.

Figure 6 . 20 :

 620 Figure 6.20: The S parameter results of BPM with di erent wire o sets.

Figure 6 . 21 :

 621 Figure 6.21: The total impedance result of BPM with di erent wire o sets.

Figure 6 . 22 :

 622 Figure 6.22: Vertical impedance measurements of pumping port with a 1 mm diameter wire which is extracted from the wire o set of 0 mm, 4 mm, 5 mm, 6 mm and 6.5 mm with polynomial t and compared with CST simulation results of dipolar and quadrupolar impedances. The simulations were performed with 5 mm o set, 1000 mm wakelength and 20 mesh cells per wavelength.

Figure 6 .

 6 Figure 6.23: The CST model of the cavity with tapers

Figure 6 .

 6 Figure 6.25: The CST model of Interaction Point center.

Figure 6 . 26 :

 626 Figure 6.26: The dipolar impedance CST simulations of the IP chamber in the horizontal plane. The simulations were performed with 10 mm o set, 1000 mm wakelength and 12 mesh cells per wavelength.

Figure 6 . 27 :Figure 6 . 28 :Figure 6 . 29 :

 627628629 Figure 6.27: The dipolar impedance CST simulations of the IP chamber in the vertical plane. The simulations were performed with 5 mm o set, 1000 mm wakelength and 12 mesh cells per wavelength.

  and 6.36, and the horizontal quadrupolar impedance simulation results in Figures 6.[START_REF] Full | Detection and clearing of trapped ions in the high current cornell photoinjector[END_REF] and 6.34. As long as the bellows were used with RF fingers, the largest contribution to the quadrupolar impedance comes from FBT. The kick factor of the transverse impedances were calculated as shown in Equation 1.34. In the storage ring, there are 4 BPM4s, 4 BPM6s, 4 BPM8s, 2 IP chambers, 15 Pumping Ports, 18 Bellows with an RF finger, 1 Septum, 1 Transverse Kicker, and 2 Cavity+Tapers. Taking the impedances of each of the above-mentioned components into account, the collective quadrupolar impedances in the vertical and the horizontal directions are shown in Figures 6.37 and 6.38:

1.36e 4 FBT

 4

Figure 6 . 24 :

 624 Figure 6.24: Up: The frequency spectrum of the dipolar impedance component of the RF cavity. Down: Enlarged section of the spectrum between 0.7 GHz and 2 GHz. The black line is the dipolar impedance which is calculated in CST PS with 2 mm bunch. The red lines represent the measured dipolar HOM frequencies.

Figure 6 . 30 :

 630 Figure 6.30: The CST model of septum.

Figure 6 . 31 :Figure 6 . 32 :

 631632 Figure 6.31: The dipolar impedance simulation of the septum in both planes. The simulations were performed with 5 mm o set for the vertical and 1 cm oset for the horizontal, with 1000 mm wakelength and 15 mesh cells per wavelength.

Figure 6 . 33 :

 633 Figure 6.33: The real part of horizontal quadrupolar impedance simulation of di erent components placed in the storage ring. The simulations were performed with 1 cm o set, 1000 mm wakelength and 15 mesh cells per wavelength.

Figure 6 . 34 :

 634 Figure 6.34: The imaginary part of quadrupolar impedance simulation of di erent components placed in the storage ring. The simulations were performed with 1 cm o set, 1000 mm wakelength and 15 mesh cells per wavelength.

Figure 6 . 35 :

 635 Figure 6.35: The real part of horizontal quadrupolar impedance simulation of di erent components placed in the storage ring. The simulations were performed with 5 mm o set, 1000 mm wakelength and 15 mesh cells per wavelength.

Figure 6 . 36 :Figure 6 . 37 :

 636637 Figure 6.36: The imaginary part of quadrupolar impedance simulation of di erent components placed in the storage ring. The simulations were performed with 5 mm o set, 1000 mm wakelength and 15 mesh cells per wavelength.

  ence le faisceau de particules, et les interactions des particules entre elles. L'objectif est d'expliquer comment une particule du faisceau influence la dynamique des autres. Les effets collectifs sont des facteurs limitant les paramètres et la qualité du faisceau. Cependant, les effets collectifs sont difficiles à analyser analytiquement avec des équipements complexes. Les programmes de simulation et les mesures complémentaires au banc doivent être utilisés pour décrire les effets collectifs pour différents composants de l'accélérateur. En somme, le bilan d'impédance, qui est l'estimation de l'impédance pour chaque composant, est crucial pour atteindre les performances voulues. Cette thèse se concentre sur l'impédance, et exprime les effets résistifs et géométriques des champs de sillage dans le domaine fréquentiel (voir définitions ci-dessous). Les particules sont accélérées dans des conditions de vide poussé afin de supprimer les effets des collisions entre les particules accélérées et les molécules d'air. Les champs de sillage sont les champs électromagnétiques (EM) créés par les particules circulant à l'intérieur de ce type de tube à vide. Plus précisément, l'optimisation de la méthode de mesure, appelée méthode du fil coaxial, est au coeur de cette thèse. Cette méthode utilise l'équivalence du champ EM entre le faisceau de particules et un courant qui circule dans un fil mince. La dérivation de l'effet d'impédance géométrique sera également effectuée, en vérifiant les résultats de fabrication des composants mécaniques. Dans cette thèse, différents effets collectifs et des exemples de limitations de ces effets seront introduits, plus tard j'entrerai dans les détails des champs de sillage et des impédances. Des informations générales sur les différents types d'effets collectifs avec des modèles théoriques seront données, les méthodes pour prédire et compenser les effets collectifs avec la simulation et les mesures seront expliquées. Ensuite, l'introduction aux notions d'impédance sera montrée à la fois théoriquement et expérimentalement avec des mesures sur fil. Ensuite, des informations générales seront données sur le projet ThomX, ainsi que sur les effets collectifs limitant cet accélérateur complexe. Les principaux composants seront analysés par des mesures filaires et des simulations dans cette thèse. Conclusion Dans cette thèse, les études d'impédance de couplage de faisceau avec des mesures à fil et des simulations de champ électromagnétique ont été présentées pour les équipements de l'accélérateur ThomX. Les principaux problèmes concernant les mesures et les simulations ont été examinés et de nombreuses précautions ont été prises pour réduire l'erreur des résultats. Les simulations CST ont été réalisées pour chaque pièce, et les modèles ont été étudiés avec soin en fonction des restrictions de calcul numérique. Les résultats ont été comparés aux structures de base et aux travaux antérieurs. La précision et la fonction des paramètres de simulation ont été discutées en détail afin de trouver un équilibre entre la puissance de calcul et la précision des mesures. Les cas de structures en forme de cavité ont été étudiés en profondeur, et les limites avec les erreurs ont été fixées. Tous les résultats ont été vérifiés par des mesures longitudinales, à deux fils et à fils mobiles. Grâce aux améliorations apportées pour réduire le niveau de bruit, il est possible de mesurer de petites impédances à l'aide du dispositif de mesure à fil. Les premières mesures ont été effectuées pour l'impédance longitudinale pour deux raisons : Premièrement, le nouveau banc devait être vérifié et la fiabilité devait être comparée aux anciennes études d'impédance longitudinale. Deuxièmement, la principale erreur sur les résultats du fil mobile provenait généralement de l'ajustement effectué. C'est pourquoi les résultats longitudinaux ont été comparés aux résultats longitudinaux du fil mobile pour vérifier l'efficacité de l'ajustement effectué. Deux méthodes différentes ont été utilisées pour étudier l'impédance transverse : Tout d'abord, des mesures à deux fils ont été effectuées sur toutes les parties de l'accélérateur pour obtenir l'impédance dipolaire. Ensuite, les mesures du fil mobile ont été effectuées, ce qui nous a permis d'obtenir l'impédance transverse totale en effectuant un ajustement des résultats. Lorsque l'impédance dipolaire des mesures à deux fils a été extraite des résultats du fil mobile, l'impédance quadrupolaire a été trouvée séparément pour chaque partie. La fiabilité des mesures par fil a été vérifiée pour de nombreux types d'objets différents, tels que les cavités ou les électrodes, et a également été confirmée par les calculs analytiques. La comparaison des mesures filaires avec les simulations nous a permis de définir les limites et la précision de l'impédance de couplage du faisceau. Un très bon accord a été trouvé entre les mesures à fil et les simulations pour les équipements de ThomX. En raison de la nature des impédances transverses, l'effet mesurable était généralement inférieur au niveau de bruit dans les travaux antérieurs, et même la vérification de la précision de la mesure de fréquence n'était pas toujours possible. Cependant, dans ce travail, même des impédances assez petites ont été mesurées avec une gamme de fréquences élevées, (pour certains cas) 0-6 GHz. Même pour les hautes fréquences, les résultats des mesures étaient très comparables aux résultats de la simulation. De plus, certains petits paramètres de conception se sont avérés importants et très efficaces sur le budget d'impédance, comme l'espace capacitif dans le FBT. Ces mesures nous ont permis de vérifier la fabrication, et de nous assurer que la conception et la réalité sont compatibles. Comme dernière étape, les facteurs de kick dus aux impédances de couplage transverses du faisceau ont été présentés avec une faible erreur autour 220 de 5% maximum sur l'échelle de fréquence. Ce travail ouvre également la perspective de travaux ultérieurs tels que l'utilisation de mesures de résonance pour définir les pertes sur le fil, et aussi la dynamique du faisceau de l'accélérateur entier peut être réalisée en utilisant un code de suivi tel que CODAL avec les résultats d'impédance étudiés. A partir de la relation entre l'impédance et la fonction de sillage telle que montrée dans l'équation 1.27, la fonction de sillage peut être extraite et utilisée dans un code de suivi. Il existe également une divergence sur le calcul des décalages d'accord et de la perte d'énergie en raison de l'interférence des champs de sillage, de la puissance de calcul et des restrictions de maillage, ainsi que de la largeur de bande de l'impédance calculée [75]. C'est pourquoi dans un dernier temps, cette étude devrait aller plus loin après la mise en service du Thomx en comparant les mesures basées sur le faisceau avec le modèle étudié pour mieux comprendre la dynamique du faisceau du ThomX.

  Table of ThomX parameters. . . . . . . . . . . . . . . . . . . . . . . . . . Table of the ThomX photoinjector and the linac parameters. . . . . . . Table of the ThomX photoinjector and the linac parameters. . . . . . .

2.3 2.4

Table 2 . 1 :

 21 The main characteristic of the X-Ray beam of ThomX with respect to di erent applications.[START_REF] Jacquet | High intensity compact compton x-ray sources: Challenges and potential of applications[END_REF]

easily. The usage parameters in numerous areas, from cultural heritage investigation to materials research, are shown in table 2.2

[START_REF] Bruni | ThomX -Conceptual Design Report[END_REF]

. It was designed and produced to generate X-Rays with energies between 50 keV and 90 keV, by changing the electrons' energy.

Table 2 .

 2 2: Table of ThomX parameters.

Table 2 .

 2 3: Table of the ThomX photoinjector and the linac parameters.

	18 MW of

the electron energy of 65 MeV, has been planned as a ThomX upgrade. All in all, the photoinjector and linac have been designed and built with the parameters in Table

2

.3

[START_REF] Garolfi | High-gradient S-band electron Linac for ThomX[END_REF]

.

Table 2 .

 2 4: Table of the ThomX photoinjector and the linac parameters.

	Parameters	Value (Units)
	Beam Energy	50 (MeV)
	Emittance (e rms , e normalized )	7 -9 (mm•mrad)
	Bunch charge Q	1 (nC)
	Number of electrons in a bunch N	6.25•10 9
	RMS bunch length at injection	6 -16 (ps)
	Revolution frequency f0	16.66 (MHz)
	Current I	16.7 (mA)
	Transverse damping time	1 (s)
	Longitudinal damping time	0.5 (s)
	Storage time before extraction	20 (ms)

Table 3 .

 3 1: Simulation time of the step in-out cavity with respect to the parameters

	67

Table 4 .

 4 2: The frequency of resonant peaks measurements and Cst results.

	Analytically Calculated Frequency CST Simulations Measurement Results
	2.51	2.45	2.54
	5.71	5.99	5.9
	8.92	9.2
	Table 4.1: The analytically calculated frequencies (obtained by using Equation 4.18),
	CST simulations, and measurement results of resonant peaks.
	CST Simulations Measurement Results
	0.46
	1.43
	2.35	2.24
	2.45	2.54
	3.11	3.25
	3.63	3.72
	4.19	4.23
	4.76	4.76
	5.37
	5.99

Table 5 .

 5 1: The calculated dipolar kick factor of di erent accelerator components.

	Real
	Imaginary

cm. Equation 3.7 shows that sum of the horizontal and vertical transverse impedance data from the MWM will give us the sum of the dipolar component

Acknowledgements

s (z) is the Gaussian temporal beam distribution with Gaussian beam sigma set. This parameter can affect the beam frequency; when the sigma gets smaller, the frequency will get higher.

• Simulated Wake Length: It is the wake potential calculation length. Impedance is obtained by Fourier transform of the wakefield in Equation 1.17. Simulated Wakelength then defines the frequency spectrum width as well. The duration of the simulation is affected proportionally by the simulated wake length. The wake length should be set to have an acceptable simulation duration and accurate resolution in the wake potential profile.

• The EM force integration gives the wake function as shown in Equation 1.14.

It depends on tests and source particles that were defined in the first section, in Figure 1.5. The integration also depends on the test particle's location. The test particle can be defined through the integration line. The coordinates of the integration path will define what kind of wakefield we have, as shown in Figure 3.1. For example, to calculate the quadrupolar component of the wakefield, the transverse offset of the test particle has to be set on the integration line, while the electron bunch is set at the center of the mechanical component. In the case of dipolar component, the source particle should be set at the offset value compared to the reference line (the axis defining the symmetry of the component, represented with a dashed line in Figure 1.5) and the test particle should be set on the integration line, at the center of the mechanical component.

The offset should be chosen carefully to avoid the higher order modes as it will be explained thoroughly in Chapter 5.

• The current injection scheme is the method which defines the modeling of the beam. There are two different types of current injection schemes: numerical wake potential and impedance calculations.

. Beam Position Monitor (BPM) Longitudinal

Measurements

The beam diagnostic is crucial for all accelerators to bring them to their nominal performances. For this reason, the THOMX storage ring contains 12 Beam Position Monitor (BPM)'s. There are 3 different types of BPM which are BPM4, BPM6 and BPM8. BPM4, BPM6 and BPM8 are classic button type BPMs with four electrodes, six electrodes and eight electrodes respectively. In figure 4.21 the 3D design of For TWMs, the choice of wire spacing (D) is crucial to having reliable results. In the literature, a lot of research has been carried out on this subject in order to verify two things: the contribution of the higher order modes with the larger D, and the effect of the error on stretching the wires with Ds too small. The general conclusion of these works [START_REF] Zanini | Electromagnetic Simulation of CERN Accelerator Components and Experimental Applications[END_REF][START_REF] Kroyer | Longitudinal and transverse wire measurements for the evaluation of impedance reduction measures on the mke extraction kickers[END_REF] are that 10% to 20% of the beam pipe diameter is well fitted for the two wires spacing.

In order to understand the effect of the simulation parameters and to test the reliability of the transverse impedance and wake field simulations and the calculations, the same configuration of the simulations on a former paper by Zanini [START_REF] Zanini | Electromagnetic Simulation of CERN Accelerator Components and Experimental Applications[END_REF] were performed as a benchmark. Also, these simulations gave us the opportunity to examine the higher order contributions of the different two wire distances.

The first simulations were done with rectangular vacuum pipe having a length of 120 mm, half-width of 3 mm and half-height of 1 mm. The design can be seen in Figure 5. In the simulations, the standard deviation of the longitudinal beam width was 1 mm and the wake length was 5000 mm. The length of the device in Zanini's paper was 120 mm, and they used 20% of the beam pipe diameter as the distance between the two wires. The simulations were done for both the dipolar and the

Thus the MWMs in both planes can be crosschecked with the TWMs for symmetrical structures. While the longitudinal results obtained in Chapter 4 can be used to extract transverse impedance from moving wire data using Equation 6.2, the longitudinal impedance results obtained by using one wire on axis measurements may also be used to crosscheck the efficiency and the accuracy of the polynomial fit.

. Reference Measurements

As for the longitudinal case and TWM, it is also crucial to perform reference measurements with the MWM. The application of the measurement procedure for the MWM is similar to those of the other ones. Five different measurements were taken with different wire offset distances from the axis such as: 0 mm, 4 mm, 5 mm, 6 mm and 6.5 mm. The procedure for reference measurement results were found to be exactly the same as for the longitudinal measurements. The error calculation was not performed at this stage in each case, however, the error calculations on the impedance results was performed later, as explained in the following section, since each impedance measurement was used, not only the mean.

The mean value of S 21 for the different wire offsets can be seen in Figure 6.2.

In theory, the reference measurements are expected to be the same for different wire position offsets up to the cutoff frequency which is around 5.7 GHz for the ThomX beam pipe. The cutoff frequency is comparable to our measurements. Up to the cutoff frequency, the standard deviation on the measurements is below 1 dB, however, after cutoff frequency, it is around 2-3 dB. was found separately for each part.

The reliability of the wire measurements were checked for many different types of objects, such as cavity-like, or with the electrodes and also supported with the analytical calculations. The comparison of the wire measurements with the simulations allowed us to define the limits and the accuracy of the beam coupling impedance. A very good agreement was found between the wire measurements and the simulations of the ThomX parts. Due to the nature of the transverse impedances, the measurable effect was generally below the noise level in former works, and even checking the frequency measurement precision was not always possible. However in this work, even quite small impedances were measured with a high frequency range, (for some cases) 0 6 GHz. Even for the high frequencies, the measurement results were highly comparable to the simulation results. Also, some small design parameters were found to be important and highly effective on the impedance budget, such as the capacitive gap in FBT. These measurements allowed us to check the manufacturing, and to make sure that the design and reality are compatible. As the last step, the kick factors were presented due to the transverse beam coupling impedances with a low error around 5% maximum on the frequency scale.

This work also opens the prospect of further works such as using resonance measurements to define the losses on the wire, and also the beam dynamics of the entire accelerator can be performed using a tracking code such as CODAL with the studied impedance results. From the relation between the impedance and the wake function as shown in Equation 1.27, the wake function can be extracted and used in a tracking code. Also there is a discrepancy on the calculation of the tune shifts and the energy loss due to the interference of the wakefields, the computational power and mesh restrictions, and the bandwidth of the calculated impedance [START_REF] Smaluk | Beam-based Impedance Measurement Techniques[END_REF]. This is why as a last step, this study should go further after the L'interaction des particules avec les composants de l'accélérateur et les particules elles-mêmes devient un sujet et comportement prédom-