Beam Diagnostics and Dynamics in Nonlinear Fields

Transcription

1 Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 583 Beam Diagnostics and Dynamics in Nonlinear Fields JIM ÖGREN ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 27 ISSN ISBN urn:nbn:se:uu:diva-33975

2 Dissertation presented at Uppsala University to be publicly eamined in Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen, Uppsala, Friday, 8 December 27 at 9:5 for the degree of Doctor of Philosophy. The eamination will be conducted in English. Faculty eaminer: Dr. Stephen Peggs (Brookhaven National Laboratory). Abstract Ögren, J. 27. Beam Diagnostics and Dynamics in Nonlinear Fields. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology pp. Uppsala: Acta Universitatis Upsaliensis. ISBN Particle accelerators are indispensable tools for probing matter at the smallest scales and the improvements of such tools depend on the progress and understanding of accelerator physics. The Compact Linear Collider (CLIC) is a proposed, linear electron positron collider on the TeV-scale, based at CERN. In such a large accelerator comple, diagnostics and alignment of the beam are crucial in order to maintain beam quality and luminosity. In this thesis we have utilized the nonlinear fields from the octupole component of the radio-frequency fields in the CLIC accelerating structures for beam-based diagnostics. We have investigated methods where the nonlinear position shifts of the beam are used to measure the strength of the octupole component and can also be used for alignment. Furthermore, from the changes in transverse beam profile, due to the nonlinear octupole field, we determine the full transverse beam matri, which characterizes the transverse distribution of the beam. In circular accelerators, nonlinear fields result in nonlinear beam dynamics, which often becomes the limiting factor for long-term stability. In theoretical studies and simulations we investigate optimum configurations for octupole magnets that compensate amplitude-dependent tune-shifts but avoid driving fourth-order resonances and setups of setupole magnets to control individual resonance driving terms in an optimal way. Keywords: Beam diagnostics, Nonlinear beam dynamics, Accelerator physics Jim Ögren, Department of Physics and Astronomy, High Energy Physics, Bo 56, Uppsala University, SE-75 2 Uppsala, Sweden. Jim Ögren 27 ISSN ISBN urn:nbn:se:uu:diva (

3 Nature, it seems, is the popular name For milliards and milliards and milliards Of particles playing their infinite game Of billiards and billiards and billiards. Piet Hein

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5 List of papers This thesis is based on the following papers, which are referred to in the tet by their Roman numerals. I II III IV V VI Measuring the full transverse beam matri using a single octupole J. Ögren, R. Ruber, V. Ziemann and W. Farabolini, Phys. Rev. ST Accel. Beams, vol. 8, issue 7, 728, July, 25. Aligning linac accelerating structures using a copropagating octupolar mode J. Ögren and V. Ziemann, Phys. Rev. Accel. Beams, vol. 2, issue, 28, October, 27. Beam-Based Alignment Studies at CTF3 Using the Octupole Component of CLIC Accelerating Structures J. Ögren, A.K. Bhattacharyya, M. Holz, R. Ruber, V. Ziemann and W. Farabolini in Proc. 8th Int. Particle Accelerator Conf. (IPAC 7), Copenhagen, Denmark, pp , May 27. Compensating amplitude-dependent tune-shift without driving fourth-order resonances J. Ögren and V. Ziemann, Nuclear Instruments and Methods in Research A, 869: 9 (27). Optimum resonance control knobs for setupoles J. Ögren and V. Ziemann, submitted to Nuclear Instruments and Methods in Research A. Surface Characterization and Field Emission Measurements of Copper Samples inside a Scanning Electron Microscope J. Ögren, S.H.M. Jafri, K. Leifer, and V. Ziemann in Proc. 7th Int. Particle Accelerator Conf. (IPAC 6), Busan, Korea, pp , May 26. Reprints were made with permission from the publishers.

6 List of other publications The following publications are not included in this thesis VII VIII IX X XI XII XIII ELEPHANT: A MATLAB-code for Hamiltonians, Lie algebra, normal form and particle tracking J. Ögren FREIA Report, 27/9. An electron energy loss spectrometer based streak camera for time resolved TEM measurements H. Ali, J. Eriksson, H. Li, S. H. M. Jafri, S. Kumar, J. Ögren, V. Ziemann and K. Leifer, Ultramicroscopy, vol. 76, pp. 5-, May, 27. A Method for Determining the Roll Angle of the CLIC Accelerating Structures From the Beam Shape Downstream of the Structure J. Ögren, W. Farabolini, and V. Ziemann, in Proc. 8th Int. Particle Accelerator Conf. (IPAC 7), Copenhagen, Denmark, pp , May 27. Wave Propagation in a Fractal Wave Guide A.K. Bhattacharyya, J. Ögren, M. Holz and V. Ziemann, in Proc. 8th Int. Particle Accelerator Conf. (IPAC 7), Copenhagen, Denmark, pp , May 27. Dielectric Laser Accelerator Investigation, Setup Substrate Manufacturing and Investigation of Effects of Laser Induced Electromigration RF Cavity Breakdown Influences M. Hamberg, E. Vargas Catalan, M. Karlsson, J. Ögren and M. Jacewicz, in Proc. 8th Int. Particle Accelerator Conf. (IPAC 7), Copenhagen, Denmark, pp , May 27. Updated baseline for a staged Compact Linear Collider edited by P.N. Burrows, P. Lebrun, L. Linssen, D. Schulte, E. Sicking, S. Stapnes and M.A. Thomson, CERN-26-4, Geneva, Switzerland, 26. Beam-based Alignment of CLIC Accelerating Structures Utilizing Their Octupole Component J. Ögren and V. Ziemann, in Proc. 7th Int. Particle Accelerator Conf. (IPAC 6), Busan, Korea, pp , May 26.

7 XIV XV XVI XVII XVIII CALIFES: A Multi-Purpose Electron Beam for Accelerator Technology Tests J. L. Navarro et al. (4 authors), in Proc. 27th Linear Accelerator Conf. (LINAC4), Geneva, Switzerland, Sept., 24. The momentum distribution of the decelerated drive beam in CLIC and the two-beam test stand at CTF3 Ch. Borgmann, M. Jacewicz, J. Ögren, M. Olvegård, R. Ruber and V. Ziemann in Proc. 5th Int. Particle Accelerator Conf. (IPAC 4), Dresden, Germany, pp , May 24. General-purpose spectrometer for vacuum breakdown diagnostics for the 2 GHz test stand at CERN M. Jacewicz, Ch. Borgmann, J. Ögren, R. Ruber and V. Ziemann, in Proc. 5th Int. Particle Accelerator Conf. (IPAC 4), Dresden, Germany, pp , May 24. Recent results from CTF3 two-beam test stand W. Farabolini, et al. (5 authors), in Proc. 5th Int. Particle Accelerator Conf. (IPAC 4), Dresden, Germany, pp , May 24. A Fully-levitated Cone-shaped Lorentz-type Self-bearing Machine with Skewed Windings J. Abrahamsson, J. Ögren and M. Hedlund, Magnetics, IEEE Transactions on, vol. 5, no. 9, pp. -9, Sept., 24.

8 The author s contribution to the papers Paper I: I did the derivations, built the simulation model and did all the simulations. I wrote the data acquisition scripts for the eperiment and performed the measurements together with WF. I analyzed the data and wrote the majority of the paper. Paper II: I did the derivations, wrote the simulation code and did all the simulations. The paper was written by me in close collaboration with VZ. Paper III: The eperiment was a joint effort between AKB, MH, WF and myself. I did the majority of the data analysis and wrote the paper. Paper IV: The idea originated from VZ and was developed as a joint effort. I did all the simulations and wrote a code that can treat Hamiltonians, Lie algebra methods and normal forms. The analysis and writing of the paper was done in collaboration. Paper V: The idea was based on earlier work by VZ and developed as a joint effort. I derived the main results and did all the simulations. I wrote the bulk of the paper. Paper VI: I wrote a LabView-based control system for interfacing all hardware and developed automatized measurement procedures. I did all the measurements together with SHMJ who helped in particular with operation of the scanning electron microscope. The data analysis and interpretation of the results were done in collaboration but the paper was written by me. Cover illustration The illustration on the cover shows a Poincaré surface of a section for an octupole followed by a rotation with tune Q =.269. The plot is generated by the author in MATLAB and displays phase space for thirteen particles with different starting amplitudes tracked for 2 turns.

9 Contents Introduction Physics motivation Particle colliders Storage rings Thesis structure The Compact Linear Collider (CLIC) Layout The main beam Vacuum breakdown studies CLIC test facility 3 (CTF3) Transverse beam dynamics Fundamentals of beam physics Linear beam dynamics Hamiltonians, Lie algebra and normal forms Beam-based diagnostics with octupoles Measuring the RF octupole component Measuring the beam matri Beam-based alignment Beam dynamics in nonlinear fields Compensating amplitude-dependent tune-shift Optimum resonances control knobs Conclusions Summary in Swedish Acronyms Acknowledgements References

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11 . Introduction This thesis includes eperimental and theoretical work on particle accelerators and beam physics. Before presenting the research I will give a brief introduction to the field of accelerator-based science and outline the motivations for building and operating particle accelerators.. Physics motivation In the beginning of the last century Ernest Rutherford used a beam of alpha particles emitted from radioactive nuclei for studying scattering in a thin gold foil. The results were not consistent with the theory of the atom at the time since some of the particles scattered at very large angles. From the results of the eperiment Rutherford rejected Thomson s "plum-model" and instead concluded that the atom is made up of a small, heavy nucleus surrounded by lighter electrons []. This was one of the very first eperiments where a beam of charged particles was used as a means of studying the matter on atomic and subatomic level. However, a beam from a radioactive source is not controllable and achieving high intensity is problematic. Thus the idea of constructing a machine to generate a particle beam, i.e. a particle accelerator, was born. The pioneering work on accelerator physics started in the 92s with Gustav Ising inventing the electron drift tube later built by Rolf Widerøe. In 929 Robert J. Van de Graaff invented a high-voltage generator and an electrostatic accelerator. Ernest Lawrence, inspired by the works of Widerøe, invented the first cyclotron in 932 which could accelerate protons to energies above MeV [2]. Since then, particle accelerators have been an indispensable part of eperimental physics and over time higher and higher energy together with higher and higher intensity was achieved. Higher energy meant probing smaller length-scales and eploring new territories and higher intensity meant faster statistics. For the past century, progress in nuclear and particle physics has been closely linked to progress in accelerator physics. A prime eample of this is the development of the Standard Model of particle physics. The Standard Model The Standard Model of particle physics [3 5] is a quantum field theory that describes matter and energy as kinematics and interactions of fundamental

12 particles. The model consists of matter particles that are fermions and forcecarrying particles that are bosons. The fermions come in two families of fundamental particles quarks and leptons and each type of fermion comes in three generations. Quarks can combine in doublets to form mesons or in triplets to form baryons, such as protons and neutrons, which make up the nuclei of atoms and constitute most of the visible matter in the universe. The other piece of the atom is the electron, which is a lepton a fundamental particle with no known internal structure. The photon is a massless boson that mediates the electromagnetic interaction. Then there are the short-ranged interactions: the strong nuclear interaction mediated by gluons and the weak nuclear interaction mediated by the W ± and Z bosons. When electromagnetism and the weak nuclear force were unified into the electroweak theory there was a problem with the theory since it predicted some known massive particles, i.e. the W ± and Z bosons, to be massless. The resolution was to introduce a new scalar field that causes breaking of the symmetry and as a consequence some of the fundamental particles would acquire mass [6 9]. The field is named the Higgs field after one of the predictors and the corresponding mechanism is named the Higgs mechanism. A quantum ecitation of the Higgs field results in yet another particle the Higgs boson which was the final piece of the Standard Model to be discovered in 22. The particles of the Standard Models are shown in Fig.. with the three generations of leptons and quarks on the left and the bosons on the right. Beyond the Standard Model Even though the Standard Model is currently the best theory that describes the universe at the fundamental level we know it to be incomplete. For one thing, the Standard Model only incorporates three out of the four known fundamental forces of nature: electromagnetic, strong nuclear and weak nuclear. Gravity is still not included in the Standard Model. There are several eperimental results that lie outside the realm of the Standard Model. Astronomical observations of galay rotational curves [], the cosmic microwave background [2] and weak gravitational lensing [3] all suggest that ordinary, visible matter cannot provide enough mass for the observed effects. This unknown, non-luminous matter is called dark matter and can not be accounted for by the Standard Model. Furthermore, in the 99s there were astronomical observations showing that the epansion of the universe is in fact speeding up. What drives this acceleration of the epansion is not known and commonly labeled dark energy [4 6]. It is estimated that only 5% of the energy and matter in the universe is ordinary, visible matter that can be accounted for by Standard Model, the rest is dark matter (27%) and dark energy (68%) [2]. 2

13 Figure.. Elementary particles of the Standard Model. The matter particles consist of three generations of quarks and three generations of leptons. The force-carrying bosons are shown on the right. Image credit: [] Another mismatch between theory and eperiment is that neutrinos are massless in the Standard Model but we know from the observation of neutrino oscillations [7, 8] that the neutrinos must have nonzero mass. Furthermore, in the Standard Model each particle has an antiparticle and it is not clear why there is only matter and no antimatter in the universe when equal amounts of the two should have been created in the Big Bang. The Standard Model cannot eplain the puzzle of the matter/antimatter asymmetry. Finally, there is also a hierarchy problem in the Standard Model. In essence the question is why the electroweak force is so many orders of magnitude stronger than gravity. For this to be compatible with the Standard Model fine-tuned cancellations of correction terms are required and this is often regarded as problematic since it seems unnatural. An attempt to resolve the hierarchy problem and potentially many of the other issues stated above is the theory of supersymmetry (SUSY) which is an etension of the Standard Model based on additional symmetries [9, 2]. SUSY models predict many additional particles to eist [2], none of which have been found eperimentally. There are also other theories that address the shortcomings of the Standard Model such as String theory and etra dimensions. So far, theories beyond the Standard Model lack empirical evidence. Ultimately, eperimental observations must guide us forward. 3

14 .2 Particle colliders Models of particle physics are tested in a controllable way by accelerating and colliding particles in an accelerator. The high energy density in the collision leads to the creation of new particles. The new particles and their decay products are measured and identified in a detector and in this way interactions between fundamental particles are studied and compared to predictions of a theory. Testing the Standard Model Despite the problems with the Standard Model described above, the theory has been very successful and stood by many tests over the years. Many particles that were predicted by the theory were later confirmed eperimentally. In 968 the first evidence of the eistence of quarks was found from deep inelastic electron proton collisions at SLAC [22,23]. The predicted electroweak bosons W ± and Z were discovered in 983 [24 26] from proton antiproton collisions at the Super Proton Synchrotron (SPS) at the European organization for nuclear research (CERN). This was followed by precision measurements of the Z boson from electron positron collisions at SLAC Linear Collider (SLC) and the Large Electron-Positron Collider (LEP) where the eistence of a fourth generation of leptons was ruled out [27]. The heaviest of the quarks, the top quark, was discovered at the Tevatron accelerator at Fermi National Laboratory in 995 [28]. The Higgs boson was the final particle of the Standard Model to be found eperimentally. The search for the Higgs boson was the main motivator for building the Large Hadron Collider (LHC) in the eisting LEP tunnel at CERN. LHC is to date the highest energy particle accelerator ever built where protons are accelerated to 6.5 TeV a million times the energy of Lawrence s first cyclotron in the 93s. In 22 it was announced that a 25 GeV Higgs-like particle had been discovered at the CMS and ATLAS eperiments at the LHC [29, 3]. So far are all the measured properties of this particle consistent with those of the Standard Model Higgs boson [3]. Linear lepton colliders With the Tevatron and LHC the energy frontier on the TeV-scale was opened. However, there has been consensus in the particle physics community that discoveries at the LHC need to be complemented by precision measurements at a lepton collider on the same energy-scale [32]. Hadrons, such as protons, are composite particles and when they collide it is really their constituents the quarks and the gluons that interact and this leads to a myriad of interactions and production of particles. Leptons, e.g. electrons and positrons, on the other hand are fundamental, point-like particles. In a lepton collider the initial 4

15 state of the collision is well-defined and the outcome of the collision is much cleaner and more easily interpreted, which makes lepton colliders much more suitable for precision measurements. Things to be studied in a high-energy lepton collider are precision measurements on the Higgs boson and the top quark (guaranteed program) and various SUSY particles (potential program). Currently there are two international collaborations studying linear highenergy electron positron colliders: the International Linear Collider (ILC) [33] and the Compact Linear Collider (CLIC) [34]. ILC is based on superconducting acceleration cavities [35] and proposed to be built in Japan. CLIC on the other hand is based on normal-conducting technology [36] and will be eplained further in the net chapter. One of the challenges with a high-energy linear collider is to preserve beam quality over a very long accelerator and this makes beam diagnostics crucial, which is one of the topics of this thesis. Why a linear accelerator? When a charge particle is accelerated perpendicular to its velocity it emits electromagnetic radiation, which is called synchrotron radiation [37]. This means that particles in a circular accelerator lose some of their energy as they are bent around the ring by magnets. The energy loss per turn, E, for a particle is proportional to the fourth power of the ratio of its energy E and mass m and inversely proportional to the bending radius ρ of the ring [38], i.e. E E 4 /(m 4 ρ). Since electrons and positrons have about /2 of the mass of the proton they are much more subjected to energy loss due to synchrotron radiation. This is the reason the electrons and positrons in LEP only reached 4.5 GeV while the protons in LHC reach 6.5 TeV despite both being accelerators in the same tunnel with a circumference of 27 km. Building a circular electron positron collider on the TeV-scale would result in a machine of gigantic proportions and hence a linear machine is the only sensible option. However, a circular accelerator is still needed as a part of a linear collider and used as a damping ring, which has the purpose of enhancing the beam quality..3 Storage rings A circular particle accelerator with the purpose of storing a circulating beam of particles is known as a storage ring. In linear colliders they are used as damping rings where the effect of synchrotron radiation is utilized to damp the oscillations of the particles. This enhances the quality of the beam and is essential for achieving the required small beam size. A related application is to use a storage ring as a dedicated synchrotron light source, such as the MAX IV Laboratory [39] in Lund, where high-brightness beams of photons emitted from a stored electron beam are guided through beamlines and utilized for studying condensed matter physics, material science and biology. Storage rings are also used as circular colliders such as previously mentioned LHC, 5

16 or the proposed Future Circular Collider (FCC), where two beams of particles are circulated in opposite directions, stored and collided. One of the challenges with iterative systems such as storage rings is beam stability. Since the purpose of the machine is to store a beam for a long time, i.e. billions of turns, the particles most be able to traverse stable trajectories. But in order to achieve high intensity, the performance of the machine needs to be pushed to the limits and this leads to issues with stability. For instance, designing a machine for high-quality beams with small beam sizes requires strong focusing which in turn increases sensitivity to nonlinear effects. Thus understanding and dealing with nonlinear dynamics is essential for enhancing performance of future storage rings. Particle accelerators have many different applications but the improvement and development of such machines are dependent on the progress and understanding of accelerator physics. A particle accelerator is a complicated system of thousands of components with the challenges to maintain, diagnose and control an ultra-relativistic plasma under etreme conditions. Making beams with smaller and smaller beams sizes, or higher and higher energies, requires additional understanding of the physics of beams. This thesis contains research that deals with the diagnostics and dynamics of beams in the presence of nonlinear fields..4 Thesis structure The thesis is structured in the following way: Chapter 2 introduces CLIC and the CLIC Test Facility 3 (CTF3) where the eperiments reported in this thesis were conducted. In chapter 3 we review the theory of linear and nonlinear transverse beam dynamics. In Chapter 4 we introduce some methods for beam-based diagnostics and summarize the most important results from Papers I III. Finally in Chapter 5 we apply the tools of nonlinear beam dynamics to circular accelerators and address tune-shift compensation and resonance control knobs, these are the topics of Papers IV and V. 6

17 2. The Compact Linear Collider (CLIC) CLIC is a proposed linear electron positron collider on the TeV energy-scale based at CERN. The CLIC study is carried out by an international collaboration and aims to provide a suitable complement to the LHC. In 22, before the announcement of the discovery of the Higgs boson, the collaboration published a conceptual design report [36]. An updated baseline design was published in 26 [4] (which is also Paper XII) where the first energy stage of 38 GeV center-mass-energy is optimized for physics on the Higgs boson and the top quark. After the initial energy stage it is possible to upgrade CLIC to a second stage of.5 TeV and a third stage of 3 TeV center-of-mass energy for physics beyond the Standard Model. 2. Layout Figure 2. shows a schematic of the 3 TeV CLIC layout that consists of two accelerator complees: the main beam and the drive beam. At the bottom of Fig. 2. we have the electron and positron injectors for the main beam. Both injectors consist of a particle source and a linear accelerator (linac) that accelerates the beam to 2.86 GeV before injecting the beam into the damping rings. The purpose of the damping rings [4] is to reduce the emittance of the electron and positron beams prior to acceleration in the main linac. Emittance will be defined in the net chapter but it can be thought of as a measure of beam quality. After the damping rings, a booster linac accelerates the beams to 9 GeV before they are transported to the far ends of the main linac. The electrons are accelerated from one end and the positrons from the other and each linac brings the particles from 9 GeV to.5 TeV by transferring energy from radio-frequency (RF) fields to the particles in tens of thousands accelerating structures. When the beams reach the center, the final focus system focuses the beams to nanometer sizes before they collide inside the detector at the interaction point. After collision the beams are transported to beam dumps where the energy is dissipated. The upper part of Fig. 2. shows the drive beam comple which is a highintensity electron beam with the purpose to create RF power for acceleration of the high-energy main beam. In the 3 TeV energy stage two drive-beam accelerators are needed. The high intensity of the drive beam is generated by successive recombinations of a 2.4 GeV electron beam. The beam from the drive beam accelerator consists of a long bunch train (48 µs) and a low 7

18 Figure 2.. The CLIC layout for 3 TeV center-of-mass energy. Lower part: electron and positron injectors followed by the damping rings. The the beams are accelerated in the main linac and collided at the interaction point. Upper part: the drive beam comple. The two 2.4 GeV beams are interleaved in delay loops and combiner rings in order to reduce the bunch spacing and increase the intensity. The RF power is then etracted in the decelerator sectors and guided to the accelerating structures in the main linacs. Image source: [4] bunch frequency (.5 GHz), which gives an average beam current of only 4.2 A. Then the beam enters the recombination comple where the bunches in the sub-trains are interleaved in a delay loop and two subsequent combiner rings. This results in a factor 24 shortening of the total pulse length, due to the reduction of bunch spacing, and a factor 24 multiplication of the intensity. After recombination the drive beam pulse is 244 ns long with an average beam current of A and 2 GHz bunch frequency and this high-intensity beam is then sent to the decelerator sections for RF power generation. The key parameters for CLIC are summarized in Table 2.. The Compact in the name of the Compact Linear Collider comes from the usage of high-gradient, normal-conducting acceleration structures. The design accelerating gradient of CLIC is MV/m, which is considerably higher than what can be achieved with super-conducting technology. Since it is a linear accelerator and the particles must be accelerated in a single pass, about 4 accelerating structures are needed to reach 3 TeV center-of-mass energy and this results in a 5 km long tunnel for the main linac. The luminosity is a measure of the number of collisions per unit area and time. Achieving high luminosity is the main objective of any particle collider since higher luminosity means higher event rates and faster statistics. The luminosity L [36, 42] is given by 8 L = H D N 2 σ σ y n b f rep (2.)

19 Table 2.. A summary of parameters for CLIC at the 3 TeV energy stage [4]. Parameter Symbol Unit Value Center-of-mass energy s TeV 3 Repetition frequency f rep Hz 5 Pulse length τ RF ns 244 Number of bunches per train n b 32 Number of particles per bunch N Bunch separation t ns.5 Bunch length σ z µm 44 Luminosity L cm 2 s 2 34 Beam size at interaction point σ /σ y nm 4/ Normalized emittance (end of linac) ε /ε y nm 66/2 Accelerating gradient G MV/m RF frequency f RF GHz 2 Length of main tunnel km 5. Total number of accelerating structures Estimated power consumption P wall MW 589 where σ,y is the transverse beam size at the interaction point, N is the number of particles per bunch, n b is the number of bunches per pulse, f rep is the repetition rate, i.e. the number of pulses per second. H D is a correction factor to account for the self-focusing effect during the collision. The target luminosity for CLIC is 2 34 cm 2 s. For a circular collider the circulating beams can be used for collisions again and again which gives a large repetition rate f rep. But a linear collider is a single-pass machine and means that every particle bunch is used only once and limits f rep, which for CLIC is set to 5 Hz. The way to compensate for the low repetition rate and still achieve high luminosity is to squeeze the beam size (σ,σ y ) down to nanometer size at the interaction point. Two-beam acceleration scheme The CLIC acceleration concept is based on a two-beam acceleration scheme where RF power is generated by the high-intensity electron beam (the drive beam) that is decelerated in a Power Etraction and Transfer Structure (PETS). The decelerator sections contain several PETS and runs in parallel with the main beam. The PETS is a structure resonant at 2 GHz that transforms the kinetic energy of the drive beam into 2 GHz RF power. It is a traveling wave structure made of copper cells with spacing adapted to 2 GHz. When an electron bunch of the drive beam passes through a PETS it ecites a wakefield and the succeeding bunches are decelerated by this wakefield and in this way energy is transferred from the beam to the RF fields. Every bunch of the drive beam travels through a sector of 492 PETS over a distance of about km 9

20 Figure 2.2. Two-beam acceleration scheme. 2 GHz RF power is generated when the electrons in the drive beam are decelerated in a series of PETS. This power is then transported to the acceleration structures where the main beam is accelerated. Image source: [36] and approimately 9% of the energy is etracted. The generated RF power is transferred from the PETS to the main beam via waveguides. Figure 2.2 shows a schematic of the two-beam acceleration scheme. A different option is to use a more conventional method for generating RF power with klystrons. Klystrons are RF power amplifiers that take a highpower DC pulse and amplify a low power RF signal. In CLIC the injectors and drive beam accelerators use klystrons but not the main linac and the motivation for using two-beam acceleration comes from the large number of accelerator structures required. For the 3 TeV stage it is estimated that about 35 klystrons would be needed and in that case the two-beam acceleration scheme is much more efficient and cost effective [36]. However, for the first energy stage of 38 GeV it could be more efficient to use klystrons [4]. Using klystrons is a way to distribute the RF power generation over many units whereas the drive beam is one single large RF power generator. 2.2 The main beam The main beam of CLIC consists of two identical accelerators, one for electrons and one for positrons, with the purpose of bringing the particles to high energy before collision. The particles are accelerated in normal-conducting, MV/m acceleration structures made of copper and since they are normalconducting, power will be lost through resistive wall heating. Therefore, in order to minimize these ohmic losses the length of the RF pulse is kept short (244 ns). This also implies a short bunch-train and narrow bunch separation which for CLIC is set to.5 ns. But the short bunch separation causes problems since the bunches can interact with each other, i.e. wakefields from 2

21 bunches in the head of the train disrupt trailing bunches. This is highly undesirable since it leads to emittance growth and luminosity reduction. To avoid issues with wakefields the beam must be well-aligned in the main linac. Alignment is a challenging part for CLIC and must be kept on the order of a few micrometer to maintain the design luminosity. The most sensitive part is the accelerating structures due to their small irises. In CLIC several accelerating structures (2 to 8) will be mounted in a CLIC module together with the corresponding number of PETS (one PETS powers two accelerating structures). The accelerating structures in a module are mounted on movable girders that allow for aligning the beam with respect to the accelerating structure by means of moving the accelerating structures themselves. In order to measure the transverse position of the beam with respect to the accelerating structure wakefield monitors [43] are installed in every second structure. The wakefield monitors measure one of the higher order modes, i.e. modes of higher frequencies, that is ecited by transverse offsets of the beam. In Papers II and III we investigate a complementary method for beam-based alignment. The accelerating structure Figure 2.3 shows a CLIC accelerating structure which is about 25 cm long and consists of 28 cells. The RF power is fed into the accelerating structure via the input coupler cell and travels through the structure and eit through the output coupler where the remaining RF power is absorbed in an RF load. The beam is timed with respect to the RF fields in such a way that it sees a longitudinal electric field and is accelerated in the forward direction. Figure 2.3 also shows a cross-section of a single accelerating cell where we see the cell itself, the small iris and four radial waveguides. The purpose of these waveguides is to reduce wakefields by damping higher order modes. The apertures of these transverse waveguides are designed in such a way that higher order modes are transmitted and absorbed in RF absorbing materials but the fundamental mode of 2 GHz remains unaffected, essentially it is a high-pass filter with cut-off frequency above 2 GHz. The purpose is again to mitigate emittance growth due to wakefields. Octupole component The four-fold symmetry from the radial waveguides of the accelerating cell in Fig. 2.3 allows for an octupole component of the RF fields. This is a multipolar component with fundamental frequency 2 GHz that co-propagates with the main accelerating field. Even though this octupole component has the same frequency, it is phase-shifted 9 with respect to the main accelerating field. The octupole component is known from simulations [44] and eperimental 2

22 Walter Wuensc Figure 2.3. Left: the CLIC accelerating structure consist of 26 regular cells plus one input coupler cell and one output coupler cell. Image source: [36]. Right: a single cell of the CLIC accelerating structure. Centered is the iris where the beam travels through. There are four radial waveguides connected to each cell for damping higherorder modes. Image courtesy of Walter Wuensch, CERN. observations, see Paper XVII. The effect on the beam is equivalent to the effect of an octupole magnet. Figure 2.4 shows a screen image from CTF3 of a beam perturbed by the octupole component in a CLIC accelerating structure. Since the octupole component is phase-shifted with 9 with respect to the main accelerating field the effect on the beam is small during normal on-crest acceleration. There are also other multipolar components of the RF fields in the CLIC accelerating structures, such as a quadrupole component from the two-fold symmetry of the input and output coupling cells. The effects on the beam from the multipolar components have been studied and found to be negligible under normal operation [45]. However, in Papers I-III we investigate methods to utilize the octupole component for diagnostics purposes. Before discussing the beam-based diagnostics we will make a small digression to Paper VI and address one of the limiting factors for high-gradient acceleration: vacuum breakdowns. 2.3 Vacuum breakdown studies The high electric fields in the acceleration structures, due to the high gradient, might induce electron emission from the metallic surface that can lead to an avalanche process where neutrals are ionized and form a plasma. This creates an arc, i.e. a conducting medium in the form of a plasma, in the otherwise insulating vacuum, and the power of the RF fields is dissipated into the wall of the structure. This collapse of the RF fields in the accelerating structure is called a breakdown, or a discharge, and constitutes a big challenge for CLIC. Breakdowns cause power loss which means that the beam is accelerated less but they can also be harmful to the accelerating structures themselves. If the beam passes through the accelerating structure during a breakdown it will receive a transverse kick due to the magnetic field created by a high 22

23 6 5 y [mm] [mm] Figure 2.4. Transverse beam profile of a beam perturbed by the octupole component of the RF fields in a CLIC accelerating structure. Image source: Paper IX. breakdown current [46]. This makes RF breakdowns problematic for CLIC since these transverse momentum kicks cause misalignments of the beam and lead to luminosity reduction. Breakdown rate is defined as the number of breakdowns per pulse and for CLIC the maimum limit of breakdown rate is set to 7. This tight requirement is to ensure less than % luminosity reduction due to breakdowns [36]. In order to test how the accelerating structures perform in terms of breakdown rate there is a lot of eperimental activity in several dedicated RF teststands at CERN where CLIC accelerating structures are tested over longer periods of time. In the beginning of operation the structures have too high breakdown rates at nominal RF pulse length and power so in order to protect the structures they are operated at reduced pulse length and power. But over time the breakdown rate decreases, a process called conditioning, and then the pulse length and power are slowly increased until finally reaching nominal values at the required breakdown rate. In addition there are also a number of smaller direct current (DC) setups with the benefit of generating similar field strengths but with much simpler infrastructure and faster repetition rates. For more information on the recent progress of high-gradient structures see [47]. Although there has been much progress in understanding breakdowns there are parts of the phenomenon that are not fully understood. It is generally believed that the onset of a breakdown is due to electron field emission [48]. To address the more fundamental aspects of vacuum breakdowns and field emission in particular, we have an Uppsala-based eperimental setup [49] that can be operated in-situ a scanning electron microscope (SEM). This is the topic of Paper VI and will be briefly presented here. 23

24 Field emission Vacuum discharges are initiated by emission of electrons under influence of high electric fields, this is called field emission. The theory of field emission dates back to 928 when Fowler and Nordheim described how electrons can escape from a cold cathode by quantum tunneling through the potential barrier [5]. The Fowler Nordheim equation gives the current density of electron emission as a function of applied electric field. Written in a commonly used form [5, 52] we have.54 6 β 2 E 2 I = A e e.4φ /2 e φ 3/2 /βe φ (2.2) where the emitted current I depends on the emission area A e, work function φ, applied electric field E and a parameter β called the field enhancement. In the original Fowler Nordheim equation there was a large discrepancy between the required field strengths from the theory compared with eperiments where field emission started at much lower field strengths. This is due to the idealized conditions of the theory, which assumes a perfect surface. The field enhancement was introduced as a quantification of the local enhancement of the electric field due to protrusions or other surface imperfections. We can rewrite (2.2) as { } I ln E 2 = m k (2.3) E which describes a straight line in the coordinates ln{ I and E. Thus we can plot data from measurements in these coordinates and from the slope k of the line we determine the field enhancement as β = φ 3/2 k. E 2 } Eperimental setup The setup at Uppsala University consists of a movable sample holder and a tungsten tip above the sample. Both the sample holder and the holder for the tip are controlled by piezo-motors with position sensors and a feedback controller that allows for nanometer precision in surface position and gap distance. A Keithley 657A electrometer is connected to the sample and tip and can source up to kv DC voltage across the gap. The electrometer can also measure the current flowing through the gap with sub-pa resolution, allowing for measurement of small field emission currents. The whole sample holder can be placed in the vacuum chamber of a SEM which provides a 5 5 mbar vacuum and the possibility to look at the sample surface with high magnification. The SEM we use is an environmental SEM Philips XL3 ESEM-FEG. The parameters of the system are summarized in Table 2.2 and Fig. 2.5 shows an image of the sample holder and a schematic of the setup. This eperimental setup is a development from a previous setup built by T. Muranaka et al. [49]. 24

25 electron beam z W tip y Cu sample Stage holder z R T y Vacuum chamber Figure 2.5. Left: the sample holder and the tungsten tip, driven by the piezo-motors. Middle: a schematic of the setup inside the vacuum chamber of the SEM. Right: the SEM used for the eperiments. Table 2.2. Parameters of the system. Parameter Value Voltage source V Current measurement resolution sub-pa Position control nm Sample diameter 2 mm Tungsten needle radius of curvature 5 µm Vacuum level in SEM 5 5 mbar The current version of the setup has low noise levels of the electrometer and ecellent position control. The strength of this setup is that we can measure locally using the small tungsten tip with only 5 µm radius of curvature together with a precise position and gap distance. Thus only a small part, about µm, of the sample is subject to high electric field. This allows us to observe changes on the surface and compare to surrounding areas not subjected to high electric field. For these eperiments we used copper samples with diameters of 2 mm provided by CERN and with similar treatment to the copper of the CLIC accelerating structures. In order to measure the field enhancement we performed voltage scans at a fied gap of 5 nm. The voltage was ramped in steps of V while monitoring the current, this was repeated times. SEM images before and after shows that a crater was formed during the measurement, see Fig For each scan, we stop the voltage ramp once a threshold current of µa is reached. This enables us to separate the tunneling and pre-breakdown regime from the breakdown regime where local melting occurs on the sample surface. From the resulting I V curves we calculate the field enhancement from a linear fit to (2.3). Figure 2.7 (left figure) shows the maimum voltage and field enhancement for the voltage scans. 25

26 Figure 2.6. Formation of a crater during voltage scans. Left: the surface before the measurements. The rugged-looking ellipsoid on the right is the tungsten tip, here at µm gap distance. Right: a crater was formed during the voltage scans. Maimum voltage [V] Field enhancement β Maimum voltage [V] Vma Fit β Fit Field enhancement β Scan step Scan step Figure 2.7. The maimum voltage reached during the scans and the measured field enhancements. Left: voltage scans. Right: voltage scans. Around scan step 25 there is a jump in maimum voltage and field enhancement. At another location we did a longer eperiment with voltage scans, see Fig. 2.7 (right figure). An interesting feature is clearly visible: at around scan step 25 there is an increase in maimum voltage and at the same time a decrease in field enhancement which suggest that the surface conditions changed. After the eperiment SEM images showed no crater formation. Thus it is possible that during the voltage scan before the increase in maimum voltage, one or several field emitters melted due to high field emission currents. This effect seems similar to conditioning and is appealing since overwhelming statistics from all long-term eperiments at CERN and other labs show that conditioning is dependent on the number of pulses and not the number of breakdowns [53]. The lack of visible surface changes or surface characteristics could also be due to limited resolution of the SEM images. This is not mainly due to the capabilities of the SEM itself but due to the large working distance needed for our setup to fit below the electron column. We have an ongoing eperiment where another high-magnification SEM is used for surface imaging and where we use markers on the surface for orientation. This makes it possible to 26

27 take before and after images with a high-magnification SEM while performing localized field emission measurements with our setup. Furthermore, we are combining our setup for field emission measurements with other surface analysis tools. The rest of this thesis will focus on beam-based diagnostics and beam dynamics. In the net section we will present the test facility at CERN where the other eperiments in this thesis took place. 2.4 CLIC test facility 3 (CTF3) In the 98s CERN initiated an eperimental research and development program for CLIC. The first eperimental facility was the CLIC Test Facility (CTF) [54, 55] which operated between and had the objective of testing a drive beam injector. The net incarnation was CLIC Test Facility 2 (CTF2) [56, 57] where the two-beam acceleration concept was tested for the first time. Finally, a third upgrade of the test facility was proposed [58, 59] with the main objectives to address some key issues for CLIC, in particular the high-intensity drive beam generation and high-gradient two-beam acceleration [36]. CTF3 was in full operation from 29 to 26 at CERN and a recent summary of the results from CTF3 is found in [6]. Figure 2.8 shows the CTF3 layout. CTF3 is essentially a scaled-down version of CLIC with a 2 MeV high-intensity electron beam called the drive beam and a low-intensity 2 MeV electron beam called the probe beam, which corresponds to the CLIC main beam. The two beams are joined in the the CLIC eperimental area (CLEX) where the beamline of the drive beam splits into two beamlines. One beamline is the Test Beamline (TBL) which is a prototype decelerator section where the electrons are decelerated in a series of PETS which allows for the energy etraction and beam stability of the drive beam to be studied [6]. The second beamline in CLEX is the the Two-Beam Test Stand (TBTS) where the drive beam runs in parallel to the probe beam and two-beam acceleration is studied, this will be eplained in more detail in the coming paragraphs. Table 2.3 lists a summary of the CTF3 parameters. Drive beam The drive beam injector consists of a high-current thermionic gun followed by a subharmonic.5 GHz buncher followed by 3 GHz bunchers. These bunchers are RF structures that create the necessary phase coding of the bunches in the drive beam pulse. A section of 3 GHz normal conducting accelerating structures accelerate the electron beam to 2 MeV before entering the recombination section. The overall length of the linac is about 7 m. Recombination of the beam takes place in a 42 m delay loop followed by an 84 m combiner ring where the intensity of the beam is increased by first a factor of 2 and then 27

28 magnetic chicane pulse compression frequency multiplication 3 GHz test stand 5 MeV e linac 3.5 A,.4 μs delay loop combiner ring m photo injector tests and laser CLIC eperimental area (CLEX) with two-beam test stand, probe beam and test beam line total length about 4 m 28 A, 4 ns Figure 2.8. Layout of the CLIC test facility 3. It is a scaled-down version of CLIC with a drive beam that starts in the upper left part, is interleaved in the delay loop and combiner ring, and then entered into CLEX where it runs in parallel to the probe beam. Image source: [36] Table 2.3. Parameters of the two beams at the CLIC test facility CTF3 [36]. Parameter Unit Drive beam Probe beam Energy MeV 2 2 Energy spread (r.m.s.) % 2 Pulse length ns Bunch frequency GHz Bunch charge nc up to Intensity (short pulse) A 28 Intensity (long pulse) A 4.3 Repetition rate Hz an additional factor of 4. The initial pulse of.2 µs and 4 A is recombined to 4 ns and 28 A. For more information on the CTF3 drive beam see [36, 6]. Probe beam The probe beam starts with an injector called Concept d Accélérateur Linéaire pour Faisceau d Electrons Sondes (CALIFES) which is 24 m long linac that accelerates electrons to about 2 MeV. The electron gun uses a photocathode and a laser with pulse length of about 6 ps and.5 GHz repetition rate. The electron bunches are then accelerated in two 3 GHz structures before entering the TBTS in CLEX. The gun, a buncher and the two accelerating structures are all powered by one single klystron. The bunch charge can be varied between.6 and.6 nc and the number of bunches per pulse can range from to 3. The CALIFES injector is eplained in more detail in [62] and Paper XIV. 28

29 Figure 2.9. Layout of the two-beam test stand. The drive beam and the probe beam enter from the right and travel to the left. 2 GHz RF power is generated when the drive beam is decelerated in the two PETS and guided to the four accelerating structures (ACS) mounted on a movable girder. There is an additional PETS before the two-beam module in order to reach nominal power. Various beam diagnostics are available such as beam position monitors (BPMs) and screens. At the end of both beamlines there is a spectrometer leg for beam energy measurement. Image source: Paper XIV. The two-beam test stand (TBTS) The main purpose of TBTS was to demonstrate the CLIC two-beam acceleration scheme. The layout is shown in Fig. 2.9 where the drive beam and probe beam enter from the right and travel in parallel. The drive beam is decelerated in two PETS in the two-beam module and there is also a PETS before that generates additional 2 GHz RF power. This power is fed-forward to the net PETS and the purpose of this is to generate CLIC nominal RF power despite the lower drive beam intensity (28 A compared to the nominal A for CLIC). At the time of the eperiment in Paper I only one PETS and two accelerating structures were installed. At the end of both the probe beam and the drive beam there are spectrometer legs that consist of a dipole magnet that bends the beam, a beam position monitor (BPM), a screen and a beam dump. From the measured bending angle the energy of the beam can be calculated. A BPM is a non-destructive electromagnetic pick-up device that measures the position of the beam centroid. In the probe beam there are also beam profile monitors [63] simply referred to as screens. A beam profile monitor consists of a fluorescent screen that emits light when the electron beam impact on it and this light is recorded by a CCD camera. The beam profile monitor after the two-beam module is the one used in the eperiments of this thesis and it consists of a 4 4 mm YAG screen tilted 45. The CCD camera record a smaller part of the center of the screen, about 9 7 mm, and markers on the screen are used for calibration and compensating the tilt of the screen. An eample of a beam image using this screen was shown previously in Fig

30 The repetition rate of the probe beam can be changed independently of the repetition rate of the drive beam. This is an important fleibility and makes it possible to operate with the repetition rate of the probe beam twice the repetition rate of the drive beam, which is particularly desirable for eperiments where we want to compare accelerated and non-accelerated beams in TBTS. In that case every other pulse of the probe beam will pass through the accelerating structures when there is RF power in the 2 GHz accelerating structures and every other pulse when there is no RF power. This eliminates many systematic errors from slow drifts of the machine. More detailed information about TBTS can be found in [64]. Before we present the beam-based diagnostics methods we need to review the theory of transverse beam dynamics and this is the topic of the net chapter. 3

31 3. Transverse beam dynamics In this chapter we present the fundamentals of beam physics with focus on transverse beam dynamics. The main goal of accelerator beam dynamics is to describe and understand the motion of particles in accelerators. More information on the topic can be found in various tetbooks such as [38, 65 69]. 3. Fundamentals of beam physics In accelerator physics it is convenient to describe an individual particle with respect to an idealized particle traversing an ideal reference orbit. We introduce a co-moving reference frame following this ideal particle and we let s be the independent, time-like variable describing the longitudinal position in the accelerator, see Fig. 3.. In this co-moving frame we describe a generic particle s coordinates as a vector = (,,y,y,τ,δ ) T (3.) where and y are the horizontal and vertical positions with respect to the reference orbit, the angles and y are the transverse momenta normalized to longitudinal momentum, i.e. = p /p z and y = p y /p z. The longitudinal coordinates are epressed as arrival time τ and relative momentum δ with respect to the ideal particle. In this thesis we will mainly focus on the transverse dynamics and then = (,,y,y ) T is an adequate description of a particle. A beam is a collection of particles that we describe as a statistical distribution characterized by its covariance matri, in the contet of accelerator physics, called the beam matri. The 4D transverse beam matri is given by σ σ 2 σ 3 σ 4 σ 2 σ σ y σ y σ = σ 2 σ 22 σ 23 σ 24 σ 3 σ 32 σ 33 σ 34 = σ σ 2 σ y σ y σ y σ y σ 2 y σ yy (3.2) σ 4 σ 42 σ 43 σ 44 σ y σ y σ y y σy 2 Figure 3.. A co-moving reference frame. 3

32 with elements given by second central moments, σ i j = ( i X i )( j X j ) where angle brackets denotes averaging over all particles and i = {,,y,y } for i =,2,3,4. We use capital letters to indicate the first moments, e.g. X i = i. Since σ i j = σ ji the matri is symmetric and elements uniquely describe the full transverse beam matri. The emittance of a beam is proportional to the area in phase space occupied by the particles and is an invariant of the motion in a conservative system. In fact, there is one conserved quantity for each degree of freedom and for the transverse motion we have two degrees of freedom and thus two emittances. For a beam without y correlations the beam matri is block diagonal and can be written as ( ) Σ σ = (3.3) Σ y where Σ and Σ y are 2 2 matrices describing the horizontal and vertical parts of the beam distribution, respectively. In that case the horizontal emittance of the beam can be calculated as ε = det(σ ) = σ 2 σ 2 σ 2 (3.4) and similarly for the vertical emittance ε y. When the transverse beam matri has nonzero y correlations we can still calculate the emittances from the 2 2 blocks, these are called the projected emittances but are not conserved quantities. However, even for the correlated case it is possible to find the two conserved quantities, called eigen-emittances, by eigenvalue decomposition of the beam matri [69]. Multipole kicks A particle with charge q and velocity v in the presence of electric and magnetic fields ( E, B) is subjected to the Lorentz force: ) F = q( E + v B. (3.5) The longitudinal electric field component of the RF fields accelerates and the transverse components of the magnetic field steer and focus the beam. A general transverse magnetic field can be epressed as a multipole epansion that comes from finding solutions to the source-free Mawell s equations [37], i.e. the magnetic field must satisfy B = and B =. If we assume no longitudinal field component (B z = ) and epress the multipole epansion in comple form [69] we can write 32 B y + ib = n= C n ( + iy) n. (3.6)

33 Eplicitly writing the first multipoles we have B y + ib = C }{{} Dipole +C ( + iy) }{{} Quadrupole +C 2 ( 2 y 2 + 2iy) }{{} Setupole +C 3 ( 3 3y 2 + i(3 2 y y 3 )) +C }{{} 4 ( y 2 + y 4 + i4( 3 y y 3 )) +... }{{} Octupole Decapole (3.7) and these are the most common types of transverse magnetic fields in a particle accelerator. Real values of C n correspond to upright magnets and comple values are equivalent to rotations of the field around the beam ais. By convention multipoles with Im(C n ) = are called normal and multipoles with Re(C n ) = are called skew. A true multipole field is assumed to have only one nonzero coefficient in the sum in (3.6). Dipole magnets are used for beam steering and quadrupole magnets are used as magnetic lenses to focus the beam. Since quadrupole fields have a linear dependence on transverse position (, y) the resulting force is linear and the resulting dynamics is linear. Then we have nonlinear fields such as setupoles, octupoles and decapoles, where the field strength depends nonlinearly on transverse position, which in turn results in nonlinear forces and nonlinear dynamics. Figure 3.2 shows the first four multipole fields from (3.7). When a particle traverses a transverse magnetic field the transverse momentum of the particle will change, in particle accelerator contets this change in angles (,y ) is called a kick. The kick can be calculated by integrating (3.5) for the given transverse magnetic field. If the magnetic element is short we can use the thin lens approimation and get = p = evb yl = B yl p z p z v (Bρ) y = p y = evb l p z p z v = B l (Bρ) (3.8) where l is the length of the magnetic element, the product B l is the integrated strength and (Bρ) is the beam rigidity, which is a common way in accelerator physics to epress a particle s momentum. In the case of a quadrupole magnet the thin lens approimation is justified if the length of the quadrupole is short compared to its focal length. If the thin lens approimation does not hold we can calculate the effect of the thick magnetic element by slicing the element in thin lenses separated by drifts, i.e. a drift kick map. In a particle accelerator the particles travel in vacuum pipes with magnets placed at certain longitudinal positions separated by field-free regions called drift spaces, this constitutes the magnetic lattice of the accelerator and defines the dynamics and the beam properties. In order to analyze the dynamics of an accelerator we want to calculate how particles move in the magnetic lattice and this is the topic of the net section. 33

34 B y [arb. units] y y B y [arb. units] - - B y [arb. units] y B y [arb. units] y Figure 3.2. Magnetic multipoles. On the left hand side we show the field lines for dipole, quadrupole, setupole and octupole fields. All the fields are normal multipoles and the field lines start and terminate at surfaces of constant magnetic scalar potential (the solid black lines). On the right we have the vertical magnetic field component as a function of horizontal position, By (), for the corresponding fields at y =. 34

35 Particle tracking The purpose of particle tracking is to calculate the motion of a particle or a beam in an accelerator. Hamiltonian systems, such as a system with static magnetic fields, have an important property: they are symplectic, which means that energy is conserved and that the total phase space volume is constant under the motion (Liouville s theorem). To the describe a particle s motion through a small part of an accelerator we use symplectic transfer maps mathematical functions that relate the particle s coordinates, e.g. (3.), at different positions s. An element s effect on a particle can be described by a map M that maps the initial coordinates i (just before the element) to the final coordinates f (just after the element) according to f = M i. (3.9) The structure of the map depends on the type of element. Linear elements, such as drift spaces and quadrupoles, have linear maps that can be fully characterized by transfer matrices and will be further eplained in the net section. Maps of nonlinear elements, however, such as setupole or octupole magnets, cannot be epressed as matrices and instead we employ Hamiltonians and tools of Lie algebra, this will be covered in Section 3.3. If we want to track through several elements we apply consecutive maps f = M N M N M 2 M i (3.) in order to get the particle s coordinates at the end of the beamline. Several maps can be combined to a single map that can represent a single element, a segment of a beamline or even the whole accelerator commonly referred to as the one-turn map. A Poincaré plot, or a Poincaré surface of a section, is a powerful way to analyze repetitive dynamical systems such as circular accelerators. It is a portrait of multiturn phase space coordinates, for instance, at a single longitudinal position. Information about the motion and analysis of the dynamics are retrieved from the phase space plots of consecutive applications of the one-turn map. 3.2 Linear beam dynamics To first order the beam dynamics of an accelerator is linear and can conveniently be described by linear maps represented by transfer matrices. For an element with transfer matri M, tracking a single particle is simply a matri multiplication with the coordinate vector: f = M i. Using the same transfer matri, we can also propagate a beam by propagating the beam matri describing the particle distribution, it can be shown that σ f = Mσ i M T. (3.) 35

36 The transfer matrices for a drift space of length L and a thin-lens quadrupole magnet with focal length f are L M drift = L M quad = f f where the minus sign in the second row of M quad indicates that the quadrupole is focusing in the horizontal plane but defocusing in the vertical plane. Both these matrices are block-diagonal which means that the motion is uncoupled and we can treat the two planes separately. Furthermore, we note that det(m) = for both of them and this is the case for symplectic maps. In 958 Courant and Snyder demonstrated strong focusing [7], which is the underlying principle of a synchrotron where, despite that quadrupole magnets are focusing in one plane and defocusing in the other, they can still combine to achieve net focusing in both planes. They used a parameterization of the transfer matri that we can apply to each of the two 2 2 blocks according to M = A RA = = ( ) ( ) β cos µ sin µ α β β β sin µ cos µ α β β ( ) (3.2) cos µ + α sin µ β sin µ +α2 β sin µ cos µ α sin µ where α,β are the Courant-Snyder parameters and µ is the phase advance. This parameterization of a linear transfer matri reveals a lot of information about the dynamics. The requirement for the parameterization to work is that we have a stable and periodic system and we find from (3.2) that the trace of M has to be less than two, Tr(M) 2, for the system to be stable. Such a system is characterized by an elliptical fied-point and the particles will trace out ellipses in phase space. In fact, the parameterization can be understood as follows: the matri A is a coordinate transformation that transforms the ellipse to a circle. This coordinate system is called normalized phase space and there the linear motion is fully captured by a simple rotation R with phase advance µ. The final step is to bring the system back to real phase space coordinates by applying the inverse transform A. In normalized phase space (, ) it is convenient to transform into action angle variables (J,ψ) where the action 2J is the radius of the circle and ψ the polar angle. Figure 3.3 shows a schematic of the A-transformation and depicts the action angle variables in normalized phase space. The action 36

37 Figure 3.3. Transformation into normalized phase space brings elliptical trajectories into circles. In normalized phase space it is convenient to epress the particle s coordinates in action angle variables (J,ψ). angle variables are defined as follows = 2J cos(ψ) = 2J sin(ψ) J = ( ) ψ = tan. (3.3) In a conservative, stable, linear system such as the magnetic lattice of an accelerator the action J is an invariant of the motion. In this section we used a map-based approach and used matrices to describe the linear motion. An alternative approach is to solve Hill s equation, which is simply a harmonic oscillator with s-dependent and periodic strength. Tune The tune, denoted Q, of an accelerator is an important concept for many stability issues and it is defined as the normalized phase advance for the one-turn map, Q = µ/2π. The tune is the number of oscillations about the design orbit and for the transverse plane we have two degrees of freedom and hence two tunes. The tune consists of a integer part and a fractional part and latter has impact on stability. For instance, the tune cannot be an integer since a single dipole-like imperfection will lead to secular growth and divergence of the particles. A quadrupolar-like imperfection will lead to secular growth if the fractional part of the tune is 2. For coupled motion problem arises when the sum of difference of the two tunes equal an integer. In general, the resonance condition for the transverse motion can be written as kq + lq y = m (3.4) where k, l and m are integers and the sum k + l gives the order of the resonance. Figure 3.4 plots the resonance condition lines in the tune plane for 37

38 Qy Q Figure 3.4. Tune diagram showing resonance condition lines up to fifth order. all resonances up to fifth order, the lines are points that satisfy (3.4) for k + l 5. The tune of an accelerator should be set to avoid resonances in order to ensure stability. However, not all resonances are harmful and not all resonances are driven by the system. In the Section 3.3 we will introduce the concept of resonance driving terms. Particles with different energy than the ideal particle, are bent differently in the quadrupole magnets and results in different focal lengths as depicted in Fig This change in focal length results in a different tune and means that the tune depends on energy, this effect is called chromaticity. Since a beam of particles unavoidably will have some spread in energy this will result in a spread in tune. To compensate for this potentially harmful feature we can utilize the nonlinear field of setupoles. Figure 3.5 shows how a setupole, placed in a dispersive section, e.g. after a bending dipole where the particles are sorted with respect to their energies, can correct the focal length and the tune. Setupoles for chromaticity correction is one of the most common applications where a nonlinear magnetic field is introduced in the accelerator by design. However, since setupole magnets have nonlinear fields we must also consider their nonlinear effect on the dynamics which is the topic of the net section. 38

39 Figure 3.5. Upper: particles with different energy are focused differently in the quadrupoles leading to an energy-dependent tune, an effect called chromaticity. Lower: by inserting a setupole magnet in a dispersive region, where the particles are sorted with respect to energy, we can compensate the chromaticity. 3.3 Hamiltonians, Lie algebra and normal forms In particle accelerators there are many sources for nonlinear effects such as multipole errors of magnets, e.g. superconducting dipoles with high fields have significant multipole components, for circular colliders there is the beambeam effect: when the beams collide they eperience a highly nonlinear field from the opposing beam. Then there are of course the dedicated magnets with nonlinear fields, put in by design, such as setupoles to compensate chromaticity. Since nonlinear effects in many cases are the limiting factor for the stability of an accelerator it is important to have a good understanding of these effects and for this we need the appropriate analytical tools. Hamiltonians and Hamilton s equations The Hamiltonian formalism is the most natural description of a mechanical system in terms of conserved quantities and resonance theory. Furthermore, systems with an evolution determined by Hamilton s equations are symplectic. More information on Hamiltonian mechanics can be found in [7] and for more information about Hamiltonian formalism and Lie algebra methods in the contet of accelerator physics see [72 74]. A Hamiltonian is a function on phase space coordinates that together with Hamilton s equations yield the equations of motion. Hamilton s equations are 39

40 Table 3.. Hamiltonians for normal and skew multipoles. Multipole normal skew Dipole k k y Quadrupole Setupole Octupole Decapole k 3 l k l 2 (2 y 2 ) k ly k 2 l 3! (3 3y 2 k ) 2 l 3! (32 y y 3 ) k 3 l 4! (43 y 4y 3 ) 4! (4 6 2 y 2 + y 4 ) k 4 l 5! (5 3 y 2 + 5y 4 ) k 4 l 5! (54 y 2 y 3 + y 5 ) conveniently epressed using the Poisson bracket as d ds = [ H,] d ds = [ H, ] (3.5) and similarly for the vertical plane. The Poisson bracket of two functions of phase space coordinates, f and g, is defined as [ f,g] = f g f g + f g y y f g y y. (3.6) A thin-lens magnet can be epressed as a Hamiltonian H = H(,y) and using (3.5) and (3.6) we get d ds = and the thin-lens kick epressed as the Poisson bracket of the Hamiltonian and the angle coordinate, = d ds = [ H, ] = H. (3.7) Table 3. lists the Hamiltonians for common multipoles. Using these Hamiltonians together with (3.7) yield the same kicks as (3.7) together with (3.8) and we identify k nl n! = Re{C nl} (Bρ) and k n l n! = Im{C nl} (Bρ). So far we have only reepressed the multipole kicks in Hamiltonian formalism. Net we will present some powerful methods that allow us to analyze nonlinear maps. Lie formalism The Lie operator is denoted with : on both sides and is defined as the Poisson bracket. The Lie operator f acting on function g is given by : f : g = [ f,g] (3.8) and thus we can epress a nonlinear multipole kick with the Hamiltonian as a Lie operator acting on, we have 4 =: H :. (3.9)

41 The Lie transformation is defined as the eponential of the Lie operator e : f : = n= n! (: f :)n (3.2) and can be thought of as a Taylor map. Powers of the Lie operators results in consecutive Poisson brackets, e.g. (: f :) 2 g = [ f,[ f,g]]. Two important tools in the Lie formalism are the similarity transformation and the Campbell-Baker-Hausdorff (CBH) formula. The similarity transformation allows Hamiltonians to be moved to other locations. Consider the simplest case where we have a Hamiltonian kick followed by a linear map M. We can epress an equivalent map with reversed order, i.e. first a Hamiltonian kick followed by the linear map M, by using the similarity transform according to M = Me : H( ): = Me : H( ): M M (3.2) = e : H(M ): M = e : H( 2): M which in words means that in order to transform the operator (H) we only need to transform the generator ( 2 = M ). If we have two Hamiltonians H A and H B at the same location we can concatenate them into an effective Hamiltonian H using the CBH formula where H = H A + H B + 2 [H A,H B ] + 2 [H A H B,[H A,H B ]] +... (3.22) Note that the order of H A and H B depends on the convention used for order of Lie transformations. Here we use the same convention for Lie transformations as for transfer maps, i.e. they are applied from right to left. By using the similarity transform and CBH formula iteratively the description of a beamline or a full accelerator can be greatly simplified. We can move all the Hamiltonian kicks from the nonlinear elements to a reference point and concatenate all of them into a single effective Hamiltonian, H, comprising a "super-kick" with the combined effect of all the nonlinear elements. In that case we write the one-turn map as a linear map M and a kick: M = e : H: M. Once a map is simplified it can be written in a non-resonant normal form according to M = e : H: R = e : K: e : C: Re :K: (3.23) where we now have assumed that we have transformed into normalized phase space and the linear map is given by a rotation R. The normal form takes into account the effect from iterative applications of the one-turn map which can be though of as self-interactions of the effective Hamiltonian. Consider a system of a single setupole and a rotation R, then the one-turn map, e : H: R, would only contain third-order terms from the third-order terms in the Hamiltonian of the setupole. Only when the multiturn effects are taken into account we see that higher-order terms are generated and that a single setupole, for instance, 4

42 Figure 3.6. Top: description of a circular accelerator as a linear map R and a single "super-kick" epressed as an effective Hamiltonian H. Bottom: a depiction of the normal form as an infinite regression of the circular system as a beamline with selfinteractions of the effective Hamiltonian ' ' Figure 3.7. Left: horizontal phase space for a single setupole followed by a phase rotation. We show the traces for particles with five different starting amplitudes. Right: the same plot after application of e :K: transformation. drives fourth-order resonances and amplitude-dependent tune-shifts, which is an effect in second-order setupole strength. Figure 3.6 shows a schematic of circular accelerator described as a linear map with a super-kick and a depiction of the normal form. The normal form is essentially a generalization of the Courant-Snyder parameterization of nonlinear maps. In (3.2) the matri A transforms the ellipse into an invariant sub-space (normalized phase space) where the map is given by a rotation R and then A transforms back. In (3.23) the Lie transformation e :K: transforms into an invariant sub-space where the nonlinear map is given by e : C: R and finally e : K: transforms back. The map e : C: R contains the invariant part of the map, in other words only action-terms if we epress the map in action angle variables and these terms corresponds to the amplitude-dependent tune-shifts. Figure 3.7 shows the e :K: transform applied to a horizontal phase space plot of a system with a single setupole. 42

43 We can find the normal form of a map by solving (3.23) order by order [75]. First we rewrite (3.23) as e : H: Re : K: R = e : K: e : C: (3.24) and identify Re : K: R as a similarity transformation of K that we can write as a linear transformation, represented by matri S, operating on the coefficients of the Hamiltonian K. We use that Re : K: R = e : SK: and end up with e : H: e : SK: = e : K: e : C: (3.25) which we can solve order by order. We can write the Hamiltonians as sums of the different orders, here keeping up to fifth order, we have H = H (3) + H (4) + H (5) K = K (3) + K (4) + K (5) C = C (3) +C (4) +C (5) SK = S (3) K (3) + S (4) K (4) + S (5) K (5) (3.26) where the (n) superscript denotes order n. Note that the second order in a Hamiltonian results in linear motion and is incorporated in the linear map R. In third order (3.25) becomes e : H(3): e : S(3) K (3): = e : K(3): e : C(3) : (3.27) and we can use CBH to obtain H (3) + S (3) K (3) = K (3) +C (3) + higher orders (3.28) where all the Poisson brackets have been neglected since they generate higher orders. All invariant parts in C occur in even orders and thus C (3) =. Then we can find the third order part of K as K (3) = ( S (3) ) H (3). (3.29) Keeping all terms up to fourth order in (3.25) we get e : H(3) H (4): e : S(3) K (3) S (4) K (4): = e : K(3) K (4): e : C(3) C (4) : (3.3) and again we apply CBH. If we only consider the fourth order terms we get H (4) + S (4) K (4) + [ H (3),S (3) K (3)] = K (4) +C (4) (3.3) 2 where we now need to keep the Poisson bracket of the third order terms since the result is fourth-order terms. We solve for C (4) and K (4) ( S (4) )K (4) +C (4) = H (4) + [ H (3),S (3) K (3)] (3.32) 2 43

44 where in fact ( S (4) ) is not invertible. We resolve this issue by etracting the invariant sub-space from the right hand side which is C (4). Let P J be a projector that projects the part containing only J 2, Jy 2 and J J y, then we have { C (4) = P J H (4) + [ H (3),S (3) K (3)]} (3.33) 2 and the remaining part constitutes K (4). Finally, we can add any Hamiltonian that is a part of the invariant sub-space to K (4) without changing (3.32) and we resolve this ambiguity by setting P J {K (4) } =, which corresponds to "fiing the gauge". This procedure can be repeated to solve for higher and higher orders in an algorithmic way. For the effective Hamiltonian or the Hamiltonian K in the normal form epression of the map, we identify the resonance driving terms by transforming to action angle variables using (3.3). As an eample, the Hamiltonian for a one-dimensional setupole in normalized phase space can be written H = k 2l 6 β 3/2 3 = k 2l 6 β 3/2 (2J) 3/2 cos 3 ψ = k 2l 24 (2Jβ)3/2 cos(3ψ) + k 2l }{{} 8 (2Jβ)3/2 cos(ψ) }{{} driving term for 3Q driving term for Q (3.34) where we identify phase-dependent terms of 3ψ and ψ. Since these perturbations make the system sensitive to tunes 3Q = integer and Q = integer they are called resonance driving terms of 3Q and Q respectively. A few words of caution. Both the Lie transformation and the CBH formula are infinite series that must be truncated when implemented on a computer. This means that the equivalent map is only an approimation of the original composite map. Furthermore, this approimative map might also violate the symplectic condition due to the truncation. Because of this the normal form is primarily used for analysis of the dynamics but element-by-element propagation of particles with symplectic maps remains the most reliable for simulations of long-term behavior. In this section we reviewed the theory of transverse beam dynamics. In Chapter 4 we will investigate how nonlinear kicks from an octupole field can be utilized as a means of beam-based diagnostics and in Chapter 5 we utilize the tools of nonlinear beam dynamics and apply them to two specific dynamical problems. 44

45 4. Beam-based diagnostics with octupoles In this chapter we eplore beam-based diagnostics methods utilizing the copropagating octupole component of the CLIC accelerating structure described in Section 2.2. In Paper I we investigate how to measure the field strength of the octupole component and how the octupole component can be used for measuring the transverse beam matri. In Papers II and III we investigate how the nonlinear kicks from the octupole component can be used as a method for beam-based alignment of the accelerating structures. All eperiments were done at the test facility CTF3 at CERN, which is described in Section Measuring the RF octupole component From (3.7) we retrieve the octupole field and together with (3.8) we can calculate the kicks (, y ). We can epress the shift in position at a distance L downstream from the octupole field as ˆ = L = C 3l (Bρ) L( 3y 2 3) (4.) where ˆ is the horizontal position of the particle perturbed by the octupole field, is the horizontal position of the particle in the accelerating structure and C 3 l is the integrated octupole strength. Individual particle positions are not measurable in a particle accelerator but we can measure the position of the centroid of the beam. The position shift of the centroid of the beam distribution is calculated by taking the epectation of (4.) and we obtain ˆX X = C3 l (Bρ) (3y2 3 ) = C 3l [ 3 y 2 3 ]. (4.2) (Bρ) If we assume a Gaussian beam distribution, the epectation values y 2 and 3 can be calculated analytically. A Gaussian beam is a reasonable assumption for many electron beams and in TBTS the beam from the laser cathode follows a Gaussian distribution to good approimation and the beam profile is preserved during the transport through the short linac. A general epectation value for a multivariate Gaussian distribution can be calculated by taking derivatives of the following generating function: ( m m m N m N = b m m 2 b m m N b m N N ) ep [ 2 b iσ i j b j + b i X i]. bi = (4.3) 45

46 Figure 4.. A schematic of the eperimental setup. The beam (traveling from right to left) is scanned transversely across the accelerating structure, and parallel to the beam ais, by using two steering magnets while observing the beam profile on a screen downstream. For a derivation of this generating function see Appendi A of Paper I. Using (4.3) we can calculate the epectation values in (4.2) and we do the same for the vertical position shift and finally end up with ˆX X = KL [X(Y 2 + σ 2y σ 2 ) X 3 ] 3 + 2Y σ y Ŷ Y = KL [Y (X 2 + σ 2 σ 2y ) Y 3 ] (4.4) 3 + 2Xσ y where we have introduced K = 3C 3 l/(bρ) which is the integrated octupole strength normalized to the beam momentum epressed as beam rigidity (Bρ). Equation (4.4) describes how the beam centroid position shifts at a distance L downstream of an octupole field and it depends on the normalized integrated octupole strength K, beam centroid position inside the octupole field (X,Y ) and transverse beam size (σ 2,σ 2 y,σ y ). Eperiment We tested the method in CTF3 and set up the probe beam as follows. First we assured that the beam was well-aligned so that a minimum number of magnets was needed for propagating the beam straight through the beamline. We used a quadrupole triplet at the very beginning of the probe beam to prepare a slowly convergent beam with a small beam size on the screen, we used the first screen after the accelerating structure (c.f. Fig. 2.9) and no magnetic elements between the accelerating structure and screen were active. Two steering magnets before the accelerating structure were used to move the beam transversely and parallel to the beam ais. Figure 4. shows a schematic of the eperimental setup. By scanning over the RF phase we first measured the on-crest acceleration in order to know the maimum energy gain, which in this case was E = 5. MeV. The phase of the RF was adjusted to the zero-crossing, i.e. where we have the maimum strength of the octupole component. The probe beam was operated at twice the repetition frequency of the drive beam, which means 46

47 Figure 4.2. Sample images from the eperiments. The beam was moved vertically across the face of the accelerating structure. On the left are screen images of the beam when there was no RF in the accelerating structure and on the right the corresponding images when there RF was in the accelerating structure. The beam on the left changes position but not shape. The beam on the right is perturbed by the octupole field and both position and shape change. that every other screen image is an image of the beam when there was RF in the accelerating structure and every other when there was no RF. We moved the beam vertically ±.5 mm across the 4 mm in diameter face of the accelerating structure and at each scan step we collected 6 screen images (8 with RF and 8 without RF). Figure 4.2 shows a few sample images from the eperiment. From the screen images we make Gaussian fits, first one-dimensional Gaussian fits to the projections and then a full two-dimensional Gaussian fit. From the fits the relevant information is etracted, such as beam centroid positions that allows us to calculate the beam position shifts. Since we collect several pulses for each scan step we calculate the average and use the standard de- A video of the screen images can be found in the Supplemental Material of the online version of Paper I: 47