University of Southern California Dissertations and Theses

Resonant excitation of plasma wakefield

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Resonant excitation of plasma wakefield

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Content RESONANT EXCITATION OF PLASMA WAKEFIELD by Yun Fang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMICAL ENGINEERING) May 2014 Copyright 2014 Yun Fang ii Dedication To my family. iii Acknowledgments I would like to thank everyone who encouraged me, supported me, and inspired me during my doctoral work. As a matter of fact, this work would not be finished without these wonderful people influencing me in every aspects of my life. Foremost, I would like to offer my most sincere gratitude to my re- search advisor and eternal mentor Dr. Patric Muggli. Dr. Muggli guided me patiently to the fantastic research areas of PWFA by explaining com- plex physics problems in a very understandable and interesting way. He inspires me to raise questions scientifically, solve problems creatively, and deliver research results accurately and efficiently. All of those are the essen- tial elements in conducting research. In addition, he always makes himself available whenever I need advice from him, even though we are in differ- ent time zone (9 hours difference) for recent years. Throughout my Ph.D. study, I am greatly influenced by his professionalism, communication and enthusiasm, which I believe are more important in developing my career in the future. I am also grateful for having an exceptional dissertation committee and would like to thank other members of the committee, Dr. Gundersen, iv Dr. Shing, Dr. Nakano as well as Dr. Mori from UCLA for their guidance and insights. Especially, deep gratitude goes to the chair of my committee, Dr. Gundersen, who offers generous support and many valuable sugges- tions for my research work. I appreciate the time spending with Dr. Gun- dersen’s Pulse Power Group, which greatly broadens my horizon on the application of plasmas. Next, I would like to thank my colleagues in PWFA group at USC, including Yi, Brian, Stephen, Xiaoying and Reza. Yi has been always a close friend and it is she who first introduced me to the plasma research group. I am really fortunate to have Brian around during most of my Ph.D. time. He offers me many useful advices and suggestions, teaches me to plan things ahead, and motivate me to achieve my goals. I would like to thank all group members in Dr. Mori’s research group. They have made my cooperation experience there very enjoyable and beneficial. Weiming, Xinlu, Peicheng, Frank all offered me very generous help in running simulation codes and trouble shooting the computer cluster, which contribute a significant part of this thesis. Many thanks also goes to my dear friends, including both old and new ones, who have added wonderful colors and flavors to my life. A lot of them are/were also in the Ph.D. program, such as Yueyang, Yining, Johnny, Tao, Haiguang, Bingbing, Yingying and Zaoshi (Amy), who give me enor- v mous help and support thoughout the years. I also want to give my thanks to the sweet couple: Meng and Shuning, who constantly remind me to cher- ish and celebrate life when I get overwhelmed with study. There are many ups and downs in the journey of pursing Ph.D. and those friends keep me companied and refreshed along the way. Finally, I would like to express my deepest appreciation to my hus- band Hu Li and my parents, Dongxiang and Meiying, who always encour- age me to pursue my dream and give their unconditional support. ”It is good to have an end to journey toward; but it is the journey that matters, in the end.” -By Ernest Hemingway vi Table of Contents Dedication ii Acknowledgments iii List of Tables x List of Figures xi Abstract xxi Chapter 1. Introduction 1 1.1 Particle Accelerators and Future Collider Design . . . . . . . . 1 1.2 Limitation of Conventional Radio-Frequency Accelerators . . . 3 1.3 Plasma Wakefield Accelerators . . . . . . . . . . . . . . . . . . 4 1.3.1 Advantage of Plasma Wakefield Accelerators . . . . . . 4 1.3.2 Basic Principles of Plasma Wakefield Accelerators . . . . 4 1.4 Current Development and Future Challenges . . . . . . . . . . 8 1.4.1 Current Development . . . . . . . . . . . . . . . . . . . . 8 1.4.2 Future Challenges for the PWFA and Dissertation Outline 10 1.5 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 14 Chapter 2. Numerical Simulation Towards Reaching High Transformer Ratio in the Weakly Nonlinear Regime of PWFA using the Ramped Bunch Scheme 16 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Ramped Bunch Scheme in Linear Regime . . . . . . . . . . . . 17 2.2.1 Overview of Linear Theory . . . . . . . . . . . . . . . . . 17 2.2.2 Reaching High Transformer Ratio in Linear Regime: Ramped Bunch Scheme . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.3 High Transformer Ratio not Maintained in Linear Regime 24 2.2.4 Emittance Preservation in Nonlinear Regime . . . . . . . 25 vii 2.3 Applying Ramped Bunch Scheme to Weakly Nonlinear Regime 27 2.3.1 Experimental Background . . . . . . . . . . . . . . . . . . 28 2.3.2 Simulation Parameters versus Current Experimental Pa- rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.1 Single Bunch . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.2 Two Bunches . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.3 Three Bunches . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 37 Chapter 3. Numerical Study of the Seeding of Self-modulation In- stability with a Low Charge (50 pC) long Electron Bunch in a Plasma 38 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Beam Parameters Available at ATF . . . . . . . . . . . . . . . . 42 3.3 Simple Model for Energy Modulation . . . . . . . . . . . . . . . 43 3.4 Estimation of Optimum Bunch Charge from Linear Theory . . 46 3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . 48 3.5.2 E z Evolution and SMI Development . . . . . . . . . . . . 49 3.5.3 Energy Modulation . . . . . . . . . . . . . . . . . . . . . 51 3.6 Seeding of Self-modulation Instability . . . . . . . . . . . . . . 54 3.7 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 54 Chapter 4. Experimental Study of the Seeding for Self-modulation Instability 56 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3 Experimental Result . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3.1 Energy Modulation . . . . . . . . . . . . . . . . . . . . . 61 4.3.2 Effect of the Ramp Profile of the Bunch Current Rise . . 65 4.4 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 66 Chapter 5. Numerical Study of the Self-modulation Instability with a High Charge (1nC) long Electron Bunch in a Plasma 68 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 viii 5.2 Beam Parameters at ATF and Simulation Parameters . . . . . . 69 5.3 Self-modulation of the Electron Bunch . . . . . . . . . . . . . . 70 5.4 Wakefield Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.5 Particles’ Phase Space . . . . . . . . . . . . . . . . . . . . . . . . 74 5.6 Energy Gain and Loss . . . . . . . . . . . . . . . . . . . . . . . . 75 5.6.1 Energy Spectra from Simulation . . . . . . . . . . . . . . 75 5.6.2 Possibility of Confirming SMI Development through Measuring Energy Spectra . . . . . . . . . . . . . . . . . 76 5.7 Measure the Radial Modulation . . . . . . . . . . . . . . . . . . 80 5.8 Hosing Instability . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.9 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 81 Chapter 6. The Effect of Plasma Radius and Profile on the Develop- ment of Self-modulation Instability 87 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2 Hints from Linear Theory . . . . . . . . . . . . . . . . . . . . . . 90 6.2.1 Linear Theory Review . . . . . . . . . . . . . . . . . . . . 90 6.2.2 Determining the ”Infinite” Plasma Radius for PWFA from 2D Linear Theory . . . . . . . . . . . . . . . . . . . 94 6.2.3 Modified Linear Theory for Finite but Uniform Plasma . 95 6.2.4 Transversely Finite and Inhomogeneous Plasma . . . . . 96 6.3 OSIRIS 2D Simulation Results with ATF Parameters . . . . . . 97 6.3.1 Finite Plasma Radius with Uniform Density . . . . . . . 100 6.3.1.1 Initial Wakefields . . . . . . . . . . . . . . . . . . 100 6.3.1.2 Evolution of the PeakE z . . . . . . . . . . . . . . 104 6.3.2 Finite Plasma Radius with Cosine Density Profile . . . . 105 6.4 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 110 Chapter 7. Conclusions/Outlook 113 Bibliography 115 Appendix A. Data Analysis of the Self Modulation Instability Ex- periment 119 A.1 Energy Spectrum Calibration . . . . . . . . . . . . . . . . . . . . 119 A.2 The Estimation of Bunch Length and Energy . . . . . . . . . . . 120 ix A.3 Energy Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Appendix B. List of Publications 126 x List of Tables Table 2.1 Bunch train parameters . . . . . . . . . . . . . . . . . . . . . . 31 Table 2.2 Square bunch train in the linear regime vs. Gaussian bunch train in the weakly nonlinear regime . . . . . . . . . . . . . . 36 Table 3.1 ATF electron bunch and plasma parameters . . . . . . . . . . 42 Table 3.2 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . 48 Table 5.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . 69 Table 6.1 Experimental Parameters of SMI Experiment at ATF and Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . 98 xi List of Figures Figure 1.1 Schematic of the plasma wakefield accelerator for:a) a sin- gle bunch, andb) a drive (D), witness (W) bunch system. The wakefields are also cartooned in the second bucket of the wake in a). Notice that we consider a negatively charged witness bunch in the second bubble of the wake- field, so in Fig.a) the left-pointing arrow indicates acceler- ation, and the outward-pointing arrow indicates focusing, and vice versa. . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Figure 2.1 The on-axis transverse component of the longitudinal wake- fieldR(0). Picture taken from Ref. [37]. . . . . . . . . . . . . 19 Figure 2.2 Longitudinal wakefields driven by ramped bunch scheme with a train of square bunches in linear regime. The red rectangles represent the drive bunches, the green rectangle represents the witness bunch, and the black lines shows the longitudinal wakefields. The arrow shows the trains travel direction, and = ctz is the transformed vari- able in the frame moving at the speed of light. The drive bunches are separated by 1:5 pe and their charge is Q = 40pC [1 : 3 : 5 : 7]. The witness bunch is in the accelerat- ing phase of the wakefield. The transformer ratio is 7:89. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Figure 2.3 Transverse wakefield driven by ramped bunch scheme with a train of square bunches in linear regime. . . . . . . . 25 xii Figure 2.4 One-time-step simulation result of a single bunch with pa- rameters given in Table : a) the 2D image of normalized beam density,b) 2D image of the normalized plasma elec- tron density,c) plasma electron density perturbation along on the axis (as the dotted line inb)), and the inset magni- fies the plasma density lineout of a small region (as indi- cated by the red circle) where the plasma electron density changes from -1 to 0,d) the longitudinal wakefield on the axisE z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Figure 2.5 One-time-step simulation result of two bunches with pa- rameters given in the text: a) the 2D image of normalized beam densityn b =n e ,b) 2D image of the normalized plasma electron density, c) plasma electron density perturbation along on the axis (as the dotted line inb)),d) the longitudi- nal wakefield on the axisE z . . . . . . . . . . . . . . . . . . . 32 Figure 2.6 One-time-step simulation result of three bunches with pa- rameters given in the text: a) the 2D image of normalized beam densityn b =n e ,b) 2D image of the normalized plasma electron density, c) plasma electron density perturbation along on the axis (as the dotted line inb)),d) the longitudi- nal wakefield on the axisE z . . . . . . . . . . . . . . . . . . . 34 Figure 2.7 Simulation result of three bunches after the propagation for 2cm with parameters given in the text:a) the 2D image of normalized beam density,b) 2D image of the normalized plasma electron density,c) plasma electron density pertur- bation along on the axis (as the dotted line in b)), d) the longitudinal wakefield on the axisE z . . . . . . . . . . . . . 35 Figure 3.1 A simple model of beam energy modulation: the red lines indicate the wakefield (E z ), the horizontal green arrows in- dicate the energy gain or loss. The original bunch is in blue rectangle and the blue ellipses indicate the energy beam- lets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 xiii Figure 3.2 a) Initial longitudinal wakefield at the beam axisE z (z = 0;r = 0) and b) initial transverse focusing field at r = r for various plasma densities (shown in Table 3.2) such that L beam = pe is equal to: 1 (red), 2 (blue), 3 (green), 4 (black), 5 (purple), 6 (orange) and 7 (magenta dotted line). . . . . . . 47 Figure 3.3 Peak accelerating field along the bunch versus propaga- tion distance in the simulation for the electron bunch with 50pC charge and plasma densities such thatL beam = pe = 1 (red), 2 (blue), 3 (green), 4 (black), 5 (purple), 6 (orange) and 7 (cyan). The exponential growth at z > 3 cm for l beam = pe > 2 indicates the growth of SMI. . . . . . . . . . . 50 Figure 3.4 Energy spectra of the electron bunch at the plasma exit in the cases of various plasma densities such thatL beam = pe = 1 to 7. The expected energy modulation peaks are clearly visible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Figure 3.5 Two dimensional beam density plots: a) Transverse mo- mentum of the bunch particles at the plasma exitz =L p = 2 cm from OSIRIS2D simulations with the experimental parameters in Table 3.1. b) Bunch density after propaga- tion of 7:5 cm in vacuum after the plasma showing that modulation in transverse momentum from Fig. a) results in a modulation of the bunch transverse bunch radius and density. The white line in Fig. b) shows the bunch density lineout atr = 74m in the bunch. . . . . . . . . . . . . . . . 53 Figure 4.1 Experimental setup for the study of SMI. Two dipole and five quadrupole magnets are arranged in a dogleg config- uration. The side graphs represent the beam energy corre- lation with the beam front labeled by F and the back by B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 xiv Figure 4.2 a) Image of the long bunch dispersed in energy a short distance downstream from the mask. Two early in time and low energy bunches not used for this experiments are shown in the front of the bunch. The white lines indicates the sum of the images over the vertical and horizontal im- age dimensions. The energy range selected by the mask is E 0 . The bunch length is L beam and the length scale for the rising edge is L. b) Ratio of the longitudinal wake- field amplitude E z for the case of a square bunch with a cos 2L forL 0 rising edge of base L (as shown in the inset) toE z0 , which would result from a per- fectly square bunch (L = 0), both calculated using 2D linear theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Figure 4.3 Energy spectra obtained at various plasma densities. Spec- tra a) with n e = 0 (no plasma) and b)-h) with increasing plasma densities between 2 10 15 - 8 10 16 cm 3 obtained in the experiment. The white lines indicates the sum of the images over the vertical image dimension (no dispersion), and the red lines show the positions of the density peaks as identified for Fig. A.6. The direction of the energy chirp is indicated on Fig. a). Figure i) shows a simulated energy spectrum for the case L beam = pe = 2, very similar to the experimental case of Fig. c). The color tables are deliber- ately chosen different to avoid possible confusion between experimental and simulation results. . . . . . . . . . . . . . 62 Figure 4.4 Energy of the peaks identified by the red lines on Figs 4.3 b) to h) as a function of plasma density. The yellow zone corresponds to the FWHM of the incoming bunch energy and suggests that the peaks are missing in the back, high energy part of the bunch forn e > 0:5 10 16 cm 3 . . . . . . . 64 xv Figure 5.1 Beam density image obtained from OSIRIS 2D simulations a) at plasma entrancez = 0, and at plasma exit for different plasma densities such thatL beam = pe is equal to: b) 1,c) 2, d) 3,e) 4 andf) 5, as shown in Table 5.1. . . . . . . . . . . . 70 Figure 5.2 a) Longitudinal field E z at r = 0. b) transverse focusing field (E r cB ) evaluated atr = r . Both fields are obtained from 2D linear theory (assuming no beam density evolu- tion) for different plasma densities such that L beam = pe is equal to: 1 (red), 2 (blue), 3 (green), 4 (black), 5 (purple), as shown in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . 71 Figure 5.3 Beam density lineout (red line), longitudinal fieldE z (blue line) atr = 0 and transverse focusing field (E r cB ) (green line) evaluated atr = r at plasma exit (z = 2cm), obtained from OSIRIS 2D simulation for different plasma densities such thatL beam = pe is equal to: a) 1, b) 2, c) 3, d) 4 and e) 5, as shown in Table 5.1 . . . . . . . . . . . . . . . . . . . . . . 72 Figure 5.4 The evolution of maximum accelerating fieldE z along the propagation distance z, obtained from OSRIS 2D simula- tion for different plasma densities such that L beam = pe is equal to: a) 1 (red curve), b) 2 (blue curve), c) 3 (green curve), d) 4 (black curve) and e) 5 (purple curve), as shown in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Figure 5.5 Normalized radial velocity of beam particles (v 2 =c) along the bunch (ctz) at: a) plasma entrance, b)-f): plasma exit obtained from OSIRIS 2D simulation for different plasma densities such thatL beam = pe is equal to: b) 1, c) 2, d) 3, e) 4 and f) 5, as shown in Table 5.1. . . . . . . . . . . . . . . . . . 75 xvi Figure 5.6 The bunch energy spectra at plasma exit (z = 2 cm) ob- tained from OSIRIS 2D cylindrical code for different plasma densities such that L beam = pe is equal to: 1 (red line), 2 (green line), 3 (blue line), 4 (black line) and 5 (purple line), as shown in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . 76 Figure 5.7 a) The longitudinal wakefield E z on the axis (r = 0) at plasma entrance (z = 0) within the bunch. The black square indicates position of the electron bunch. b) The evolution of maximumEz on the axis (r = 0) along the propagation distance; c) The energy spectrum of the bunch particles at the plasma exit from OSIRIS2D. . . . . . . . . . . . . . . . . 83 Figure 5.8 a) Evolution of the accelerating wakefield structure E z within the bunch along the propagation distance z. The red rectangle shows the longitudinal position of the bunch and the white dotted lines show the ”trajectory” of the peak accelerating and decelerating field respectively. b) The shifting of the position of the peak accelerating field along the propagation distancez, and the fitted line is given as: z(cm) = 5:5467 0:0072932 (um), yielding v = c (1 + 1 z= ) 0:986c. . . . . . . . . . . . . . . . . . . . . . 84 xvii Figure 5.9 Energy gain/loss by particles along the bunch () after 2cm plasma, obtained by integrating the acceleration field (as shown in Fig. 5.8 a)) over the propagation distance z. 0 indicates the position in the bunch where particles gain the most energy; b) the accelerating field on the axis at position 0 of the bunch (E z ( 0 ;r = 0)) along the propa- gation distancez (red line) and the maximumE z field on the axis along z (blue line); c) Longitudinal wakfield E z on axis (r = 0) at plasma entrance z = 0 (blue line) and plasma exitz = 2cm (red line). Note thatE z atz = 0 (blue line) is multiplied by 50 in order to make it visible on the same scale. The dotted lines show the accelerating peaks of the respective fields, and the distance between the two dotted lines directly shows the wakefield dephasing. The red rectangles show the longitudinal position of the bunch. 85 Figure 5.10 Beam image after 25cm free propagation in vacuum after the plasma for different plasma densities such thatL beam = pe is equal to: a) 1, b) 2, c) 3, d) 4 and e) 5, as shown in Table 5.1. The white lines indicate the density lineout at the axis. . 86 Figure 5.11 Beam density image at plasma exit obtained from OSIRIS 3D code at different plasma densities such thatL beam = pe is equal to: a) 2 , and b) 5, as shown in Table 5.1. . . . . . . . . 86 Figure 6.1 a): The transverse dependence term of the longitudinal wakefield on the axis R(0) (as shown in Eq. 6.5) versus k p r ,b): the normalized transverse dependency termR(r)=R(r = 0) versus the normalized radiusr= r fork p r = 1:57 (red curve), 1 (blue curve) and 0:79 (black curve), c): the ”in- finite” plasma radius versus the plasma skin depthc=! pe , both normalized with r . . . . . . . . . . . . . . . . . . . . . 93 Figure 6.2 A schematic plot of the plasma density profile. . . . . . . . 96 xviii Figure 6.3 The transverse dependence termR(r) calculated from the above modified linear theory (dotted line) versus the sim- ulation result (solid line) for different plasma radius with densities such thata)k p r = 1:57,b)k p r = 0:79. . . . . . . . 99 Figure 6.4 R(r = 0) for various plasma radii R p , normalized with theR(r = 0) of the ”infinite” radiusR p1 , fork p r = 0:79 (black curve) and k p r = 1:57 (red curve), obtained from the numerical calculation of Eq. 6.10. . . . . . . . . . . . . . 99 Figure 6.5 dR(r)=dr calculated from the above modified linear theory (dotted line) versus the simulation result (solid line) for different plasma radius with densities such thata)k p r = 1:57,b)k p r = 0:79. . . . . . . . . . . . . . . . . . . . . . . . . 100 Figure 6.6 The plasma return current image at the plasma entrance (z = 0) (a)) and the lineout at highest current density for different plasma radii such that R p = r equal to 4:2 (black curve), 3 (orange curve), 2 (red curve), 1:5 (blue curve), 1 (green curve) fork p r = 1:57 (b)) andk p r = 0:79 (c)). . . . 103 Figure 6.7 Maximum acceleration gradientE z (logarithmic scale) along the propagation distancez for different plasma radii such that R p = r equal to 4:2 (black curve), 3 (orange curve), 2 (red curve), 1:5 (blue curve), 1 (green curve) fora): k p r = 1:57 andb):k p r = 0:79 . . . . . . . . . . . . . . . . . . . . . . 104 Figure 6.8 Comparison of maximum E z along z between uniform plasma (black curve) and plasma with cosine profile (red curve) for the casek p r = 1:57 with different plasma radii such thatR p = r equal to 4:2 (a)), 3 (b)), 2 (c)), 1:5 (d)), 1 (e)). 106 Figure 6.9 Comparison of maximum E z along z between uniform plasma (black curve) and plasma with cosine profile (red curve) for the casek p r = 0:79 with different plasma radii such thatR p = r equal to 4:2 (a)), 3 (b)), 2 (c)), 1:5 (d)), 1 (e)). 106 xix Figure 6.10 Comparison of maximum focusing fieldE r cB along the radius r at the plasma entrance (z = 0) between uniform plasma (black curve) and plasma with cosine profile (red curve) for the casek p r = 1:57 with different plasma radiii such thatR p = r equal to 4:2 (a)), 3 (b)), 2 (c)), 1:5 (d)), 1 (e)). 107 Figure 6.11 Comparison of maximum focusing fieldE r cB along the radius r at the plasma entrance (z = 0) between uniform plasma (black curve) and plasma with cosine profile (red curve) for the casek p r = 0:79 with different plasma radiii such thatR p = r equal to 4:2 (a)), 3 (b)), 2 (c)), 1:5 (d)), 1 (e)). 108 Figure 6.12 2D density plot of plasma focusing fieldE r cB at plasma entrance (z = 0) generated by the electron bunch propagat- ing in a plasma with radiusR p = 1:5 r and cosine density profile. The plasma density is such thatk p r = 1:57. . . . . 109 Figure A.1 a) Sum of five spectrometer images of the bunch with the currents specified in the text. b) Spectrometer energy cali- bration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Figure A.2 Spectrometer energy resolution estimate. . . . . . . . . . . . 121 Figure A.3 Right: energy spectrum of run number 84 , plasma off, event number 35. Left: the beam parameters based on the calibration shown above. . . . . . . . . . . . . . . . . . . . . 122 Figure A.4 Energy Spectra obtained at various plasma densities through various delay times between the plasma discharge and the bunch arrival time (decreasing delay, 100ns step, increas- ing density). Figure a) is for plasma off. The white lines are the energy spectra, and the red lines indicate the positions of the density peaks. . . . . . . . . . . . . . . . . . . . . . . 123 xx Figure A.5 Energy of the peaks identified by the red lines on Figs. A.4 b) to h) as a function of plasma density. The yellow zone corresponds to the FWHM of the incoming bunch energy. . 124 Figure A.6 Energy of the peaks identified by the red lines on Figs. A.4 b) to h) as a function of plasma density. The yellow zone corresponds to the FWHM of the incoming bunch energy. . 125 xxi Abstract Particle accelerators are the main tool for discovering new elemen- tary particles. Plasma based accelerator (PWFA) has been proven a very at- tractive new acceleration technique due to the large acceleration gradient it has reached (> 50GV=m), which is two to three orders higher than the con- ventional radio frequency accelerators. PWFA is essentially an energy trans- former transferring the energy from the drive bunches to witness bunches. For a future more compact and more affordable linear electron/positron collider, such an accelerator will require drive bunches with small longitu- dinal size (on the order of 100um) and multi-kilojules of energy to access the new physics at the energy frontier. Proton bunches produced at CERN have been proven as potential drivers for PWFA due to the many tens of kilojules energy they carry (10 11 particles, 3:5 7 TeV per particle). However, the CERN proton bunches are too long ( 12 cm) to drive the wakefield effi- ciently. It has been proposed that a long particle bunch (protons, electrons, positrons, ... ) traveling in dense plasmas is subject to self-modulation in- stability (SMI), which transversely modulates a long bunch into multiple short bunches (on the scale of plasma wavelength) and therefore results in high acceleration amplitudes through resonant excitation. In this thesis, we xxii demonstrate the first experimental evidence for the seeding of SMI with an electron bunch. We also use numerical simulations to study the SMI devel- opment with a higher-charge electron bunch and propose a possible exper- iment to demonstrate the transverse modulation directly in experiments. Moreover, we investigate with simulations the effect of transverse plasma radius on the SMI development, which is an important factor to consider when designing plasmas for future SMI and SMI-based experiments. Besides efficient drivers such as high-energy proton bunches, the PWFA also requires high transformer ratio (an indication of energy trans- fer efficiency) so that the witness bunch can gain energy efficiently from the drive bunch. In this thesis, we explore the possibility of reaching high transformer ratio in the weakly nonlinear PWFA regime so that the witness bunch particles can gain many times the energy of the drive bunch particles in a single acceleration stage. 1 Chapter 1 Introduction 1.1 Particle Accelerators and Future Collider Design Mankind has never stopped seeking the elementary constituents of matter, and the most powerful tools that have ever been employed are the particle accelerators [1]. These accelerators use strong electric fields to ac- celerate charged particles such as electrons, positrons and protons to high energies, which then collide with each other at near the speed of light. By observing the behavior of the products of those collisions, the elementary particles can be analyzed and new fundamental laws can be established [2]. In the past 70 years, the development of radio-frequency (RF) acceler- ators has greatly pushed the frontiers for the fundamental particle research. Most of the quark structures inside protons and neutrons were identified in the 1970s and 1980s with high-energy electrons on solid targets at the Stan- ford Linear Accelerator Center (SLAC) [3]. The carriers of the electroweak force, W and Z bosons, were discovered in 1983 at CERNs Super Proton Synchrotron (SPS). The heaviest elementary particle known to date, the top quark was discovered only in 1995 at Fermi lab Tevatron accelerator with a mass of 171GeV=c 2 [4]. Recently, the predicted mass carrier particle the 2 ”Higgs Boson” with a mass around 126GeV was confirmed through ATLAS and CMS experiment at CERN’s Large Hadron Collider (LHC). A future particle collider must be designed to satisfy certain require- ments in order to pinpoint even smaller structure of the micro-world, and we take the lepton (electron-positron) collider as an example: First, the particle energy needs to be at least 250 GeV . For comparison, the largest electron-positron collider to date, CERNs Large Electron-Positron (LEP), ac- celerated particles up to 105 GeV . Equally important is the luminosity L, or the number of collisions per cross-sectional area, defined as: L = f N 2 A , where f is the repetition rate of the collisions (per turn for circular col- liders), N is the number of particles each beam, and A is the area of each beam. An extremely high collision rate is desired because not all the col- lisions produce the same effects, and the type of collision we are trying to study is extremely rare. Therefore we need a very large number of ordinary collisions (hence lots of luminosity) to see just a few of the interesting ones. In fact, some of the most interesting types of collisions are so rare that they may occur only once every few months with the parameters of today’s ac- celerators. If the luminosity can be enhanced by two orders of magnitude, we might be able to see the desired collisions every few hours, and that would make the discovery much easier. The definition of luminosity indi- cates that achieving high luminosity requires small bunch transverse size, 3 many particles per bunch, and many bunch crossings per unit time. Small bunch transverse size can be translated into low emittance (a certain phase space volume occupied by the beam and is a measure of beam transverse temperature) and high brightness corresponding to low energy spread. To sum up, a future lepton collider requires particle bunches with high energy (> 250 GeV ), high charge, low emittance, low energy spread and many beam crossings in order to produce new physics. 1.2 Limitation of Conventional Radio-Frequency Accelera- tors Despite the progress of the traditional radio-frequency accelerators, their length, the complexity and price of the accelerator structure have sky- rocketed along with their energy. As the worlds largest and most powerful accelerator, the Large Hadron Collider (LHC) at CERN is 27km in circum- ference with 93000 magnets and cost more than 10 billion dollars. In order to satisfy the requirements of future lepton colliders, the next linear accelerator for electrons and positrons, International Linear Collider (ILC), is expected to be more than 40km-long and may be of even higher cost. Such high com- plexity and cost is due to the low accelerating gradient (10 100MeV=m) in conventional accelerators. The radio-frequency accelerating electric field amplitude is limited to the breakdown field at which it begins ripping elec- trons off from the surrounding metal material [5]. Therefore, in order to 4 produce new physics at an affordable price, a fundamental revolution in the accelerator technology that can generate much higher accelerating gra- dient as well as good beam quality (high charge, low emittance, low energy spread) is urgently needed. 1.3 Plasma Wakeeld Accelerators 1.3.1 Advantage of Plasma Wakeeld Accelerators The most revolutionary approach that has been proposed so far is to build plasma wakefield accelerators. The extremely attractive feature of such accelerators is the high accelerating gradients they produce, on the order of 10100GeV=m, which are two to three orders of magnitude higher than the conventional RF accelerators. In addition, the plasma accelerating cavities consist of already ionized (or partially ionized) particles, so they do not suffer from the breakdown damage like the traditional metallic cavities. Therefore PWFA, if successful, can reduce the acceleration distance down to hundreds of meters scale and greatly decrease the construction cost of a collider. 1.3.2 Basic Principles of Plasma Wakeeld Accelerators Plasmas, composed of quasineutral gas of charged and neutral par- ticles, exhibit collective behavior. Unlike other states of matter where the particles and molecules are distributed in incoherent way, inside a plasma its billions of free electrons can be manipulated together and forced to act 5 coherently, which is the physical reason why plasmas can support much higher accelerating gradients. Plasma wakefields can be excited by either a single, intense laser pulse (a scheme known as the ”Laser Wakefield Ac- celerator” or ”LWFA”) or by a relativistic particle bunch (a scheme known as the ”Plasma Wakefield Accelerator” or ”PWFA”). We focus on the PWFA throughout this thesis. The generation of the high-amplitude wakefield is as follows: as the relativistic charged particle bunch propagates in the plasma, the plasma electrons are displaced by then space charge electric field of the particle bunch, and the much heavier plasma ions stay almost static. When the particle bunch moves further in the plasma, the displaced plasma elec- trons come back to the axis, due to the positive space charge of the plasma ions left behind the bunch, and then overshoot. As a result, an oscillating plasma wave is formed behind the propagating particle bunch. This oscilla- tion sets up a coherent plasma density fluctuation, which in turn leads to the periodic longitudinal (accelerating and decelerating) and transverse (focus- ing and defocusing) electric field propagating at the velocity of drive bunch. Figure 1.1a) shows the wakefield excitation schematically [6]. If a relativis- tic witness beam is placed in the proper accelerating phase of the plasma wakefield, as illustrated in Fig. 1.1b), it is accelerated by the electric field and reaches high energy. The field structure sketched in the second electron bubble of Figure 1.1a) shows that in the front of the bubble these fields are 6 decelerating for negatively charged particles and accelerating in the back. The transverse fields are focusing. Energy can therefore be extracted from a drive bunch placed in the front of the bubbles and transferred to a wit- ness bunch placed in the back of the bubbles. The transverse focusing field can maintain both bunches to a small transverse size against their natural tendency to diverge due to their emittance or transverse temperature. It is worth noting that for relativistic bunches, the dephasing distance between the bunch particles are given as: L dephase = 1 2 L p ( is the Lorentz factor, L p is the plasma length and << ). Therefore no dephasing happens even for very large energy gain of the witness bunch, which could be many times the incoming energy of the drive and witness bunch. Note also that the electrostatic ion plasma waves also exits [7]. However, their character- istic wave frequency is ! pi = q me m i ! pe !pe 43 , where ! pe = (n e e 2 = 0 m e ) 1=2 is the electron-plasma wave frequency (n e the unperturbed plasma density, m e the electron mass,m i the ion mass, and 0 = 8:85 10 12 Fm 1 and is the vacuum permittivity. Therefore ions can be considered as immobile (at the scale of interest, i.e., 1=! pe ). Note also that the role of the plasma is to ”convert” the transverse fields of the laser pulse or particle bunch (space charge) into longitudinal fields. This is the same function as that of resonant (waveguide) cavities in RF accelerators supporting TM modes. Longitudi- nal fields parallel to the particles motion are of course necessary to transfer 7 energy (accelerate) to the particles. Figure 1.1: Schematic of the plasma wakefield accelerator for: a) a single bunch, and b) a drive (D), witness (W) bunch system. The wakefields are also cartooned in the second bucket of the wake ina). Notice that we con- sider a negatively charged witness bunch in the second bubble of the wake- field, so in Fig. a) the left-pointing arrow indicates acceleration, and the outward-pointing arrow indicates focusing, and vice versa. A rough estimate of the wakefield amplitude can be obtained through Maxwells equation for the electrostatic field: 5 ! E = 0 ; (1.1) where E is the electric field, is the charge density, and 0 is the vacuum permittivity. Assume a one dimensional perturbation of the charge density n: =ene ikpz , wheree is the electron charge,n is the plasma electron density perturbation,k p =! pe =v b = ! pe =c is the plasma wave number (v b is the velocity of beam particles,c is light velocity andv b = c in the relativistic case). The accelerating electric field can be written as ! E z = ^ zE 0 e ikpz with 5 = ^ z@=@z. For a density perturbation on the order of plasma density, e.g. nn e ,E 0 is calculated as: eE 0 mc! pe p ne 10 16 cm 3 10 GeV m [8], so for 8 a neutral plasma densityn e = 10 16 cm 3 , the equation gives the accelerat- ing electric field of 10GV=m. HereE 0 is known as the wave-breaking field E wb , which is obtained when the charge density perturbation is equal to the plasma density. This extremely large wakefield amplitude is the most im- portant feature that has motivated the plasma accelerator community over the last several decades to investigate the details of using plasma as accel- erating structures. It is worth noting here that the PWFA is an energy transformer that transfers the energy of an drive bunch to a trailing bunch. Hence it still needs a pre-created efficient drive bunch to provide the initial energy to be extracted. Therefore instead of replacing existing accelerators, the PWFA extends them to higher energies. 1.4 Current Development and Future Challenges 1.4.1 Current Development The beam-plasma interaction research was started from the linear regime where the beam density is much smaller than the plasma density (n b n e ). In 1979, Tajima and Dawson initiated the idea of using a laser pulse to drive the relativistic electrostatic waves in plasma [9]. In 1987, Chen [10, 11] and his colleagues analyzed the excitation of plasma waves by relativistic beam. In the following year, Rosenzweig et al. demonstrated experimentally for the first time energy gain of a trailing electron bunch in 9 the wakefield excited by a preceding driving electron bunch [12]. Those ex- periments were also verified in the mildly nonlinear regime in 1989 [13]. It was in 1991 that the so-called ”blowout regime” of the plasma wakefield accelerators was identified through simulations [14, 15]. In this highly non- linear regime the drive bunch is much denser than the plasma (n b >> n e ), and its radius (generally) much smaller than the plasma skin depth (c=! pe ). As a result, it depletes all the electrons in a region surrounding the drive bunch and an uniform ion column is formed [16]. Simulations demon- strated that operating in the blowout region is extremely advantageous for the witness bunch because of the radially uniform accelerating gradient and linear focusing force along the radius, which is an extremely desirable situ- ation that is not possible in the linear regime (see next Chapter). In the last two decades, the PWFA community has set forth to demonstrate that theP- WFA can operate in the blowout regime over meter scale distances [17], and this is only made possible by unique properties of the SLAC electron and positron bunches: high energy (28:5 and 42GeV ) and charge ( 3nC), low normalized emittance (< 5 10 5 mmmrad), and ultra-short bunch length (< 20 um). With such drive bunches, the experimental success includes the transverse beam-envelope dynamic studies by Clayton et al. [18], the first demonstration of a positron bunch-driven PWFA by Blue et al. [19], 156MV=m acceleration of electron beam over 1:4m-long PWFA module by 10 Muggli et al. [20], the first multi-GeV energy gain by Hogan et al. [21], the production of ultra-short trapped electron bunch by Oz et al. [22, 23], as well as the energy doubling of 42GeV incoming electrons in a plasma only 85-cm long reported by Blumenfeld et al. [24]. Although the energy spread was an undesirable 100% in this last experiment, an accelerating gradient of more than 50 GV=m sustained over a meter-scale plasma has been es- tablished and it is a great success. Note that USC’s contribution has been essential for all these results. 1.4.2 Future Challenges for the PWFA and Dissertation Outline A PWFA must be designed to meet the requirements of a future col- lider, as mentioned above, to produce new physics. Again, take the lepton (positron-electron) collider as the example. In order for the trailing bunch to reach the required high energy as in a meters scale distance, an average ac- celerating gradient of 50100GeV=m is necessary for the PWFA , which has already been demonstrated in experiments. Based on this starting point, the following major challenges needs to be overcome before realizing the dream of building the more compact plasma based collider: 1. Wake excitation from a high efficiency driver, i.e. high charge, high en- ergy, small transverse and longitudinal sizes (high brightness). 2. Enhancement of the transformer ratio (an indication of energy transfer efficiency and is defined in Chapter 2). 11 3. High energy acceleration of a trailing electron beam with high charge, low energy spread and low emittance. Actually, most of the challenges could be addressed or resolved through resonant excitation of the wakefields, either by creating a train of short bunches (on the scale of plasma wavelength) or by the self-modulation in- stability (called SMI, and this will be addressed in Chapters 3 through 6). Those are the main focus of this thesis. Regarding challenge 1, as we mentioned before, PWFA is an energy transformer and it needs a pre-existing, efficient drive bunch to provide the initial energy to be extracted. Energy conservation imposes of course that the energy gained by the accelerated bunch can only be at most equal to the energy loss of the drive bunch, even if the energy gain per particle of the witness bunch can be many times the energy loss per particle of the drive bunch. In the case of a PWFA, the best case isE w N w L p =N d E d L p , where E w andE d denote the accelerating field of the witness bunch and decelerat- ing field of the drive bunch, respectively ,N w andN d denote particle num- ber of the witness bunch and drive bunch, respectively, andL p denotes the plasma length. Therefore, in order to obtain a high energy witness bunch, a drive bunch with high energy and high charge is required. The expression of E wb as given in the previous section shows that a high wave breaking 12 limit is reached only in high density plasmas (currently 10 17 10 18 cm 3 available). It has been proven that plasma-based particle accelerators are most effectively driven by relativistic particle bunches or laser pulses ap- proximately one plasma period long:L beam . pe . Hence the drive bunch is also required to be short (< 100um). Besides, as B. Allen recently demon- strated experimentally, the transverse size of a bunch needs to be smaller than or on the order of the plasma skindepth (c=! pe ) to avoid the current filamentation instability [11, 25]. In order to meet the high-energy requirement of the future lepton col- lider, a new scheme has been proposed that utilizes theTeV proton bunch produced by LHC at CERN as the drive bunch [26]. By assuming the bunch length to be 100 um, simulations have shown the possibility of producing aTeV electron bunch in a single acceleration stage only a few hundreds of meters long. In this case the drive proton bunch carries multiple kilojules of energy and only part of the energy is extracted to accelerate thee bunch. The remaining energy can be recycled for later acceleration. However, ultra- short proton bunches are not available today. The alternative approach is to propagate the long LHC bunch in dense plasma and let transverse focus- ing force modulate the beam density [27, 28]. As a result, multiple beam- lets are formed, leading to resonant excitation and hence large acceleration amplitude. This instability is called the self-modulation instability (SMI). 13 In Chapter 3, we use numerical simulations to demonstrate the seeding of SMI with currently available experimental parameters. Then in Chapter 4 we present the first experimental results for SMI seeding, an indirect but very important evidence of the driving of wakefields with period shorter than the drive bunch length. We study the physics of seeding with electron bunches, but the results can be equally applied to other particles species (positrons, protons, ...). In Chapter 5 we study numerically the develop- ment of the SMI for a higher charge bunch. Based on the simulation results, we discuss the possibility of carrying out experiments to confirm the trans- verse self modulation instability, which is the direct evidence of SMI devel- opment. In Chapter 6 we address the effect of transverse plasma density profile on the SMI development, an important consideration for the design of plasma sources for future SMI experiments. Regarding challenge 2, the scheme of multiple bunches can enhance the transformer ratio [29, 30, 31, 32], and hence multiply the energy gain of the witness bunch in a single acceleration stage. This multiple-bunches scheme is made possible by P . Muggli’s recent demonstration [33, 34] of a masking technique at BNL-ATF (Accelerator Test Facility at Brookhaven National Laboratory) to partially block sections of a long bunch with a cor- rected energy-spread. This scheme is capable of generating microbunches with subpicosecond spacing. We explore the possibility of reaching high 14 transformer ratio in the weakly nonlinear PWFA regime with the ramped bunch scheme, as will be shown in Chapter 2 with simulations. On one hand, it achieves the high transformer ratio reachable in the linear regime. On the other hand, due to the linear focusing force in nonlinear regime, the bunchs transverse size can be maintained during the propagation, which is a favorable situation for acceleration. This scheme will be tested with experiments in the near future. Regarding challenge 3, a witness bunch that is much shorter than the plasma wavelength needs to be placed in the accelerating and focusing phase and accelerated with high gradient. Such an experiment has been performed by E. Kallos (USC) et al. at the Accelerator Test at Brookhaven National Lab (BNL) [36]. It shows that a trailing bunch shorter than half the plasma wavelength is generated and then accelerated at 150MV=m loaded gradients, for the fist time. This result, if successful, combined with SLAC high acceleration gradient, could be the solution for challenge 3. 1.5 Chapter Conclusions Particle accelerators are the main tool for discovering new elemen- tary particles. Future lepton colliders require high energy (> 250GeV ), high luminosity (10 34 cm 2 s 1 ) as well as low emittance and low energy spread, and all those requirements have to be achieved in a relatively short distance and at an affordable price. However, conventional accelerators can meet the 15 physics requirements only through very long ( 40km) and expensive ma- chines, due to their relatively low acceleration gradients (10 100MeV=m). Plasma-based accelerators are able to overcome this limitation and have been demonstrated to be the potential alternative to the conventional accel- erators. In this dissertation, we explore through numerical simulations the possibility of reaching high transformer ratio in weakly nonlinear regime of PWFA. Also, we study through both simulations and experiments the physics of SMI, which could lead to the possibility of using a high-energy proton bunch to drive the wakefield. In addition, we investigate through simulations the effect of transverse plasma radius on SMI development. We address the physics and the principles of the PWFA by considering and us- ing low energy electron beams. Those studies all contribute to fulfilling the in the dream of building future much more compact lepton colliders. 16 Chapter 2 Numerical Simulation Towards Reaching High Trans- former Ratio in the Weakly Nonlinear Regime of PWFA using the Ramped Bunch Scheme 2.1 Introduction The transformer ratioR is defined as the ratio of the maximum ac- celerating wakefield behind a bunch or bunch trainE + max and the maximum decelerating wakefield inside the bunch (or train)E max . It is an important indication of energy transfer efficiency from the drive bunch to the wake- field by means of plasma oscillations. The importance of transformer ra- tio can be illustrated by considering the following scenario: a drive bunch with the incoming energy per particleW 0 excites the wakefields and has a maximum decelerating fieldE max inside the bunch. A witness bunch trails behind and gains energy by sampling the wakefield with a maximum ac- celerating field E + max . If the drive bunch keeps propagating until it loses all its energy, the total propagation distance is L = W 0 =E max . Then the peak energy gain per particle for the witness bunch is: W = E + max L = E + max W 0 =E max = RW 0 . Therefore maximizing the transformer ratio is equivalent to maximizing the energy gain per particle for the witness bunch, 17 which is important for the design of future compact accelerators. There are a number of ways to enhance the transformer ratio. Two examples are the ramped bunch scheme and tailoring the bunch shape, and we focus on the former in this chapter. Note that for a single drive bunch with symmetric current profile, the transformer ratio is no larger than 2 (R 2) [35]. 2.2 Ramped Bunch Scheme in Linear Regime 2.2.1 Overview of Linear Theory In principle, the linear theory is valid when the plasma density per- turbation is small compared to the unperturbed plasma density n e . Since the density perturbation is typically on the order of peak beam densityn b0 , the condition of quasi-neutrality is equivalent ton b0 <<n e . First, let us have a brief review of the wakefields excited by a relativistic electron bunch with a certain density profile. Define the bunchs electron density distribution as: n b ( =ctz;r) =n b k ()n b? (r); (2.1) where =ctz is the transformed variable in the frame moving at the speed of light. The longitudinal wakefield and transverse wakefield excited by the electron bunch in a 2D cylindrically symmetric geometry can be expressed as [38]: W z (;r) =E k ()R(r) = e 0 Z 1 n b k ( 0 )cosk p ( 0 )d 0 R(r); (2.2) 18 W r (;r) = e 0 k p Z 1 n b k ( 0 )sink p ( 0 )d 0 dR(r) dr ; (2.3) where E k () and R(r) are the longitudinal and unitless transverse depen- dencies, respectively (Note that thisR(r) is different from the transformer ratioR). Their expressions are given as follows: E k = e 0 Z 1 n b k ( 0 )cosk p ( 0 )d 0 ; (2.4) R(r) =k p 2 R r 0 r 0 dr 0 n b? (r 0 )I 0 (k p r 0 )K 0 (k p r) +k p 2 R 1 r r 0 dr 0 n b? (r 0 )I 0 (k p r)K 0 (k p r 0 ) (2.5) whereI 0 andK 0 are the zeroth order modified Bessel functions of the first and second kind, respectively. The transverse component R(r) is maximum at r = 0 and with a transversely Gaussian beam profilen b? (r) =exp( r 2 2r 2 ) it can be written as: R(0) =k p 2 I 0 (0) R 1 0 r 0 dr 0 n b? (r 0 )K 0 (k p r 0 ) = 8 > > > < > > > : k p 2 r 2 0:05797ln(k p r ) ; k p r 1 k 2 p 2 r 2 e k 2 p 2 r 2 E 1 k 2 p 2 r 2 ; 8k p r 1; k p r 1 (2.6) Here E 1 (b) = R 1 1 e bt t dt is the exponential integral of the first kind. Fig- ure. 2.1 shows R(0) increases as a function of the normalized transverse size (k p r ) [37]. Equations 2.2 through 2.5 completely determine the wake- 19 Figure 2.1: The on-axis transverse component of the longitudinal wakefield R(0). Picture taken from Ref. [37]. fields excited by a particle bunch with a density much lower than the initial plasma density, which is the linear regime of the PWFA. 2.2.2 Reaching High Transformer Ratio in Linear Regime: Ramped Bunch Scheme The ramped bunch scheme [29, 30, 31, 32] was initially proposed in linear regime where the longitudinal electric field excited by individ- ual bunches add each other [39]. The scheme can be illustrated as follows: For a train of equidistant bunches, the maximum transformer ratio can be achieved when the subsequent bunch is placed in the previous bunchs ac- celerating field, and also the charge ratios of the bunches are tuned so that 20 the peak decelerating wakefields inside each bunch are equal. The separa- tions and charge ratio depend on the shape (width, density profile) of the bunches. The most interesting case is the one that maximizes the trans- former ratio of the single bunch, e.g. a square bunch with electron density profilen b as follows: n b =n b0 (L beam ) exp( r 2 2 2 r0 ); (2.7) where n b0 is peak density and n b0 << n e , is the Heaviside function ( = 1 when 0L beam , and = 0 otherwise), =ctz, and r0 is the rms of the initial transverse size of the bunch. Assume the beam length is chosen such thatk p L beam = , by plugging the density defined by Eqn. 2.7 into Eqns. 2.2 and 2.3, we obtain the longitudinal wakefields: W z (;r) = 8 < : en b0 0 kp sink p R(r) 0L beam en b0 0 kp sink p sink p (L beam ) R(r) >L beam (2.8) and transverse wakefields: W r (;r) = 8 < : en b0 0 k 2 p (1cosk p ) dR(r) dr 0L beam en b0 0 k 2 p cosk p (L beam )cosk p dR(r) dr >L beam (2.9) Equations 2.8 and 2.9 show that for the longitudinally square bunch profile, the wakefields scale as: W z (;r)/n b0 =k p R(r)/n b0 =n 1=2 e R(r); (2.10) 21 and W r (;r)/n b0 =k 2 p dR(r)=dr/n b0 =n e dR(r)=dr; (2.11) The maximum decelerating wakefield inside the bunchE max and maximum after the bunchE + max are respectively: E max = en b0 0 k p R(0);E + max = 2en b0 0 k p R(0); (2.12) which gives the transformer ratioR = 2. In this case, the wake inside the bunch is symmetric with respect to its center and the equations are greatly simplified. Note that electrons are negatively charged, so positive fields decelerate and negative fields accelerate them, and this explains the signs ofE max andE + max . Now consider a train consisting of such bunches with the sameL beam and r . In order to achieve high overall transformer ratio, one can show that the charge per bunch and their location scale in a simple way [37]: Q 1 :Q 2 :Q 3 ::::Q m = 1 : 3 : 5 :::: 2m 1; (2.13) 2 1 = 3 2 =::: = m m1 = 1:5 pe ; (2.14) wherem indicates themth bunch, pe is the plasma wavelength pe = 2 c=! pe . The above equations show that the charge scales linearly with bunch number m and the bunches must be placed 1:5 plasma wavelength apart. 22 And as a result, all the peak decelerating wakefields under each bunch are equal. The maximum wakefield left after the mth bunch E + (max)m can be obtained by superimposing the longitudinal wakefield of each bunch (given by Eqn. 2.12): E + (max)m =m 2en b1 0 k p R(0); (2.15) wheren b1 is the peak density for the first bunch. Equation 2.12 and 2.15 give the overall transformer ratio as: R tot = 2m (2.16) Therefore, in principle, the energy gain of the witness bunch can be up to W =RW 0 = 2mW 0 (therefore, due to energy conservation, the charge of the witness bunch can at most be 1 2m of the drive bunch charge). This equation indicates that the overall transformer ratio scales with the number of bunches, which is the ultimate goal for the ramped bunch scheme. Figure 2.2 gives an example of this scheme for four drive bunches in a plasma with the density of 1:8 10 16 cm 3 . The bunches are pe long, separated by 1:5 plasma wavelengths, and their charge scales as: 40 pC [1 : 3 : 5 : 7]. All the bunches experience the same (peak) decelerating wake- field and the overall transformer ratio reaches 7:89, almost 4 times more than that of a single bunch (R = 2), the value predicted by the Eqn. 2.16. The slight difference comes from the numerical calculation. Note that for 23 Figure 2.2: Longitudinal wakefields driven by ramped bunch scheme with a train of square bunches in linear regime. The red rectangles represent the drive bunches, the green rectangle represents the witness bunch, and the black lines shows the longitudinal wakefields. The arrow shows the trains travel direction, and = ctz is the transformed variable in the frame moving at the speed of light. The drive bunches are separated by 1:5 pe and their charge is Q = 40 pC [1 : 3 : 5 : 7]. The witness bunch is in the accelerating phase of the wakefield. The transformer ratio is 7:89. 24 maximum efficiency the decelerating fieldE should be constant along the bunch. This is possible with a single shaped bunch. 2.2.3 High Transformer Ratio not Maintained in Linear Regime No dephasing occurs in the above scheme since the bunch particles are relativistic. While the ramped bunch scheme ensures that every bunch experience the same decelerating field, it is worthwhile to take a look at the transverse fields. Transverse wakefields focus/defocus bunch particles, change the bunches density and therefore the wakefield they drive. Hence the evolution of the bunches due to transverse fields determines whether high transformer ratio can be maintained along the propagation. Again superimposing the transverse field for each bunch as given in Eqn. 2.9, we obtain the transverse fields at the center ofmth bunch as follows [37]: W r (;r) = en b0 0 k 2 p (m 1) 2 +sink p ( cm ) (1cosk p ) dR(r) dr ; (2.17) where cm is the center of the mth bunch. It is interesting to observe that although this scheme is designed such that the longitudinal wakefield is identical under each bunch. On the other hand, the transverse wakefields are not the same for each bunch, as shown in Fig. 2.3, but scale quadrati- cally withm, the number of the bunches, as shown in Eqn. 2.17. The fact that later bunches experience stronger focusing force leads to a more rapid 25 density increase than the earlier bunches, which in turn breaks the initial balance that every bunch experiences the same decelerating field (E z n b , see Eqn. 2.8). Therefore, the high transformer ratio cannot be maintained through the long acceleration distance in the linear regime due to the trans- verse evolution of the bunch. Note that the fields of Figures 2.2 and 2.3 are those at the plasma entrance (no propagation, no evolution). ! ! Figure 2.3: Transverse wakefield driven by ramped bunch scheme with a train of square bunches in linear regime. 2.2.4 Emittance Preservation in Nonlinear Regime If the beam density is increased to be much larger than the plasma density (n b >n e ), the beam-plasma interaction enters the nonlinear regime, also called blowout regime [40, 41]. All the nearby plasma electrons are de- 26 pleted by the bunch head and a pure ion column is formed around the core of the bunch. Unlike in the linear regime where the transverse focusing field is proportional todR(r)=dr (as shown in Eqn. 2.11), the transverse wakefield in nonlinear regime scales linearly with the radius asE r = (en e =2 0 )r, and E r is constant along the propagating direction. The fieldE r is obtained sim- ply from Gauss’ law applied to to the pure ion column of densityn i =n e . The focusing strength is defined from Newton’s Equation as: K =F r =(r mc 2 ) =n e e 2 =(2 0 mc 2 ) = k 2 p 2 ; (2.18) where is bunch particles relativistic Lorentz factor. Therefore,K is a con- stant for a given plasma density. With the focusing force linearly increasing with radius the transverse Gaussian shape of the electron bunch with nor- malized emittance N is preserved. The envelope equation for the transverse rms size in thex ory direction of an transversely Gaussian electron bunch is given as [43]: d 2 x;y dz 2 +K x;y = 2 Nx;y 2 3 x;y ; (2.19) where Nx;y is the normalized emittance ( x;y = ) of the bunch inx ory di- rection, and x;y is the geometric emmittance. This equation reveals that the transverse size evolution is determined by focusing force and the beam emittance, which are respectively the second term on the left and the term on the right of the Eqn. 2.19. The variation of the transverse size is mini- 27 mized ifd 2 x;y =d 2 = 0, which yields a matched beam emittance given as: Nx;y = p K 2 x;y = p =2k p 2 x;y (2.20) If we assume that at the plasma entrance, the beam is focused so thatd x;y =dz = 0 atz = 0, with the matched beam emittance, the beam size remains the ini- tial transverse size x 0 ;y 0 along the plasma, and hence the beam density is maintained the same during the propagation, which is an extremely favor- able situation in plasma wakefield accelerator. Since there is no transverse size evolution in this case the density of each bunch is modulated along their propagation and so are the wakefields and the initial transformer ra- tio. 2.3 Applying Ramped Bunch Scheme to Weakly Nonlinear Regime As discussed in previous sections, the high transformer ratio achieved in linear regime with a ramped bunch train cannot be maintained over the propagation distance due to the transverse size evolution of the beam. In this section, we aim at finding an optimal regime that combines both the advantages of linear and non-linear regimes of PWFA. On one hand, the longitudinal electric field excited by individual bunches adds as in the lin- ear regime, and the transformer ratio can be maximized (i.e. much larger than 2) by applying the ramped bunch scheme. On the other hand, the 28 bunches create large wakefield independent of transverse sizes evolution while propagating through the plasma as in the non-linear regime. In prin- ciple, such a scheme can multiply the energy of the witness bunch following the drive bunch train in a single PWFA stage. This motivation brings us to the weakly nonlinear regime, an intermediate between linear and nonlinear regimes, where the beam density is comparable to the plasma density, i.e., n b n e . We start with the experimental parameters that are available at ATF and then use simulations to optimize the beam parameters for ramped bunch scheme, such as the bunch separation and charge ratio, in order to reach high transformer ratio in this weakly nonlinear regime. Note that the same idea was simultaneously proposed in Ref. [42] . These experiments are in the early stage at ATF. 2.3.1 Experimental Background Muggli et al. have successfully generated trains of electron micro bunches with adjustable subpicosecond spacing in ATF experiments [33, 34], which can be applied to the generation of wakefields for PWFA. The masking technique employed in the experiment allows for easy tailoring of the bunch width, relative positioning and the charge ratio. Based on this technique, a train of bunches as required in the ramped bunch scheme can be produced. If we are able to use simulations to demonstrate that ramped bunch scheme can achieve a high transformer ratio in weakly nonlinear 29 regime, then in principle, we can plan to test it with experiments at ATF. This is why we consider these particular parameters. 2.3.2 Simulation Parameters versus Current Experimental Parameters The micro bunches produced at BNL-ATF typically have a transverse size of 100um, mean energy of 59MeV and normalized emittance of 1mm mrad [33, 34]. In order to reach the weakly nonlinear regime ( n b n e ), we assume the bunch can be focused down to 5 um, which is possible in the experiment. Detailed simulation parameters are shown in Table 2.1. Note that we use longitudinally Gaussian bunches in simulation instead of square bunches, and it is due to the fact that the Gaussian shape is a better approximation for the bunches available in experiments. The simulations are performed with a highly efficient, fully relativis- tic, three- dimensional particle-in-cell code QuickPic, which uses a moving window that propagates at velocityc. The resolutions are chosen such that k p z = 0:035 for the longitudinal andk p x = k p y = 0:016 for transverse cell sizes, with 2 2 2 plasma and beam particles per cell. 2.4 Simulation Results 2.4.1 Single Bunch Using the parameter listed in the right hand side column of Table 2.1, Fig. 2.4 shows the one-time-step simulation results for a single bunch propagating in the plasma. The bunch travels in thez direction from right 30 Figure 2.4: One-time-step simulation result of a single bunch with param- eters given in Table : a) the 2D image of normalized beam density, b) 2D image of the normalized plasma electron density, c) plasma electron den- sity perturbation along on the axis (as the dotted line in b)), and the inset magnifies the plasma density lineout of a small region (as indicated by the red circle) where the plasma electron density changes from -1 to 0, d) the longitudinal wakefield on the axisE z . 31 Table 2.1: Bunch train parameters ATF Simulations Current Typical “Gaussian” z 25 um Energy 59 MeV n e 1:24 10 16 cm 3 r 100 um 5 um Charge 40pC/bunch 40pC for 1st bunch Spacing pe 1:5 pe n b /n e 0.005 (n b n e ) 2(n b >n e ) Regime linear nonlinear to left as indicated by the arrow. Remember thatzct is the transformed variable in the frame moving at the speed of light. Figure 2.4 a) shows the 2D (ZX) image of the normalized density (n b =n e , n e is the unper- turbed plasma density) of a single Gaussian bunch. Figure 2.4b) shows the 2D image of the normalized plasma electron density, and as expected, the plasma electrons surrounding the bunch are depleted (as indicated by the red bubble where the plasma electron density is low) and after the bunch passes, the displaced plasma electrons come back to the axis (as indicated by the blue color) and overshoot, and therefore an oscillation wave is formed behind the bunch. Figure 2.4 c) shows the plasma electron density along on the axis, and the inset magnifies the plasma density lineout of a small region (as indicated by the red circle) where the plasma electron density changes from1 to 0 (0 means that all electrons are depleted), indicating that the blowout is reached. Figure 2.4 d) shows the longitudinal wake- 32 field (on the axis) E z , and the transformer ratio R 1 can be obtained as: R 1 = E + 1 =E 1 = 0:56=0:38 1:5. Note that here the transformer ratio for one bunch is smaller than 2, as given by the square bunch in previous sec- tions. The reason is that in these simulations, Gaussian bunches are used instead of square bunches. 2.4.2 Two Bunches Figure 2.5: One-time-step simulation result of two bunches with parameters given in the text: a) the 2D image of normalized beam densityn b =n e ,b) 2D image of the normalized plasma electron density,c) plasma electron density perturbation along on the axis (as the dotted line inb)),d) the longitudinal wakefield on the axisE z . Using the ramped bunch scheme in linear regime as the guideline, 33 we add a second bunch with the same bunch shape (Gaussian) and sizes ( r and z ). In order to obtain as high overall transformer ratio as possible and also to keep the decelerating field to be the same, we optimize the spacing between bunches () as well as charge ratio (Q 2 =Q 1 ) by performing mul- tiple simulations. The optimumparameters are found to be: = 1:5 pe , andQ 2 =Q 1 = 2:7 , and the one-time-step simulation corresponding to Fig. 2.5 is shown in Fig. 2.5, which gives the maximum transformer ratio as: R 2 = E + 2useful =max(E 1 ;E 2 ) = 1:08=0:38 2:8, which is close to 1:5 2. The difference between E 1 and E 2 is less than 2%, an indication that the two bunches are losing energy at very similar rates. This result is compa- rable with the ramped bunch scheme in linear theory where the spacing is 1:5 pe , the charge ratio is 3, and the difference is due to different bunch shape (square vs. Gaussian) as well as different beam-plasma interaction regime (linear vs. weakly nonlinear). Note that we used E + 2useful instead of the peak accelerating wakefield due to the appearance of spike in theE z curve. 2.4.3 Three Bunches The third bunch is added to investigate whether the transformer ra- tio continues to increase. The maximum transformer ratio is achieved when the third bunch is placed 1:32 pe behind the second one, and the charge ratio Q 3 =Q 1 = 4:2, close to 1:5 3, and the one times step simulation is 34 Figure 2.6: One-time-step simulation result of three bunches with parame- ters given in the text: a) the 2D image of normalized beam densityn b =n e , b) 2D image of the normalized plasma electron density,c) plasma electron density perturbation along on the axis (as the dotted line inb)),d) the lon- gitudinal wakefield on the axisE z . 35 shown on Fig. 2.6. The maximum transformer ratio (R 3 ) is obtained as: R 3 = E + 3useful =max(E 1 ;E 2 ;E 3 ) = 1:61=0:38 4:2, which is also close to 3R 1 . Figure 2.7: Simulation result of three bunches after the propagation for 2cm with parameters given in the text: a) the 2D image of normalized beam density, b) 2D image of the normalized plasma electron density, c) plasma electron density perturbation along on the axis (as the dotted line inb)),d) the longitudinal wakefield on the axisE z . All previous simulations only show the transformer ratio after a sin- gle time step propagation, which is immediately after the bunches enter the plasma (as was calculated in the linear regime case, as shown in Figs 2.2 and 2.3). It is interesting to investigate whether this high transformer ratio can be maintained over much longer distance, i.e. over the 2 cm plasma 36 Table 2.2: Square bunch train in the linear regime vs. Gaussian bunch train in the weakly nonlinear regime Linear Weakly Nonlinear R1 (TR for a single bunch) 2 1:5 Charge Ratio 1 : 3 : 5 1 : 2:7 : 4:2 Bunch Separation 1:5 pe 1:35 pe Overall Transformer Ratio 6 4:2 Transverse Size Preservation No Yes Maximum Acceleration Gradient 300MV=m 1:6GV=m capillary as available at ATF. Figure 2.7 shows the same plots as on Fig. 2.6, but after a 2cm propagation in the plasma. The transformer ratio is found to decrease from 4:2 to 4:0, i.e., stays essentially constant during the accel- eration and propagation over 2cm. In addition, the maximum accelerating field remains at very large amplitude (1:5 GV=m) even with the ATF pa- rameters. The advantages of the linear regime (addition of wakefields) and the nonlinear regime (large acceleration gradient, linear radial focusing and emittance preservation) can be obtained at the same time in this weakly nonlinear regime. The comparison between the ramped bunch scheme in the linear regime and in the weakly nonlinear regime is summarized in Ta- ble 2.2. However, simulations performed so far show that when adding the fourth bunch, the transformer ratio does not increase anymore, because the density of the fourth bunch becomes much higher than the plasma density so that the beam-plasma interaction becomes highly nonlinear, and there- 37 fore the wakefield does not add linearly any more. 2.5 Chapter Conclusions We have shown in initial simulations that a transformer ratio larger than 2 can be reached in the weakly nonlinear regime of the PWFA with the bunch train scheme. In this regime the plasma wakefields reach large amplitudes (> 1GV=m) and the transformer ratio can be maintained over a plasma length equal to that in the experiments planned at the ATF. These results have been published in Ref. [44] by Y. Fang et al.. Further simu- lations are needed to demonstrate that this high transformer ratio can be maintained over a much longer distance, over which the bunches energy is completely depleted (estimated to be 58:2 MeV=0:38 GV=m 15 cm). However, for these low energy bunches, large energy lost could lead to dephasing since at low energy the electrons are not relativistic any more. The study must be extended to the case of a high-energy collider beam for which dephasing is not an issue. This scheme relies on the masking tech- nique to shape a long bunch into a train of small bunches, resulting in the resonant excitation the wakefields. However, the occurrence of the self- modulation instability (SMI) can naturally modulates the beam density at the plasma wavelength scale and also lead to the resonant excitation of the plasma wakefields, a scheme that is introduced in the following chapters. 38 Chapter 3 Numerical Study of the Seeding of Self-modulation Instability with a Low Charge (50pC) long Electron Bunch in a Plasma 3.1 Introduction Plasma-based particle accelerators are most effectively driven by rel- ativistic particle bunches or laser pulses approximately one plasma period long: L beam . pe = 2c=! pe , where pe = 2c !pe is the plasma wavelength. Present drivers are very short,< 30um or 100fs, thereby allowing for op- eration at large plasma densities and very large accelerating gradients. As mentioned previously, the accelerating field amplitude is on the order of the non-relativistic cold wave breaking field [8]E wb = 2mec 2 e 1 pe orE wb (GV=m) 10 (n e =10 16 cm 3 ) 1=2 , exceeding 10 GV=m with n e > 10 16 cm 3 . Plasma- based acceleration is therefore a very attractive possible new acceleration technique for a future more compact and more affordable linear electron/posi- tron collider. Such an accelerator will need to produce bunches with multi- kilojoules of energy (e.g. 210 10 e =e + at 500GeV ) and extremely low emit- tance to access new physics at the energy frontier ( 1TeV=particle) and to reach the required luminosity. However, present electron bunch and laser 39 pulse drivers carry less than 100 Joules, thereby limiting the energy gain by the accelerated bunch to< 100 Joules and would require many acceleration stages to reach the desired energy. Proton bunches produced in circular accelerators can carry many tens of kilo Joules. These bunches with 10 11 protons are routinely pro- duced, for example at CERN by the Super Proton Synchrotron (SPS, 450GeV ) and the Large Hadron Collider (LHC, 3:5-7 TeV ). Principle studies [26] showed that a single plasma section driven by a 1TeV , 100m-long bunch with 10 11 protons could accelerate a 10 GeV incoming electron bunch to 600 GeV in 500 m of plasma with n e = 6 10 14 cm 3 . However, the CERN proton bunches are = 12 cm-long, and short proton bunches with high charge are not currently available. It was recently proposed that long charged particle bunches travel- ing in dense plasmas (L beam >> pe / n 1=2 e ) are subject to a transverse two-stream self-modulation instability (SMI) and can drive wakefields to large amplitudes [27]. The radius of the long bunch and thereby also its density become modulated at the scale of plasma period by the transverse focusing/defocusing component of the wakefields. This periodic density modulation (n b ()/ 1= 2 r (), r the bunch transverse size) provides feed- back for the SMI to grow and saturate, and thereafter resonantly drive the wakefields to amplitudes (/ n b ) that can approach E wb . The instability is 40 convective and grows both along the bunch () and along the plasma (z), as illustrated by the number of e-folding growth for a flat-top bunch [28, 45]: G = = 3 3=2 4 ( n b0 m e n e M b ) 1=3 (k p ) 1=3 (k p z) 2=3 ; (3.1) where k p is the wave number and k p = c=! pe , 1 kpr 2 6 , r 0 the initial bunch radius,n b0 the initial beam density,M b the bunch particles’ mass. Note that the self-modulation of a particle bunch is completely anal- ogous to that of a laser pulse or photon beam [46]. Long particle bunches propagating in plasmas are also subject to another transverse two-stream instability, the hosing instability [47, 48]. This instability is radially asym- metric and has a growth rate comparable to that of the SMI [28, 45]. It could therefore disrupt the bunch propagation before the SMI can grow and sat- urate, and before externally injected particles can gain large amount of en- ergy from the longitudinal wakefields component. Numerical simulations indicate that the seeding of the SMI may miti- gate the development of the hosing instability [49]. Seeding also reduces the plasma length needed for the SMI to reach saturation. A number of methods could be used to seed the SMI, including driving low amplitude wakefields with a short laser pulse or particle bunch preceding the long bunch, using a relativistic ionization front co-propagating with the long bunch [50] and shaping the long bunch with a sharp (when compared to pe ) current rise at 41 its front, the method used here. In this chapter, we would like to take advantage of the experimen- tal parameters that is available at BNL-ATF to study the physics of SMI numerically. Although periodic transverse focusing fields are the driving component for the SMI to develop, it is not possible to measure directly the amplitude of transverse focusing force given the current experimen- tal conditions at ATF. However, longitudinal wakefields can be measured by observing the energy gain and loss by bunch particles in experiment. Wakefields theory implies the existence of corresponding transverse focus- ing/defocusing components [51]. Therefore, the observation of the peri- odic energy gain/loss of the bunch particles is an indirect but very impor- tant evidence of the seeding of SMI. The relativistic electron bunch avail- able at ATF is shaped with a sharp rising edge and has a correlated energy spread. In this chapter, we explore the possibility of demonstrating the SMI seeding through the periodic energy modulation of a long electron bunch propagating in a dense plasma. We estimate the approximate bunch charge required for optimum visibility of energy spectra modulation based on a simple energy-modulation model and the linear theory of PWFA. An SMI seeding experiment is then proposed to to be carried out at ATF. 42 Table 3.1: ATF electron bunch and plasma parameters charge 50pC-1nC current profile ”Square” beam lengthL beam 960um r 120um n b0 3:6 10 12 -7:2 10 13 cm 3 mean energyE 0 58:3MeV ( 114) correlated energy spread E 0 0.48 MeV N 13mmmrad plasma lengthL p 2cm plasma densityn e 10 15 10 17 cm 3 3.2 Beam Parameters Available at ATF Table 3.1 shows the electron beam and plasma parameters that are available at ATF. The present experimental condition allows for the cre- ation of a square bunch profile by using a rectangular mask using a method demonstrated at ATF [33, 34]. Therefore, in the simulation, the electron den- sity profile is assumed to be: n b =n b0 (L beam ) exp r 2 2 2 r ; (3.2) wheren b0 is the initial peak density, is the Heaviside function. The electron bunch has a mean energy of = 58:3 MeV and an ap- proximately linear correlated energy spread (chirp) of about 1% of the in- coming energy. For the experiment we choose that the bunch front has the lowest energy. The charge of the bunch is adjustable from a few tens of pic- ocoulombs to 1nC. The experimental set-up will be introduced in detail 43 in the next chapter. 3.3 Simple Model for Energy Modulation Figure 3.1: A simple model of beam energy modulation: the red lines in- dicate the wakefield (E z ), the horizontal green arrows indicate the energy gain or loss. The original bunch is in blue rectangle and the blue ellipses indicate the energy beamlets. As the bunch enters the plasma its space charge field displaces the plasma electrons and drives the plasma wakefields. In the linear regime of the PWFA (wheren b n e ), the wakefields are sinusoidal, as shown by Eqns. 2.2 and 2.3 in Chapter 2. The wakefields have a period given by the plasma wavelength and have both longitudinal and transverse com- ponents. The transverse component creates a periodic modulation of the 44 bunch transverse momentum and of the bunch radius and charge density if strong enough or the plasma long enough. The longitudinal component leads to periodic energy loss and gain by particles along the bunch. Figure 3.1 shows the energy bunching process schematically. In this simple model, we consider an electron bunch with an incoming energy-position-time cor- relation, with lower energy in the front, as is the case in the experiment. In the figure, the bunch travels from right to left. As the particles located at the crest of decelerating field lose the maximum energy, their energies con- verge to that of the particles that are one quarter of wavelength in front of them (whose initial energy is lower and do not lose or gain energy). Sim- ilarly, the energy of particles sitting in the peak accelerating increases and becomes closer (in energy, not in time!) to that of particles that are one quarter of wavelength behind them. The longitudinal plasma wakefields can therefore cause the beam particles to bunch in energy, which can di- rectly lead to observable peaks in the bunch energy spectrum, as shown in Fig. 3.1. In our experiment where particles in the front have lower energy, the positions of energy bunching along the bunch are those corresponding to the odd zeros of the periodic longitudinal wakefields withE z = 0 at the sharp front. This means that there is always a energy bunch forming at the front. Therefore, when the plasma density is such that the bunch isl plasma wavelengths long,l + 1 peaks form in the energy spectrum (for simplicity 45 l is considered integer here). The bunching in energy due to the longitudi- nal plasma wakefield component is similar to that occurring in the inverse free electron laser [52] or in dielectric loaded wakefield accelerators [53]. For the optimum bunching at the plasma exit, the particles’ energy gain or loss must be on the order of E l = E 0 =4l. Here E 0 is the correlated energy spread. Assuming constant wakefields over the plasma length L p (no growth of SMI) the optimum average electric field amplitude is thus E z = 1 4 E 0 elLp . In this case particles that lose or gain energy meet exactly at the ”positions” of the odd zeros of the wakefield. Higher or lower wake- fields (or plasma length) lead to less visible peaks. This expression shows that anE z decreasing with increasingn e is necessary to preserve the energy bunching visibility. For example, given our beam and plasma parameters, E z values of 5:2; 3:0; 2:0; 1:5; 1:2; 1:0; 0:9MV=m are best for integer values l = 1 to 7, respectively. Note that over the 2cm plasma length and with the MV=m amplitudes considered here, no significant longitudinal dephasing between bunch particles is expected since L dephase = 1 2 L p << pe ( is the Lorentz factor and 114, = = 1%) and there is no longitudinal bunching. Therefore the bunching is purely in energy. 46 3.4 Estimation of Optimum Bunch Charge from Linear The- ory As shown in Table 3.1, the peak density of the electron bunch is much smaller than the plasma density range in the experiment (n b <<n e ). There- fore, the beam plasma interaction is in the linear regime of the PWFA. Ac- cording to 2D linear theory, for an electron bunch with density profile as in Eqn. 3.2, the initial (z = 0cm) longitudinal wakefieldW z on the beam axis (r = 0) and the transverse focusing fieldW r atr = r inside the bunch are given in Eqns. 2.8 and 2.9 in the previous chapter: W z (; 0) = en b0 0 k p sin(k p )R(0)/n b0 n 1=2 e R(0) (3.3) W r (; r ) =E r cB = en b0 0 k 2 p (1cos(k p )) dR( r ) dr / n b0 n e R 0 ( r ); (3.4) wherek p is the plasma wave number andk p = 2l=L beam ,R(0) is the unit- less transverse factor R(0) = k 2 p R 1 0 e r 2 =2 2 r K 0 (2lr=L beam )rdr, as given in Chapter 2. In this case the longitudinal wakefield amplitude within the bunch becomes: E z = lQ 0 L 2 beam 2 r Z 1 0 e r 2 =2 2 r K 0 (2lr=L beam )rdr (3.5) It is worth noting that bothR(0) anddR( r )=dr are increasing func- tions ofk p r , the bunch transverse size r relative to the plasma skin depth 47 -4 -2 0 2 4 0200 400 600 800 -1 0 1 2 3 4 0200 400 600 800 Figure 3.2:a) Initial longitudinal wakefield at the beam axisE z (z = 0;r = 0) andb) initial transverse focusing field atr = r for various plasma densi- ties (shown in Table 3.2) such thatL beam = pe is equal to: 1 (red), 2 (blue), 3 (green), 4 (black), 5 (purple), 6 (orange) and 7 (magenta dotted line). (c=! pe ). Based on Eqn. 3.4 and 3.5, the wakefields amplitudes can be calcu- lated for the electron bunch with various charge between 50 pC and 1 nC in various plasma densities such thatl = 1 to 7. Figure 3.2 shows the line- out of the longitudinal field E z (r = 0;) (Fig. a)) and transverse focus- ing field E r cB at r = r (Fig. b)) excited by a 50 pC electron bunch at different plasma densities. Figure a) shows that for this chosen charge, E z (r = 0) = 3:7; 3:2; 2:6; 2:1; 1:8; 1:5; 1:3 MV=m for l = L beam = pe = 1 to 7, respectively. Note that these values decrease asn e (orl) increases and are close to the ones for optimum bunching visibility. Therefore, the 50pC bunch is optimal for generating visible energy spectra modulation at the 48 Table 3.2: Simulation Parameters l Plasma Densities (cm 3 ) n b ne (for 50pC bunch) 1 1:21 10 15 2:96 10 3 2 4:85 10 15 7:41 10 4 3 1:09 10 16 3:29 10 4 4 1:94 10 16 1:85 10 4 5 3:03 10 16 1:18 10 4 6 4:37 10 16 0:82 10 4 7 5:94 10 16 0:60 10 4 plasma exit based on the predication of 2D linear theory. TheE z field de- creases with increasingn e orl because the decreasing trend of the termn 1=2 e dominates that of the increasing termR(0), and the wakefields are less ef- fectively driven at larger density. The transverse focusing strength also de- creases with increasing l due to the more rapid decrease in n e 1 than the increase ofdR( r )=dr. As a result, it decreases even faster thanE z withl, as seen in Fig. 3.2. Such decrease strongly impacts the growth of the instability, as will be demonstrated later. 3.5 Simulation Results 3.5.1 Simulation Parameters Numerical simulations with the experimental beam and plasma pa- rameters are performed to interpret the nonlinear development of SMI and to validate the simple model of energy modulation. For the reasons stated in the above paragraph, a bunch charge of 50pC is chosen in the simulations so that the overall visibility of the energy spectra modulation is maintained as 49 n e is varied. We assume a step function current rise time at the bunch front. Simulations are performed with plasma densities varying from 1:2 10 15 to 5:9 10 16 cm 3 corresponding to integer values ofl = L beam = pe from 1 to 7, as shown in Table 3.2. The plasma density range chosen here is similar to that available in the experiment at ATF. Note that although the plasma length in the experiment is 2 cm, we propagate the bunch in plasma for 20cm in the simulations in order to understand the physics involved in the development of the SMI. We perform the 2D cylindrically symmetric simulations using the fully relativistic, massively parallel particle-in-cell code OSIRIS [54]. The 2D simulation uses a moving window that propagates at velocity c, with res- olutionsk p z = 0:05 andk p r = 0:02 for the longitudinal and transverse cell sizes respectively and with 22 plasma and beam particles per cell. We use quadratic particle shapes and current smoothing. Convergence stud- ies have shown that those parameters are sufficient to model the physics presented here. 3.5.2 E z Evolution and SMI Development Figure 3.3 shows the evolution of maximum accelerating field E z along the bunch with a charge of 50pC as a function of propagation distance z for the seven different plasma densities. First, it shows that overz = 2cm, the experimental plasma length, there is no significant SMI growth at any 50 0 2 4 6 8 10 02468 10 Ez_7lines E z (MV/m) z (cm) L beam / pe =1 3 2 4 5 6 7 Figure 3.3: Peak accelerating field along the bunch versus propagation dis- tance in the simulation for the electron bunch with 50pC charge and plasma densities such thatL beam = pe = 1 (red), 2 (blue), 3 (green), 4 (black), 5 (pur- ple), 6 (orange) and 7 (cyan). The exponential growth at z > 3 cm for l beam = pe > 2 indicates the growth of SMI. 51 plasma density considered here, meaning that the E z remains at its initial value. This is consistent with the assumption of the simple energy modu- lation model. Second, it shows that the initial values E z (z = 0) decrease from 4 to 1MV=m asn e increases. Both theE z (z = 0) amplitudes and their trend are in excellent agreement with those deduced here above from the 2D PWFA linear theory. A lower transverse force (not shown here, but similar to Fig. 3.2b)) also leads to a decreasing SMI growth rate whenl in- creases from 2 to 7, as visible on Fig. 3.3. Note that thel = 1 case has smaller growth rate (e.g. compared to thel = 2 case) despite its larger transverse focusing force because there is no feedback for the SMI to grow in this short bunch case [27, 28, 45]. 3.5.3 Energy Modulation Here we analyze the simulated energy spectra of the bunch after the 2cm propagation in plasma to validate thesimple energy modulation model presented above. Figure 3.4 shows the bunch particles’ energy spectra generated from the above simulations after 2cm propagation in plasma. As predicted, the spectra reveal that energy self-modulation generates two to eight peaks along the bunch as the plasma density increases such that l varies from 1 to 7. In all figures the self-modulation peaks are clearly visible, suggesting that the energy gain and loss by the particles remain close to that for opti- 52 Figure 3.4: Energy spectra of the electron bunch at the plasma exit in the cases of various plasma densities such thatL beam = pe = 1 to 7. The expected energy modulation peaks are clearly visible. mum bunching at all plasma densities. Note that the apparent transverse modulation suggested by the spectra of Fig. 3.4 only originates from the energy modulation and not from radial modulation due to SMI. The trans- verse size of the bunch does not change since the SMI does not grow over the first 2cm, as shown by Fig. 3.3. 53 Figure 3.5: Two dimensional beam density plots:a) Transverse momentum of the bunch particles at the plasma exit z = L p = 2 cm from OSIRIS2D simulations with the experimental parameters in Table 3.1. b) Bunch den- sity after propagation of 7:5 cm in vacuum after the plasma showing that modulation in transverse momentum from Fig. a) results in a modulation of the bunch transverse bunch radius and density. The white line in Fig. b) shows the bunch density lineout atr = 74m in the bunch. 54 3.6 Seeding of Self-modulation Instability Figure 3.3 shows that with the 50pC bunch, the SMI does not grow significantly over the 2 cm propagation in the plasma. However, over the 2cm plasma length the transverse wakefield component leads to a periodic modulation of the bunch radial momentum, as shown in Fig. 3.5a) for the case ofL beam = pe = 2. Further numerical calculation of the ballistic propaga- tion of the particles in vacuum downstream from the plasma shows radial modulation of the bunch, as shown in Fig. 3.5b) at a distance of 7:5cm. The radial modulation could be measured using, for example, a transverse de- flecting cavity with sub-picosecond time resolution. Such a device is avail- able, for example, at the DESYPITZ laboratory where similar experiment will be conducted. 3.7 Chapter Conclusions In this chapter, we identified a method to evidence the driving of wakefield with multiple periods along a charged particle bunch: the for- mation of periodic energy bunching. We demonstrated from linear PWFA theory that with the ATF experiment parameters, a bunch charge of 50pC is ideal to maintain the visibility of the bunching as the the plasma density and therefore the the number of modulation periods increases (l = L beam = pe n 1=2 e ). We used simulations to confirm this choice of bunch charge and to determine that the SMI does not grow over the 2cm plasma length. How- 55 ever, PWFA theory [51] and simulations indicate the necessary coexistence of longitudinal and transverse fields, which leads to the beam also obtain- ing a spatially correlated transverse momentum spread along the plasma. This transverse momentum is the seed for the SMI development. Therefore, the periodic energy modulation of the incoming bunch acts as an indirect but important evidence for SMI seeding. These results have been publish in Ref. [55]. They motivate the experiments described in the next chapter. 56 Chapter 4 Experimental Study of the Seeding for Self-modulation Instability 4.1 Introduction In this chapter, we present the first experimental evidence that a charged particle bunch drives wakefields with multiple periods of the rel- ativistic plasma wave or wake. We use a 58 MeV , 1mm-long electron bunch with 50pC of charge. As mentioned previously, the bunch is shaped with a sharp rising edge that drives initial wakefield amplitudes in the MV=m range. These wakefields act as a seed for the SMI [56]. The number of wake periods within the bunch is varied between 1 and 7 by varying the plasma den- sity from 10 15 to 10 17 cm 3 while keeping other parameters fixed. The effect of the longitudinal wakefields on the bunch entering the plasma with a time-energy correlation is observed as periodic modulation of the final energy spectrum. This effect can be observed thanks the careful choice of experimental parameters. Such effect is in excellent agreement with linear theory estimates and simulation results as shown in the previous chapter. The seeding of the SMI by the sharp rising edge of the bunch is confirmed 57 by the observation that the initial phase of the wakefields is fixed by the bunch rising edge, a condition necessary to deterministically inject a wit- ness bunch in the accelerating and focusing phase of the wakefield in an accelerator experiment. We use a long electron bunch to study the physics of SMI seeding, but the results equally apply to bunches of other particle species (positrons, protons, ...). Details about the experiment are described below. 4.2 Experimental Setup Figure 4.1: Experimental setup for the study of SMI. Two dipole and five quadrupole magnets are arranged in a dogleg configuration. The side graphs represent the beam energy correlation with the beam front labeled by F and the back by B. The experiment is performed at the Brookhaven National Laboratory Accelerator Test Facility (ATF). A radio-frequency (RF) photo-injector and 58 two S-band accelerator sections produce the = 58MeV electron bunch with a charge adjustable from a few tens of picocoulombs to 1nC. The bunch travels along the accelerator sections off the crest of the sinusoidal RF wave so that it acquires an approximately linear correlated energy spread (chirp) of about 1% of the incoming energy. The particles energy can either increase or decrease along the bunch, depending on whether the bunch is accelerated before or after the crest of the RF wave. For these experiments the chirp is chosen such that the bunch front has the lowest energy (see Fig. 4.2). The bunch then enters a magnetic dogleg section where it is dispersed in energy in the horizontal plane. The combination of linear chirp and energy dispersion allows for the shaping in time (or equivalently in = ct z or energy) of the bunch current profile, as was demonstrated in Refs [33] and [34] using a variable width slit and a mask. The present experiments use a rectangular mask to produce a bunch with a square current temporal profile. Figure 4.2 a) shows the image of the bunch dispersed in energy (and therefore in time) a short distance downstream from the mask. The length of the square bunch is inferred from its accelerating phase relative to the RF crest (9 o ), its mean energy (E 0 = 58:3 MeV ) and selected energy spread (E 0 = 0:48 MeV ). The extracted bunch length is L beam = 960 um. The current rise profile is fitted withcos 2L forL 0 with character- 59 !"#$%&'()*$+$ 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ,-$ " . /" .0 $ 12/3 4* $$ 5" 0 6$2 ,*78$ 52$ 7-$ $%)9:;$ <7=>$ ":*)'?$ @&8*6$!" Figure 4.2: a) Image of the long bunch dispersed in energy a short distance downstream from the mask. Two early in time and low energy bunches not used for this experiments are shown in the front of the bunch. The white lines indicates the sum of the images over the vertical and horizontal image dimensions. The energy range selected by the mask is E 0 . The bunch length isL beam and the length scale for the rising edge is L. b) Ratio of the longitudinal wakefield amplitudeE z for the case of a square bunch with a cos 2L forL 0 rising edge of base L (as shown in the inset) to E z0 , which would result from a perfectly square bunch (L = 0), both calculated using 2D linear theory. 60 istic width L, as shown in the inset of Fig. 4.2 b). The ratio of current rise length L over the overall bunch length is: L=L beam = 0:06. Once exiting the dogleg the bunch propagates towards the plasma and the imaging mag- netic spectrometer. The spectrometer is calibrated by varying the current in the dispersive dipole magnet and the resolution is< 0:03MeV , sufficient to resolve the 0:1MeV smallest features observed below. The bunch emit- tances and transverse sizes at the plasma entrance are determined by mea- suring the beam transverse size along the beam line for a well-known con- figuration of the magnetic elements. The bunch is focused near the plasma entrance with transverse sizes x = y = 120m and the normalized emit- tance is N = 13mmmrad. Note that this emittance value is much larger than that the accelerator typically produces ( N < 2mmmrad). This emit- tance growth is caused by a 2m nitrocellulose pellicle placed immediately after the dogleg to isolate the high vacuum of the RF gun from the tempo- rary pressure rise created by hydrogen gas puff associated with the plasma discharge. The plasma source is a capillary discharge [57] with lengthL p = 2cm and radius = 500m. The applied voltage is 15kV and the discharge cur- rent 700A. The capacitance of the charging system isC 1:8nF . The plasma density is measured at three longitudinal locations along the capil- lary by time resolving the Stark broadening of theH line at 656:28nm [58]. 61 Measurements at these three locations show that the relative density varia- tions are less than 10%. The overall plasma density is varied by changing the relative delay between the electrical discharge time and the arrival time of the bunch. The actual density for a given delay is inferred from the expo- nential fit to the measured density time evolution. A similar procedure has already been used in recent current filamentation instability experiments [25]. 4.3 Experimental Result 4.3.1 Energy Modulation A bunch charge of 50pC is chosen in the experiment so that the en- ergy modulation from the longitudinal wakefields remains within the in- coming correlated energy spectrum and the overall visibility of the energy spectra modulation is maintained as the plasma density is varied, as ex- plained in the previous chapter. Figure 4.3 shows a typical experimental energy spectrum in a null case with no plasma (Fig. a)), and seven en- ergy spectra (Figsb) toh)) acquired with plasma densities increasing from 2 10 15 to 8 10 16 cm 3 . The values ofL beam = pe labelled in the figures are calculated as the ratio of the maximum initial energy spread (E 0 ) and the average energy difference between the neighboring peaks E l on Figsb) toh). The spectra reveal that energy self-modulation generates two to seven peaks along the bunch as the plasma density is increased, corresponding to 62 Figure 4.3: Energy spectra obtained at various plasma densities. Spectra a) withn e = 0 (no plasma) and b)-h) with increasing plasma densities between 2 10 15 - 8 10 16 cm 3 obtained in the experiment. The white lines indicates the sum of the images over the vertical image dimension (no dispersion), and the red lines show the positions of the density peaks as identified for Fig. A.6. The direction of the energy chirp is indicated on Fig. a). Figure i) shows a simulated energy spectrum for the caseL beam = pe = 2, very similar to the experimental case of Fig. c). The color tables are deliberately chosen different to avoid possible confusion between experimental and simulation results. 63 1:4 to 7:5 plasma periods according to the E l measurements. These measured spectra exhibit remarkable similarities with the sim- ulated ones of Fig. 3.4 in Chapter 3, e.g., Fig. 4.3 i) for l = 2). In addition to the evident energy modulation, the spectra of Fig. 4.3 also show a loss of charge along the bunch that is at present not understood. This leads to the observed number of peaks in Figs c) to h) to bel rather thanl + 1, wherel is the integer part of the number of the plasma periods (see Fig. 4.4 for the missing peak in the yellow region) since the last peak is missing. However this does not change the conclusions reached here. In all figures the self- modulation peaks are clearly visible, suggesting again that the energy gain and loss by the particles remain close to that for optimum bunching at all plasma densities. This confirms that the wakefield amplitude does decrease with increasing n e , and its average values along the plasma are compara- ble to those estimated here above. It is also consistent with the wakefields (or the SMI) not growing over the plasma length, but remaining close to their initial value. Therefore, the energy gain or loss by the particles can be calculated as:W = R Lp 0 E z dz E z0 L p . Figure 4.4 shows the energies at which the density peaks appear in the spectra of Fig. 4.3 as a function of plasma density. The plasma density is deduced from the value ofL beam = pe as mentioned above (n e 1= 2 pe = E 0 E l L beam 2 ). The densities obtained from the bunch energy modulation are 64 !"#$ !%#& !%#' !%#! !%#" ()*&( &+ ,*&( &+ +*&( &+ %*&( &+ -./012*34/56 . / 378 9' 6 Figure 4.4: Energy of the peaks identified by the red lines on Figs 4.3 b) to h) as a function of plasma density. The yellow zone corresponds to the FWHM of the incoming bunch energy and suggests that the peaks are missing in the back, high energy part of the bunch forn e > 0:5 10 16 cm 3 . 65 in excellent agreement with those determined from the Stark broadening of theH line [58]. Note that the fact that the peak labelled #1 does not shift in energy with varyingn e confirms that this peak is at the front of the bunch. For this negative energy chirp, the first peak forms at the very front of the long bunch with the sharp rising edge since the front particles are in the energy loss phase of the wakefields. This would not be the case with the opposite chirp. The bunch front therefore acts as a starting point or phase reference for the wakefields along the bunch. This is particularly important in an accelerator experiment in order to deterministically inject a witness bunch into the accelerating/focusing wake phase. 4.3.2 Eect of the Ramp Prole of the Bunch Current Rise We now estimate the possible effect of the experimental bunch cur- rent rise length on the initial longitudinal wakefield amplitude using 2D linear theory for cylindrically symmetric bunches. We consider the ramp profile to be a cosine function of L as mentioned earlier (see Fig. 4.2 b) inset). We integrate the expression for the longitudinal electric field [38] to obtain the maximum amplitudeE z;max ofE z (;r = 0) along the bunch with 0 L pe . We plotE z;max =E z0 versus L= pe , whereE z0 is the longi- tudinal wakefield amplitude excited by an infinitely sharp bunch (L = 0). The result is shown in Fig. 4.2 b). With L beam = pe varying from 1 to 7 in our experiment, the ratio L= pe ranges approximately from 0:06 to 0:42. 66 Figure 4.2 b) shows that over such a range the effect of the ramp length is small, i.e.,E z;max =E z0 > 0:8. Therefore the bunch rise length of the experi- ment can be considered as short when compared to pe and can be neglected to interpret the results presented here. The rising edge of the bunch is sharp enough to effectively seed the wakefields and therefore the SMI, in agree- ment with the experimental results. 4.4 Chapter Conclusions In this chapter, we have demonstrated experimentally for the first time that an electron bunch with a sharp rising edge can drive wakefields with multiple plasm periods. The effect of these wakefields is observed as the formation of corresponding peaks in the bunch’s correlated energy spectrum. Linear PWFA implies that transverse wakefields with the same period are also excited . These act as a seed for the SMI. Observation of the energy modulation is the first but indirect evidence of the seeding of wakefields for SMI development. A number of SMI experiments are con- templated or planned at major facilities, such as AWAKE at CERN [59, 60], Fermi National Laboratory, E209 at SLAC National Accelerator Laboratory [49], DESY, etc.. All of them will rely on seeding to observe the instabil- ity, some to deterministically inject external electrons in the wakefields or to mitigate the occurrence of the hose instability [47, 48]. The results pre- sented here are an important seed for these major experiments and have 67 been submitted toPhysicalReviewLetter for publication [61]. Although sim- ulation and experimental results indicate that with the chosen experimental parameters (Q = 50pC) the SMI does not grow over the 2cm plasma length, simulations show that with a bunch with higher charge (e.g. 1nC) the SMI does reach saturation at the plasma exit. This result will be demonstrated in Chapter 5. 68 Chapter 5 Numerical Study of the Self-modulation Instability with a High Charge (1 nC) long Electron Bunch in a Plasma 5.1 Introduction Previous simulations presented in Chapter 3 and experimental re- sults presented in Chapter 4 have studied the seeding of the SMI develop- ment by the low chage (50pC) electron bunch available at Accelerator Test Facility (ATF) of Brookhaven National Laboratory (BNL). The results show that the bunch drives multiple periods of the longitudinal plasma wake- field and the associated transverse wakefields.However, the 50 pC bunch does not experience significant SMI growth over the propagation distance of 2cm, the plasma length at ATF. In this chapter, we study numerically the development of the SMI for a 1 nC bunch. The initial wakefields and the SMI growth rate [28, 45] depend on the bunch density or charge. There- fore growth of the SMI may be expected with the higher charge bunch, even in the short 2cm plasma. Based on the simulation results, we discuss the possibility of carrying out experiments at ATF to confirm the transverse self modulation instability through measuring the energy spectrum at the 69 Table 5.1: Simulation parameters l Plasma Densitiesn e n b ne 1 1:21 10 15 cm 3 5:92 10 2 2 4:85 10 15 cm 3 1:48 10 2 3 1:09 10 16 cm 3 6:58 10 3 4 1:94 10 16 cm 3 3:70 10 3 5 3:03 10 16 cm 3 2:36 10 3 plasma exit, which is the direct evidence of SMI development. Note that when the SMI reaches saturation the bunch radius and density are also pe- riodically self modulated. Observation of the radial modulation using, for example, a streak camera or a transverse deflecting cavity would be direct evidence of the SMI occurrence. 5.2 Beam Parameters at ATF and Simulation Parameters The electron beam parameters are the same as shown in Table 3.2 , except a bunch charge of 1nC is used in the following simulations instead of 50pC. The plasma densities are chosen such thatl =L beam = pe = 1; 2; 3; 4; 5, as shown in Table 5.1. When first entering the plasma, the beam density is much smaller than the plasma density (n b =n e < 0:06), indicating that the beam plasma interaction is in linear regime for all plasma densities (as was the case in the 50 pC bunch). In simulations, we propagate the electron bunch for 6cm in order to observe the SMI development, although the cap- illary length is limited to 2cm at ATF. 70 5.3 Self-modulation of the Electron Bunch Figure 5.1: Beam density image obtained from OSIRIS 2D simulations a) at plasma entrance z = 0, and at plasma exit for different plasma densities such that L beam = pe is equal to: b) 1, c) 2, d) 3, e) 4 and f) 5, as shown in Table 5.1. Figure 5.1 shows beam density images at the plasma entrance (Fig. 5.1 a) and at plasma exit (Figs b to f)) for different plasma densities such that l = L beam = pe is varied from 1 to 5 (plasma densities as shown in Ta- ble. 3.2). It is clear that due to the SMI development, the long bunch (Fig. 5.1 a)) is radially modulated into several beamlets on a longitudinal scale of the plasma wavelength, and the number of beamlets varies from 1 to 5 as the ratio of L beam = pe increases from 1 to 5, as shown in Figs 5.1 b) to f). For the plasma densities such thatL beam = pe = 1; 2; 3 (Figs 5.1 b), c), d) respectively), the colorbars reveal that at the plasma exit, the on-axis beam 71 plasma density ratio reachesn b =n e 1, indicating the possibility of reach- ing the weakly nonlinear regime or quasi nonlinear regime of the PWFA [63] discussed previously. 5.4 Wakeeld Evolution Figure 5.2: a) Longitudinal field E z at r = 0. b) transverse focusing field (E r cB ) evaluated atr = r . Both fields are obtained from 2D linear theory (assuming no beam density evolution) for different plasma densities such thatL beam = pe is equal to: 1 (red), 2 (blue), 3 (green), 4 (black), 5 (purple), as shown in Table 5.1. As mentioned above, when the beam first enters the plasma, its den- sity is uniform along = ctz (Fig. 5.1 a)) and the interaction with the plasma is in the linear regime (n b =n e << 1). The initial (z = 0cm) longitudi- nal wakefieldE z on the beam axis (r = 0) and the transverse focusing field (E r cB ) atr = r inside the bunch can be estimated from the 2D linear theory [38]. The lineout of the initialE z and (E r cB ) is very similar to the 50pC case as in Chapter 3, except that the amplitudes are 20 times higher 72 (E z , (E r cB )/ n b0 / Q) (see Figure 5.2) as expected. As the plasma density increases, the amplitude of the longitudinal wakefieldE z decreases from 73 to 36MV=m, and that of the focusing field (E r cB ) decreases from 70 to 15MV=m. Such large amplitude transverse focusing fields modulate the bunch periodically and lead to the development of SMI. The wakefield amplitudes at the plasma entrance (z = 0cm) obtained from the 2D OSIRIS simulations agree very well with the above estimate of linear theory (as will be shown in the following section). !"## !$## !%## !&## # &## %## #'## $## (## )&## !"## !$## !%## !&## # &## %## $## #'## $## (## )&## !"### !$## # $## "### #%## &## '## "(## !"#$$ !"$$$ !#$$ $ #$$ "$$$ $%$$ &$$ '$$ "($$ !"#$$ !"$$$ !#$$ $ #$$ "$$$ $%$$ &$$ '$$ "($$ Figure 5.3: Beam density lineout (red line), longitudinal fieldE z (blue line) atr = 0 and transverse focusing field (E r cB ) (green line) evaluated at r = r at plasma exit (z = 2cm), obtained from OSIRIS 2D simulation for different plasma densities such thatL beam = pe is equal to: a) 1, b) 2, c) 3, d) 4 and e) 5, as shown in Table 5.1 . Figure 5.3 shows the on-axis lineout of beam density in arbitrary 73 units (red curve), longitudinal fieldE z atr = 0 (blue curve), and the trans- verse focusing field (E r cB ) atr = r (green curve) after 2cm propagation in plasma for the 5 plasma density cases. It shows that the multiple beam- lets (red negative peaks) excite the wakefields resonantly and therefore the accelerating fieldE z reaches large amplitudes (e.g. 1GV=m in Fig. d)). Note that due to the fact that the wakefields phase velocity is smaller than that of the bunch during the SMI growth, the number ofE z or (E r cB ) pe- riods is no longer integer as at the plasma entrance, where the modulation has not occurred yet. It is also worth mentioning that althoughn b =n e 1 at the plasma exit, as shown by the color bars of Fig. 5.1, the peak longi- tudinal wakefieldE z remains small (< 5%) compared to the wave-breaking amplitudeE wb =mc! pe =e. Figure 5.4 shows the peak accelerating fieldE z along the bunch ver- sus the propagation distancez. The value of initialE z (atz = 0) decreases from 80 to 40 MV=m as the plasma density increases, which is in good agreement with the estimates from the above 2D linear theory. The SMI grows,E z saturates and the saturation distances increases from 1:8 to 2:4 cm with increasing plasma density, close to the plasma capillary length. Therefore, given the plasma capillary length of 2cm in experiment, the electron bunch in principle have reached the SMI saturation when it exits the plasma. 74 Figure 5.4: The evolution of maximum accelerating fieldE z along the prop- agation distancez, obtained from OSRIS 2D simulation for different plasma densities such thatL beam = pe is equal to: a) 1 (red curve), b) 2 (blue curve), c) 3 (green curve), d) 4 (black curve) and e) 5 (purple curve), as shown in Table 5.1. 5.5 Particles' Phase Space Figure 5.5 shows the bunch particles’ radial velocity along the bunch after the 2cm propagation distance. The radial velocity is periodically mod- ulated due to the alternating transverse focusing and defocusing fields, and it can be negative (motion towards the axis in 2D), positive (away form the axis), or close to zero (remain on the axis). The radial velocity perturba- tion amplitude decreases with plasma density, which is again due to the smaller transverse focusing field at higher plasma density, as shown in Fig. 5.3. Note that in these simulations the bunch parameters are kept constant 75 whilen e changes. Figure 5.5: Normalized radial velocity of beam particles (v 2 =c) along the bunch (ctz) at: a) plasma entrance, b)-f): plasma exit obtained from OSIRIS 2D simulation for different plasma densities such thatL beam = pe is equal to: b) 1, c) 2, d) 3, e) 4 and f) 5, as shown in Table 5.1. 5.6 Energy Gain and Loss 5.6.1 Energy Spectra from Simulation Figure 5.6 shows the particles’ energy spectra at the plasma exit for different plasma densities. The energies range from 56 to 60MeV , with symmetric energy gain and loss of about 2MeV . The broad continuous en- ergy spectra are the result of the bunch particles sampling all the phases of the wakefield and of the energy gain/loss of the particles being much larger than the incoming correlated energy spread (E growthspectrum >> E 0 ). Since the width of the spectra ( 4 MeV maximum) is much larger than the electron bunch’s incoming energy spread ( 0:48MeV ), It is impossible 76 to distinguish any modulation features such as those that were observed in the 50pC charge case in Chapter 3, which was used as the indirect evidence of multiple-period wakefields excitation and of SMI seeding. Figure 5.6: The bunch energy spectra at plasma exit (z = 2 cm) obtained from OSIRIS 2D cylindrical code for different plasma densities such that L beam = pe is equal to: 1 (red line), 2 (green line), 3 (blue line), 4 (black line) and 5 (purple line), as shown in Table 5.1. 5.6.2 Possibility of Conrming SMI Development through Measuring Energy Spectra We here further analyze energy change of 1 nC bunch to examine the possibility of measuring the energy spectra experimentally to confirm the development of SMI. As an example, the plasma density is chosen such thatL beam = pe = 2. 77 We first assume that there is no SMI growth during the 2cm propa- gation for the 1nC bunch, and therefore the energy gain/loss at the plasma exit can be estimated by multiplying the initial E z amplitude (at z = 0) with 2 cm. Figure 5.7 a) shows the initial longitudinal wakefield E z on the axis (r = 0) at the plasma entrance (z = 0) obtained from simulation, with an acceleration amplitudeE z0 67MV=m, which is within 5% differ- ence with the estimation of linear theory ( 64MV=m). Therefore, assum- ing no SMI growth and hence no E z amplitude evolution along the 2 cm plasma, the energy gain/loss at the plasma exit would be E nogrowth = E z0 2 cm 1:33 MeV . Figure 5.7 b) shows the evolution of maximum longitudinal field on the axisE zmax along the propagation distance. The ex- ponential growth and saturation ofE zmax indicate the development of SMI with the 1nC bunch. Since the SMI is a convective instability, the maximum E z is expected to be at the back of the bunch, e.g. at the acceleration peak of the second wakefield bucket in this case. Assuming no dephasing of the plasma wave with respect to the square bunch during the propagation, the maximum energy gain can be estimated as the integral of E zmax over the propagation distance, resulting in E growthintegration 3:52 MeV . Figure 5.7c) shows the energy spectrum of the electron bunch at the plasma exit, obtained from the OSIRIS-2D simulation. With the mean incoming energy of 58:35MeV , the maximum energy gain is estimated as E growthspectrum 78 60:98 58:35 = 2:63MeV , more than 2 times larger than the value obtained assuming no SMI growth. We hence conclude that it is experimentally pos- sible to confirm the development of SMI through the measuring of the en- ergy gain/loss at the plasma exit. It is also worth noting that the particles’ actual maximum energy gain/loss E growthspectrum is 25% smaller than the integration of the maximum E z result E growthintegration . This clearly in- dicates the occurrence of dephasing of the wakefields with respect to the bunch during the development of SMI, which will be further examined in the following section. Note that the conclusion also applies to energy loss, although the figures are not shown here. To confirm the dephasing of the plasma wave with respect to the electron bunch during the propagation, Fig. 5.8a) shows the evolution of accelerating wakefield structure E z . During the first 0:5 cm propagation, there is no significant growth of SMI (as can be observed in Fig. 5.7b)) , and therefore the wakefield phase velocity is close to the beam velocityv b (c). Betweenz = 0:5 1:5cm, the beam density becomes modulated due to the periodic focusing/defocusing field and hence SMI develops, resulting in a lower wakfield phase velocity v . The phase velocity of the longitudinal plasma wavev can be deduced from the slope of maximumE z in thez and (=ctz) plane (as shown schematically by the dotted line in Fig. 5.8)a), yielding v = c (1 + 1 z= ) [49]. Consequently, a smaller negative slope 79 (z=) means a smaller phase velocity. Figure 5.8b) shows the linear fitting to obtain the slopez=, and therefore the relative phase velocity. Remem- ber that the simulation window moves at the speed of light. The wakefield phase velocity is therefore estimated as 0:986 c. For z = 1:5 2:0 cm, the SMI saturates and the wake phase velocity almost reachesc again. Figure 5.9 a) shows the energy gain of the beam particles along , obtained by integrating the accelerating field in Fig. 5.8 over the propa- gation distancez, yielding the approximate energy gain/loss of the bunch particles during the 2 cm propagation ( R E z dz P E z z in simulations). We can therefore identify the 0 position where the bunch particles gain the most energy. This is indicated by the red dotted line in the Fig. 5.9a). Fig- ure 5.9b) shows the evolution of the accelerating wakefield experienced by the the particles at 0 (red curve), compared to the maximum accelerating wakefield (blue curve) along the bunch. The fact that the former amplitude is smaller than the later one explains the energy difference between the in- tegration from maximumE z and the actual energy spectrum of the particles at the plasma exit, as mentioned above. Figure 5.9c) shows the longitudinal fieldE z on the axis along at the plasma entrancez = 0 (blue line,50) and at the plasma exitz = 2cm (red line). The shift of the peak accelerating field between z = 0 and z = 2 cm clearly confirms the dephasing. It is worth noting that the dephasing occurs mostly over the first plasma period. 80 The dephasing occurring during the SMI development is due to the evolution of the bunch density created by the periodic focusing/defocuisng force, which is=2 out of phase withE z . The modulated bunch density then leads to the reset of theE z field and hence the dephasing of the wakefield occurs. This evolution is similar to that observed when attempting to main- tain the large transformer ratio in the linear wakefield regime (shown in Chapter 2). Therefore, in principle, the experimental setup at ATF allows us to identify the SMI development of the 1nC bunch through the energy mea- surement at the plasma exit. We emphasize again that simulations indicate that because of the evolution of the wakefields phase velocity during the SMI growth, the effective energy gain/loss is smaller than expected from the simple integration of the peakE z field over the propagation distance. 5.7 Measure the Radial Modulation In the previous section, we demonstrated that it is possible to con- firm the SMI development through energy measurement, which is an in- direct but important evidence of SMI. In this section, we explore the pos- sible diagnostics to measure the radial modulation caused by SMI directly. Figure 5.10 shows the electron bunch density after free streaming in vac- uum for 25cm downstream from the plasma. The radial modulation pattern is clearly visible and could be detected using coherent transition radiation 81 (CTR) diagnostics or with a transverse deflecting cavity. It is worth noting that the particles’ distribution in Fig. 5.10 is very similar to the pattern of the particles with almost zero radial velocity in Fig. 5.5. This is because only those particles that remain around the axis create a large density near the axis after the 25cm of free streaming, and other particles with large pos- itive or negative radial velocity move far way from the axis, resulting in low local particle density regions. 5.8 Hosing Instability 3D OSIRIS simulations are performed to investigate the develop- ment of hosing instability when the electron bunch propagates in the plasma, and the result is shown in Fig. 5.11. The figure show that the bunch par- ticle distribution remains symmetric about the propagation axis, therefore for both plasma densities such thatL beam = pe is equal to 2 (Fig. 5.11 a)), or 5 (Fig. 5.11 b)), no hosing is observed. This is consistent with the previous report that a sharp current-rising bunch can seed the SMI efficiently and therefore mitigate the development of hosing instability [49]. 5.9 Chapter Conclusions To summarize, this chapter studies the the development of SMI using the 1 nC electron bunches available at ATF. Simulation results show that the radial modulation of the bunch is reached due to the SMI growth after 82 the electron bunch propagated into 2cm of plasmas with different densities. Measurement of this modulation would be direct evidence of the occurrence of SMI. The radial modulation of the electron bunch remains after 25 cm propagation in vacuum downstream the plasma, where CTR diagnostic can be placed and therefore detect the SMI directly. No hosing instability is observed in the simulation. The bunch particles’ energy gain/loss is much larger than the incoming energy spread, and therefore the spectra lose the energy modulation features observed in the previous chapter with the 50pC bunch. However, simulations show that the resulting energy spectra is 2 times larger than the case assuming no SMI growth. This indicates the possibility of confirming the SMI development through the energy spectra measurement at the plasma exit. Those results have been published in Ref. [62] and [64]. 83 0 100 200 300 0123456 E z (MV/m) z (cm) -80 -40 0 40 80 0400 800 1200 Figure 5.7: a) The longitudinal wakefieldE z on the axis (r = 0) at plasma entrance (z = 0) within the bunch. The black square indicates position of the electron bunch. b) The evolution of maximumEz on the axis (r = 0) along the propagation distance; c) The energy spectrum of the bunch particles at the plasma exit from OSIRIS2D. 84 0 0.3 0.6 0.9 1.2 1.5 450 540 630 720 810 900 Figure 5.8: a) Evolution of the accelerating wakefield structure E z within the bunch along the propagation distance z. The red rectangle shows the longitudinal position of the bunch and the white dotted lines show the ”tra- jectory” of the peak accelerating and decelerating field respectively. b) The shifting of the position of the peak accelerating field along the propagation distancez, and the fitted line is given as:z(cm) = 5:54670:0072932(um), yieldingv =c (1 + 1 z= ) 0:986c. 85 !" # " $ #%## &## "$## !"## !$## # $## #$## "## %&## ! "!! #!! $!! !!%& ""%& # ' ( )*+,-./ ()*0./ Figure 5.9: Energy gain/loss by particles along the bunch () after 2 cm plasma, obtained by integrating the acceleration field (as shown in Fig. 5.8 a)) over the propagation distancez. 0 indicates the position in the bunch where particles gain the most energy; b) the accelerating field on the axis at position 0 of the bunch (E z ( 0 ;r = 0)) along the propagation distance z (red line) and the maximum E z field on the axis along z (blue line); c) Longitudinal wakfield E z on axis (r = 0) at plasma entrance z = 0 (blue line) and plasma exitz = 2cm (red line). Note thatE z atz = 0 (blue line) is multiplied by 50 in order to make it visible on the same scale. The dotted lines show the accelerating peaks of the respective fields, and the distance between the two dotted lines directly shows the wakefield dephasing. The red rectangles show the longitudinal position of the bunch. 86 Figure 5.10: Beam image after 25cm free propagation in vacuum after the plasma for different plasma densities such that L beam = pe is equal to: a) 1, b) 2, c) 3, d) 4 and e) 5, as shown in Table 5.1. The white lines indicate the density lineout at the axis. !""# Figure 5.11: Beam density image at plasma exit obtained from OSIRIS 3D code at different plasma densities such thatL beam = pe is equal to: a) 2 , and b) 5, as shown in Table 5.1. 87 Chapter 6 The Eect of Plasma Radius and Prole on the Development of Self-modulation Instability 6.1 Introduction As mentioned in previous chapters, the study of the self-modulation instability is becoming an attractive research topic, due to the prospect of us- ing long ( 12cm) proton bunches produced at CERN and carrying many kilojules of energy as driver for a plasma wakefield accelerator (PWFA). Recently, we have demonstrated experimentally for the first time that the sharp-rising edge of a long electron electron bunch can effectively seed the SMI [61]. An important factor that was recognized to affect the development of SMI is the longitudinal plasma density profile, which directly determines the growth rate of SMI [48, 65]. Plasmas available for PWFA experiments may have longitudinal and transverse density profiles that could affect the outcome of an experiment. The effect could be positive. For example, den- sity ramp at the entrance of the plasma may be used to slowly focus the bunch transverse profile to a size close to a matched radius in the short bunch PWFA case [66]. Also the dynamic evolution of the plasma radial profile is used in LWFA experiments to guide the pulse and drive wakefield 88 over distances much longer than the beam incoming Rayleigh length, us- ing the hydrodynamic expansion of a laser-produced cylindrical spark [67] and several types of capillary discharge [68]. There are also cases where the finite plasma radius or transverse plasma profile could play a negative role. For plasmas ionized by a particle bunch, the plasma radius is confined by the radial electric field being above the vapor ionization threshold [69]. Plasmas produced by capillary discharges have both a finite transverse size and possibly an inhomogeneous transverse density profile during the dis- charge (e.g. maximum on axis and zero on the wall). When particle bunches are injected into such plasmas, it is possible that the plasma electron pertur- bation is not zero at the transverse plasma boundary, or the perturbation amplitude is different within the non-uniform plasma compared to the uni- form density case. Therefore the resulting plasma wakefield and the growth of SMI can also be different in those finite size, real plasmas from those in infinite and homogenous plasmas. However, previous PWFA particle-in-cell simulations generally ig- nore the effect of plasma transverse profile and assume ”ideal plasma”. These plasmas have a uniform radial density and their transverse dimen- sion is large enough so that the plasma electron perturbation approaches zero at the transverse boundary, which is an approximation for infinite plasma radius. 89 In this chapter, we use both analytical and numerical approaches to study the effect of plasma transverse density profiles that are present and may affect SMI and SMI-based PWFA experiments. The transverse plasma profile can impact three aspects of the SMI development: the amplitude of the seeding wakefields, including both longitudinal and transverse compo- nents, the growth rate and the saturated fields of the instability since, as mentioned here above the growth rate is driven by the transverse focus- ing/defocusing wakefields. The results in this chapter demonstrate that such transverse plasma density profiles could in fact play a positive role in the SMI development. Although the initial longitudinal wakefield is generously lower in such plas- mas, the increased focusing force could favor the development trend of SMI, i.e., higher growth rate, larger saturation amplitude of acceleration gradient and shorter propagation distance to reach saturation. This is particularly possible when there are large number of plasma periods within the drive bunch, resulting in the convective growth of the already increased focus- ing force. Therefore, in order to reach larger acceleration amplitudes and shorter propagation distances, future SMI simulation should take into con- sideration transverse plasma profiles similar to those realized in some ex- periments. The simulation results also provide an insight into the plasma design requirements for future large-scale SMI experiments, such as AWAKE 90 at CERN [59, 60] and E209 experiment at SLAC [49]. We first consider 2D PWFA linear theory to investigate analytically the possible effect of transverse plasma density profile on the initial wake- field excitation. Then we use the 2D particle-in-cell simulation code OSIRIS [54] to study the wakefield evolution. Here we assume two different situa- tions for the transverse plasma density profile: 1) finite radius (as opposed to ”infinite” plasma) with uniform density, describing plasmas generated, for example, by field ionization with intense laser pulse or a dense charged particle bunch (by ”finite”, we mean on the order of the plasma skindpeth c=! pe or of the bunch radius r ); 2) finite radius with non-uniform density profile, e.g. decreasing with radius, a good approximation for capillary dis- charge plasmas in their late time evolution. 6.2 Hints from Linear Theory 6.2.1 Linear Theory Review For the currently planned or contemplated SMI experiments, when the drive bunch first enters the plasma, the beam density is much smaller than the plasma density. The wakefields are thus described by the 2D PWFA linear theory [38]. As introduced in previous chapters, assuming an infi- nite transverse plasma radius, the longitudinal wakefieldW z and transverse wakefieldW r excited by an electron bunch with separable density profile: 91 n b (;r) =n b k ()n b? (r) can be written as: W z (;r) = e 0 Z 1 n b k ( 0 )cosk p ( 0 )d 0 R(r); (6.1) W r (;r) = e 0 k p Z 1 n b k ( 0 )sink p ( 0 )d 0 dR(r) dr ; (6.2) In these expressions R(r) describes the transverse (radial) dependency of the wakefield and is given by: R(r) =k p 2 K 0 (k p r) R r 0 r 0 dr 0 n b? (r 0 )I 0 (k p r 0 ) +k p 2 I 0 (k p r) R 1 r r 0 dr 0 n b? (r 0 )K 0 (k p r 0 ); (6.3) and dR(r) dr =k 3 p K 1 (k p r) Z r 0 r 0 dr 0 n b? (r 0 )I 0 (k p r 0 )+k 3 p I 1 (k p r) Z 1 r r 0 dr 0 n b? (r 0 )K 0 (k p r 0 ); (6.4) whereI 0;1 ,K 0;1 are the modified Bessel functions of zeroth and first order. For a transversely Gaussian bunchn b? (r) =e r 2 2 2 r , the on-axis valueR(r = 0) can be written as: R(0) =k p 2 I 0 (0) Z 1 0 r 0 dr 0 n b? (r 0 )K 0 (k p r 0 ) = 8 > > > < > > > : k p 2 r 2 0:05797ln(k p r ) ; k p r 1 k 2 p 2 r 2 e k 2 p 2 r 2 E 1 k 2 p 2 r 2 ; 8k p r 1; k p r 1 (6.5) HereE 1 (b) = R 1 1 e bt t dt is the exponential integral of the first kind. When a relativistic electron bunch travels in a overdense plasma (n b << n e ), the space charge field of the electron beam is cancelled by the plasma, 92 and therefore the radial fieldW r becomes the Lorentz force resulting from the magnetic component term (v b B ) and can be calculated from the elec- tron beam and induced plasma return currentj p (;r). This focusing effect is the basis of overdense plasma lenses and has been observed experimen- tally with electron bunches that are short when compared with the plasma wavelength [70, 71]. The focusing field can be written as: W r (;r) =v b (B b B p ) = v b 0 r Z r 0 r 0 dr 0 j b (;r 0 )j p (;r 0 ) ; (6.6) wherej b (;r) =en b (;r)v b , for a relativistic bunchv b c,n b (;r) =n bk ()n b? (r), and j p (;r) =en b (;r)v b R(r); (6.7) where R(r) is given in Eqn. 6.3. Then Eqn. 6.6 can be written as: W r (;r) ec 2 0 r n bk () Z r 0 r 0 dr 0 n b? (r 0 ) ec 2 0 r n bk () Z r 0 r 0 dr 0 n b? (r 0 )R(r 0 ); (6.8) Therefore, for a relativistic electron bunch with a given density profile, the first term of Eqn. 6.6 is fixed, and therefore the focusing field depends on the return currentj p , which in turn is determined by the transverse dependence termR(r). 93 0 0.2 0.4 0.6 0.8 1 012345 0 0.2 0.4 0.6 0.8 1 02468 10 Figure 6.1: a): The transverse dependence term of the longitudinal wake- field on the axisR(0) (as shown in Eq. 6.5) versusk p r ,b): the normalized transverse dependency term R(r)=R(r = 0) versus the normalized radius r= r fork p r = 1:57 (red curve), 1 (blue curve) and 0:79 (black curve),c): the ”infinite” plasma radius versus the plasma skin depthc=! pe , both normal- ized with r . 94 6.2.2 Determining the "Innite" Plasma Radius for PWFA from 2D Linear Theory For an electron bunch with Gaussian transverse density profile:n b? (r) = exp( r 2 2 2 r ), R(0) in Eqn. 6.5 is an increasing function of r and can be com- puted numerically for differentk p r values, as shown in Fig. 6.1a). Figure 6.1b) shows the transverse dependence termR(r) (normalized toR(0)) ver- sus the radiusr (normalized to beam radius r ) for different regimes such thatk p r = 0:79 (black line), 1 (blue line) and 1:57 (red line). The reason for choosing these values will be explained later. It is worth noting thatR(r) ap- proaches 0 faster at largerk p r value. This is due to the fact that the plasma electrons respond to the electron bunch over a radius on the order of skin depth (c=! pe ). Therefore for fixed beam radius, plasma perturbation radius decreases in the smallerc=! pe (i.e. largerk p r ) cases. For the practical pur- pose of particle-in-cell simulations or experiments, we here considerR p1 to be the ”infinite” radius if the transverse dependence termR(R p1 ) is smaller than 5% ofR(0), and therefore the ”infinite” plasma radiusR p1 , for a bunch with given transverse size r can be determined for different plasma den- sities or plasma skin depthc=! pe . For a bunch with fixed transverse size r traveling in plasma, as the plasma density is decreased and therefore the plasma skin depth c=! pe increases, the resulting ”infinite” plasma radius R p1 increases linearly withc=! pe , as shown in Fig. 6.1c). The linear rela- tionship is fitted asR p = r = 2:1 + 1:8 c=! pe , indicating that the ”infinite” 95 plasma radius is approximately 2:1 r whenc=! pe << r , and 1:8c=! pe when c=! pe >> r . Such finding provides guidance for determining the minimum plasma radius required in experiments. 6.2.3 Modied Linear Theory for Finite but Uniform Plasma When the plasma radius becomes finite, the above equations are no longer valid. We therefore modify the linear theory with the assumption that the transverse dependence term R(r) becomes 0 at r = R p instead of 1, and we obtain the following solution for the radial dependency: R(r) m =k 2 p (K 0 (k p r) +I 0 (k p r)) R r 0 r 0 dr 0 n b? (r 0 )I 0 (k p r 0 )+ k p 2 I 0 (k p r) R Rp r r 0 dr 0 n b? (r 0 )(K 0 (k p r 0 ) +I 0 (k p r 0 )); (6.9) where = K 0 (k p R p )=I 0 (k p R p ). Therefore the transverse depen- dency term on the axis can be written as: R(0) m =k 2 p I 0 (0) Z Rp 0 r 0 dr 0 n b? (r 0 )(K 0 (k p r 0 ) +I 0 (k p r 0 )); (6.10) Then taking the derivative of R(r) with respect to r, we obtain: dR(r) dr m =k p 3 (K 1 (k p r) +I 1 (k p r)) R r 0 r 0 dr 0 n b? (r 0 )I 0 (k p r 0 ) +k 3 p I 1 (k p r) R Rp r r 0 dr 0 n b? (r 0 ) (K 0 (k p r 0 ) +I 0 (k p r 0 )) (6.11) Then the longitudinal and transverse wakefields can be obtained by substi- tuting Eqns. 6.9 and 6.11 into Eqns. 6.1 and 6.2, respectively. Later, we will show the calculation results of the wakefields amplitude. 96 ! " !#"!! "# $%& # '$ % Figure 6.2: A schematic plot of the plasma density profile. 6.2.4 Transversely Finite and Inhomogeneous Plasma As mentioned above, inhomogeneous transverse plasma density also needs to be considered to better describe some experimental conditions, e.g. plasmas generated by capillary discharges. After the discharge current, the plasma created in the capillary decays by diffusion to the walls and by recombination. While recombination may dominate at early times (or high densities), diffusion dominate at later times when the plasma density reaches low levels [7]. In this case, the radial plasma density in the cylindri- cal capillary can be described as a Bessel function of the first order with zero density on the wall. Such profile can be well approximated by the cosine function shown in Fig. 6.2, which can be written as: n e = n e0 cos( r 2Rp ), with plasma densityn e0 on the axis and 0 on the capillary wall. In principle, the cosine plasma profile would result in smaller initialE z amplitude whenR p 97 is smaller than the ”infinite” radius, and this is because the plasma density perturbation is limited at larger radius where the plasma density becomes smaller than the electron beam density. However, for a cosine transverse density profile, it is difficult to derive analytically the exact solution for the wakefields. Therefore, we rely on OSIRIS simulation results to gain insight into the effect of such plasma profile on the focusing force and on the SMI development. 6.3 OSIRIS 2D Simulation Results with ATF Parameters The 2D cylindrically symmetric particle-in-cell code OSIRIS is used to perform simulations to validate the above modified linear theory, and to study the development of SMI in a plasma with transverse density pro- file. As an example, we use again the beam and plasma parameters from BNL-ATF (Accelerator Test Facility at Brookhaven National Laboratory), as shown in Table 6.1. Simulations with ideal plasmas presented in the previ- ous chapter have shown that the 1nC bunch available at ATF can reach the SMI saturation level at the exit of the 2cm long plasma [62]. Therefore a 1nC bunch charge is used in the following simulations. For the plasma wakefield excitation, four size parameters play important roles: the beam transverse size r , beam length z , the plasma skin depthc=! pe and the plasma radius R p . Our purpose here is to study the effect of finite plasma radiusR p with different plasma densities, and therefore different plasma skin depth with 98 Table 6.1: Experimental Parameters of SMI Experiment at ATF and Simula- tion Parameters charge 1nC current profile ”Longitudinally Square” beam lengthL beam 960um r 120um n b 7:2 10 13 cm 3 N 13mmmrad mean energy 58:3MeV correlated energy spread 0:48MeV plasma lengthL p 2cm plasma radiusR p 500um plasma density 4:85 10 15 cm 3 1:21 10 15 cm 3 k p r 1:57 0:79 n b =n e 0:0148 0:0593 fixed bunch parameters. As shown in Table 6.1, the plasma densities n e are chosen as 4:85 10 15 cm 3 and 1:21 10 15 cm 3 so that r =(c=! pe ) (or k p r ) = 1:57 (> 1) and 0:79 (< 1), respectively, and 2) the bunch length is an integer number of plasma wavelength, andl = L beam = pe = 2 and 1, re- spectively. This latter choice is related to the SMI process we are studying in which the bunch self-modulates radially with a period equal to the plasma wakefield period. For convenience, we will denote the two different plasma densities cases ask p r = 1:57 andk p r = 0:79. Note thatk p z also changes as the plasma density varies. 99 Figure 6.3: The transverse dependence termR(r) calculated from the above modified linear theory (dotted line) versus the simulation result (solid line) for different plasma radius with densities such thata)k p r = 1:57,b)k p r = 0:79. 0 0.2 0.4 0.6 0.8 1 0123456 Figure 6.4:R(r = 0) for various plasma radiiR p , normalized with theR(r = 0) of the ”infinite” radiusR p1 , fork p r = 0:79 (black curve) andk p r = 1:57 (red curve), obtained from the numerical calculation of Eq. 6.10. 100 !"# # "# $# %# &# #" $%& !"# # "# $# %# &# '# #" $%& Figure 6.5:dR(r)=dr calculated from the above modified linear theory (dot- ted line) versus the simulation result (solid line) for different plasma radius with densities such thata)k p r = 1:57,b)k p r = 0:79. 6.3.1 Finite Plasma Radius with Uniform Density 6.3.1.1 Initial Wakeelds As suggested by Fig. 6.1c), the ”infinite” plasma radii fork p r = 1:57 andk p r = 0:79 are 3:2 r and 4:2 r , respectively. We therefore choose trans- verse plasma dimension to be 4:2 r for both k p r cases, and decrease the plasma radius in steps, assuming vacuum outside the plasma. The plasma radius is decreased from 4:2 to 3, 2, 1:5, and 1 r to observe the effect on the plasma wakefield. The terms R(r) and dR(r) dr can be obtained from the wakefields amplitudes from simulations results for a very short propaga- tion distance (no r evolution) and from Eqns. 6.1 and 6.2. Alternatively, these two terms can also be calculated numerically from Eqns. 6.9 and 6.11. 101 Figure 6.3 shows that for various plasma radiiR(r), obtained from the sim- ulation (solid line), has remarkable similarities with that obtained through the numerical calculation (dotted line). As expected, for a certain radius r, the term R(r) decreases as the plasma radius decreases, due to the fact that less bunch particles are in the plasma and drive the wakefield, and also that less plasma electrons respond to the drive bunch. Figure 6.4 illustrates how the finite radius R p affects R(r = 0), showing the change of initial E z (r = 0) amplitude as R p becomes smaller than the infinite radius R p1 . For thek p r = 1:57 case (red curve), the decrease of plasma radius results in less decrease ofR(0) and henceE z (r = 0) than in thek p r = 0:79 case (black curve), consistent with the fact that the plasma electron perturbation is less sensitive to the decrease of plasma radius if the plasma skin depth (c=! pe ) is relatively smaller. Figure 6.5 shows similar consistency between the numerical calcula- tions from Eqn. 6.11 (dotted line) and the simulation results (solid line) for thedR(r)=dr term, which is proportional to the transverse focusing force, as shown in Eqn. 6.2. It clearly indicates that in bothk p r cases, for a certain radial positionr inside the bunch and plasma (e.gr = 60um orr= r = 0:5 for Fig. a), andr = 120um orr= r = 1 for Fig. b)), the smaller the plasma radius the higher the focusing field. As indicated by Eqn. 6.8, the focusing force depends only on the amplitude of return current. Figure 6.6a) is an 102 example of a 2D image of the longitudinal plasma current density (or re- turn current) for the case of k p r = 1:57 at the plasma entrance, and Fig. b) and c) are the line-out from the 2D image along the transverse position at the peak density for the cases k p r = 1:57 (e.g. 360 um as shown in Fig. 6.6a)) andk p r = 0:79, respectively. Both Fig. b) and c) show that as the plasma radius decreases, the plasma return current density j p also decreases, which then leads to a lower azimuthally defocusing magnetic field from the plasma. Therefore, the net increase in focusing field is due to the decrease of the plasma defocusing field, which in turn originates from a decrease of the plasma return current amplitude and hence a smaller de- focusing azimuthal B field from the plasma. Also note that as Eqn. 6.7 indicates, the amplitude of return current as shown in Fig. 6.6a) andb) is proportional to the transverse dependence term R(r) as shown in Fig. 6.3a) andb), respectively. To summarize for this section, we demonstrate that an electron bunch propagating in a plasma with smaller radius generates a lower initial E z field. However, it also excites a higher initial total focusing fieldE r cB , leading to a higher SMI growth rate in the early stage, as will be shown later. 103 0 0.01 0.02 0 1 2 3 4 0 200 400 600 800 1000 1200 0 100 200 300 400 500 0 2 4 6 8 10 12 14 16 18 x 10 −3 a)' 0 400 800 1200 0 250 500 r''(um)' ctQz'(um)' 3' 1' 1.5' Plasma'Return'Current'' '''''''''''''(arb.'unit)' r/σ r ' b) 2 R p /σ r =4.2' 0 0.01 0.02 0.03 0.04 0 1 2 3 4 r/σ r ' Plasma'Return'Current'' '''''''''(arb.'unit)' c) R p /σ r =4.2' 2' 3' 1.5' 1' 0 200 400 600 800 1000 1200 0 100 200 300 400 500 0 2 4 6 8 10 12 14 16 18 x 10 −3 0 8 4 12 16 10 Q3 arb.'unit Figure 6.6: The plasma return current image at the plasma entrance (z = 0) (a)) and the lineout at highest current density for different plasma radii such thatR p = r equal to 4:2 (black curve), 3 (orange curve), 2 (red curve), 1:5 (blue curve), 1 (green curve) fork p r = 1:57 (b)) andk p r = 0:79 (c)). 104 !" #" $" %" &" '" (" )" *"" !"" "* !# $% & $ ! !" #$% &' #! %! '! (! "!! $!! &!! )!! $ ! $ ! " $ " # Figure 6.7: Maximum acceleration gradientE z (logarithmic scale) along the propagation distance z for different plasma radii such that R p = r equal to 4:2 (black curve), 3 (orange curve), 2 (red curve), 1:5 (blue curve), 1 (green curve) fora):k p r = 1:57 andb):k p r = 0:79 . 6.3.1.2 Evolution of the Peak E z It is important to note that while the transverse wakefields drive the SMI, larger longitudinal wakefield amplitudes are interesting for particle acceleration. We therefore look at the effect of the plasma radius and density profile on the evolution of longitudinal fields. Figure 6.7 illustrates the evolution of the peak E z (in logarithmic scale) along the propagation distancez when the electron bunch propagates in plasmas with different plasma radii and densities such thatk p r = 1:57 (Fig. a)) andk p r = 0:79 (Fig. b)). The initial accelerating field (E z (z = 0)) decreases as the plasma radius becomes smaller. However, the growth rate 105 of the instability (slope of the curve between z = 0:5 cm and z = 1 cm on the two figures) increases; so that the saturated field value does not change much from that in the ”infinite ” radius plasma (k p r = 4:2). It is worth noting that as plasma radius decreases, the peak saturation ampli- tude decreases ink p r = 0:79 case (Fig. b)), but peaks atR p = 1:5 r in the k p r = 1:57 case (Fig.a)). Such difference can be explained by the following two reasons: 1) the initialE z is less sensitive to the decrease of plasma ra- dius in the largerk p r case, as shown in Fig. 6.4; 2) within the bunch, there are two plasma periods fork p r = 1:57 but only one period fork p r = 0:79, and therefore the SMI development of the former one leads to convective growing of the focusing force and hence even larger SMI growth rate. The saturation distance also decreases for smallerR p as a result of larger focus- ing force. Therefore, the largerk p r above 1, and the more plasma periods within the bunch, the more unlikely it is that a finite plasma radius has a negative effect on the saturatedE z amplitude. 6.3.2 Finite Plasma Radius with Cosine Density Prole In this section, we use OSIRIS 2D simulations to study the effect on the SMI development when the plasma has a finite plasma radius with co- sine density profile, again with the ATF beam and plasma parameters. Since no analytical description is available, we rely only on simulation results. Figures 6.8 and 6.9 show the comparison forE z between the uniform 106 0123456 0 100 200 300 400 500 600 700 E zmax (MV/m) z (cm) Figure 6.8: Comparison of maximumE z alongz between uniform plasma (black curve) and plasma with cosine profile (red curve) for the casek p r = 1:57 with different plasma radii such that R p = r equal to 4:2 (a)), 3 (b)), 2 (c)), 1:5 (d)), 1 (e)). ! "! #!! #"! $!! !#$%&"' ( ) *+,-./0 1*+2/0 Figure 6.9: Comparison of maximumE z alongz between uniform plasma (black curve) and plasma with cosine profile (red curve) for the casek p r = 0:79 with different plasma radii such that R p = r equal to 4:2 (a)), 3 (b)), 2 (c)), 1:5 (d)), 1 (e)). 107 Figure 6.10: Comparison of maximum focusing field E r cB along the radius r at the plasma entrance (z = 0) between uniform plasma (black curve) and plasma with cosine profile (red curve) for the casek p r = 1:57 with different plasma radiii such thatR p = r equal to 4:2 (a)), 3 (b)), 2 (c)), 1:5 (d)), 1 (e)). 108 01 234 0 30 60 90 120 Figure 6.11: Comparison of maximum focusing field E r cB along the radius r at the plasma entrance (z = 0) between uniform plasma (black curve) and plasma with cosine profile (red curve) for the casek p r = 0:79 with different plasma radiii such thatR p = r equal to 4:2 (a)), 3 (b)), 2 (c)), 1:5 (d)), 1 (e)). 109 Figure 6.12: 2D density plot of plasma focusing field E r cB at plasma entrance (z = 0) generated by the electron bunch propagating in a plasma with radius R p = 1:5 r and cosine density profile. The plasma density is such thatk p r = 1:57. (black curve) and cosine plasma density profile (red curve) with various plasma radiiR p for the casek p r = 1:57 andk p r = 0:79, respectively. For bothk p r values with different plasma radii, the cosine plasma profile leads to smaller initialE z amplitude, and this is because the plasma density per- turbation is limited by the lower plasma density at larger radius. Such effect is more significant in the case ofk p r = 0:79 (as shown in Fig. 6.9), where the plasma skin depth is larger compared to the beam transverse size. The cases of cosine plasma profile exhibit larger (fork p r = 1:57 in Fig. 6.8 ) or similar growth rate (fork p r = 0:79 in Fig. 6.9 ) ofE z during the the SMI develop- ment, which is again due to the larger or similar focusing force compared 110 to the uniform plasma cases, as shown in Figs 6.10 and 6.11, respectively. Note that due to the cosine transverse density profile, the wakefield period increases along the plasma radius r, resulting in the shift of the focusing fieldE r cB on image of the 2D density plot, as shown in Fig. 6.12. There- fore, the amplitude ofE r cB alongr in Fig. 6.10 and Fig. 6.11 is taken as the maximum value among the longitudinal positionctz for eachr value. The saturatedE z amplitudes in the cosine plasma profile cases remain close to the uniform plasma density cases, although they become smaller ( 25%) whenk p r = 0:79, due to the smaller initialE z field. Therefore, simulations show that the electron bunch propagating in a cosine plasma profile results in a smaller initialE z field but similar growth rate, when compared to the uniform density plasma. The resulting satu- ratedE z fields are not significantly affected by the cosine density profile. 6.4 Chapter Conclusions In summary, this chapter investigates the effect of plasmas with fi- nite radius and inhomogeneous transverse density profiles on the wake- field excitation and the SMI development in overdense plasmas. The simu- lation results show that such plasmas generate larger focusing force for the propagating electron beam, and therefore higher growth rate for the SMI. Although the initial E z wakefield is lower in such plasmas, the increased focusing force can dominate the development trend of the SMI, i.e. larger 111 saturatedE z amplitude can be reached. This effect is more significant when there is a larger number of plasma periods within the drive bunch and when the bunch radius is larger than the plasma skin depth (k p r > 1), resulting in more convective growth of the already increased focusing force. These results may at first be surprising since one would expect a plasma radius not large when compared to the bunch radius or the plasma skin depth would have a negative effect on the development of the SMI. However, these results are consistent with well known characteristics of overdense plasma focusing. That is that the focusing is caused by the re- duced ”defocusing” effect of the plasma return current on the relativistic bunch. It is clear however that too small a plasma radius (R p r ) even- tually impedes the development of SMI. These results are important for the planning of SMI and SMI-based PWFA experiments. Indeed, very large radius plasmas are not necessary for the SMI to develop, and neither is a constant density across the plasma radius, at least until R p r . This is particularly true whenk p r 1. Therefore, capillary discharges with small radius ( r ) and inherent decreasing radial plasma density profile may be considered appropriate for experiments. Note that in general it is easier to reach higher plasma densities in smaller radius capillaries. These results may also have favorable implications for experiments in which the plasma radius is limited by laser ionization (limited laser pulse energy[60], focusing 112 with axicon lens with a small focal radius [72]), or by the bunch self-fields [69]. A manuscript on those result has been prepared and will soon be sub- mitted toPhysicsofPlasmas for publication [73]. 113 Chapter 7 Conclusions/Outlook In this thesis, we have demonstrated experimentally for the first time that energy modulation is an indirect but important evidence for the seed- ing of the transverse self-modulation instability by the sharp-rising edge of an electron bunch. This evidence is observed as periodic energy bunch- ing after the plasma. A number of SMI experiments are contemplated or planned at major facilities (CERN, Fermi National Laboratory, SLAC Na- tional Accelerator Laboratory, DESY, etc.) [49, 59]. All of them will rely on seeding to observe the instability, some to deterministically inject external electrons in the wakefields or to mitigate the occurrence of the hose insta- bility [47, 48]. The results presented here are an important seed for these major experiments. Although simulation and experimental results indicate that with the chosen experimental parameters (Q = 50 pC) the SMI does not grow over the 2cm plasma length, simulations show that with a bunch with higher charge (1nC) the SMI does reach saturation at the plasma exit. We demonstrated that the occurrence of SMI can in principle be inferred from the measurement of the bunch energy spectrum. Simulation results show that plasma with finite radius and inhomogeneous transverse density 114 profiles generates larger focusing force for the propagating electron beam, and therefore higher growth rate for the SMI. Although the initialE z wake- field is lower in such plasmas, the increased focusing force could dominate the development trend of SMI, i.e. larger saturated E z amplitude can be reached. These results have favorable implications for the plasma design of future SMI experiments. We identified that the bunch transverse evolution when in the linear PWFA regime limits the effective transformer ratio. This evolution is similar to that drives the self modulation instability. We thus have shown in initial simulations that a transformer ratio larger than 2 can be reached in the weakly nonlinear regime of the PWFA with the ramped bunch scheme. In this regime the plasma wakefields reach large amplitude (> 1 GV=m) and the transformer ratio can be maintained over a plasma length equal to that in the experiments planned at the ATF. Such a scheme, if successful, can multiply the energy gain of the witness bunch in a single acceleration stage. Using SMI to drive wakefields was only recently proposed. We per- formed the first SMI-related experiments taking advantage of the ATF ex- perimental setup. There is no doubt that many SMI experiments will be performed and that many more simulations will be necessary to fully un- derstand the SMI physics and the outcome of these experiments. This is a wide-open research field. 115 Bibliography [1] W. K. H. Panofsky et al., Reviews of Modern Physics 71, no.2, S121 (1999). [2] R. M. Barnett, et al., Review of Particle Physics, Physical Review D 54, no.1 issue 1 (1996). [3] J. J. Aubert,etal., Physical Review Letters 33, no.23, 1404 (1974). [4] C. D. F. Collaboration et al., Physical Review Letters 74, no.14, 2626 (1995). [5] H. H. Braun,etal., Physical Review Letters 90, no.22, 224801 (2003). [6] P . Mugglietal., C. R. Physique 10, 116-129 (2009). [7] F. F. Chen, second edition, Volume 1, p.163-165. [8] J. M. Dawson, Phys. Rev. 113, 383 (1959). [9] T. 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Fang.etal., to be submitted toPhysicsofPlasmas. 119 Appendix A Data Analysis of the Self Modulation Instability Ex- periment A.1 Energy Spectrum Calibration The energy spectrum is calibrated by varying the current in the dipole of the spectrometer that disperse the bunch. The ATF calibration is :E(MeV ) = I(A) 1:8). The spectrometer was operated atI 0 = 31:911A in the dipole, plus 0:232A in a trim coil during the measurement. Images of the bunch with the mask in were acquired to determine the calibration and to estimate the resolution by looking at the small energy feature. Figure A.1 shows the sum of five images take withI 0 0:6;0:3: + 0:0: + 0:3: + 0:6A. The image withI 0 + 0:232A is not shown, but is listed in the calibration. By following the small bunch feature, the energy calibration can be determined, as shown in Fig. A.1 The energy resolution can be determined from the narrowest fea- tures on one of the calibration images (see Fig.A.2). Measuring the FWFM of the peak , dividing by 2 to have the FWHM and by 2.4548 to obtain the rms of an equivalent Gaussian distribution , the resolution is estimated to be<= 0:0261MeV since we have not tried to make smaller energy features. The energy resolution can be determined from the narrowest features 120 !" !"#! !$ !$#! !% !%#! !& '('' )'' "'' %''*''' +,-,!!#)'&,.,'#''//!*"0,,,1-,'#&&$%&, 23456+,7,849: ;<04= ('*('*(",!"#$%&'()*+,-.#/0123)4$5 Figure A.1: a) Sum of five spectrometer images of the bunch with the cur- rents specified in the text. b) Spectrometer energy calibration. on one of the calibration images (see Fig.A.2). Measuring the FWFM of the peak , dividing by 2 to have the FWHM and by 2.4548 to obtain the rms of an equivalent Gaussian distribution , the resolution is estimated to be <= 0:032MeV since we have not tried to make smaller energy features. A.2 The Estimation of Bunch Length and Energy Based on the above calibration, we can estimate the bunch length. Assume that the bunch (center) rides the rf wave with amplitude A 0 at a phase 0 , and then it wil gain an average energy of : E 0 = A 0 cos(). In the experiment, the values ofE 0 and 0 are respectively 57:9MeV and9 o , and therefore the amplitudeA 0 is determined as: A 0 = 58:6MeV . We now es- timate measure the bunch energy spectrumE from one of the plasma-off images, as shown for image 35 on Fig. A.3: E = 0:48MeV . The first parti- 121 Figure A.2: Spectrometer energy resolution estimate. cle, at low energy is accelerated by followingE 0 +E=2 =A 0 cos( 0 =2). Therefore,=2 =3:2434 o . SInce the rf frequency in the accelerating cavity isf rf = 2:856GHz, corresponding to a of 360 o , therefore the bunch length is = =360=f rf = 3:2ps in this case. This is shorter than usually quoted, but may be compatible with the fact that the charge is low ( 50pC/bunch), and space charge effects in the gun when the bunch is not yet relativistic maybe weaker in lengthening the bunch than at the usual higher charge (500pC to 1000pC). A.3 Energy Spectra Next, we can measure the energy of the peaks on the data with vari- ous plasma densities, which are the results of varying delay times between the plasma charge and the bunch arrival time. Figure A.4 shows the energy peaks with delay time decreasing and therefore plasma density increasing 122 Figure A.3: Right: energy spectrum of run number 84 , plasma off, event number 35. Left: the beam parameters based on the calibration shown above. from b) to h). The peak energy of different events (plasma densities) are plotted in Fig. A.5 a), and average energy spacing between neighboring peaks is shown in Fig. A.5 b). This energy period can be converted to the corresponding plasma wavelength in each event using the long bunch’s pa- rameter 3:2 ps=0:48MeV , which is the indication of the plasma density in each event. Figure A.6 shows the energy peaks versus the plasma densities. 123 " 0* " 0+ $ $ ,.!/ ,/!/ ,/!* # $ ' -' ()' # $ $ $ $ $ $ $ $ $ " 0( " 0* " 0) " 0+ " 0, " 0- " 0. $ " 0( $ " 0. Figure A.4: Energy Spectra obtained at various plasma densities through various delay times between the plasma discharge and the bunch arrival time (decreasing delay, 100 ns step, increasing density). Figure a) is for plasma off. The white lines are the energy spectra, and the red lines indicate the positions of the density peaks. 124 !"#$ !%#& !%#' !%#! !%#" '( ') '% (* (+ ,-./01234.56 ,7.-829 !"!# !"$ !"$# !"% !"%# !"& !"&# &' &( &) '! '% *+,-./012,34 *5,+607 Figure A.5: Energy of the peaks identified by the red lines on Figs. A.4 b) to h) as a function of plasma density. The yellow zone corresponds to the FWHM of the incoming bunch energy. 125 !"#$ !%#& !%#' !%#! !%#" ()*&( &+ ,*&( &+ +*&( &+ %*&( &+ -./012*34/56 . / 378 9' 6 Figure A.6: Energy of the peaks identified by the red lines on Figs. A.4 b) to h) as a function of plasma density. The yellow zone corresponds to the FWHM of the incoming bunch energy. 126 Appendix B List of Publications Y Fang et al., ”The Effect of Plasma Radius and Profile on the De- velopment of Self-modulation Instability”, submitted toPhysicsofPlasmas ( Nov. 2013). Y Fang et al., ”Seeding of the Self-modulation Instability of a long Electron Bunch in a Plasma” submitted toPhysicalReviewLetter for publica- tion (Sep. 2013). Y. Fangetal. ”Possibility of confirming SMI through energy spectrum with the 1nC ATF”, to appear in Proceedings of Particle Accelerator Conference 2013 (IEEE, Pasadena, CA, 2013). J. Vieira, Y. Fangetal.,”Transverse self-modulation of ultra-relativistic lepton beams in the plasma wakefield accelerator, ” Physics of Plasmas 19, 063105 (2012). Y. Fang et al. ”Numerical Study of Self Modulation Instability of 127 ATF Electron Beam”, Proceedings of International Particle Accelerator Confer- ence2012 (IEEE, New Orleans, USA, p.2809, 2012). Y. Fangetal. ”Numerical Study of Self Modulation Instability of 1nC Electron Bunch at ATF”, AIP Conf. Proc. 1507, pp. 559 (2012). Y. Fang et al. ”Numerical Study of Plasma Wakefields excited by a Train of Electron Bunches”,ProceedingsofParticleAcceleratorConference2011 (IEEE, New York City, NY, USA, p.392, 2011).

Abstract (if available)

Abstract Particle accelerators are the main tool for discovering new elementary particles. Plasma based accelerator (PWFA) has been proven a very attractive new acceleration technique due to the large acceleration gradient it has reached (> 50 GV/m), which is two to three orders higher than the conventional radio frequency accelerators. PWFA is essentially an energy transformer transferring the energy from the drive bunches to witness bunches. For a future more compact and more affordable linear electron/positron collider, such an accelerator will require drive bunches with small longitudinal size (on the order of 100 um) and multi‐kilojules of energy to access the new physics at the energy frontier. Proton bunches produced at CERN have been proven as potential drivers for PWFA due to the many tens of kilojules energy they carry (10¹¹ particles, 3.5 – 7 TeV per particle). However, the CERN proton bunches are too long (≈ 12 cm) to drive the wakefield efficiently. It has been proposed that a long particle bunch (protons, electrons, positrons, ...) traveling in dense plasmas is subject to self-modulation instability (SMI), which transversely modulates a long bunch into multiple short bunches (on the scale of plasma wavelength) and therefore results in high acceleration amplitudes through resonant excitation. In this thesis, we demonstrate the first experimental evidence for the seeding of SMI with an electron bunch. We also use numerical simulations to study the SMI development with a higher‐charge electron bunch and propose a possible experiment to demonstrate the transverse modulation directly in experiments. Moreover, we investigate with simulations the effect of transverse plasma radius on the SMI development, which is an important factor to consider when designing plasmas for future SMI and SMI‐based experiments. ❧ Besides efficient drivers such as high‐energy proton bunches, the PWFA also requires high transformer ratio (an indication of energy transfer efficiency) so that the witness bunch can gain energy efficiently from the drive bunch. In this thesis, we explore the possibility of reaching high transformer ratio in the weakly nonlinear PWFA regime so that the witness bunch particles can gain many times the energy of the drive bunch particles in a single acceleration stage.

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Creator Fang, Yun (author)

Core Title Resonant excitation of plasma wakefield

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Chemical Engineering

Publication Date 02/03/2014

Defense Date 11/26/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag accelerator,OAI-PMH Harvest,plasma,self-modulation instability,wakefield

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Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Muggli, Patric (committee chair), Gundersen, Martin (committee member), Mori, Warren (committee member), Nakano, Aiichiro (committee member), Shing, Katherine (committee member)

Creator Email yunf@usc.edu,yunf13@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-362522

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