ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2017

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1583

Beam Diagnostics and Dynamics inNonlinear Fields

JIM ÖGREN

ISSN 1651-6214ISBN 978-91-513-0121-1urn:nbn:se:uu:diva-330975

Dissertation presented at Uppsala University to be publicly examined in Polhemsalen,Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 8 December 2017 at 09:15 forthe degree of Doctor of Philosophy. The examination will be conducted in English. Facultyexaminer: Dr. Stephen Peggs (Brookhaven National Laboratory).

AbstractÖgren, J. 2017. Beam Diagnostics and Dynamics in Nonlinear Fields. Digital ComprehensiveSummaries of Uppsala Dissertations from the Faculty of Science and Technology 1583. 87 pp.Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-513-0121-1.

Particle accelerators are indispensable tools for probing matter at the smallest scales and theimprovements 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 theTeV-scale, based at CERN. In such a large accelerator complex, diagnostics and alignmentof the beam are crucial in order to maintain beam quality and luminosity. In this thesis wehave utilized the nonlinear fields from the octupole component of the radio-frequency fieldsin the CLIC accelerating structures for beam-based diagnostics. We have investigated methodswhere the nonlinear position shifts of the beam are used to measure the strength of the octupolecomponent and can also be used for alignment. Furthermore, from the changes in transversebeam profile, due to the nonlinear octupole field, we determine the full transverse beam matrix,which characterizes the transverse distribution of the beam.

In circular accelerators, nonlinear fields result in nonlinear beam dynamics, which oftenbecomes the limiting factor for long-term stability. In theoretical studies and simulations weinvestigate optimum configurations for octupole magnets that compensate amplitude-dependenttune-shifts but avoid driving fourth-order resonances and setups of sextupole magnets to controlindividual 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, Box 516, UppsalaUniversity, SE-751 20 Uppsala, Sweden.

© Jim Ögren 2017

ISSN 1651-6214ISBN 978-91-513-0121-1urn:nbn:se:uu:diva-330975 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-330975)

Nature, it seems, is the popular nameFor milliards and milliards and milliardsOf particles playing their infinite gameOf billiards and billiards and billiards.

—Piet Hein

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Measuring the full transverse beam matrix using a single octupoleJ. Ögren, R. Ruber, V. Ziemann and W. Farabolini,Phys. Rev. ST Accel. Beams, vol. 18, issue 7, 072801, July, 2015.

II Aligning linac accelerating structures using a copropagatingoctupolar modeJ. Ögren and V. Ziemann,Phys. Rev. Accel. Beams, vol. 20, issue 10, 102801, October, 2017.

III Beam-Based Alignment Studies at CTF3 Using the OctupoleComponent of CLIC Accelerating StructuresJ. Ögren, A.K. Bhattacharyya, M. Holz, R. Ruber, V. Ziemann andW. Faraboliniin Proc. 8th Int. Particle Accelerator Conf. (IPAC’17), Copenhagen,Denmark, pp. 371–373, May 2017.

IV Compensating amplitude-dependent tune-shift without drivingfourth-order resonancesJ. Ögren and V. Ziemann,Nuclear Instruments and Methods in Research A, 869:1–9 (2017).

V Optimum resonance control knobs for sextupolesJ. Ögren and V. Ziemann,submitted to Nuclear Instruments and Methods in Research A.

VI Surface Characterization and Field Emission Measurements ofCopper Samples inside a Scanning Electron MicroscopeJ. Ögren, S.H.M. Jafri, K. Leifer, and V. Ziemannin Proc. 7th Int. Particle Accelerator Conf. (IPAC’16), Busan, Korea,pp. 283-285, May 2016.

Reprints were made with permission from the publishers.

List of other publicationsThe following publications are not included in this thesis

VII ELEPHANT: A MATLAB-code for Hamiltonians, Lie algebra, nor-mal form and particle trackingJ. ÖgrenFREIA Report, 2017/09.

VIII An electron energy loss spectrometer based streak camera for timeresolved TEM measurementsH. Ali, J. Eriksson, H. Li, S. H. M. Jafri, S. Kumar, J. Ögren, V. Ziemannand K. Leifer,Ultramicroscopy, vol. 176, pp. 5-10, May, 2017.

IX A Method for Determining the Roll Angle of the CLIC AcceleratingStructures From the Beam Shape Downstream of the StructureJ. Ögren, W. Farabolini, and V. Ziemann,in Proc. 8th Int. Particle Accelerator Conf. (IPAC’17), Copenhagen,Denmark, pp. 368-370, May 2017.

X Wave Propagation in a Fractal Wave GuideA.K. Bhattacharyya, J. Ögren, M. Holz and V. Ziemann,in Proc. 8th Int. Particle Accelerator Conf. (IPAC’17), Copenhagen,Denmark, pp. 3128-3130, May 2017.

XI Dielectric Laser Accelerator Investigation, Setup Substrate Manu-facturing and Investigation of Effects of Laser Induced Electromi-gration RF Cavity Breakdown InfluencesM. Hamberg, E. Vargas Catalan, M. Karlsson, J. Ögren and M. Jacewicz,in Proc. 8th Int. Particle Accelerator Conf. (IPAC’17), Copenhagen,Denmark, pp. 3286-3288, May 2017.

XII Updated baseline for a staged Compact Linear Collideredited by P.N. Burrows, P. Lebrun, L. Linssen, D. Schulte, E. Sicking,S. Stapnes and M.A. Thomson,CERN-2016-004, Geneva, Switzerland, 2016.

XIII Beam-based Alignment of CLIC Accelerating Structures UtilizingTheir Octupole ComponentJ. Ögren and V. Ziemann,in Proc. 7th Int. Particle Accelerator Conf. (IPAC’16), Busan, Korea,pp. 280-282, May 2016.

XIV CALIFES: A Multi-Purpose Electron Beam for Accelerator Tech-nology TestsJ. L. Navarro et al. (14 authors),in Proc. 27th Linear Accelerator Conf. (LINAC14), Geneva, Switzer-land, Sept., 2014.

XV The momentum distribution of the decelerated drive beam in CLICand the two-beam test stand at CTF3Ch. Borgmann, M. Jacewicz, J. Ögren, M. Olvegård, R. Ruber andV. Ziemannin Proc. 5th Int. Particle Accelerator Conf. (IPAC’14), Dresden, Ger-many, pp. 62-64, May 2014.

XVI General-purpose spectrometer for vacuum breakdown diagnosticsfor the 12 GHz test stand at CERNM. Jacewicz, Ch. Borgmann, J. Ögren, R. Ruber and V. Ziemann,in Proc. 5th Int. Particle Accelerator Conf. (IPAC’14), Dresden, Ger-many, pp. 3668-3670, May 2014.

XVII Recent results from CTF3 two-beam test standW. Farabolini, et al. (15 authors),in Proc. 5th Int. Particle Accelerator Conf. (IPAC’14), Dresden, Ger-many, pp. 1880-1882, May 2014.

XVIII A Fully-levitated Cone-shaped Lorentz-type Self-bearing Machinewith Skewed WindingsJ. Abrahamsson, J. Ögren and M. Hedlund,Magnetics, IEEE Transactions on, vol. 50, no. 9, pp. 1-9, Sept., 2014.

The author’s contribution to the papersPaper I: I did the derivations, built the simulation model and did all the simu-lations. I wrote the data acquisition scripts for the experiment and performedthe measurements together with WF. I analyzed the data and wrote the major-ity of the paper.

Paper II: I did the derivations, wrote the simulation code and did all the sim-ulations. The paper was written by me in close collaboration with VZ.

Paper III: The experiment was a joint effort between AKB, MH, WF andmyself. 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, Liealgebra methods and normal forms. The analysis and writing of the paper wasdone in collaboration.

Paper V: The idea was based on earlier work by VZ and developed as a jointeffort. I derived the main results and did all the simulations. I wrote the bulkof the paper.

Paper VI: I wrote a LabView-based control system for interfacing all hard-ware and developed automatized measurement procedures. I did all the mea-surements together with SHMJ who helped in particular with operation of thescanning electron microscope. The data analysis and interpretation of the re-sults were done in collaboration but the paper was written by me.

Cover illustrationThe illustration on the cover shows a Poincaré surface of a section for an oc-tupole followed by a rotation with tune Q = 0.269. The plot is generated bythe author in MATLAB and displays x–x′ phase space for thirteen particleswith different starting amplitudes tracked for 2000 turns.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Physics motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Particle colliders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Storage rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 The Compact Linear Collider (CLIC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 The main beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Vacuum breakdown studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 CLIC test facility 3 (CTF3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Transverse beam dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1 Fundamentals of beam physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Linear beam dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Hamiltonians, Lie algebra and normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Beam-based diagnostics with octupoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Measuring the RF octupole component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Measuring the beam matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Beam-based alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Beam dynamics in nonlinear fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.1 Compensating amplitude-dependent tune-shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 Optimum resonances control knobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

1. Introduction

This thesis includes experimental and theoretical work on particle acceleratorsand beam physics. Before presenting the research I will give a brief introduc-tion to the field of accelerator-based science and outline the motivations forbuilding and operating particle accelerators.

1.1 Physics motivationIn the beginning of the last century Ernest Rutherford used a beam of alphaparticles emitted from radioactive nuclei for studying scattering in a thin goldfoil. The results were not consistent with the theory of the atom at the timesince some of the particles scattered at very large angles. From the resultsof the experiment Rutherford rejected Thomson’s "plum-model" and insteadconcluded that the atom is made up of a small, heavy nucleus surrounded bylighter electrons [1]. This was one of the very first experiments where a beamof charged particles was used as a means of studying the matter on atomic andsubatomic level. However, a beam from a radioactive source is not controllableand achieving high intensity is problematic. Thus the idea of constructing amachine to generate a particle beam, i.e. a particle accelerator, was born.

The pioneering work on accelerator physics started in the 1920s with Gus-tav Ising inventing the electron drift tube later built by Rolf Widerøe. In 1929Robert J. Van de Graaff invented a high-voltage generator and an electro-static accelerator. Ernest Lawrence, inspired by the works of Widerøe, in-vented the first cyclotron in 1932 which could accelerate protons to energiesabove 1 MeV [2]. Since then, particle accelerators have been an indispensablepart of experimental physics and over time higher and higher energy togetherwith higher and higher intensity was achieved. Higher energy meant probingsmaller length-scales and exploring new territories and higher intensity meantfaster statistics. For the past century, progress in nuclear and particle physicshas been closely linked to progress in accelerator physics. A prime exampleof this is the development of the Standard Model of particle physics.

The Standard ModelThe Standard Model of particle physics [3–5] is a quantum field theory thatdescribes matter and energy as kinematics and interactions of fundamental

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particles. The model consists of matter particles that are fermions and force-carrying particles that are bosons. The fermions come in two families offundamental particles—quarks and leptons—and each type of fermion comesin three generations. Quarks can combine in doublets to form mesons or intriplets to form baryons, such as protons and neutrons, which make up the nu-clei of atoms and constitute most of the visible matter in the universe. Theother piece of the atom is the electron, which is a lepton—a fundamental par-ticle with no known internal structure. The photon is a massless boson thatmediates the electromagnetic interaction. Then there are the short-ranged in-teractions: the strong nuclear interaction mediated by gluons and the weaknuclear interaction mediated by the W± and Z0 bosons.

When electromagnetism and the weak nuclear force were unified into theelectroweak theory there was a problem with the theory since it predicted someknown massive particles, i.e. the W± and Z0 bosons, to be massless. The res-olution was to introduce a new scalar field that causes breaking of the sym-metry and as a consequence some of the fundamental particles would acquiremass [6–9]. The field is named the Higgs field after one of the predictorsand the corresponding mechanism is named the Higgs mechanism. A quan-tum excitation of the Higgs field results in yet another particle—the Higgsboson—which was the final piece of the Standard Model to be discovered in2012. The particles of the Standard Models are shown in Fig. 1.1 with thethree generations of leptons and quarks on the left and the bosons on the right.

Beyond the Standard ModelEven though the Standard Model is currently the best theory that describes theuniverse at the fundamental level we know it to be incomplete. For one thing,the Standard Model only incorporates three out of the four known fundamentalforces of nature: electromagnetic, strong nuclear and weak nuclear. Gravity isstill not included in the Standard Model.

There are several experimental results that lie outside the realm of the Stan-dard Model. Astronomical observations of galaxy rotational curves [11], thecosmic microwave background [12] and weak gravitational lensing [13] allsuggest that ordinary, visible matter cannot provide enough mass for the ob-served effects. This unknown, non-luminous matter is called dark matter andcan not be accounted for by the Standard Model. Furthermore, in the 1990sthere were astronomical observations showing that the expansion of the uni-verse is in fact speeding up. What drives this acceleration of the expansionis not known and commonly labeled dark energy [14–16]. It is estimated thatonly 5% of the energy and matter in the universe is ordinary, visible matterthat can be accounted for by Standard Model, the rest is dark matter (27%)and dark energy (68%) [12].

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Figure 1.1. Elementary particles of the Standard Model. The matter particles consistof three generations of quarks and three generations of leptons. The force-carryingbosons are shown on the right. Image credit: [10]

Another mismatch between theory and experiment is that neutrinos aremassless in the Standard Model but we know from the observation of neutrinooscillations [17, 18] that the neutrinos must have nonzero mass. Furthermore,in the Standard Model each particle has an antiparticle and it is not clear whythere is only matter and no antimatter in the universe when equal amounts ofthe two should have been created in the Big Bang. The Standard Model cannotexplain the puzzle of the matter/antimatter asymmetry. Finally, there is alsoa hierarchy problem in the Standard Model. In essence the question is whythe electroweak force is so many orders of magnitude stronger than gravity.For this to be compatible with the Standard Model fine-tuned cancellations ofcorrection terms are required and this is often regarded as problematic since itseems unnatural.

An attempt to resolve the hierarchy problem and potentially many of theother issues stated above is the theory of supersymmetry (SUSY) which isan extension of the Standard Model based on additional symmetries [19, 20].SUSY models predict many additional particles to exist [21], none of whichhave been found experimentally. There are also other theories that address theshortcomings of the Standard Model such as String theory and extra dimen-sions. So far, theories beyond the Standard Model lack empirical evidence.Ultimately, experimental observations must guide us forward.

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1.2 Particle collidersModels of particle physics are tested in a controllable way by acceleratingand colliding particles in an accelerator. The high energy density in the colli-sion leads to the creation of new particles. The new particles and their decayproducts are measured and identified in a detector and in this way interactionsbetween fundamental particles are studied and compared to predictions of atheory.

Testing the Standard ModelDespite the problems with the Standard Model described above, the theoryhas been very successful and stood by many tests over the years. Many par-ticles that were predicted by the theory were later confirmed experimentally.In 1968 the first evidence of the existence of quarks was found from deep in-elastic electron–proton collisions at SLAC [22,23]. The predicted electroweakbosons W± and Z0 were discovered in 1983 [24–26] from proton–antiprotoncollisions at the Super Proton Synchrotron (SPS) at the European organizationfor nuclear research (CERN). This was followed by precision measurementsof the Z0 boson from electron–positron collisions at SLAC Linear Collider(SLC) and the Large Electron-Positron Collider (LEP) where the existence ofa fourth generation of leptons was ruled out [27]. The heaviest of the quarks,the top quark, was discovered at the Tevatron accelerator at Fermi NationalLaboratory in 1995 [28].

The Higgs boson was the final particle of the Standard Model to be foundexperimentally. The search for the Higgs boson was the main motivator forbuilding the Large Hadron Collider (LHC) in the existing LEP tunnel at CERN.LHC is to date the highest energy particle accelerator ever built where protonsare accelerated to 6.5 TeV—a million times the energy of Lawrence’s first cy-clotron in the 1930s. In 2012 it was announced that a 125 GeV Higgs-likeparticle had been discovered at the CMS and ATLAS experiments at the LHC[29, 30]. So far are all the measured properties of this particle consistent withthose of the Standard Model Higgs boson [31].

Linear lepton collidersWith the Tevatron and LHC the energy frontier on the TeV-scale was opened.However, there has been consensus in the particle physics community that dis-coveries at the LHC need to be complemented by precision measurements ata lepton collider on the same energy-scale [32]. Hadrons, such as protons, arecomposite particles and when they collide it is really their constituents—thequarks and the gluons—that interact and this leads to a myriad of interactionsand production of particles. Leptons, e.g. electrons and positrons, on theother hand are fundamental, point-like particles. In a lepton collider the initial

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state of the collision is well-defined and the outcome of the collision is muchcleaner and more easily interpreted, which makes lepton colliders much moresuitable for precision measurements. Things to be studied in a high-energylepton collider are precision measurements on the Higgs boson and the topquark (guaranteed program) and various SUSY particles (potential program).

Currently there are two international collaborations studying linear high-energy electron–positron colliders: the International Linear Collider (ILC)[33] and the Compact Linear Collider (CLIC) [34]. ILC is based on super-conducting acceleration cavities [35] and proposed to be built in Japan. CLICon the other hand is based on normal-conducting technology [36] and will beexplained further in the next chapter. One of the challenges with a high-energylinear collider is to preserve beam quality over a very long accelerator and thismakes beam diagnostics crucial, which is one of the topics of this thesis.

Why a linear accelerator? When a charge particle is accelerated perpen-dicular to its velocity it emits electromagnetic radiation, which is called syn-chrotron radiation [37]. This means that particles in a circular accelerator losesome of their energy as they are bent around the ring by magnets. The energyloss per turn, ∆E, for a particle is proportional to the fourth power of the ratioof its energy E and mass m and inversely proportional to the bending radius ρ

of the ring [38], i.e. ∆E ∝ E4/(m4ρ). Since electrons and positrons have about1/2000 of the mass of the proton they are much more subjected to energy lossdue to synchrotron radiation. This is the reason the electrons and positrons inLEP only reached 104.5 GeV while the protons in LHC reach 6.5 TeV despiteboth being accelerators in the same tunnel with a circumference of 27 km.Building a circular electron–positron collider on the TeV-scale would result ina machine of gigantic proportions and hence a linear machine is the only sen-sible option. However, a circular accelerator is still needed as a part of a linearcollider and used as a damping ring, which has the purpose of enhancing thebeam quality.

1.3 Storage ringsA circular particle accelerator with the purpose of storing a circulating beamof particles is known as a storage ring. In linear colliders they are used asdamping rings where the effect of synchrotron radiation is utilized to dampthe oscillations of the particles. This enhances the quality of the beam andis essential for achieving the required small beam size. A related applicationis to use a storage ring as a dedicated synchrotron light source, such as theMAX IV Laboratory [39] in Lund, where high-brightness beams of photonsemitted from a stored electron beam are guided through beamlines and utilizedfor studying condensed matter physics, material science and biology. Storagerings are also used as circular colliders such as previously mentioned LHC,

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or the proposed Future Circular Collider (FCC), where two beams of particlesare circulated in opposite directions, stored and collided.

One of the challenges with iterative systems such as storage rings is beamstability. 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 needsto 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 requiresstrong focusing which in turn increases sensitivity to nonlinear effects. Thusunderstanding and dealing with nonlinear dynamics is essential for enhancingperformance of future storage rings.

Particle accelerators have many different applications but the improvementand development of such machines are dependent on the progress and under-standing of accelerator physics. A particle accelerator is a complicated systemof thousands of components with the challenges to maintain, diagnose andcontrol an ultra-relativistic plasma under extreme conditions. Making beamswith smaller and smaller beams sizes, or higher and higher energies, requiresadditional understanding of the physics of beams. This thesis contains re-search that deals with the diagnostics and dynamics of beams in the presenceof nonlinear fields.

1.4 Thesis structureThe thesis is structured in the following way: Chapter 2 introduces CLIC andthe CLIC Test Facility 3 (CTF3) where the experiments reported in this the-sis were conducted. In chapter 3 we review the theory of linear and nonlin-ear transverse beam dynamics. In Chapter 4 we introduce some methods forbeam-based diagnostics and summarize the most important results from Pa-pers I–III. Finally in Chapter 5 we apply the tools of nonlinear beam dynam-ics to circular accelerators and address tune-shift compensation and resonancecontrol knobs, these are the topics of Papers IV and V.

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2. The Compact Linear Collider (CLIC)

CLIC is a proposed linear electron–positron collider on the TeV energy-scalebased at CERN. The CLIC study is carried out by an international collabora-tion and aims to provide a suitable complement to the LHC. In 2012, beforethe announcement of the discovery of the Higgs boson, the collaboration pub-lished a conceptual design report [36]. An updated baseline design was pub-lished in 2016 [40] (which is also Paper XII) where the first energy stage of380 GeV center-mass-energy is optimized for physics on the Higgs boson andthe top quark. After the initial energy stage it is possible to upgrade CLIC to asecond stage of 1.5 TeV and a third stage of 3 TeV center-of-mass energy forphysics beyond the Standard Model.

2.1 LayoutFigure 2.1 shows a schematic of the 3 TeV CLIC layout that consists of twoaccelerator complexes: the main beam and the drive beam. At the bottom ofFig. 2.1 we have the electron and positron injectors for the main beam. Bothinjectors consist of a particle source and a linear accelerator (linac) that ac-celerates the beam to 2.86 GeV before injecting the beam into the dampingrings. The purpose of the damping rings [41] is to reduce the emittance of theelectron and positron beams prior to acceleration in the main linac. Emittancewill be defined in the next chapter but it can be thought of as a measure ofbeam quality. After the damping rings, a booster linac accelerates the beamsto 9 GeV before they are transported to the far ends of the main linac. Theelectrons are accelerated from one end and the positrons from the other andeach linac brings the particles from 9 GeV to 1.5 TeV by transferring energyfrom radio-frequency (RF) fields to the particles in tens of thousands accel-erating structures. When the beams reach the center, the final focus systemfocuses the beams to nanometer sizes before they collide inside the detector atthe interaction point. After collision the beams are transported to beam dumpswhere the energy is dissipated.

The upper part of Fig. 2.1 shows the drive beam complex which is a high-intensity electron beam with the purpose to create RF power for accelerationof the high-energy main beam. In the 3 TeV energy stage two drive-beamaccelerators are needed. The high intensity of the drive beam is generatedby successive recombinations of a 2.4 GeV electron beam. The beam fromthe drive beam accelerator consists of a long bunch train (148 µs) and a low

17

Figure 2.1. The CLIC layout for 3 TeV center-of-mass energy. Lower part: electronand positron injectors followed by the damping rings. The the beams are acceleratedin the main linac and collided at the interaction point. Upper part: the drive beamcomplex. The two 2.4 GeV beams are interleaved in delay loops and combiner ringsin order to reduce the bunch spacing and increase the intensity. The RF power is thenextracted in the decelerator sectors and guided to the accelerating structures in themain linacs. Image source: [40]

bunch frequency (0.5 GHz), which gives an average beam current of only4.2 A. Then the beam enters the recombination complex where the bunchesin the sub-trains are interleaved in a delay loop and two subsequent combinerrings. This results in a factor 24 shortening of the total pulse length, due tothe 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 beamcurrent of 100 A and 12 GHz bunch frequency and this high-intensity beamis then sent to the decelerator sections for RF power generation. The keyparameters for CLIC are summarized in Table 2.1.

The ’Compact’ in the name of the Compact Linear Collider comes from theusage of high-gradient, normal-conducting acceleration structures. The designaccelerating gradient of CLIC is 100 MV/m, which is considerably higher thanwhat can be achieved with super-conducting technology. Since it is a linearaccelerator and the particles must be accelerated in a single pass, about 140000accelerating structures are needed to reach 3 TeV center-of-mass energy andthis results in a 50 km long tunnel for the main linac.

The luminosity is a measure of the number of collisions per unit area andtime. Achieving high luminosity is the main objective of any particle collidersince higher luminosity means higher event rates and faster statistics. Theluminosity L [36, 42] is given by

L = HDN2

σxσynb frep (2.1)

18

Table 2.1. A summary of parameters for CLIC at the 3 TeV energy stage [40].

Parameter Symbol Unit Value

Center-of-mass energy√

s TeV 3Repetition frequency frep Hz 50Pulse length τRF ns 244Number of bunches per train nb 312Number of particles per bunch N 3.7×109

Bunch separation ∆t ns 0.5Bunch length σz µm 44

Luminosity L cm−2s−1 2×1034

Beam size at interaction point σx/σy nm 40/1Normalized emittance (end of linac) εx/εy nm 660/20

Accelerating gradient G MV/m 100RF frequency fRF GHz 12Length of main tunnel km 50.1Total number of accelerating structures 143328Estimated power consumption Pwall MW 589

where σx,y is the transverse beam size at the interaction point, N is the numberof particles per bunch, nb is the number of bunches per pulse, frep is the rep-etition rate, i.e. the number of pulses per second. HD is a correction factor toaccount for the self-focusing effect during the collision. The target luminosityfor CLIC is 2× 1034 cm−2s−1. For a circular collider the circulating beamscan be used for collisions again and again which gives a large repetition ratefrep. But a linear collider is a single-pass machine and means that every par-ticle bunch is used only once and limits frep, which for CLIC is set to 50 Hz.The way to compensate for the low repetition rate and still achieve high lu-minosity is to squeeze the beam size (σx,σy) down to nanometer size at theinteraction point.

Two-beam acceleration schemeThe CLIC acceleration concept is based on a two-beam acceleration schemewhere RF power is generated by the high-intensity electron beam (the drivebeam) that is decelerated in a Power Extraction and Transfer Structure (PETS).The decelerator sections contain several PETS and runs in parallel with themain beam. The PETS is a structure resonant at 12 GHz that transforms thekinetic energy of the drive beam into 12 GHz RF power. It is a traveling wavestructure made of copper cells with spacing adapted to 12 GHz. When anelectron bunch of the drive beam passes through a PETS it excites a wakefieldand the succeeding bunches are decelerated by this wakefield and in this wayenergy is transferred from the beam to the RF fields. Every bunch of the drivebeam travels through a sector of 1492 PETS over a distance of about 1 km

19

Figure 2.2. Two-beam acceleration scheme. 12 GHz RF power is generated when theelectrons in the drive beam are decelerated in a series of PETS. This power is thentransported to the acceleration structures where the main beam is accelerated. Imagesource: [36]

and approximately 90% of the energy is extracted. The generated RF poweris transferred from the PETS to the main beam via waveguides. Figure 2.2shows a schematic of the two-beam acceleration scheme.

A different option is to use a more conventional method for generating RFpower with klystrons. Klystrons are RF power amplifiers that take a high-power DC pulse and amplify a low power RF signal. In CLIC the injectorsand drive beam accelerators use klystrons but not the main linac and the mo-tivation for using two-beam acceleration comes from the large number of ac-celerator structures required. For the 3 TeV stage it is estimated that about35000 klystrons would be needed and in that case the two-beam accelerationscheme is much more efficient and cost effective [36]. However, for the firstenergy stage of 380 GeV it could be more efficient to use klystrons [40]. Us-ing klystrons is a way to distribute the RF power generation over many unitswhereas the drive beam is one single large RF power generator.

2.2 The main beamThe main beam of CLIC consists of two identical accelerators, one for elec-trons and one for positrons, with the purpose of bringing the particles to highenergy before collision. The particles are accelerated in normal-conducting,100 MV/m acceleration structures made of copper and since they are normal-conducting, power will be lost through resistive wall heating. Therefore, inorder 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 separa-tion which for CLIC is set to 0.5 ns. But the short bunch separation causesproblems since the bunches can interact with each other, i.e. wakefields from

20

bunches in the head of the train disrupt trailing bunches. This is highly unde-sirable since it leads to emittance growth and luminosity reduction.

To avoid issues with wakefields the beam must be well-aligned in the mainlinac. Alignment is a challenging part for CLIC and must be kept on the orderof a few micrometer to maintain the design luminosity. The most sensitivepart is the accelerating structures due to their small irises. In CLIC severalaccelerating structures (2 to 8) will be mounted in a CLIC module togetherwith the corresponding number of PETS (one PETS powers two acceleratingstructures). The accelerating structures in a module are mounted on movablegirders that allow for aligning the beam with respect to the accelerating struc-ture by means of moving the accelerating structures themselves. In order tomeasure the transverse position of the beam with respect to the acceleratingstructure wakefield monitors [43] are installed in every second structure. Thewakefield monitors measure one of the higher order modes, i.e. modes ofhigher frequencies, that is excited by transverse offsets of the beam. In Pa-pers II and III we investigate a complementary method for beam-based align-ment.

The accelerating structureFigure 2.3 shows a CLIC accelerating structure which is about 25 cm long andconsists of 28 cells. The RF power is fed into the accelerating structure via theinput coupler cell and travels through the structure and exit through the outputcoupler where the remaining RF power is absorbed in an RF load. The beamis timed with respect to the RF fields in such a way that it sees a longitudinalelectric field and is accelerated in the forward direction.

Figure 2.3 also shows a cross-section of a single accelerating cell wherewe see the cell itself, the small iris and four radial waveguides. The purposeof these waveguides is to reduce wakefields by damping higher order modes.The apertures of these transverse waveguides are designed in such a way thathigher order modes are transmitted and absorbed in RF absorbing materialsbut the fundamental mode of 12 GHz remains unaffected, essentially it is ahigh-pass filter with cut-off frequency above 12 GHz. The purpose is again tomitigate emittance growth due to wakefields.

Octupole componentThe four-fold symmetry from the radial waveguides of the accelerating cell inFig. 2.3 allows for an octupole component of the RF fields. This is a multipolarcomponent with fundamental frequency 12 GHz that co-propagates with themain accelerating field. Even though this octupole component has the samefrequency, it is phase-shifted 90◦ with respect to the main accelerating field.The octupole component is known from simulations [44] and experimental

21

Walter Wuensch, CERN

Figure 2.3. Left: the CLIC accelerating structure consist of 26 regular cells plus oneinput coupler cell and one output coupler cell. Image source: [36]. Right: a singlecell of the CLIC accelerating structure. Centered is the iris where the beam travelsthrough. There are four radial waveguides connected to each cell for damping higher-order modes. Image courtesy of Walter Wuensch, CERN.

observations, see Paper XVII. The effect on the beam is equivalent to the effectof an octupole magnet. Figure 2.4 shows a screen image from CTF3 of a beamperturbed by the octupole component in a CLIC accelerating structure.

Since the octupole component is phase-shifted with 90◦ with respect to themain accelerating field the effect on the beam is small during normal on-crestacceleration. There are also other multipolar components of the RF fields inthe CLIC accelerating structures, such as a quadrupole component from thetwo-fold symmetry of the input and output coupling cells. The effects on thebeam from the multipolar components have been studied and found to be neg-ligible under normal operation [45]. However, in Papers I-III we investigatemethods to utilize the octupole component for diagnostics purposes.

Before discussing the beam-based diagnostics we will make a small di-gression to Paper VI and address one of the limiting factors for high-gradientacceleration: vacuum breakdowns.

2.3 Vacuum breakdown studiesThe high electric fields in the acceleration structures, due to the high gradient,might induce electron emission from the metallic surface that can lead to anavalanche process where neutrals are ionized and form a plasma. This createsan arc, i.e. a conducting medium in the form of a plasma, in the otherwiseinsulating vacuum, and the power of the RF fields is dissipated into the wallof the structure. This collapse of the RF fields in the accelerating structure iscalled a breakdown, or a discharge, and constitutes a big challenge for CLIC.Breakdowns cause power loss which means that the beam is accelerated lessbut they can also be harmful to the accelerating structures themselves.

If the beam passes through the accelerating structure during a breakdownit will receive a transverse kick due to the magnetic field created by a high

22

x [mm]

0 2 4 6 8

y [m

m]

0

1

2

3

4

5

6

Figure 2.4. Transverse beam profile of a beam perturbed by the octupole componentof the RF fields in a CLIC accelerating structure. Image source: Paper IX.

breakdown current [46]. This makes RF breakdowns problematic for CLICsince these transverse momentum kicks cause misalignments of the beam andlead to luminosity reduction. Breakdown rate is defined as the number ofbreakdowns per pulse and for CLIC the maximum limit of breakdown rateis set to 10−7. This tight requirement is to ensure less than 1% luminosityreduction due to breakdowns [36].

In order to test how the accelerating structures perform in terms of break-down rate there is a lot of experimental activity in several dedicated RF test-stands at CERN where CLIC accelerating structures are tested over longerperiods of time. In the beginning of operation the structures have too highbreakdown rates at nominal RF pulse length and power so in order to protectthe structures they are operated at reduced pulse length and power. But overtime the breakdown rate decreases, a process called conditioning, and thenthe pulse length and power are slowly increased until finally reaching nominalvalues at the required breakdown rate. In addition there are also a number ofsmaller direct current (DC) setups with the benefit of generating similar fieldstrengths but with much simpler infrastructure and faster repetition rates. Formore information on the recent progress of high-gradient structures see [47].

Although there has been much progress in understanding breakdowns thereare parts of the phenomenon that are not fully understood. It is generallybelieved that the onset of a breakdown is due to electron field emission [48].To address the more fundamental aspects of vacuum breakdowns and fieldemission in particular, we have an Uppsala-based experimental setup [49] thatcan be operated in-situ a scanning electron microscope (SEM). This is thetopic of Paper VI and will be briefly presented here.

23

Field emissionVacuum discharges are initiated by emission of electrons under influence ofhigh electric fields, this is called field emission. The theory of field emissiondates back to 1928 when Fowler and Nordheim described how electrons canescape from a cold cathode by quantum tunneling through the potential bar-rier [50]. The Fowler–Nordheim equation gives the current density of electronemission as a function of applied electric field. Written in a commonly usedform [51, 52] we have

I = Ae1.54×106β 2E2

φe10.41φ−1/2× e−6.53×103φ3/2/βE (2.2)

where the emitted current I depends on the emission area Ae, work functionφ , applied electric field E and a parameter β called the field enhancement.In the original Fowler–Nordheim equation there was a large discrepancy be-tween the required field strengths from the theory compared with experimentswhere field emission started at much lower field strengths. This is due to theidealized conditions of the theory, which assumes a perfect surface. The fieldenhancement was introduced as a quantification of the local enhancement ofthe electric field due to protrusions or other surface imperfections. We canrewrite (2.2) as

ln{

IE2

}= m− k

E(2.3)

which describes a straight line in the coordinates ln{

IE2

}and 1

E . Thus we canplot data from measurements in these coordinates and from the slope k of theline we determine the field enhancement as β = 6.53×103φ3/2

k .

Experimental setupThe setup at Uppsala University consists of a movable sample holder and atungsten tip above the sample. Both the sample holder and the holder for thetip are controlled by piezo-motors with position sensors and a feedback con-troller that allows for nanometer precision in surface position and gap distance.A Keithley 6517A electrometer is connected to the sample and tip and cansource up to 1 kV DC voltage across the gap. The electrometer can also mea-sure the current flowing through the gap with sub-pA resolution, allowing formeasurement of small field emission currents. The whole sample holder canbe placed in the vacuum chamber of a SEM which provides a 5×10−5 mbarvacuum and the possibility to look at the sample surface with high magnifica-tion. The SEM we use is an environmental SEM Philips XL30 ESEM-FEG.The parameters of the system are summarized in Table 2.2 and Fig. 2.5 showsan image of the sample holder and a schematic of the setup. This experimentalsetup is a development from a previous setup built by T. Muranaka et al. [49].

24

Vacuum chamber

electron beam

x

y

z

T

R

Stage holder

x

yz

W tip

Cu sample

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: theSEM used for the experiments.

Table 2.2. Parameters of the system.

Parameter Value

Voltage source 0–1000VCurrent measurement resolution sub-pAPosition control 1 nmSample diameter 12 mmTungsten needle radius of curvature 5 µmVacuum level in SEM 5×10−5 mBar

The current version of the setup has low noise levels of the electrometer andexcellent position control. The strength of this setup is that we can measure lo-cally using the small tungsten tip with only 5 µm radius of curvature togetherwith a precise position and gap distance. Thus only a small part, about 10×10µm, of the sample is subject to high electric field. This allows us to observechanges on the surface and compare to surrounding areas not subjected to highelectric field.

For these experiments we used copper samples with diameters of 12 mmprovided by CERN and with similar treatment to the copper of the CLIC ac-celerating structures. In order to measure the field enhancement we performedvoltage scans at a fixed gap of 500 nm. The voltage was ramped in steps of1 V while monitoring the current, this was repeated 10 times. SEM imagesbefore and after shows that a crater was formed during the measurement, seeFig. 2.6. For each scan, we stop the voltage ramp once a threshold current of1 µA is reached. This enables us to separate the tunneling and pre-breakdownregime from the breakdown regime where local melting occurs on the samplesurface. From the resulting I–V curves we calculate the field enhancementfrom a linear fit to (2.3). Figure 2.7 (left figure) shows the maximum voltageand field enhancement for the 10 voltage scans.

25

Figure 2.6. Formation of a crater during voltage scans. Left: the surface before themeasurements. The rugged-looking ellipsoid on the right is the tungsten tip, here at10 µm gap distance. Right: a crater was formed during the voltage scans.

Scan step0 20 40 60 80 100

Ma

xim

um

vo

lta

ge

[V

]

0

100

200

300

400

Fie

ld e

nh

an

ce

me

ntβ

0

5

10

15

20

25

30VmaxFitβFit

Scan step

2 4 6 8 10

Ma

xim

um

vo

lta

ge

[V

]

0

50

100

150

200

250

Fie

ld e

nh

an

ce

me

nt

β

0

10

20

30

40

50

60

70

Figure 2.7. The maximum voltage reached during the scans and the measured fieldenhancements. Left: 10 voltage scans. Right: 100 voltage scans. Around scan step 25there is a jump in maximum voltage and field enhancement.

At another location we did a longer experiment with 100 voltage scans, seeFig. 2.7 (right figure). An interesting feature is clearly visible: at around scanstep 25 there is an increase in maximum voltage and at the same time a de-crease in field enhancement which suggest that the surface conditions changed.After the experiment SEM images showed no crater formation. Thus it is pos-sible that during the voltage scan before the increase in maximum voltage, oneor several field emitters melted due to high field emission currents. This effectseems similar to conditioning and is appealing since overwhelming statisticsfrom all long-term experiments at CERN and other labs show that conditioningis dependent on the number of pulses and not the number of breakdowns [53].

The lack of visible surface changes or surface characteristics could also bedue to limited resolution of the SEM images. This is not mainly due to thecapabilities of the SEM itself but due to the large working distance neededfor our setup to fit below the electron column. We have an ongoing experi-ment where another high-magnification SEM is used for surface imaging andwhere we use markers on the surface for orientation. This makes it possible to

26

take before and after images with a high-magnification SEM while perform-ing localized field emission measurements with our setup. Furthermore, weare combining our setup for field emission measurements with other surfaceanalysis tools.

The rest of this thesis will focus on beam-based diagnostics and beam dy-namics. In the next section we will present the test facility at CERN where theother experiments in this thesis took place.

2.4 CLIC test facility 3 (CTF3)In the 1980s CERN initiated an experimental research and development pro-gram for CLIC. The first experimental facility was the CLIC Test Facility(CTF) [54, 55] which operated between 1990-1995 and had the objective oftesting a drive beam injector. The next incarnation was CLIC Test Facility 2(CTF2) [56, 57] where the two-beam acceleration concept was tested for thefirst 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 particularthe high-intensity drive beam generation and high-gradient two-beam accel-eration [36]. CTF3 was in full operation from 2009 to 2016 at CERN and arecent summary of the results from CTF3 is found in [60].

Figure 2.8 shows the CTF3 layout. CTF3 is essentially a scaled-down ver-sion of CLIC with a 120 MeV high-intensity electron beam called the drivebeam and a low-intensity 200 MeV electron beam called the probe beam,which corresponds to the CLIC main beam. The two beams are joined in thethe CLIC experimental area (CLEX) where the beamline of the drive beamsplits into two beamlines. One beamline is the Test Beamline (TBL) which isa prototype decelerator section where the electrons are decelerated in a seriesof PETS which allows for the energy extraction and beam stability of the drivebeam to be studied [61]. The second beamline in CLEX is the the Two-BeamTest Stand (TBTS) where the drive beam runs in parallel to the probe beamand two-beam acceleration is studied, this will be explained in more detail inthe coming paragraphs. Table 2.3 lists a summary of the CTF3 parameters.

Drive beamThe drive beam injector consists of a high-current thermionic gun followed bya subharmonic 1.5 GHz buncher followed by 3 GHz bunchers. These bunch-ers are RF structures that create the necessary phase coding of the bunchesin the drive beam pulse. A section of 3 GHz normal conducting acceleratingstructures accelerate the electron beam to 120 MeV before entering the recom-bination section. The overall length of the linac is about 70 m. Recombinationof the beam takes place in a 42 m delay loop followed by an 84 m combinerring where the intensity of the beam is increased by first a factor of 2 and then

27

30 GHz test stand 150 MeV e– linac

magnetic chicane pulse compression frequency multiplication

photo injector tests and laser CLIC experimental area (CLEX) with

two-beam test stand, probe beam and

test beam line

28 A, 140 ns

total length about 140 m

10 m

delay loop

combiner ring

3.5 A, 1.4 μs

Figure 2.8. Layout of the CLIC test facility 3. It is a scaled-down version of CLICwith a drive beam that starts in the upper left part, is interleaved in the delay loopand combiner ring, and then entered into CLEX where it runs in parallel to the probebeam. 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 120 200Energy spread (r.m.s.) % 2 1Pulse length ns 140–1100 0.6–150Bunch frequency GHz 1.5–15 1.5Bunch charge nC up to 3 0.05–0.6Intensity (short pulse) A 28 1Intensity (long pulse) A 4 0.13Repetition rate Hz 0.8–5 0.8–5

an additional factor of 4. The initial pulse of 1.2 µs and 4 A is recombined to140 ns and 28 A. For more information on the CTF3 drive beam see [36, 60].

Probe beamThe probe beam starts with an injector called Concept d’Accélérateur Linéairepour Faisceaux d’Electrons Sondes (CALIFES) which is 24 m long linac thataccelerates electrons to about 200 MeV. The electron gun uses a photocathodeand a laser with pulse length of about 6 ps and 1.5 GHz repetition rate. Theelectron bunches are then accelerated in two 3 GHz structures before enteringthe TBTS in CLEX. The gun, a buncher and the two accelerating structures areall powered by one single klystron. The bunch charge can be varied between0.06 and 0.6 nC and the number of bunches per pulse can range from 1 to 300.The CALIFES injector is explained in more detail in [62] and Paper XIV.

28

Figure 2.9. Layout of the two-beam test stand. The drive beam and the probe beamenter from the right and travel to the left. 12 GHz RF power is generated when thedrive beam is decelerated in the two PETS and guided to the four accelerating struc-tures (ACS) mounted on a movable girder. There is an additional PETS before thetwo-beam module in order to reach nominal power. Various beam diagnostics areavailable such as beam position monitors (BPMs) and screens. At the end of bothbeamlines 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 accelera-tion scheme. The layout is shown in Fig. 2.9 where the drive beam and probebeam enter from the right and travel in parallel. The drive beam is deceler-ated in two PETS in the two-beam module and there is also a PETS beforethat generates additional 12 GHz RF power. This power is fed-forward to thenext PETS and the purpose of this is to generate CLIC nominal RF powerdespite the lower drive beam intensity (28 A compared to the nominal 100 Afor CLIC). At the time of the experiment in Paper I only one PETS and twoaccelerating structures were installed.

At the end of both the probe beam and the drive beam there are spectrom-eter legs that consist of a dipole magnet that bends the beam, a beam positionmonitor (BPM), a screen and a beam dump. From the measured bending anglethe energy of the beam can be calculated. A BPM is a non-destructive elec-tromagnetic pick-up device that measures the position of the beam centroid.In the probe beam there are also beam profile monitors [63] simply referred toas screens. A beam profile monitor consists of a fluorescent screen that emitslight when the electron beam impact on it and this light is recorded by a CCDcamera. The beam profile monitor after the two-beam module is the one usedin the experiments of this thesis and it consists of a 40×40 mm YAG screentilted 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 com-pensating the tilt of the screen. An example of a beam image using this screenwas shown previously in Fig. 2.4.

29

The repetition rate of the probe beam can be changed independently of therepetition rate of the drive beam. This is an important flexibility and makesit possible to operate with the repetition rate of the probe beam twice the rep-etition rate of the drive beam, which is particularly desirable for experimentswhere 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 acceler-ating structures when there is RF power in the 12 GHz accelerating structuresand every other pulse when there is no RF power. This eliminates many sys-tematic errors from slow drifts of the machine. More detailed informationabout TBTS can be found in [64].

Before we present the beam-based diagnostics methods we need to reviewthe theory of transverse beam dynamics and this is the topic of the next chap-ter.

30

3. Transverse beam dynamics

In this chapter we present the fundamentals of beam physics with focus ontransverse beam dynamics. The main goal of accelerator beam dynamics is todescribe and understand the motion of particles in accelerators. More infor-mation on the topic can be found in various textbooks such as [38, 65–69].

3.1 Fundamentals of beam physicsIn accelerator physics it is convenient to describe an individual particle withrespect to an idealized particle traversing an ideal reference orbit. We intro-duce a co-moving reference frame following this ideal particle and we let sbe the independent, time-like variable describing the longitudinal position inthe accelerator, see Fig. 3.1. In this co-moving frame we describe a genericparticle’s coordinates as a vector

~x =(x,x′,y,y′,τ,δ

)T (3.1)

where x and y are the horizontal and vertical positions with respect to the ref-erence orbit, the angles x′ and y′ are the transverse momenta normalized tolongitudinal momentum, i.e. x′ = px/pz and y′ = py/pz. The longitudinalcoordinates are expressed as arrival time τ and relative momentum δ with re-spect to the ideal particle. In this thesis we will mainly focus on the transversedynamics and then~x = (x,x′,y,y′)T is an adequate description of a particle.

A beam is a collection of particles that we describe as a statistical distri-bution characterized by its covariance matrix, in the context of acceleratorphysics, called the beam matrix. The 4D transverse beam matrix is given by

σ =

σ11 σ12 σ13 σ14σ21 σ22 σ23 σ24σ31 σ32 σ33 σ34σ41 σ42 σ43 σ44

=

σ2

x σxx′ σxy σxy′

σx′x σ2x′ σx′y σx′y′

σyx σyx′ σ2y σyy′

σy′x σy′x′ σy′y σ2y′

(3.2)

Figure 3.1. A co-moving reference frame.

31

with elements given by second central moments, σi j =⟨(xi−Xi)(x j−X j)

⟩where angle brackets denotes averaging over all particles and xi = {x,x′,y,y′}for i = 1,2,3,4. We use capital letters to indicate the first moments, e.g.Xi = 〈xi〉. Since σi j = σ ji the matrix is symmetric and 10 elements uniquelydescribe the full transverse beam matrix.

The emittance of a beam is proportional to the area in phase space occupiedby the particles and is an invariant of the motion in a conservative system. Infact, there is one conserved quantity for each degree of freedom and for thetransverse motion we have two degrees of freedom and thus two emittances.For a beam without x–y correlations the beam matrix is block diagonal andcan be written as

σ =

(Σx 00 Σy

)(3.3)

where Σx and Σy are 2×2 matrices describing the horizontal and vertical partsof the beam distribution, respectively. In that case the horizontal emittance ofthe beam can be calculated as

εx =√

det(Σx) =√

σ2x σ2

x′−σ2xx′ (3.4)

and similarly for the vertical emittance εy. When the transverse beam matrixhas nonzero x–y correlations we can still calculate the emittances from the2× 2 blocks, these are called the projected emittances but are not conservedquantities. However, even for the correlated case it is possible to find the twoconserved quantities, called eigen-emittances, by eigenvalue decomposition ofthe beam matrix [69].

Multipole kicksA particle with charge q and velocity~v in the presence of electric and magneticfields (~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 thetransverse components of the magnetic field steer and focus the beam. A gen-eral transverse magnetic field can be expressed as a multipole expansion thatcomes from finding solutions to the source-free Maxwell’s equations [37], i.e.the magnetic field must satisfy ∇ · ~B = 0 and ∇× ~B =~0. If we assume nolongitudinal field component (Bz = 0) and express the multipole expansion incomplex form [69] we can write

By + iBx =∞

∑n=1

Cn−1(x+ iy)n−1. (3.6)

32

Explicitly writing the first multipoles we have

By + iBx = C0︸︷︷︸Dipole

+C1(x+ iy)︸ ︷︷ ︸Quadrupole

+C2(x2− y2 +2ixy)︸ ︷︷ ︸Sextupole

+C3(x3−3xy2 + i(3x2y− y3))︸ ︷︷ ︸Octupole

+C4(x4−6x2y2 + y4 + i4(x3y− xy3))︸ ︷︷ ︸Decapole

+ . . .

(3.7)and these are the most common types of transverse magnetic fields in a parti-cle accelerator. Real values of Cn−1 correspond to upright magnets and com-plex values are equivalent to rotations of the field around the beam axis. Byconvention multipoles with Im(Cn−1) = 0 are called normal and multipoleswith Re(Cn−1) = 0 are called skew. A true multipole field is assumed to haveonly one nonzero coefficient in the sum in (3.6). Dipole magnets are used forbeam steering and quadrupole magnets are used as magnetic lenses to focusthe beam. Since quadrupole fields have a linear dependence on transverse po-sition (x,y) the resulting force is linear and the resulting dynamics is linear.Then we have nonlinear fields such as sextupoles, octupoles and decapoles,where the field strength depends nonlinearly on transverse position, which inturn results in nonlinear forces and nonlinear dynamics. Figure 3.2 shows thefirst four multipole fields from (3.7).

When a particle traverses a transverse magnetic field the transverse momen-tum of the particle will change, in particle accelerator contexts this change inangles (x′,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 wecan use the thin lens approximation and get

∆x′ =px

pz=−evByl

pzv=−

Byl(Bρ)

∆y′ =py

pz=

evBxlpzv

=Bxl(Bρ)

(3.8)

where l is the length of the magnetic element, the product Bxl is the integratedstrength and (Bρ) is the beam rigidity, which is a common way in acceleratorphysics to express a particle’s momentum. In the case of a quadrupole magnetthe thin lens approximation is justified if the length of the quadrupole is shortcompared to its focal length. If the thin lens approximation does not hold wecan calculate the effect of the thick magnetic element by slicing the elementin thin lenses separated by drifts, i.e. a drift–kick map.

In a particle accelerator the particles travel in vacuum pipes with magnetsplaced at certain longitudinal positions separated by field-free regions calleddrift spaces, this constitutes the magnetic lattice of the accelerator and definesthe dynamics and the beam properties. In order to analyze the dynamics ofan accelerator we want to calculate how particles move in the magnetic latticeand this is the topic of the next section.

33

x

-1 0 1

By

[arb

. units]

-1

0

1

x

-1 0 1

y

-1

0

1

x

-1 0 1

By

[arb

. units]

-1

0

1

x

-1 0 1

y

-1

0

1

x

-1 0 1

By

[arb

. units]

-1

0

1

x

-1 0 1

y

-1

0

1

x

-1 0 1

By

[arb

. units]

-1

0

1

x

-1 0 1

y

-1

0

1

Figure 3.2. Magnetic multipoles. On the left hand side we show the field lines fordipole, quadrupole, sextupole and octupole fields. All the fields are normal multipolesand 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 asa function of horizontal position, By(x), for the corresponding fields at y = 0.

34

Particle trackingThe purpose of particle tracking is to calculate the motion of a particle ora beam in an accelerator. Hamiltonian systems, such as a system with staticmagnetic fields, have an important property: they are symplectic, which meansthat energy is conserved and that the total phase space volume is constantunder the motion (Liouville’s theorem). To the describe a particle’s motionthrough a small part of an accelerator we use symplectic transfer maps—mathematical functions that relate the particle’s coordinates, e.g. (3.1), atdifferent positions s. An element’s effect on a particle can be described bya map M that maps the initial coordinates ~xi (just before the element) to thefinal coordinates~x f (just after the element) according to

~x f = M ·~xi. (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 char-acterized by transfer matrices and will be further explained in the next section.Maps of nonlinear elements, however, such as sextupole or octupole mag-nets, cannot be expressed as matrices and instead we employ Hamiltoniansand tools of Lie algebra, this will be covered in Section 3.3.

If we want to track through several elements we apply consecutive maps

~x f = MN ·MN−1 · · ·M2 ·M1 ·~xi (3.10)

in order to get the particle’s coordinates at the end of the beamline. Severalmaps can be combined to a single map that can represent a single element, asegment of a beamline or even the whole accelerator—commonly referred toas the one-turn map.

A Poincaré plot, or a Poincaré surface of a section, is a powerful way to an-alyze repetitive dynamical systems such as circular accelerators. It is a portraitof multiturn phase space coordinates, for instance x–x′, at a single longitudinalposition. Information about the motion and analysis of the dynamics are re-trieved from the phase space plots of consecutive applications of the one-turnmap.

3.2 Linear beam dynamicsTo first order the beam dynamics of an accelerator is linear and can conve-niently be described by linear maps represented by transfer matrices. For anelement with transfer matrix M, tracking a single particle is simply a matrixmultiplication with the coordinate vector: ~x f = M~xi. Using the same trans-fer matrix, we can also propagate a beam by propagating the beam matrixdescribing the particle distribution, it can be shown that

σ f = MσiMT . (3.11)

35

The transfer matrices for a drift space of length L and a thin-lens quadrupolemagnet with focal length f are

Mdrift =

1 L 0 00 1 0 00 0 1 L0 0 0 1

Mquad =

1 0 0 0− 1

f 1 0 00 0 1 00 0 1

f 1

where the minus sign in the second row of Mquad indicates that the quadrupoleis focusing in the horizontal plane but defocusing in the vertical plane. Boththese matrices are block-diagonal which means that the motion is uncou-pled and we can treat the two planes separately. Furthermore, we note thatdet(M) = 1 for both of them and this is the case for symplectic maps.

In 1958 Courant and Snyder demonstrated strong focusing [70], which isthe underlying principle of a synchrotron where, despite that quadrupole mag-nets are focusing in one plane and defocusing in the other, they can still com-bine to achieve net focusing in both planes. They used a parameterization ofthe transfer matrix that we can apply to each of the two 2×2 blocks accordingto

M = A−1RA =

( √β 0

− α√β

1√β

)(cos µ sin µ

−sin µ cos µ

) 1√β

0α√

β

√β

=

(cos µ +α sin µ β sin µ

−1+α2

βsin µ cos µ−α sin µ

) (3.12)

where α,β are the Courant-Snyder parameters and µ is the phase advance.This parameterization of a linear transfer matrix reveals a lot of informationabout the dynamics. The requirement for the parameterization to work is thatwe have a stable and periodic system and we find from (3.12) that the traceof M has to be less than two, Tr(M) ≤ 2, for the system to be stable. Such asystem is characterized by an elliptical fixed-point and the particles will traceout ellipses in phase space. In fact, the parameterization can be understood asfollows: the matrix A is a coordinate transformation that transforms the ellipseto a circle. This coordinate system is called normalized phase space and therethe linear motion is fully captured by a simple rotation R with phase advanceµ . The final step is to bring the system back to real phase space coordinatesby applying the inverse transform A−1.

In normalized phase space (x, x′) 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 anddepicts the action–angle variables in normalized phase space. The action–

36

Figure 3.3. Transformation into normalized phase space brings elliptical trajectoriesinto circles. In normalized phase space it is convenient to express the particle’s coor-dinates in action–angle variables (J,ψ).

angle variables are defined as followsx =√

2J cos(ψ)

x′ =√

2J sin(ψ)

↔

J =

x2 + x′2

2

ψ = tan−1(

x′

x

) . (3.13)

In a conservative, stable, linear system such as the magnetic lattice of an ac-celerator the action J is an invariant of the motion.

In this section we used a map-based approach and used matrices to describethe linear motion. An alternative approach is to solve Hill’s equation, whichis simply a harmonic oscillator with s-dependent and periodic strength.

TuneThe tune, denoted Q, of an accelerator is an important concept for many sta-bility issues and it is defined as the normalized phase advance for the one-turnmap, Q = µ/2π . The tune is the number of oscillations about the design orbitand for the transverse plane we have two degrees of freedom and hence twotunes. The tune consists of a integer part and a fractional part and latter hasimpact on stability. For instance, the tune cannot be an integer since a sin-gle dipole-like imperfection will lead to secular growth and divergence of theparticles. A quadrupolar-like imperfection will lead to secular growth if thefractional part of the tune is 1

2 . For coupled motion problem arises when thesum of difference of the two tunes equal an integer. In general, the resonancecondition for the transverse motion can be written as

kQx + lQy = m (3.14)

where k, l and m are integers and the sum |k|+ |l| gives the order of the res-onance. Figure 3.4 plots the resonance condition lines in the tune plane for

37

Qx

0

1

1

5

1

4

1

3

2

5

1

2

3

5

2

3

3

4

4

5

1

1

Qy

0

1

1

5

1

4

1

3

2

5

1

2

3

5

2

3

3

4

4

5

1

1

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 (3.14) for|k|+ |l| ≤ 5. The tune of an accelerator should be set to avoid resonancesin order to ensure stability. However, not all resonances are harmful and notall resonances are driven by the system. In the Section 3.3 we will introducethe concept of resonance driving terms.

Particles with different energy than the ideal particle, are bent differentlyin the quadrupole magnets and results in different focal lengths as depicted inFig. 3.5. This change in focal length results in a different tune and means thatthe tune depends on energy, this effect is called chromaticity. Since a beamof particles unavoidably will have some spread in energy this will result ina spread in tune. To compensate for this potentially harmful feature we canutilize the nonlinear field of sextupoles. Figure 3.5 shows how a sextupole,placed in a dispersive section, e.g. after a bending dipole where the particlesare sorted with respect to their energies, can correct the focal length and thetune.

Sextupoles for chromaticity correction is one of the most common appli-cations where a nonlinear magnetic field is introduced in the accelerator bydesign. However, since sextupole magnets have nonlinear fields we must alsoconsider their nonlinear effect on the dynamics which is the topic of the nextsection.

38

Figure 3.5. Upper: particles with different energy are focused differently in thequadrupoles leading to an energy-dependent tune, an effect called chromaticity.Lower: by inserting a sextupole magnet in a dispersive region, where the particlesare sorted with respect to energy, we can compensate the chromaticity.

3.3 Hamiltonians, Lie algebra and normal formsIn particle accelerators there are many sources for nonlinear effects such asmultipole errors of magnets, e.g. superconducting dipoles with high fieldshave significant multipole components, for circular colliders there is the beam-beam effect: when the beams collide they experience a highly nonlinear fieldfrom the opposing beam. Then there are of course the dedicated magnetswith nonlinear fields, put in by design, such as sextupoles to compensate chro-maticity. Since nonlinear effects in many cases are the limiting factor for thestability of an accelerator it is important to have a good understanding of theseeffects and for this we need the appropriate analytical tools.

Hamiltonians and Hamilton’s equationsThe Hamiltonian formalism is the most natural description of a mechanicalsystem 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 [71] and formore information about Hamiltonian formalism and Lie algebra methods inthe context of accelerator physics see [72–74].

A Hamiltonian is a function on phase space coordinates that together withHamilton’s equations yield the equations of motion. Hamilton’s equations are

39

Table 3.1. Hamiltonians for normal and skew multipoles.

Multipole normal skew

Dipole k0x k0y

Quadrupole k1l2 (x2− y2) k1lxy

Sextupole k2l3! (x

3−3xy2) k2l3! (3x2y− y3)

Octupole k3l4! (x

4−6x2y2 + y4) k3l4! (4x3y−4xy3)

Decapole k4l5! (x

5−10x3y2 +5xy4) k4l5! (5x4y−10x2y3 + y5)

conveniently expressed using the Poisson bracket as

dxds

= [−H,x]dx′

ds=[−H,x′

](3.15)

and similarly for the vertical plane. The Poisson bracket of two functions ofphase space coordinates, f and g, is defined as

[ f ,g] =∂ f∂x

∂g∂x′− ∂ f

∂x′∂g∂x

+∂ f∂y

∂g∂y′− ∂ f

∂y′∂g∂y

. (3.16)

A thin-lens magnet can be expressed as a Hamiltonian H = H(x,y) and using(3.15) and (3.16) we get dx

ds = 0 and the thin-lens kick expressed as the Poissonbracket of the Hamiltonian and the angle coordinate,

∆x′ =dx′

ds=[−H,x′

]=−∂H

∂x. (3.17)

Table 3.1 lists the Hamiltonians for common multipoles. Using these Hamil-tonians together with (3.17) yield the same kicks as (3.7) together with (3.8)and we identify knl

n! = Re{Cnl}(Bρ) and knl

n! = − Im{Cnl}(Bρ) . So far we have only re-

expressed the multipole kicks in Hamiltonian formalism. Next we will presentsome powerful methods that allow us to analyze nonlinear maps.

Lie formalismThe Lie operator is denoted with : on both sides and is defined as the Poissonbracket. The Lie operator f acting on function g is given by

: f : g = [ f ,g] (3.18)

and thus we can express a nonlinear multipole kick with the Hamiltonian as aLie operator acting on x′, we have

∆x′ =:−H : x′. (3.19)

40

The Lie transformation is defined as the exponential of the Lie operator

e: f : =∞

∑n=0

1n!

(: f :)n (3.20)

and can be thought of as a Taylor map. Powers of the Lie operators results inconsecutive Poisson brackets, e.g. (: f :)2g = [ f , [ f ,g]].

Two important tools in the Lie formalism are the similarity transformationand the Campbell-Baker-Hausdorff (CBH) formula. The similarity transfor-mation allows Hamiltonians to be moved to other locations. Consider thesimplest case where we have a Hamiltonian kick followed by a linear map M.We can express an equivalent map with reversed order, i.e. first a Hamiltoniankick followed by the linear map M, by using the similarity transform accordingto

M = Me:−H(~x1): = Me:−H(~x1):M−1M

= e:−H(M~x1):M = e:−H(~x2):M(3.21)

which in words means that in order to transform the operator (H) we only needto transform the generator (~x2 = M~x1). If we have two Hamiltonians HA andHB at the same location we can concatenate them into an effective HamiltonianH using the CBH formula where

H = HA +HB +12[HA,HB]+

112

[HA−HB, [HA,HB]]+ . . . (3.22)

Note that the order of HA and HB depends on the convention used for order ofLie transformations. Here we use the same convention for Lie transformationsas for transfer maps, i.e. they are applied from right to left. By using the sim-ilarity transform and CBH formula iteratively the description of a beamline ora full accelerator can be greatly simplified. We can move all the Hamiltoniankicks from the nonlinear elements to a reference point and concatenate all ofthem into a single effective Hamiltonian, H, comprising a "super-kick" withthe combined effect of all the nonlinear elements. In that case we write theone-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 formaccording to

M = e:−H:R = e:−K:e:−C:Re:K: (3.23)

where we now have assumed that we have transformed into normalized phasespace and the linear map is given by a rotation R. The normal form takes intoaccount the effect from iterative applications of the one-turn map which can bethough of as self-interactions of the effective Hamiltonian. Consider a systemof a single sextupole and a rotation R, then the one-turn map, e:−H:R, wouldonly contain third-order terms from the third-order terms in the Hamiltonian ofthe sextupole. Only when the multiturn effects are taken into account we seethat higher-order terms are generated and that a single sextupole, for instance,

41

Figure 3.6. Top: description of a circular accelerator as a linear map R and a single"super-kick" expressed as an effective Hamiltonian H. Bottom: a depiction of thenormal form as an infinite regression of the circular system as a beamline with self-interactions of the effective Hamiltonian.

x

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x'

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x'

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 3.7. Left: horizontal phase space for a single sextupole followed by a phaserotation. 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 isan effect in second-order sextupole strength. Figure 3.6 shows a schematic ofcircular accelerator described as a linear map with a super-kick and a depictionof the normal form.

The normal form is essentially a generalization of the Courant-Snyder para-meterization of nonlinear maps. In (3.12) the matrix A transforms the ellipseinto an invariant sub-space (normalized phase space) where the map is givenby a rotation R and then A−1 transforms back. In (3.23) the Lie transfor-mation e:K: transforms into an invariant sub-space where the nonlinear mapis given by e:−C:R and finally e:−K: transforms back. The map e:−C:R con-tains the invariant part of the map, in other words only action-terms if weexpress the map in action–angle variables and these terms corresponds to theamplitude-dependent tune-shifts. Figure 3.7 shows the e:K: transform appliedto a horizontal phase space plot of a system with a single sextupole.

42

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−1 = e:−K:e:−C: (3.24)

and identify Re:−K:R−1 as a similarity transformation of K that we can write asa linear transformation, represented by matrix S, operating on the coefficientsof the Hamiltonian K. We use that Re:−K:R−1 = 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 ofthe 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 aHamiltonian 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 higherorders. All invariant parts in C occur in even orders and thus C(3) = 0. Thenwe can find the third order part of K as

K(3) = (1−S(3))−1H(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): (3.30)

and again we apply CBH. If we only consider the fourth order terms we get

H(4)+S(4)K(4)+12

[H(3),S(3)K(3)

]= K(4)+C(4) (3.31)

where we now need to keep the Poisson bracket of the third order terms sincethe result is fourth-order terms. We solve for C(4) and K(4)

(1−S(4))K(4)+C(4) = H(4)+12

[H(3),S(3)K(3)

](3.32)

43

where in fact (1− S(4)) is not invertible. We resolve this issue by extractingthe invariant sub-space from the right hand side which is C(4). Let PJ be aprojector that projects the part containing only J2

x , J2y and JxJy, then we have

C(4) = PJ

{H(4)+

12

[H(3),S(3)K(3)

]}(3.33)

and the remaining part constitutes K(4). Finally, we can add any Hamiltonianthat is a part of the invariant sub-space to K(4) without changing (3.32) and weresolve this ambiguity by setting PJ{K(4)} = 0, which corresponds to "fixingthe gauge". This procedure can be repeated to solve for higher and higherorders in an algorithmic way.

For the effective Hamiltonian or the Hamiltonian K in the normal form ex-pression of the map, we identify the resonance driving terms by transformingto action–angle variables using (3.13). As an example, the Hamiltonian for aone-dimensional sextupole in normalized phase space can be written

H =k2l6

β3/2x3 =

k2l6

β3/2(2J)3/2 cos3

ψ

=k2l24

(2Jβ )3/2 cos(3ψ)︸ ︷︷ ︸driving term for 3Q

+k2l8(2Jβ )3/2 cos(ψ)︸ ︷︷ ︸driving term for Q

(3.34)

where we identify phase-dependent terms of 3ψ and ψ . Since these perturba-tions make the system sensitive to tunes 3Qx = integer and Qx = integer theyare called resonance driving terms of 3Qx and Qx respectively.

A few words of caution. Both the Lie transformation and the CBH formulaare infinite series that must be truncated when implemented on a computer.This means that the equivalent map is only an approximation of the origi-nal composite map. Furthermore, this approximative map might also violatethe symplectic condition due to the truncation. Because of this the normalform is primarily used for analysis of the dynamics but element-by-elementpropagation of particles with symplectic maps remains the most reliable forsimulations of long-term behavior.

In this section we reviewed the theory of transverse beam dynamics. InChapter 4 we will investigate how nonlinear kicks from an octupole field canbe utilized as a means of beam-based diagnostics and in Chapter 5 we utilizethe tools of nonlinear beam dynamics and apply them to two specific dynami-cal problems.

44

4. Beam-based diagnostics with octupoles

In this chapter we explore beam-based diagnostics methods utilizing the co-propagating octupole component of the CLIC accelerating structure describedin Section 2.2. In Paper I we investigate how to measure the field strengthof the octupole component and how the octupole component can be used formeasuring the transverse beam matrix. In Papers II and III we investigate howthe nonlinear kicks from the octupole component can be used as a method forbeam-based alignment of the accelerating structures. All experiments weredone at the test facility CTF3 at CERN, which is described in Section 2.4.

4.1 Measuring the RF octupole componentFrom (3.7) we retrieve the octupole field and together with (3.8) we can cal-culate the kicks (∆x′,∆y′). We can express the shift in position at a distance Ldownstream from the octupole field as

x− x = L∆x′ =C3l(Bρ)

L(3xy2− x3) (4.1)

where x is the horizontal position of the particle perturbed by the octupolefield, x is the horizontal position of the particle in the accelerating structureand C3l is the integrated octupole strength. Individual particle positions arenot measurable in a particle accelerator but we can measure the position of thecentroid of the beam. The position shift of the centroid of the beam distributionis calculated by taking the expectation of (4.1) and we obtain

X−X =

⟨C3l(Bρ)

(3xy2− x3)

⟩=

C3l(Bρ)

[3⟨xy2⟩−⟨x3⟩] . (4.2)

If we assume a Gaussian beam distribution, the expectation values⟨xy2⟩

and⟨x3⟩

can be calculated analytically. A Gaussian beam is a reasonable assump-tion for many electron beams and in TBTS the beam from the laser cathodefollows a Gaussian distribution to good approximation and the beam profileis preserved during the transport through the short linac. A general expecta-tion value for a multivariate Gaussian distribution can be calculated by takingderivatives of the following generating function:⟨

xm11 xm2

2 . . .xmNN

⟩=

(∂ m1

∂bm11

∂ m2

∂bm22

. . .∂ mN

∂bmNN

)exp[

12

biσi jb j +biXi

]∣∣∣∣bi=0

.

(4.3)

45

Figure 4.1. A schematic of the experimental setup. The beam (traveling from right toleft) is scanned transversely across the accelerating structure, and parallel to the beamaxis, by using two steering magnets while observing the beam profile on a screendownstream.

For a derivation of this generating function see Appendix A of Paper I. Using(4.3) we can calculate the expectation values in (4.2) and we do the same forthe vertical position shift and finally end up with

X−X = KL[

X(Y 2 +σ2y −σ

2x )−

X3

3+2Y σxy

]Y −Y = KL

[Y (X2 +σ

2x −σ

2y )−

Y 3

3+2Xσxy

] (4.4)

where we have introduced K = 3C3l/(Bρ) which is the integrated octupolestrength normalized to the beam momentum expressed as beam rigidity (Bρ).Equation (4.4) describes how the beam centroid position shifts at a distance Ldownstream of an octupole field and it depends on the normalized integratedoctupole strength K, beam centroid position inside the octupole field (X ,Y )and transverse beam size (σ2

x ,σ2y ,σxy).

ExperimentWe tested the method in CTF3 and set up the probe beam as follows. First weassured that the beam was well-aligned so that a minimum number of mag-nets was needed for propagating the beam straight through the beamline. Weused a quadrupole triplet at the very beginning of the probe beam to preparea slowly convergent beam with a small beam size on the screen, we used thefirst screen after the accelerating structure (c.f. Fig. 2.9) and no magnetic ele-ments between the accelerating structure and screen were active. Two steeringmagnets before the accelerating structure were used to move the beam trans-versely and parallel to the beam axis. Figure 4.1 shows a schematic of theexperimental setup.

By scanning over the RF phase we first measured the on-crest accelerationin order to know the maximum energy gain, which in this case was ∆E =5.1 MeV. The phase of the RF was adjusted to the zero-crossing, i.e. wherewe have the maximum strength of the octupole component. The probe beamwas operated at twice the repetition frequency of the drive beam, which means

46

Figure 4.2. Sample images from the experiments. The beam was moved verticallyacross the face of the accelerating structure. On the left are screen images of the beamwhen there was no RF in the accelerating structure and on the right the correspondingimages when there RF was in the accelerating structure. The beam on the left changesposition but not shape. The beam on the right is perturbed by the octupole field andboth position and shape change.

that every other screen image is an image of the beam when there was RF in theaccelerating structure and every other when there was no RF . We moved thebeam vertically±1.5 mm across the 4 mm in diameter face of the acceleratingstructure and at each scan step we collected 16 screen images (8 with RF and8 without RF). Figure 4.2 shows a few sample images from the experiment1.

From the screen images we make Gaussian fits, first one-dimensional Gaus-sian fits to the projections and then a full two-dimensional Gaussian fit. Fromthe fits the relevant information is extracted, such as beam centroid positionsthat allows us to calculate the beam position shifts. Since we collect severalpulses for each scan step we calculate the average and use the standard de-

1A video of the screen images can be found in the Supplemental Material of the online versionof Paper I: http://link.aps.org/supplemental/10.1103/PhysRevSTAB.18.072801

47

viation as uncertainty, this can be motivated since the shot-to-shot variationwas larger than the uncertainties in individual Gaussian fits. Finally, from thedata of the measured beam position shifts we look for a best fit by minimizingχ2

shi f ts defined as

χ2shi f ts =

n

∑i=1

[(Xi−Xi

)meas−(Xi−Xi

)theo

∆(Xi−Xi)

]2

+n

∑i=1

[(Yi−Yi

)meas−(Yi−Yi

)theo

∆(Yi−Yi)

]2 (4.5)

where (Xi−Xi)meas denotes the measured horizontal position shifts and ∆(Xi−

Xi) denotes the standard deviation of the position shift, at each scan step. Thetheoretical expressions (Xi−Xi)

theo and (Yi−Yi)theo are based on (4.4) and

depend on the unknown variables: integrated octupole strength, transverseposition and beam size inside the accelerating structure. For the beam sizeinside the accelerating structure we use the beam size on the screen measuredwhen there was no RF, which is reasonable because of the slowly convergingbeam that we set up. Since we know the transverse position of the beam on thescreen when no RF fields are present and we assume that the position on thescreen is the position inside the accelerating structure offset by some offsets band c. Thus, based on (4.4) we make the following ansatz(Xi−Xi

)theo= a[(X +b)((Y + c)2 +σ

2y −σ

2x )−

(X +b)3

3+2(Y + c)σxy

]+d

(4.6)where a = KL contains the sought-after integrated octupole strenght, b and cdenote horizontal and vertical offsets, and d represents a horizontal bias. Wehave a similar expression for the vertical position shift.

Figure 4.3 plots the measured beam position shifts with error bars togetherwith the resulting fits. We scan the beam vertically and as expected from (4.4)we observe a third-order dependence in vertical position shift and a quadraticdependence for the horizontal position shift. The results from the fit are pre-sented in Table 4.1. We get an integrated strength of the octupole componentof C3l = 14±2 kTm/m3, which is comparable to 16.4 kTm/m3 expected fromsimulations. As estimation of error bars for the fit parameters we varied oneparameter, while keeping the other parameters constant, until the value of χ2

was doubled. We note that from the fit we also get the transverse offsets of thebeam with respect to the center of the octupole field, b and c. This sparked theidea of utilizing the octupole component for a beam-based alignment method,which is the topic of a later section.

When the beam hits upon the screen we essentially integrate over the dis-tribution and the only observables of the transverse beam matrix are σ2

x , σ2y

and σxy, i.e. the horizontal and vertical widths, and the x–y correlation. As

48

−2 −1 0 1 2−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Vertical position [mm]

Sh

ift

in h

orizo

nta

l p

ositio

n [

mm

]

Data

Fit

−2 −1 0 1 2−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Vertical position [mm]

Sh

ift

in v

ert

ica

l p

ositio

n [

mm

]

Data

Fit

Figure 4.3. The resulting position shifts at the location of the screen for differentvertical positions plotted together with the fit.

we scanned the beam across the octupole field of the accelerating structure weobserved that the beam size on the screen also changed. This is of course notsurprising but gave us the idea that more information about the beam matrixof the beam could be extracted.

4.2 Measuring the beam matrixA small beam in a multipole field experiences locally a linear gradient similarto a quadrupole field. The gradient is position dependent and thus moving thebeam transversely inside a nonlinear multipole field is similar to changing thestrength of a quadrupole field. In Section 3.2 we saw that the focusing froma thin quadrupole can be calculated with a matrix containing the inverse ofthe focal length. Similarly, for a multipole field we can express a generalizedfocal length as

1fxx

=− l(Bρ)

∂By

∂x(4.7)

49

Table 4.1. Fit results.Parameter Symbol Value Unit

χ2 per degrees of freedom χ2N 0.28 no unit

Integrated octupole strength C3l 14 ± 2 kTm/m3

Horizontal offset b -0.47 ± 0.05 mmVertical offset c -0.25 ± 0.08 mmHorizontal bias d 0.09 ± 0.02 mmVertical bias e 0.06 ± 0.03 mm

where the double indices xx should be interpreted as the horizontal focal lengthas a function of horizontal position. But for a general magnetic multipolewe have By = By(x,y) and thus we also have 1/ fxy, which is the horizontalfocal length as a function of vertical position. If we assume a thin octupolefield followed by a drift of length L we can write the linear map as a matrixM = MLMoctu and we have

M =

1 L 0 00 1 0 00 0 1 L0 0 0 1

1 0 0 01fxx

1 1fxy

00 0 1 01fyx

0 1fyy

1

=

1+ L

fxxL L

fxy0

1fxx

1 1fxy

0Lfyx

0 1+ Lfyy

L1fyx

0 1fyy

1

.

(4.8)The position-dependent inverse focal lengths for the octupole field can be cal-culated by taking derivatives of Bx and By and we get

1fxx

=−l(Bρ)

∂By

∂x

∣∣∣∣x=Xy=Y

= K(Y 2−X2)

1fxy

=−l(Bρ)

∂By

∂y

∣∣∣∣x=Xy=Y

= 2KXY

1fyy

=l

(Bρ)

∂Bx

∂y

∣∣∣∣x=Xy=Y

= K(X2−Y 2)

1fyx

=l

(Bρ)

∂Bx

∂x

∣∣∣∣x=Xy=Y

= 2KXY

(4.9)

where the derivatives have been evaluated at the position of the beam centroid.This linearization of the octupole field around the local position of the beamcentroid results in a position-dependent quadrupole-like focusing of the beam.Since we have a linear map we can propagate the beam matrix from rightbefore the octupole to the position on the screen using (3.11). From this we

50

get the observable horizontal beam width on the screen as σ2x = σ(1,1) as

σ2x =

[KL(X2−Y 2)−1

]2σ11 +2L

[1+KL(Y 2−X2)

]σ12

+4KLXY[1+KL(Y 2−X2)

]σ13 +L2 [σ22 +4KXY (σ23 +KXY σ33)] .

(4.10)Thus we have the horizontal width at the location of the screen as a functionof octupole strength, distance and beam centroid position at the octupole field,which we assume all to be known, and the unknown elements of the beammatrix right before the octupole, i.e. σ11,σ12 and so on. We get similar ex-pressions for σy2 and σxy from the (3,3) and (1,3) elements of σ .

When we make the linear approximation of the octupole focusing we in-troduce some error. This error should be small provided that the beam size issmall compared to the curvature of the field. However, for a Gaussian beamwe can calculate the full analytical expression without approximations. Thehorizontal beam width is given by

σ2x =

⟨x2⟩−〈x〉2 (4.11)

where x is given by x = x+Lx′+ C3l(Bρ)L

(3xy2− x3

). This results in a 6th order

expression with many terms but since all expectation values are polynomialswith a Gaussian distribution we can calculate them using (4.3). We let K =C3l(Bρ) and finally get

σ2x = σ11 +L(2σ12 +Lσ22)+6KL

[2σ

213 +2Lσ13σ23 +2XY (σ13 +Lσ23)

+Y 2 (σ11 +Lσ12)−(X2−σ33 +σ11

)(σ11 +Lσ12)

]+3K2L2 [12XY 3

σ13 +36σ33σ213 +3σ11

(X4 +Y 4)+9σ11σ

233

−24σ11σ213−6σ

211σ33 +5σ

311−12XY σ13

(X2−5σ33 +3σ11

)+6Y 2 (4σ

213 +X2 (2σ33−σ11)+3σ11σ33−σ

211)

+6X2 (σ

233−4σ

213−σ11σ33 +2σ

211)].

(4.12)and with the same procedure we get similar expressions for σ2

y and σxy. Thedifference now is that we have a nonlinear dependence of the unknown beammatrix elements σi j, c.f. with (4.10) where we only have linear dependenceon σi j.

To determine the elements of the beam matrix we apply fits to the mea-sured transverse beam profile (σ2

x ,σ2y ,σxy) from the same data set as in the

Section 4.1. We define a χ2σxx to be minimized for the horizontal width:

χ2σxx =

n

∑i=1

[σmeas

xx −σ theoxx (εx,βx,αx,εy,βy,αy,r,κ,ψ,φ)

∆σ ixx

](4.13)

where σmeasxx is the measured horizontal beam width on the screen when the

beam was perturbed by the octupole component in the accelerating structure.

51

The expression for σ theoxx can be taken from either (4.10) or (4.12). Similarly,

we also define χ2σyy and χ2

σxy and make the fit to minimize the sum χ2f ocus =

χ2σxx +χ2

σyy +χ2σxy .

An issue when making a fit directly to the beam matrix elements σi j isthat the resulting beam matrix might not be positive definite, which it hasto be in order to represent a physical distribution. To circumvent this prob-lem we use a parameterization of the sigma matrix and fit to the parameters(εx,βx,αx,εy,βy,αy,r,κ,ψ,φ) instead. This parameterization ensures a posi-tive definite beam matrix. More information and derivation of the parameteri-zation can be found in Appendix B in Paper I.

Figure 4.4 shows the measured beam width on the screen as a function ofvertical beam centroid position together with the linear fit and the full ana-lytical fit. The fit problem is non-trivial, even when we assume the linearapproximation due to the parameterization of the beam matrix elements. Weused a random seed as start values for 100 iterations and among the convergentiterations we selected the solution with minimum χ2

f ocus. The resulting valuesof the parameters (εx,βx, . . .) varied considerably but yielded the same beammatrix. The problem of fitting the full analytical expressions is even worse.This time we could not find convergent solutions using the parameterization.Instead we used the fit result from the linear fit as start values and did a fit di-rectly to the beam matrix elements. As seen in Fig. 4.4 using the full analyticalexpression gave a small improvement to the fit.

To estimate the error bars we varied one parameter at the time while keep-ing other parameters constant and finding values that yielded a factor 4 in-crease in χ2

f ocus. We used a factor 4 instead of a factor 2 as in Section 4.1since we have twice the number of fit parameters. The resulting beam matrixelements, projected- and eigenemittances are presented in Table 4.2. To cross-check we compared the average measured beam sizes on the screen when therewas no RF in the accelerating structure with the result we get if we propa-gate the resulting beam matrix with a drift of length L, i.e. σ = MLσ f itMT

L .The resulting beam widths were (measured/propagated): σx = 0.11/0.13 mm,σy = 0.23/0.20 mm and σxy = 0.009/0.021 mm2 where we note a good agree-ment for the horizontal beam widths and almost no x–y correlation in bothcases.

Here we have shown how the full transverse beam matrix can be measuredusing a single octupole and a screen and we applied the method using theoctupole component of the CLIC accelerating structures. In the CLIC mainlinac this method might be difficult to use if there are no means of measuringthe transverse beam profile. However, one could foresee the possibility to havea screen somewhere prior to the entrance of the main linac.

As noted in the previous section from the nonlinear kicks from the octupolecomponent we also extracted information about the transverse offset of the

52

−2 −1 0 1 20

0.1

0.2

0.3

0.4

Vertical position [mm]

Ho

rizo

nta

l b

ea

m s

ize

[m

m2]

Data

Fit − linear

Fit − full

−2 −1 0 1 20

0.1

0.2

0.3

0.4

Vertical position [mm]

Ve

rtic

al b

ea

m s

ize

[m

m2]

Data

Fit − linear

Fit − full

−2 −1 0 1 20

0.1

0.2

Vertical position [mm]

Co

rre

latio

n t

erm

[m

m2]

Data

Fit − linear

Fit − full

Figure 4.4. The measured horizontal and vertical beam widths and the x–y correlationfor different vertical beam positions plotted together with the fits. The full analyticalexpression gave a small improvement of the fit compared to the linear approximation.

53

Table 4.2. Beam matrix fit results.

Parameter Symbol Value Unit

χ2 per DOF χ2N 7.0 no unit

Beam matrix elements σx 0.59±0.01 mmσxx′ -0.050±0.01 mm-mradσx′ 0.085±0.002 mradσy 0.28±0.02 mmσyy′ -0.016±0.001 mm-mradσy′ 0.067±0.004 mradσxy -0.034±0.008 mm2

σxy′ 0.026±0.001 mm-mradσx′y 0.005±0.001 mm-mradσx′y′ -0.0038±0.0003 mrad2

Eigenemittances ε1,norm 4.0 mm-mradε2,norm 0.8 mm-mrad

Projected emittances εx,norm 2.5 mm-mradεy,norm 3.6 mm-mrad

accelerating structure. This gave us the idea to utilize this for beam-basedalignment and this is the topic of next section.

4.3 Beam-based alignmentAlignment of the accelerating structures is crucial for CLIC since misalign-ments excites wakefields and emittance growth of the beam. Here we inves-tigate the feasibility of utilizing the nonlinear kicks from the octupole com-ponent of the RF fields in the CLIC accelerating structures as a beam-basedalignment method. The advantage of such a method is that it requires no addi-tional equipment.

MethodFollowing the notation used in Papers II and III we can express the positionshifts of the beam centroid ∆X = X −X and ∆Y = Y −Y due to an octupolekick in complex form as

∆X− i∆Y = KL⟨(x+ iy)3

⟩(4.14)

where K = C3l(Bρ) is the integrated octupole strength normalized to beam energy.

Please note that here we use a different sign-convention compared to the previ-ous sections, this corresponds to an electron moving in the positive z-direction.

54

If the beam has a horizontal offset X and vertical offset Y the beam positionshifts can be calculated as

∆X− i∆Y = KL⟨[(

x− X)+ i(y− Y

)]3⟩. (4.15)

If we expand (4.15) and calculate the expectation values we end up with

∆X− i∆Y −KL(X + iY )3 = KL{

3(X + iY

)(σ

2y −σ

2x −2iσxy

)−(X + iY

)3

+3[(

X + iY)2−

(σ

2y −σ

2x −2iσxy

)](X + iY )−3

(X + iY

)(X + iY )2

}(4.16)

which is similar to what we had in previous section but now the offsets X andY are included. The expression in (4.16) can be written as a linear equation

z = k1 + k2 (X + iY )+ k3 (X + iY )2 (4.17)

where the parameters (k1,k2,k3) are defined as

k1 = KL[3(X + iY

)(σ

2y −σ

2x −2iσxy

)−(X + iY

)3]

k2 = 3KL[(

X + iY)2−

(σ

2y −σ

2x −2iσxy

)]k3 =−3KL

(X + iY

) (4.18)

and the left hand side is given by

z = ∆X− i∆Y −KL(X + iY )3 . (4.19)

We assume that the integrated octupole strength and the distance from theoctupole field to the location of the measurement of the positions shifts, i.e.KL, are known. Then the sought-after offsets X and Y are directly retrievablefrom k3, such that

X =−Re(k3)

3KL

Y =− Im(k3)

3KL.

(4.20)

If we make a series of N measurements of the beam position shifts at differenttransverse positions (X j + iYj) inside the octupole field we can write the fitproblem in matrix form as

z1z2......

zN

=

1 (X1 + iY1) (X1 + iY1)

2

1 (X2 + iY2) (X2 + iY2)2

......

......

......

1 (XN + iYN) (XN + iYN)2

k1

k2k3

. (4.21)

55

Figure 4.5. Schematic of the alignment procedure. The beam is moved transverselywith respect to the octupole field in the accelerating structure by either steeringthe beam with magnets or by moving the girder which the accelerating structure ismounted on. A beam position monitor (BPM) is used for measuring the beam posi-tion at a distance L from the accelerating structure.

The measurement procedure is as follows. Move the beam transversely withrespect to the octupole field, i.e. the accelerating structure. This can be doneeither by steering the beam with magnets or, better, by moving the accelerat-ing structure itself. In CLIC the accelerating structures will be mounted onmovable girders in two-beam modules [36] and the beam will first be alignedwith respect to the quadrupole magnets and then aligned to the acceleratingstructures by moving the girders. Moving the accelerating structures insteadof steering the beam has the additional benefit of not introducing dispersiveeffects. The beam position can be measured using a BPM after the accelerat-ing structure and the position shifts are measured by comparing beam positionwhen there is RF power and when there is no RF power in the acceleratingstructure. This can be achieved by operating the drive beam accelerator athalf the repetition rate or by using RF ON/OFF mechanism of the PETS [76].Figure 4.5 shows a schematic of the setup.

In order to assess the feasibility of using this method for the CLIC mainlinac we need to investigate the achievable tolerances. The fit problem in 4.21is a linear fit in parameters (k1,k2,k3) and therefore we can propagate the er-rors in the parameters (∆X ,∆Y ,X ,Y,KL) by calculating the covariance matrix,for a detailed presentation c.f. Paper II. Errors in ∆X ,∆Y come from uncer-tainties in the measured position shifts, this is ultimately limited by

√2σBPM

where σBPM is the BPM resolution. The error in KL comes predominatelyfrom uncertainty in the integrated strength, which in turn comes from RF jit-ter. Assuming a scan procedure where the girder is moved ±1 mm with 20scan steps horizontally and 20 scan steps vertically, the achievable tolerancefor the offsets are close to 1 µm. Here we have assumed the tolerances forCLIC in the beginning of the main linac which are summarized in Table 4.3.The 1 µm tolerance is encouraging since it is below the required alignment tol-erance of 5 µm for CLIC and comparable to the required resolution of 3.5 µmfor the wakefield monitors [43].

In CLIC a single PETS feeds RF power to two accelerating structures thatare mounted together. Thus a minimum of two structures are powered at the

56

Table 4.3. Assumed values, errors and tolerances [40].

Parameter Unit Value

BPM resolution σBPM nm 50Integrated octupole strength, C3l kTm/m3 73.5Distance to BPM, L m 3Half-distance between the centers of twoaccelerating structures, ∆L m 0.125Error in position shift, σ

∆X , σ∆Y nm 71

Error in KL, σKL rel. 0.1%

Figure 4.6. If two accelerating structures are powered at the same time we need todisentangle the offsets from nonlinear kicks due to two octupole fields.

same time and the measured octupole kicks are due to two octupole fields,see Fig. 4.6. If the distance between the two octupole fields, ∆L, is smallcompared to the distance to the BPM, L, we can approximate the two kicks tobe independent. Then we simply add the two kicks and (4.16) becomes

∆X− i∆Y −2KL(X + iY )3

= KL{

3(

1+∆LL

)ZAc+3

(1− ∆L

L

)ZBc−

(1+

∆LL

)Z3

A−(

1− ∆LL

)Z3

B

+ 3[(

1+∆LL

)Z2

A +

(1− ∆L

L

)Z2

B−2c](X + iY )

−3[(

1+∆LL

)ZA +

(1− ∆L

L

)ZB

](X + iY )2

}(4.22)

where we have introduced ZA = XA + iYA (offset in the first octupole field),ZB = XB + iYB (offset in the second octupole field) and c = σ2

y −σ2x − 2iσxy.

This is again a fit in the form of (4.17) and the fit parameters k2 and k3 are nowgiven by

k2 = 3KL[(

1+∆LL

)Z2

A +

(1− ∆L

L

)Z2

B−2c]

k3 =−3KL[(

1+∆LL

)ZA +

(1− ∆L

L

)ZB

] (4.23)

57

and contains the two offsets. Due to the quadratic dependence on ZA and ZBin k2 we cannot solve for the individual offsets. Instead we can write thesum of the two offsets as Z = ZA + ZB and we let ∆Z′ =

∣∣ZB− ZA∣∣ denote the

magnitude of the difference of the offsets. Then we have

Z =−k3

3KL± ∆L

L∆Z′

∆Z′ =

[

1−(

∆LL

)2]−1[

2k2

3KL−(

k3

3KL

)2

+4c

]12

.

(4.24)

Expressed in this way the misalignment of the two accelerating structures isdivided into two parts: a global misalignment Z and a relative misalignment∆Z′ which measures the magnitude of the angle between the two structures.Now the alignment is carried out by first minimizing ∆Z′, i.e. ensuring relativealignment, and then minimizing the global offset Z. We note that if ∆Z′ = 0then ZA = ZB = Z/2.

We follow the same procedure to analyze the tolerance for the global andrelative offsets. Since the fit problem is identical to what we had before, wehave the same errors in k2 and k3 but now we have to propagate these errors toZ and ∆Z′. The expression for ∆Z′ contains a square root which means that theerror increases as ∆Z′ decreases. Thus, at some point the error will be equalto the value and we find that to happen at 26 µm and this is the most precisewe can reduce the relative offset. The global offset has an error of similarmagnitude as before and is found to be 1.5 µm.

ExperimentWe tested this method at CTF3 during 2016 when a prototype CLIC two-beammodule was installed in TBTS . At the time four accelerating structures weremounted on the girder but RF power was only delivered to two of the fourstructures. The girder is controlled by 6 stepper motors and can be moved in5 degrees of freedom, i.e. all rotations and x–y translation. By moving thegirder parallel2 to the beam axis we can test the proposed alignment methodfor CLIC.

We set up the probe beam to operate at a repetition frequency twice the rep-etition frequency of the drive beam. Downstream of the accelerating structurewe measured the beam on a screen—same screen as in the experiments in theprevious sections. For this experiment, using a BPM would also have beenpossible but at the time none of the suitable BPMs was sufficiently calibrated.From the measured RF power we rescale the integrated octupole strength C3l

2We also did a measurement where we rolled the girder and from the beam shape of a beamperturbed by the octupole component we tried to determine the roll angle of the acceleratingstructure. This is the topic of Paper IX.

58

Table 4.4. Experimental parameters.

Parameter Unit Value

Distance, L mm 5106Beam energy MeV 194CLIC nominal power, Pnom MW 46Average RF power, Pave MW 18.6Uncertainty RF power MW 1.8Integrated octupole strength (Pnom), C3l kT/m2 73Girder position interval µm ±1000

Figure 4.7. The measured position shifts at different girder positions. Left: the girderwas moved horizontally and we observe the expected third-order dependence for thehorizontal position shifts. Right: the girder was moved vertically and now we observea third-order dependence for the vertical position shift.

at nominal power as√

Pave/Pnom in order to get KL. The experimental param-eters are summarized in Table 4.4.

The girder was first moved horizontally and then vertically over a total of 21different girder positions. At each girder position we collected a total 40 pulses(i.e. 20 pulses with RF and 20 without RF ). From the screen images of thetransverse beam profile we extracted the beam centroid position from Gaussianfits. For each girder position we calculated the average beam positions andused the standard deviation as a measure of the uncertainty. Figure 4.7 showsthe resulting position shifts for the different girder positions and the resultingfit.

Finally, applying the fitting procedure described above we end up with re-sulting global offset of the two structures:

Xmeas = 392±53µm

Ymeas = 6±92µm

where the uncertainties come from error propagation. The two main sourcesof uncertainty in this measurement were uncertainty in the RF power due tofluctuations in the delivered RF power from the drive beam and uncertainty in

59

the measured beam positions. We observe a larger uncertainty in the verticaloffset than the horizontal offset, which is due to a larger uncertainty in thevertical beam position, compare the error bars in Fig. 4.7. This was due toa larger vertical beam jitter of the probe beam at the time of the experiment,which was traced back to jitter from the laser of the photoinjector. When weaverage the position over many pulses this jitter results in larger error bars.

There are practical considerations for using this method in CLIC . For onething as the energy of the main beam increases the effect of the octupole kickswill be smaller and smaller, which will increase the uncertainty of the method.One way might be to run the whole accelerator on zero-crossing. Even if thismethod would not fully replace the wakefield monitors it nonetheless servesas an important complement. This method could be particularly useful duringcommissioning of the main linac as a cross-check of wakefield monitor signalsand furthermore it might also be useful as a backup method during normaloperation in the event of failure of a wakefield monitor.

In this chapter we investigated several beam-based diagnostic methods uti-lizing the nonlinear kicks from the octupole component of the RF fields of theCLIC accelerating structure. The methods also extend to scenarios with nor-mal octupole magnets and need not be octupole kicks from RF fields. In ad-dition to there are many other potential applications for the technology basedon 12 GHz CLIC accelerating structures [77] where these methods also mightuseful.

60

5. Beam dynamics in nonlinear fields

In this chapter we apply the methods from Section 3.3 to solve some specificdynamical problems for circular machines when nonlinear magnetic fields arepresent. In Paper IV we investigate optimum placement for octupoles wherethey compensate amplitude-dependent tune-shifts but do not drive fourth-orderresonances. In Paper V we look for optimal setups that allow for independentcontrol of different resonance driving terms and we use sextupoles as an ex-ample. The simulations presented in this section are done in a code that canhandle Hamiltonians, Lie algebra, normal forms and do particle tracking. Thiscode is written in MATLAB [78] and more information is found in Paper VII.

5.1 Compensating amplitude-dependent tune-shiftWhen sextupoles are placed in a circular accelerator to compensate chromatic-ity (c.f. Fig. 3.5) they have the undesired side-effect of introducing anothertune-shift: amplitude-dependent tune-shift, which may push particles on largeramplitudes onto resonances and decrease the dynamic aperture of the ma-chine. A small dynamic aperture is problematic since injection becomes dif-ficult and this is why octupole magnets are introduced in a number of storagerings to compensate the amplitude-dependent tune-shift. It should be notedthat there are ways to use several families of sextupoles and place them insuch a way that for instance both chromaticity and amplitude-dependent tune-shift are compensated. However, for a lattice such as MAX IV 3 GeV stor-age ring it was found beneficial to optimize the sextupoles with regards toother constraints and add octupoles to the design for compensating amplitude-dependent tune-shift [39]. When octupoles are introduced they also drive ad-ditional resonances, in particular they drive fourth-order resonances to firstorder in octupole strength, which is undesirable since it may again reduce re-gion of stability. Here we investigate octupole configurations that address theamplitude-dependent tune-shift without driving fourth-order resonances.

Octupole placementLet us first consider two pairs of one-dimensional octupoles, in normalizedphase space, separated by phase advances φ1 and φ2 with respect to a refer-ence point in the middle, see Fig. 5.1. The Hamiltonian for an octupole is

61

Figure 5.1. Two octupole pairs separated by phase advances φ1 and φ2 with respect toa reference point in the middle.

given by H = kx4 where the excitation k is related to the integrated octupolestrength by k = k3l/4!. We consider both octupoles of the inner pair to havethe same strength k1 and move them to the reference point via the similaritytransformation. We consider only the fourth-order part of the effective Hamil-tonian and use only the first order in the CBH formula. This gives us

H = k1(xcosφ1 + x′ sinφ1)4 + k1(xcosφ1− x′ sinφ1)

4

= 2k1{

x4 cos4φ1 +6x2x′2 cos2

φ1 sin2φ1 + x′4 sin4

φ1} (5.1)

where we observe that terms involving x3x′ and xx′3 cancel due to symme-try. Then we add a second pair with strength k2 and phase advance φ2 to thereference point and calculate the new H. The amplitude-dependent tune-shiftcomes from the invariant part of the Hamiltonian: the terms that only dependon the action J, as defined in (3.13). For this octupole configuration to onlyaddress amplitude-dependent tune-shift and not drive any resonances we re-quire that the effective Hamiltonian only depends on action variable J. Sincethe action is proportional to (x2 + x′2) and the octupole Hamiltonian is fourthorder in phase space variables, this requirement is equivalent to having theHamiltonian in the form

(2J)2 = (x2 + x′2)2 = x4 +2x2x′2 + x′4. (5.2)

In Paper IV we find a condition for the ratio between the octupole excitations,k1/k2, and the phase advances, (φ1,φ2), that achieves a Hamiltonian in the cor-rect form. We find a particularly simple configuration where φ1 = 0 and thusthe inner pair can be replaced with a single octupole. We end up with a tripletof three octupoles with equal strengths and 60◦ phase advance in between.

A real transverse octupole field is two-dimensional and we must use theHamiltonian H = k(x4−6x2y2+y4). If we consider a 60◦ triplet again we findthat it drives all three amplitude-dependent tune-shift terms J2

x , J2y and JxJy as

desired but in addition also a phase-dependent term cos(2ψx−2ψy) which willdrive the 2Qx− 2Qy resonance. In other words, the 60◦ triplet addresses allamplitude-dependent tune-shift terms but drives one fourth-order resonance.We can cancel the remaining 2Qx− 2Qy resonance driving term by adding asecond triplet with the same octupole strengths, a configuration we named asix-pack. The condition for this cancellation is that the horizontal and vertical

62

Figure 5.2. The six-pack configuration consists of two 60◦ triplets. If all octupoleshave the same strength and the horizontal and vertical phase advance between thetriplets differ by 90◦ all fourth order resonances cancel.

phase advances between the two triplets differ by 90◦. Figure 5.2 shows aschematic of the six-pack configuration.

A triplet or a six-pack affects all three amplitude-dependent tune-shift termsand the ratio between these terms depend on the ratios of the horizontal andvertical beta functions at the location of the octupoles. In order to have in-dependent control of the three terms we need three triplets or three six-packsplaced at locations with different ratios of the beta functions. This can eas-ily be achieved by placing the octupoles at different locations in a 60◦ FODOcell.

SimulationTo confirm the features of the different octupole configuration we set up asimple simulation where we have a section with the octupole configurationfollowed by a rotation in normalized phase space. We test the 3 triplets andthe 3 six-packs configurations, and for comparison, also a setup that consists ofthree single octupoles. For the three configurations, we track a single particleand scan over horizontal tunes while keeping the vertical tune fixed. For eachtracking we calculate the smear [79] as

S =

√〈J2〉−〈J〉2

〈J〉2(5.3)

which is the normalized rms of the action. Close to a resonance, the motionof the particle becomes irregular and the smear increases. In order to identifythe resonance we plot the smear together with a tune diagram in Fig. 5.3. Asexpected the three octupoles drive several fourth-order resonances, the tripletsdrive only the 2Qx−2Qy resonance and the six-packs do not drive any of thefourth order resonances. The configurations behave as expected and the nextstep is to test them in a more realistic case.

We set up an extended simulation model consisting of a racetrack latticewith two 180◦ bending arcs and two straight sections. Each arc is made of 9FODO-cells and each FODO-cell contains two 10◦ bending dipoles and twosextupoles. First we simulate without the sextupoles and by varying parti-cle energy and calculating the tunes we retrieve the chromaticity. Then we

63

Qx

0 0.1 0.2 0.3 0.4 0.5

Qy

0.36

0.37

0.38

0.39

0.4

0.41

0.42

0.43

0.44

3 octupoles

3 triplets

3 six-packs

Figure 5.3. We simulate the different configurations for different horizontal tuneswhile keeping the vertical tune fixed, in other words scanning across a horizontal linein the tune diagram (left). By calculating the smear and plotting together with the tunediagram (right) we can identify the resonances corresponding to the different peaks.

set the sextupoles to compensate the chromaticity, which can be done analyt-ically [66]. The purpose is to get realistic sextupole strengths. We have twostraight sections connecting the two arcs, one we use as a trombone where wecan adjust the overall tune of the lattice and one section containing the differ-ent octupole configurations. Figure 5.4 shows a schematic of the simulationmodel.

Using the procedure described in Section 3.3, i.e. iterative use of (3.21)and (3.22), we can move all the sextupoles to the reference point and describethe whole lattice as e−:H: R where R is linear map in normalized phase spaceand H contains the effective Hamiltonian for all the sextupoles. The nextstep is to write this map in a normal form according to (3.23) and from thefourth-order part of C we get the amplitude-dependent tune-shifts from all thesextupoles. Then we find the required octupole strengths needed to compen-sate these terms. We simulate with three different configurations: 3 octupoles,3 triplets and 3 six-packs.

In order to test the compensation we perform multi-turn single particletracking for particles with different amplitudes. We choose starting ampli-tudes with x = y. From the turn-by-turn horizontal and vertical positions wecalculate the tunes from a discrete Fourier transform [80]. For each startingamplitude we track 215 turns. Figure 5.5 shows the resulting tunes at differentamplitudes for the case when there were no octupoles and the three cases withdifferent octupole configurations for compensating the amplitude-dependenttune-shifts. We also plot the tune-shift, including second-order amplitude-dependent tune-shift, expected from the normal form analysis. It is clear thatusing the octupoles compensates the amplitude-dependent tune-shifts but wealso observe that using only three octupoles made dynamic aperture worse,

64

Arc

Trombone

NPS

RPSReference

point.

Octupole section

Figure 5.4. The racetrack lattice used in the simulation consists of two 180◦-bendingarcs and two straight sections. Each arc consists of 9 FODO-cells with bending dipolesand sextupoles for chromaticity correction. The straight sections are simulated innormalized phase space (NPS) and the arcs in real phase space (RPS). One straightsection is used for the octupoles and the other section is used as a "trombone" wherethe overall tune of the machine can be adjusted.

i.e. the red crosses end at much lower amplitudes than the sextupoles. Addingonly three octupoles made more harm than good whereas the three triplets andthe three six-packs compensated the tune-shifts and improved stability.

Figure 5.6 shows the dynamic aperture in the upper-half x–y plane, withsextupoles only and with the three octupole configurations, again it is clearthat three octupoles reduced stability region. Furthermore, there is not a sub-stantial improvement of using six-packs compared to using triplets. Thus weconclude that in this particular case the cancellation of the remaining reso-nance was not important. However, this is not in general the case, in this sim-ulation the resonances driven by the sextupoles dominated and the additionalresonance from the triplets compared to the six-packs made little difference.For comparison we simulated the same octupole configurations with the sameexcitations but with all sextupoles turned off, see right figure in Fig. 5.6. Nowthe progressive improvement of the dynamic aperture is clearly visible.

In this section we investigated setups that adjust the amplitude-dependenttune-shift without driving fourth order resonances. In the next section we in-vestigate different setups where, instead of cancelling, we control resonancedriving terms independently. In particular, we look for sextupole configura-tions with optimum control of the third order resonance driving terms.

5.2 Optimum resonances control knobsWhen sextupoles are introduced in a circular accelerator they drive third-orderresonances to first order in sextupole excitation. In this section we investigatesetups that allow for independent control of each individual resonance driv-

65

Amplitude x0 0.02 0.04 0.06 0.08 0.1 0.12

Horizonta

l tu

ne

0.4

0.405

0.41

0.415

0.42

0.425

0.43

0.435

No octupolesNormal Form3 octupolesNormal Form3 tripletsNormal Form3 six-packsNormal Form

Amplitude x0 0.02 0.04 0.06 0.08 0.1 0.12

Vert

ical tu

ne

0.36

0.365

0.37

0.375

0.38

0.385

No octupolesNormal Form3 octupolesNormal Form3 tripletsNormal Form3 six-packsNormal Form

Figure 5.5. The horizontal and vertical tunes from multi-turn tracking at differentx = y starting amplitudes. We simulated without octupoles and with the three differentoctupole configurations. The solid lines show the expected tune-shifts from normalform analysis. When there are no octupoles we observe a considerable tune-shift withamplitude due to the sextupoles. Using only three octupoles reduced stability whereasthe three triplets and three six-packs configurations enhanced stability.

x

-0.2 -0.1 0 0.1 0.2

y

0

0.05

0.1

0.15

0.2 No octupoles

3 octupoles

3 triplets

3 six-packs

x

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

y

0

0.05

0.1

0.15

0.2

0.25

3 octupoles

3 triplets

3 six-packs

Figure 5.6. Left: Dynamic apertures in the upper-half of the x–y plane without oc-tupoles and with the three different octupole configurations. Right: the same octupoleconfigurations but all the sextupoles turned off. Now the improvement of using six-packs over triplets is more substantial.

66

ing term (RDT), in particular we look for optimum setups with orthogonalresonance control knobs.

Optimum placement for sextupolesThe method for determining the RDTs for a one-dimensional sextupole waspresented in (3.34) and here we do the same but for a sextupole moved viathe similarity transform. The Hamiltonian for a sextupole in normalized phasespace is given by H = kβ 3/2x3, where we define k = k2l

6 for brevity. We movethe Hamiltonian to a new location separated by phase advance φ and with thenew coordinate x = xcosφ + x′ sinφ we have

H = kβ3/2(xcosφ + x′ sinφ)3. (5.4)

This expression can be expanded and we use trigonometric identities such ascos3(φ) = 1

4 [cos(3φ)+3cos(φ)] in order to avoid powers of the trigonometricfunctions. We use (3.13) to transform into action–angle variables and, finally,we have the Hamiltonian at the new location given by

H =k4

[cos(3φ)(2Jβ )3/2 cos(3ψ)+ sin(3φ)(2Jβ )3/2 sin(3ψ)

+3cos(φ)(2Jβ )3/2 cos(ψ)+3sin(φ)(2Jβ )3/2 sin(ψ)] (5.5)

where we note that we have four RDTs: cosine and sine of the 3Q and Qresonances. In order to individually control these four terms we need a min-imum of four sextupoles. Figure 5.7 shows a schematic of a setup with foursextupoles placed at phase advances φ1, φ2 and φ3 to the reference point thatwe choose to be at the location of the fourth sextupole. We propagate the sex-tupoles to the reference point as in (5.5) and the resulting third-order part ofthe Hamiltonian is given by first-order CBH, which means that we only haveto add the four contributions. We assume that all four sextupoles are placed atlocations with equal beta functions and, as an example, for the cos(3ψ) termwe find

Hcos(3ψ) =14(2Jβ )3/2 cos(3ψ) [k1 cos(3φ1)+ k2 cos(3φ2)+ k3 cos(3φ3)+ k4]

(5.6)and similarly for the other terms. In fact, the coefficients for the third-orderterms in the Hamiltonian can be written as a linear system ~C = M~k where thevector ~C contains the coefficients of the different terms, M is a response matrixthat contains the trigonometric functions and~k is a vector with the sextupoles

67

Figure 5.7. Four sextupoles separated by phase advances φ1, φ2 and φ3 to the referencepoint, which is chosen to be the location of the fourth sextupole.

strengths. Explicitly we haveC{1

4(2Jβ )3/2 cos(3ψ)}C{1

4(2Jβ )3/2 sin(3ψ)}C{3

4(2Jβ )3/2 cos(ψ)}C{3

4(2Jβ )3/2 sin(ψ)}

=

cos(3φ1) cos(3φ2) cos(3φ3) 1sin(3φ1) sin(3φ2) sin(3φ3) 0cos(φ1) cos(φ2) cos(φ3) 1sin(φ1) sin(φ2) sin(φ3) 0

k1k2k3k4.

(5.7)

In order to find sextupole strengths that result in desired coefficients of theRDTs we have to invert the system in (5.7). We realize that the condition ofthis problem depends on the condition number of the response matrix M. Thusthere are certain phase advances φ1, φ2 and φ3 that are better suited for achiev-ing control of the independent RDTs. We find that the optimal placement ofthese four sextupoles is φ1 = 135◦, φ2 = 90◦ and φ3 = 45◦, which means thatthe separation between consecutive sextupoles is 45◦. These phase advancesresults in the following response matrix

M =

1√2

0 − 1√2

11√2−1 1√

20

− 1√2

0 1√2

11√2

1 1√2

0

(5.8)

which has condition number equal to one and implies that all columns androws of the matrix are orthogonal. Furthermore, this means that the eigenval-ues of the matrix are equal and that the same magnitude of a resonance requirethe same magnitude of ~k, in other words, all resonances are treated equallywell.

Here we showed the principle of finding optimal resonance control knobsfor one-dimensional sextupoles. In Paper V we extend the analysis to two-dimensional sextupoles and in that case there are ten RDTs but two have thesame phase-dependence and become linearly dependent if we assume equalbeta functions. Therefore, we only consider eight RDTs and a setup consistingof eight sextupoles to control them independently. Following the procedureabove we find an optimal setup based on φx = 135◦ and φy = 45◦ phase ad-

68

vance between consecutive sextupoles. In order to test this sextupole schemeand the ability to control the third-order RDTs independently we do trackingsimulations.

SimulationWe set up a simulation where we use the same racetrack model as in Sec. 5.1,c.f. Fig. 5.4, but we place our eight sextupoles in the straight section instead ofoctupoles. The sextupoles for chromaticity compensation in the arcs will drivethird-order resonances and by using the similarity transformation we move allsextupoles to the reference point and add all the contributions to the third orderresonances. Then we can set the eight sextupoles in the straight section byinverting ~C = M~k and control the RDTs independently. In particular, we canchange the magnitude and turn off an individual RDT.

We adjust the phase advance in the trombone to set the tunes to Qx = 0.447and Qy = 0.293, which is close to the Qx + 2Qy resonance. For each step inthe simulation we track 11 particles at different starting amplitudes along thex = y line for 1000 turns. Figure 5.8 shows Poincaré surfaces of a section, i.e.horizontal and vertical phase space portraits, together with bar plots displayingthe magnitude of the different RDTs. In the first row of Fig. 5.8 we havenot added any correction, i.e. the eight sextupoles in the straight section areall turned off and what we see is the resonances driven by the sextupoles inthe arcs. In the second line we set the eight sextupoles to turn off the 3Qxresonance and observe only small effect on the dynamics. In the third row wehalf the magnitude of the Qx + 2Qy resonance and, finally, in the fourth rowwe turn it off completely. This has a distinct impact on the dynamics and thephase space portraits in the fourth row shows circular trajectories.

We demonstrated that we can change the amplitude of a resonance drivingterm but since we can control the cosine and sine terms independently, we canalso control the phase of the resonances. In Paper V we show a simulationwhere the phase of a single RDT is varied while keeping all others RDTsfixed. In this section we investigate optimal schemes for controlling third-order resonances using sextupoles. However, the reasoning about resonancecontrol knobs extends to other multipoles. We are working on finding optimalschemes for skew sextupoles, octupoles and decapoles.

In this chapter we have investigate how nonlinear fields interact and howthey affect the dynamics. By placing, for instance, sextupoles or octupolesin certain ways we can achieve either cancellation of or optimum control ofRDTs.

69

x-0.02 0 0.02

x'

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

y-0.02 0 0.02

y'

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

J x(1/

2) J y[Q x

-2Qy]

J x(1/

2) J y[Q

x]

J x(3/

2) [Q x]

J x(1/

2) J y[Q x

+2Q y

]

J x(3/

2) [3Qx]

Ampl

itude

[m-1

/2]

0

1

2

3

4

5

6

x-0.02 0 0.02

x'

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

y-0.02 0 0.02

y'

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

J x(1/

2) J y[Q x

-2Qy]

J x(1/

2) J y[Q

x]

J x(3/

2) [Q x]

J x(1/

2) J y[Q x

+2Q y

]

J x(3/

2) [3Qx]

Ampl

itude

[m-1

/2]

0

1

2

3

4

5

6

x-0.02 0 0.02

x'

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

y-0.02 0 0.02

y'

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

J x(1/

2) J y[Q x

-2Qy]

J x(1/

2) J y[Q

x]

J x(3/

2) [Q x]

J x(1/

2) J y[Q x

+2Q y

]

J x(3/

2) [3Qx]

Ampl

itude

[m-1

/2]

0

1

2

3

4

5

6

x-0.02 0 0.02

x'

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

y-0.02 0 0.02

y'

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

J x(1/

2) J y[Q x

-2Qy]

J x(1/

2) J y[Q

x]

J x(3/

2) [Q x]

J x(1/

2) J y[Q x

+2Q y

]

J x(3/

2) [3Qx]

Ampl

itude

[m-1

/2]

0

1

2

3

4

5

6

Figure 5.8. Portraits showing x–x′ and y–y′ phase spaces for 11 particles startingat different amplitudes and tracked for 1000 turns. The amplitudes of the differentresonances are shown in the bar plot on the right. We change the magnitude of asingle resonance while keeping all others fixed.

70

6. Conclusions

Particle accelerators are important tools for experimental physics, in particularparticle physics, but have also many other applications. The development andimprovement of such machines rely on progress and understanding of accel-erator physics. To control a beam of particles under extreme conditions it isparamount to have information about the beam, which is the topic of beamdiagnostics. In circular accelerators beams are stored for a long time, whichrequires long-term stability and understanding of beam dynamics. To first or-der dynamics in accelerators is linear, but as the performance of the machinesis enhanced, the nonlinear effects become more and more important. Thisstudy was set out to investigate beam diagnostics and dynamics in nonlinearfields.

The Compact Linear Collider (CLIC) is a proposed, linear electron–positroncollider based on normal-conducting accelerating structures with high gradi-ent. Due to a four-fold symmetry from radial waveguides connected to thecells of the accelerating structure there is a co-propagating octupole compo-nent of the RF fields. This means, in addition to the accelerating field we alsohave a nonlinear field that is shifted in phase by 90◦. We made use of thisco-propagating nonlinear field to extract information about the beam.

The beam-based diagnostics methods were tested at the CLIC test facilityCTF3 at CERN. We measured beam position shifts from the nonlinear kicksdue the octupole field while varying transverse beam position and from thiswe determined the octupole component and compared to expected values fromsimulations. Interestingly, it turned out that much more information could beextracted from these measurements. From the measured beam widths we madefits to analytical expressions and could determine the full transverse beam ma-trix. By using nothing more than an octupole field, two steering magnets anda screen, the transverse distribution of the beam was fully characterized.

Beam alignment is essential for CLIC since the beam is sensitive to wake-fields due to the short bunch spacing. Misalignments cause deterioration ofbeam quality and ultimately loss in luminosity and performance of the col-lider. The beam must be aligned inside the accelerating structures with mi-crometer precision. By utilizing the nonlinear kicks from the octupole com-ponent we can determine the misalignment and we investigated the usage ofsuch a method for CLIC and tested it experimentally at CTF3. We demon-strated how nonlinear fields can be used for beam diagnostics but nonlinearfields also affect the dynamics of particle accelerators, which is the topic ofthe second part.

71

The second part of this thesis is about nonlinear beam dynamics for circu-lar accelerators. Making use of the powerful tools of Hamiltonian formalism,Lie algebra and normal forms it is possible to analyze the nonlinear effectson the dynamics from multipole magnetic fields. We investigate different set-ups where octupole magnets are placed in such a way that they only addressthe amplitude-dependent tune-shifts but do not drive additional fourth-orderresonances. In a second example we instead looked for setups where differ-ent resonance driving terms are controlled independently. As an example wefound optimal setups for sextupoles with orthogonal knobs for the differentthird-order resonance driving terms.

Nonlinear effects become prominent as we push the limits of particle ac-celerators and thus nonlinear magnets are needed for compensation. Whendesigning future accelerators these magnets should be placed in an optimumway. By carefully placing the nonlinear magnets, they can be arranged in sucha way that many undesired nonlinear effects cancel, as we showed with theoctupoles for tune-shift compensation, or, conversely, arranged in such a waythat different resonances can be controlled independently, as in our exampleusing sextupoles. This kind of knowledge can guide us when designing thenonlinear dynamics of an accelerator in ways a blind global optimizer can-not. There might be many different constraints to take into account but it isclear that learning more about how multipole magnets interact and affect thedynamics allows us to make wiser choices when designing new particle accel-erators.

72

Summary in SwedishSammanfattning på svenska

Avhandlingens titel på svenska: Stråldiagnostik och stråldynamik i icke-linjärafält. Forskningen innefattar både experimentellt och teoretiskt arbete inomacceleratorfysik.

Partikelfysik och partikelacceleratorerI början av det förra seklet använde Ernest Rutherford en stråle av alfa-partiklarfrån en radioaktiv källa för att studera hur dessa partiklar sprids när de träf-far en tunn guldfilm. Resultaten visade att alfa-partiklarna ibland spreds medstora vinklar och från detta kunde Rutherford dra slutsatsen att atomen är upp-byggd av en liten, tung kärna omgiven av lätta elektroner. Detta experimentvar det första i sitt slag där en partikelstråle användes för att studera subatomärfysik. Men en radioaktiv källa är svår att kontrollera och det är problematisktatt nå hög intensitet. Då föddes idén att konstruera en dedikerad maskin—enpartikelaccelerator.

Partikelacceleratorer har många olika tillämpningar men har framförallt an-vänts till att studera universums materia på minsta skala. Den så kallade stan-dardmodellen för partikelfysik formulerades successivt under 1900-talets an-dra hälft och växte fram i ett växelspel mellan teori och experiment. Modellerför partikelfysik kan testas genom att accelerera partiklar och låta dem kollid-era på ett kontrollerat sätt. I den höga energitätheten från kollisionen bildasnya partiklar. Dessa partiklar och deras sönderfallsprodukter mäts och stud-eras i en detektor och på detta sätt kan interaktionerna mellan fundamentalapartiklar studeras i detalj. Existens av vissa partiklar förutsågs från teorin ochkunde sedan sökas i partikelkollisioner. Allt eftersom pusslet har lagts har vigått till högre och högre energier genom att bygga större och större acceler-atorer. Världens hittills största partikelaccelerator är Large Hadron Collider(LHC) med 27 km i omkrets belägen vid det europeiska forskningscentretCERN. Den senaste partikeln, och sista pusselbiten i den nuvarande formen avstandardmodellen, är Higgsbosonen som upptäcktes 2012 från proton–proton-kollisioner vid LHC.

Även om standardmodellen är en lyckad teori vet vi att den är ofullständig.Det finns ett flertal astronomiska observationer som inte kan förklaras medden vanliga "synliga" materian. Dessa okända entiteter kallas mörk materiaoch mörk energi och ligger utanför standardmodellen. Ett annat exempel på

73

problem med standardmodellen är att neutriner förutsägs vara masslösa fastvi vet från observationer att neutriner har massa. Det finns många teoretiskalösningar på standardmodellens tillkortakommanden och ett exempel är super-symmetri som är en utökning av standardmodellen baserad på extra symme-trier. Denna teori förutsäger existens av flera partiklar men än så länge haringa kunnat observeras experimentellt.

För att följa upp mätningarna vid LHC och framförallt kunna studera Higgs-bosonen i detalj behövs precisionsmätningar. Hadronkolliderare, som LHC, ärbra verktyg för att hitta nya partiklar eftersom hadroner, som exempelvis pro-toner, är sammansatta partiklar och det är kvarkarna och gluonerna som inter-agerar vid kollisionerna. Leptoner, som till exempel elektroner och positroner,är å andra sidan fundamentala, punktformiga partiklar och lämpar sig bät-tre för precisionsmätningar. En kandidat för att göra precisionsmätningar påHiggsbosonen är Compact Linear Collider (CLIC) som är en föreslagen lin-jär elektron–positron-kolliderare vid CERN. Varför en linjär accelerator? Närladdade partiklar accelereras vinkelrätt mot sin färdriktning utsänds elektro-magnetisk strålning som kallas synkrotronstrålning, vilket innebär att partik-lar i en cirkulär accelerator förlorar energi när de böjs i magneterna. Dettaär en effekt starkt beroende av partiklarnas energi och massa. Lätta partiklarså som elektroner utsänder väldigt lätt synkrotronstrålning och det är därförsvårt att nå hög energi med en cirkulär accelerator och det är då fördelaktigtatt istället bygga en linjär accelerator. Nackdelen med en linjär accelerator äratt partiklarna bara används en gång. I CLIC, med en total längd på ca 50km, accelereras partikelstrålarna i tiotusentals accelerationsstrukturer där en-ergi från elektromagnetiska vågor i radiofrekvensområdet ("RF-vågor") över-förs till elektronerna och positronerna.

Cirkulära acceleratorer med syftet att lagra en partikelstråle kallas lagrings-ringar och har många användningsområden. De används till exempel i lin-jära kolliderare för att dämpa de transversala oscillationerna i strålen och påså sätt höja strålkvaliteten innan strålen accelereras i den linjära accelera-torn. Andra användningsområden för lagringsringar är att använda dem somsynkrotronkällor, som till exempel MAX IV-laboratoriet i Lund, där relativis-tiska elektroner lagras i en cirkulär accelerator och utsänder synkrotronstrål-ning när deras bana böjs av magneterna. Denna strålning leds sedan ut genomstrålrör och används för att studera till exempel materialfysik. Lagringsringaranvänds också som cirkulära kolliderare som till exempel tidigare nämndaLHC vid CERN.

Acceleratorfysik handlar om att hantera, diagnosticera och kontrollera enstråle av ultra-relativistiska laddade partiklar under extrema omständigheter.I partikelacceleratorer används transversala magnetfält för att styra och kon-trollera partikelstrålen. Generellt kan ett transversalt magnetfält skrivas somen multipol-utveckling där magnetfältet beskrivs som en summa beståendeav ett konstant fält (dipol), ett linjärt fält (kvadrupol) och icke-linjära fält(sextupol, oktupol och högre ordningar). Dipolmagneter används till att böja

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strålen och kvadrupolmagneter, likt optiska linser, används till att fokuserastrålen. Icke-linjära magnetfält används i partikelacceleratorer till exempelför att kompensera kromatiska (energiberoende) effekter. Som titeln angerhandlar arbetet i denna avhandling om stråldiagnostik och stråldynamik i icke-linjära fält.

StråldiagnostikStråldiagnostik handlar om att mäta och karakterisera en partikelstråle, vilketär nödvändigt för att kunna styra en accelerator på ett optimalt sätt. I avhand-lingens första del har vi studerat hur den icke-linära oktupolkomponenten hosRF-vågorna i accelerationsstrukturer för CLIC kan användas för strålbaseraddiagnostik.

I accelerationsstrukturerna för CLIC finns fyra transversala vågledare medsyfte att dämpa högre frekvensmoder av RF-vågorna i strukturen. Dessa våg-ledare ger upphov till en fyrfaldig symmetri som i sin tur gör att även enoktupolkomponent av RF-vågorna propagerar i strukturen. Detta fält är icke-linjärt vilket ger upphov till icke-linjära krafter som verkar på partiklarna ochvi har studerat hur detta kan användas för stråldiagnostik.

Vid CERN finns testanläggningen CTF3 vars ändamål är forskning ochutveckling av acceleratorfysik och teknologi för CLIC. Där finns en liten ac-celerator, CALIFES, som kan accelerera en elektronstråle till ca 200 MeV.Därefter följer en sektion med en CLIC-modul med CLIC accelerationsstruk-turer och diverse diagnostiska verktyg. Till exempel finns en fluorescerandeskärm efter accelerationsstrukturen som gör det möjligt att observera strålenstransversala profil vilken ger information om strålens centrumposition, ho-risontell och vertikal bredd samt korrelation. Alla experiment som presenterasi denna avhandling utfördes vid CTF3.

Styrkan hos oktupolkomponenten var känd från RF-simuleringar men hadealdrig mätts experimentellt. Genom att flytta strålen transversellt inuti ac-celerationsstrukturen och mäta strålens position på skärmen med och utanRF kunde vi mäta hur strålcentrums positionsförändring beror av transversellposition. För en elektronstråle som följer en Gaussisk fördelning (normal-fördelning) kan vi beräkna analytiskt hur positionen för strålcentrum förän-dras på grund av ett oktupolfält. Genom att anpassa mätdata till de analytiskauttrycken kunde vi bestämma oktupolkomponentens styrka och jämföra medsimuleringar.

När strålen flyttas transversellt ändras inte bara strålens position utan ocksåstrålens storlek. Denna observation gav oss idén till att utnyttja det icke-linjära oktupolfältet för att extrahera mer information om strålen. En par-tikelstråle är en ansamling partiklar och beskrivs som en statistisk distributionkarakteriserad av en kovariansmatris som i acceleratorsammanhang kallas förstrålmatris. Från analytiska beräkningar hur strålens bredd ändras från ett

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oktupolfält kunde vi göra anpassningar till mätdata och bestämma den full-ständiga transversala strålmatrisen inklusive alla korrelationer.

För att behålla strålkvaliteten genom den långa acceleratorn i CLIC måsteaccelerationsstrukturerna vara upplinjerade och strålen centraliserad med pre-cision på storleksordningen mikrometer. Från de icke-linjära positionsförän-dringarna av strålens centrum från oktupolkomponenten får vi också ut cen-trum för oktupolfältet. Detta gav oss idén att använda oktupolkomponentenför att centralisera strålen i accelerationsstrukturerna. Strålens position inutiaccelerationsstrukturen kan ändras genom att flytta strålen eller genom att fly-tta accelerationsstrukturerna och från anpassning hittar vi strukturens centrum.Vi undersökte metodens potential för CLIC och testade den experimentellt vidtestanläggningen CTF3.

Genom att observera hur en partikelstråle påverkas när den flyttas i ett icke-linjärt fält kan vi alltså få fram information om strålens egenskaper. Men icke-linjära fält påverkar också dynamiken för en partikelaccelerator och detta ärextra känsligt för cirkulära acceleratorer där vi vill kunna lagra partiklar underlång tid. Avhandlingens andra del handlar om stråldynamik under inflytandeav icke-linjära effekter.

Icke-linjär stråldynamikAllt eftersom prestanda för cirkulära acceleratorer höjs, till exempel om vi gårtill högre och högre energi eller till mindre och mindre strålstorlek, blir icke-linjära effekter alltmer avgörande. I många fall är icke-linjära effekter denbegränsande faktorn för dynamisk stabilitet. Källorna till icke-linjära effekterär många: det kan vara imperfektioner i dipolfält, interaktion mellan de tvåkolliderande strålarna i fallet cirkulär kolliderare eller icke-linjära magnetfältsom en del av designen av en accelerator.

För att förstå varför icke-linjära effekter är problematiskt för stabilitetenbehöver vi förstå vad det engelska begreppet tune för en cirkulär accelera-tor innebär. Tune anger hur många oscillationer en partikel gör omkring enidealiserad omloppsbana under ett varv. Tune består av ett heltal och en deci-maldel där den senare har stor betydelse för dynamisk stabilitet. Till exempelså kan inte tune vara ett heltal eftersom det skulle innebära att en konstant im-perfektion leder till växande oscillationer och tillslut förloras partikeln då denträffar strålröret. Tune med decimaldel 0,5 måste också undvikas eftersom vidå är känslig för linjära imperfektioner. Transversellt har vi horisontell ochvertikal tune och även summan eller differensen av dessa får inte vara heltal.

Sextupolmagneter används ofta i cirkulära acceleratorer för att kompenserakromatiska effekter men sextupolmagneter ger upphov till andra icke-linjäraeffekter, till exempel drivs tredje ordningens resonanser. Dessutom är cirkuläraacceleratorer iterativa system vilket innebär att även högre ordningens effekterdrivs från en sextupol. Detta gör att sextupoler driver fjärde ordningens effek-

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ter och till exempel: amplitudberoende förändring av tune, vilket innebär attpartiklar som oscillerar med större amplitud får annan tune och riskerar atthamna på en resonans. Amplitudberoende förändring av tune är problematiskteftersom det begränsar dynamisk stabilitet.

Vi har använt olika analytiska verktyg (Hamiltonianer, Lie algebra ochnormalform) för att analysera hur icke-linjära magnetiska fält påverkar dy-namiken för cirkulära acceleratorer. Mer specifikt har vi undersökt hur oktupol-magneter kan placeras i en lagringsring på ett sådant sätt att de påverkaramplitudberoende förändring av tune utan att driva fjärde ordningens reso-nanser. Vi åstadkommer detta genom att placera oktupolmagneterna på ett sättatt de resonansdrivande termerna tar ut varandra. I detta arbete använde vi enanalytisk kod som kan hantera Hamiltonianer, Lie algebra och normalformeroch vi testade konfigurationerna i simuleringar.

I förra exemplet var det önskvärt att inte driva resonanser men vi har ocksåundersökt fall där olika resonansdrivande termer istället kan kontrolleras in-dividuellt. Framförallt har vi hittat optimala konfigurationer med sextupol-magneter där vi kan kontrollera tredje ordningens resonanser. Vi formulerarsystemet som ett linjärt ekvationssystem där koefficienterna för de resonans-drivande termerna skrivs som en matris beroende av sextupolernas placeringaroch en vektor med sextupolernas styrkor. En optimal konfiguration ger en ma-tris med konditionstal 1 och genom att placera sextupolerna på ett specifiktsätt kan vi åstadkomma detta.

Genom att placera icke-linjära magneter i partikelacceleratorer på ett smartsätt kan vi uppnå utsläckning av vissa icke-linjära effekter, som i exempletmed oktupolmagneterna som inte driver fjärde ordningens resonanser, elleruppnå optimal kontroll över enskilda resonansdrivande termer, som i exempletmed sextupoler. I verkliga partikelacceleratorer finns förstås många praktiskabegränsningar och andra villkor att möta. Men genom att lära oss mer omhur icke-linjära magneter interagerar och påverkar stråldynamiken kan vi görasmartare val när vi konstruerar nya acceleratorer.

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78

Acronyms

BPM beam position monitor.CALIFES Concept d’Accélérateur Linéaire pour Faisceaux d’Electrons Son-

des.CBH Campbell-Baker-Hausdorff.CERN the European organization for nuclear research.CLEX the CLIC experimental area.CLIC the Compact Linear Collider.CTF3 the CLIC Test Facility 3.DC direct current.ILC the International Linear Collider.LEP the Large Electron-Positron Collider.LHC the Large Hadron Collider.linac linear accelerator.PETS Power Extraction and Transfer Structure.RDT resonance driving term.RF radio-frequency.SEM scanning electron microscope.SLC SLAC Linear Collider.SPS the Super Proton Synchrotron.SUSY supersymmetry.TBL the Test Beamline.TBTS the Two-Beam Test Stand.

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Acknowledgements

I started my doctoral studies in September 2013 and the past four years havebeen rewarding and taught me a lot. Here I like to take the opportunity tothank some people that, in one way or another, have made this possible.

First and foremost, I would like to thank my thesis advisor Volker. Over theyears we have developed a well-functioning and productive collaboration. Ourweekly discussions have involved topics from electronics to finance, program-ming, good books and everything in between. Your interest and knowledge insuch a broad range of fields and your strong physics intuition remain inspira-tional.

I thank my thesis advisor Roger for all the help and support, especially for allthe help at CERN. We have had many interesting discussions and I thank youfor all valuable input.

I like to thank Klaus, my third advisor, for many interesting discussions aboutelectron microscopy, field emission studies and the world of the very small.

I thank Marek for guiding me during my first trips to CERN and helping outwith measurements both at CTF3 and with the SEM in Uppsala. Furthermore,many thanks for all the times you shared your Linux-wizardry and helped mewith computer-related problems.

During my time as a PhD student I have had the opportunity to make numeroustrips to CERN and I would like to extend my gratitude to Roberto and everyoneat CTF3 for help and support during my stays. In particular I want to thankWilfrid—it is hard to see how any of the experiments that I did at CTF3 wouldhave been possible without you. I would also like to thank Daniel, Andrea,Walter and Alexej for interesting discussions. Finally, I thank Panos and Dariofor all the nice dinners and showing me that there are other things to do inGeneva besides work.

I would like to thank all the people at the high energy physics, FREIA andnuclear physics divisions at Uppsala University. In particular the "old crew":Rickard, Henric, Li, Lena, Daniel, Andrea and Maja, who all made me feelwelcome and at home when I first started. I also thank Mikael, Max, Petar,Myrto, Joakim, Lisa, Alex, Elin, Mathias, Alan and Georgii. A special thanks

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to office mates Michael and Krish for all the interesting discussions, nice after-works and trips to CERN and IPAC.

Thanks go to Eric, Nico and David from Team Ångström in the 2017 UppsalaKorpen chess competition for many enjoyable "chess lunches".

Jag har turen att ha så många fina vänner. Stort tack till Studentvägen-gängetoch den utökade familjen: Peter, Johan, Micke, Rebecka, Linn, Fischer, Jonas,Madde, Emma och Eric. Tack till handelsmännen från öster: Sixten och Arvid.Tack till Anna och Mauro för all god dryck, mat och umgänge. Tack tilljärngänget från Gångviken: Felipe, Mario och Robert.

Mormor, mor, Joel, Gunilla, Bengt och Katrin, stort tack för ert stöd genomåren och för att ni alltid trott på mig.

Slutligen vill jag tacka Tove. För ditt tålamod, din kärlek och för att du finns.

Jim Ögren,Uppsala, October 2017

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